Kaluza-Klein type cosmological model of the universe by namrata jain

KALUZA-KLEIN TYPE COSMOLOGICAL MODEL OF THE UNIVERSE THESIS Submitted for the Degree of

DOCTOR OF PHILOSOPHY IN PHYSICS (Faculty of Science)

NAMRATA I. JAIN

Under the guidance of

Supervisor PROF. S. S. BHOGA Department of Physics

Co-Supervisor PROF. G. S. KHADEKAR Department of Mathematics

RASHTRASANT TUKADOJI MAHARAJ NAGPUR UNIVERSITY, NAGPUR March - 2015

DECLARATION

I, hereby declares that the piece of work presented in this thesis “Kaluza-Klein type Cosmological model of the Universe” has been carried out by me under the supervision of Dr. S. S. Bhoga and cosupervision of Dr. G.S. Khadekar from March 2010 to March 2015. This work or any part of this work is based on original research and has not been submitted to any other University or Institution for the award of a Degree, a Diploma or a Certificate.

Namrata I. Jain Place: Nagpur Date :

CERTIFIED

Dr. G.S. Khadekar Professor Department of Mathematics RTM Nagpur University Nagpur-33, India

(S.S.BHOGA) Professor Department of Physics RTM Nagpur University Nagpur-33, India

RTM NAGPUR UNIVERSITY D E PA RT M EN T O F PH Y S I C S University Campus, Amravati Road NAGPUR 440033

Fax:(0712)2532841

e-mail: msrl.physics1@gmail.com

CERTIFICATE This is to certify that we have supervised the work of Mrs. Namrata Jain on “Kaluza-Klein type Cosmological model of the Universe” for the degree of DOCTOR OF PHILOSOPHY in the faculty of Science, Rashtrasant Tukadoji Maharaj Nagpur University, Nagpur. We find that the work is comprehensive and complete in all respect for the submission to the University.

(G. S. Khadekar )

(S. S. BHOGA)

Professor Department of Mathematics RTM Nagpur University Nagpur-33, India

Professor Department of Physics RTM Nagpur University Nagpur-33, India

ACKNOWLEDGEMENTS It is on occasions such as these that one realizes the poverty of vocabulary that one lives with. I am handicapped with this infirmity as I acknowledge, with immense gratitude, the contributions of various individuals to my effort in completing this thesis. My guide, Dr. S.S. Bhoga, was a beacon of hope as I floundered, especially in the initial stages, overwhelmed by the waves of the ocean of research that I had stepped into. He was patient, despite my frequent listing, untiringly guiding me back on course with meticulous attention to detail while ensuring that I did not stray from adherence to the tenets of scientific research. My gratitude to him is eternal. To my co-guide, Dr. G.S. Khadekar, I am similarly grateful having found his suggestions invaluable and his guidance extremely helpful for which he carved out time despite being beset with an extremely busy schedule. To the University Grants Commission I am deeply indebted for its enshrined policy of Faculty Improvement Programme (XII Plan) and consequently granting me its Teacher Fellowship because of which I could complete this onerous task with ease – helping me to become a more knowledgeable and better accomplished teacher, and a wiser person. I would like to place on record the silent but monumental contribution of the staff of the libraries of Inter University Center for Astrophysics and Astronomy (IUCAA), Tata Institute of Fundamental Research (TIFR), Indian Institute of Technology (IIT), Jawaharlal Nehru University (JNU) library of Mumbai University who provided easy access to their valuable libraries and the academic treasures within enabling me to complete the present research work. Dr. Farookh Rahaman, Associate Professor, Department of Mathematics, Jadavpur University, Dr. Saibal Ray, Associate Professor, Department of Physics, Government Institute of Engineering and Ceramic Technology KolKata, Dr. Anirudh Pradhan, Professor, Department of Mathematics Hindu post graduate college, Gazipur, were individually extremely helpful who took pains to examine my research work and suggest modifications as and when necessary and contributed freely of their minds and their time. I raise a salute to them. I thank Dr. T.P. Ghule, Principal M.D. College, staff members of Department of physics, registrar, librarian of M.D. College for their continuous support and total cooperation in completion of this study. To Mr. Vijay Choudhary, Mr. Mahesh Bansod and Mr. Dilip Patle research scholars at Materials Science Research Laboratory (MSRL) and all the staff members of Department of Physics, RTM Nagpur University, Nagpur, I thank them for their help in various ways, which led to the culmination of this thesis. I acknowledge with sincere gratitude the patience and perseverance exhibited by Mr. Aloysius Lobo while he painstakingly made the effort for my thesis work to progress

smoothly. My sincere thanks to Dr. Deepak Vasule and his family, Dr. Prakash Jain, Dr. Jayashree Jain, Mrs. Fatima Master and her family for their wonderful hospitality and also helping me in various ways during the tenure of this work.

I am indebted to my parents who encouraged my academic pursuits, my husband for being patient and steadfast while I juggled managing the family, my responsibilities in the college and completing this study; and my two daughters who managed their lives smilingly and ensured that they did not get in the way of completion of this research work. Last, but not the least, I thank God, who in his munificence, sent his angels – a few who I have mentioned above, to see the completion of PhD, which he was overseeing.

Namrata Jain

CONTENTS A.

Figure caption

Table caption

Preface

iii

Introduction

II.

The Friedman- Robertson- Walker

Cosmological model: A study III.

Kaluza-Klein Cosmology

IV.

Kaluza-Klein Cosmological model and Implications

of time varying Cosmological Constant on the model V.

Kaluza-Klein Cosmological model in presence of

101

Quark matter with decay lambda VI.

Kaluza-Klein Cosmological model, string, SQM

121

and time varying lambda VII. Kaluza-Klein Bulk viscous Cosmological model

140

with time varying Gravitational Constant and time varying Cosmological Constant VIII. The early universe and Kaluza-Klein Cosmological model

166

IX.

Kaluza-Klein model of the universe : A Future Scope

193

List of Publications/Presentations

212

LIST OF FIGURES and TABLES [A] FIGURES Figure caption

page no.

Fig.II.1

Bundle of intersecting world lines

Fig. II.2

Bundle of non-intersecting world lines

Fig. II.3

Types of universe- open, flat and close models

of the universe P, Q, R corresponds to present epochs [1] Fig.II.4

Dynamics of the universe for k = -1, 0, 1

and positive and negative lambda [1] Fig.II.5

Time evolution of the universe –

accelerating and decelerating universe Fig.III.1

Hosepipe structure of Kaluza Klein type

space –time [3] Fig.IV.1

Pie diagram for distribution of matter

in the universe [17] Fig.IV.2

Graph of density v/s cosmic time

Fig.IV.3

Graph of q(t) v/s cosmic time

Fig.VI.1

A and B combined to form X particle

122

which decayed to C and D. i

Fig.VI.2

Exchange of particle during

122

interaction of Aand B. Fig. VI.3

Exchange of photon during the interaction

123

Between two photons. Fig.VII.1

Plot of q(t) v/s t [for G = G0/H]

158

Fig. VII.2

Plot of H(t) V/s t [for G = G0/H]

158

Fig. VII.3

Plot of q(t) v/s t [for G = G0H]

159

Fig. VII.4

Plot of H(t) V/s t [for G = G0H]

159

Fig. VIII.1 Schematic of thermal history of the universe

169

Fig.VIII.2

First order phase transitions: V() v/s 

173

Fig.VIII.3

Dependence of effective potential on size

189

of extra dimension in Kaluza-Klein model

[B] TABLES Table II.1 List of Time varying Cosmological Constant [9]

Table IV.1 Equation of state (EOS) and phase of the universe

PREFACE Since ancient times humankind has been fascinated by heavenly bodies and the universe itself and he began to study it, initially through observation, realizing that there was a pattern. This fascination grew as Man realized that these heavenly bodies had an impact on him, and in a way their movements regulated his life and this fascination continues even today as man uncovers new secrets that the Universe possesses. Cosmology began with sky-watching and star-gazing and a growing realization that there were infinite stars, some planets and many other heavenly objects. Amazing discoveries continue to be made even today and cosmologists believe there is much more to discover. I was attracted to the heavens when read about the solar system and later sought information on as many heavenly bodies. But the idea to undertake research started after the studies on Albert Einstein’s special theory of relativity and learnt that despite the wide acceptance it was not really comprehensive and there were issues still be resolved (like the Twin paradox, etc.). The study of the general theory of relativity has then inspired to take up and understand tensor calculus by which a foundation has been laid down for the research in cosmology. Though bit complicated, to go through details of Riemannian geometry, based on tensor calculus, with several efforts, slowly it diverted my mind from regular studies in physics. It was a matter of time before I finally dived into the sea of cosmological research. In this venture, perused scientific literature by Gupta and Gupta, Jayant Narlikar and Hoyle, Robert Penrose, etc. along with similar books, many published scientific papers to keep abreast on what was the current areas in cosmological research. It was discovered that higher dimensional cosmology, specifically KaluzaKlein cosmology attracted the most. It is an elegant theory uniting gravitation, electromagnetism and particle physics. Though it is five-dimensional cosmology it addresses the evolution of the universe in a comprehensive manner. Cosmology studies on the universe with the help of theoretical models have been set up with the help of Einstein Field Equations. Those theoretical models are called a cosmological model, a key factor in cosmology. While searching current research on cosmology, higher dimensional cosmology has gained attention or rather more specifically Kaluza-Klein cosmology. Numerous published literatures have helped to take a close look at Kaluza-Klein cosmology. Although Kaluza-Klein cosmology is the five dimensional cosmology but it can be dealt with the evolution of the universe in a more comprehensive way.

iii

The Kaluza-Klein cosmological model is preferred over other models because it is simple to deal with, it has wider scope and it also explains the studies in many different ways. On the other hand, the Friedmann models though capable of explaining various features successfully; it suffers from inadequate explanation of certain features i.e. accelerated expansion, puzzle of dark energy and dark matter, etc. Contemporary studies in cosmology have explained the early universe scenario, lambda decay cosmology, space-time-matter concept and so on. The Cosmological Constant Λ, an important parameter in cosmology, was first introduced, though later discarded, by the eminent scientist Albert Einstein. It is a key ingredient in the present cosmological model. Being a representative of vacuum energy density, it has a prime role in resolving current problems of cosmology. Is the Cosmological Constant a really constant? This question arises due to the Cosmological Constant Problem (CCP). The CCP is the large difference of about 120 orders between the present observational value and that predicted by particle physics. The answer to this problem is to introduce time-dependent lambda in the cosmological model. Another important fact about Λ is its small positive value. Since the last two decades, a variation of Λ has been introduced to various cosmological models to deal with the present universe scenario. Numerous available literatures urged and inspired us to undertake the present work. Cosmological models (flat universe) with different forms of time-varying cosmological constant i.e.

,  ~  , where a is scale factor, have been

the most common studies currently undertaken and extensively studied in the scientific community in different contexts. These models (either 4D or 5D) appear to have solutions to the issues of present universe up to a certain extent. Hence, it is feasible to take up further studies on similar issues. The

four-dimensional

model

with

generalized

lambda,

i.e.,

was first introduced by Carvalho and Lima. The generalized Λ, worked upon by a few cosmologists who found it helpful in investigating the evolution of the universe. A generalized time- dependent Λ and its implications on the Kaluza-Klein cosmological model is the initial task of the present thesis. Implications of the generalized lambda demonstrate the importance of an extra dimension in the study of cosmology. iv

Reference to available literature on quarks, string theory and viscous models has helped in enhancing the present study. The early universe scenario was quite an interesting research area since it is necessary to understand the evolution of the universe from an anisotropic phase to the present isotropic phase. The comprehensive efforts were undertaken to have future aspect on the present study. A discussion on the future aspects may help for the further progress and broaden our view for acquiring more knowledge on dark energy, dark matter, matter creation, etc. which are latest subjects in cosmology.

Namrata Jain Author

CHAPTER I INTRODUCTION

I.1

HISTORICAL DEVELOPMENT OF COSMOLOGY

I.2

MODERN COSMOLOGY

I.3

COSMOLOGICAL CONSTANT AND ITS SIGNIFICANCE

I.4

HIGHER DIMENSIONAL AND MODERN COSMOLOGY

I.5

KALUZA-KLEIN COSMOLOGY AND EXTRA DIMENSION

I.6

KALUZA-KLEIN COSMOLOGY AND COSMOLOGICAL

CONSTANT I.6.1

Kaluza-Klein Cosmological model with lambda and

Quarks I.6.2

Kaluza-Klein model and strings

I.6.3

Kaluza-Klein model with viscous medium and time

varying constants I.7

KALUZA-KLEIN COSMOLOGICAL MODEL AND

FUTURE ASPECTS I.8

STATEMENT OF PROBLEMS

REFERENCES

I.1

HISTORICAL DEVELOPMENT OF COSMOLOGY Cosmology originates from Greek word Kosmos means world and logia

means the study of, which takes us to the study of origin, evolution and fate of the universe. Christian Wolff’s first coined this word in 1730 in his book Cosmologia Generalis, which has a long history involving science, philosophy, esotericism and religion. Ancient history of the world has instigated cosmology with many aspects by many religions followed all over the world. Stone age man worshipped celestial objects as the symbol of supernatural power for their safety. These celestial objects have also thought of influencing the calamities on earth. Since cosmology has origin in Greece, its philosophers, Pythagoras and his followers Aristotle and Plato have changed the scenario of the world. These philosophers have dealt with astronomical studies, which initiated the journey of cosmology. Contemporarily, in our country Aryabhata, Varahmihira and Bhaskara made ingenious observations regarding celestial objects and stars, and their physical aspects. Ptolemy and Aristarchus in third century B.C. had proposed an erroneous theory about dynamics of solar system, which later on has been corrected by Copernicus, a Bishop in fourteenth century. The modified model has been the fact now wherein sun is at center and other planets are revolving around it. These studies have explored and still it is a part of astronomy. Since ancient times, the sun, the moon, and the stars have fascinated man. Wondering how they appear, disappear, twinkles and shot across the sky. These have fuelled his curiosity to learn more about the universe. Consequently, investigations on how the universe came into and evolved. This fascination with the universe, its origin and its elements led to formal schools of thought and study, and evolved its own terminology. Nevertheless, cosmogony is a term applied to the origin of the universe, its study and theories on the origin and development of the universe. Cosmology is also a study of the universe as perceived by the senses, and the concept of the universe arrived at, through this study. Mankind credits Galileo as the pioneer and Newton as a worthy successor to this cosmic investigation. But it has been the Einstein’s general theory of relativity that lent an impetus to theoretical studies of cosmology. In 1609, the Italian astronomer and physicist, Galileo (1542-1642) his pioneering work has built and improved the telescopes for astronomical use. He observed the mountainous configuration of the moon, the phases of Venus, Jupiter’s satellites and the existence 1

of sunspots. He, also, discovered that the moon shines due to reflected sunlight, and that the Milky Way is made up of countless stars. Galileo accepted the Copernican theory of the solar system where the earth moves about the sun. This model had then contradicted the prevailing theory where the earth is fixed and is the centre about which the universe revolves. But not until his astronomical discoveries those gave concrete and visible confirmation of the theory he took a decided position in its favor. In 1630, he wrote ‘Dialogue on the Two Chief Systems of the World’. According to which the Copernican system had brilliantly been expounded and defended. Unfortunately, this work had been condemned and forbidden its sale. Galileo has applied mathematics, careful observation, experimentation and inductive reasoning to the understanding of physical phenomena. Therefore, he is considered to be the first modern physical scientist. The English physicist Newton, a versatile genius in late sixteenth century made a landmark discovery on invisible power so called gravitational attraction has proved it to be universal. Newton’s laws of motion, universal law of gravitation, relativity theory, and theory of colors have been the milestones in the field of physics. These have not only been applicable to the bodies on the earth, but also to the heavenly objects. Newton was also a great Astronomer. Using reflected telescope, he could prove Copernicus theory, Kepler’s laws for solar system and an important fact that most of the mass is concentrated at the Sun, center of the Solar system. In 1692, Newton was credited with being the first to make a scientific attempt to explain the inception of the universe. He theorized that in the beginning, particles of matter have been distributed uniformly throughout limitless space. Latter they have been attracted to each other according to the law of gravitation. The resulting clots of matter in space attracted other particles and eventually grew into the sun and other stars. Other scientists had subsequently disproved this mathematically. Satisfactory theories on the evolution of the stars and the universe had not been proposed until late 17th century. After Newton’s discoveries, the field of astronomy got boosted with the real facts about the sun, stars; and considerable work has been carried out in observational astronomy. Astrophysics and astronomy have been now the fields of physical cosmology. Physical aspects of the stellar system and planetary behavior have been extensively studied in these fields. Apart from the studies on planets and stars, scientists’ quest about the universe began in early nineteenth century. Both observational and theoretical advances have guided study of the universe thereafter. 2

Theoretical study of the universe really took off in 1917 when Einstein, the genius German physicist described the universe by means of a simplified model. 1905 has been the remarkable year, because Einstein published three papers, one on photoelectric effect, another on special theory of relativity and a paper on a relation E = mc2 proving equivalence of mass and energy. These papers have been considered as scientific miracles, a revolutionary concept that resulted into present modern physics. In 1915, he proposed on general theory of relativity that provided a revolutionary concept of gravitation. This theory featured accelerated frame of reference, bending of light at heavenly heavy objects and space-time geometry concept based on new mathematics postulating principle of equivalence. The concept of combining space and time as dimensions has founded the basis of theoretical cosmology, which is indeed the relativistic cosmology.

I.2

MODERN COSMOLOGY In 1917, Einstein, using tensor calculus, Riemannian geometry and a

modification of the Schwarzschild metric has developed field equations leading to a mathematical model. According to the published mathematical model, the universe is static and filled with matter. A few months later, in the same year, de Sitter obtained a cosmological model, which represents nonstatic empty universe. Both the models had the shortcomings, which have been realized due to the discovery of an expanding universe filled with different kinds of matter through indigenous observations by Hubble and Humason in 1920s. They observed red shift in spectral lines and found linear relationship between red shift and distance between earth and galaxy. All these parameters have enabled to get velocity of galaxy proportional to the earth- galaxy distance. Constant of proportionality is therefore called Hubble’s constant [1]. A few years before Hubble’s observations, Friedmann [2] has theoretically explained expansion of the universe, which went unnoticed, however. Friedmann set up a simple model of the universe by assuming homogeneous and isotropic universe. According to Friedmann, the universe is filled with stars, planets, galaxies, clusters of galaxies etc.. It, thus, behaves like a perfect fluid and at any cosmic time it is isotropic and homogeneous. Later on, this had become the basis of cosmic principle. In 1927 Lemaitre, a Belgian priest, an astronomer and periodic professor of physics, has independently arrived at a conclusion similar to that of Friedmann’s. He published 3

his findings in a journal of the Scientific Society of Brussels. Robertson and Walker independently have set up the cosmological model, similar to Friedmann model satisfying cosmic principle. Einstein Field Equations (EFE) based on Ricci Tensor, played a major role in deriving and setting up of mathematical model. Theoretical and observational studies of the universe have progressed rapidly since then and reached to a level where the secretes of the universe can be unfolded. There are several models now-a-day, which have been undertaken as the research work.

These models include spherical

symmetric models, Bianchi models, Friedmann-Robertson-Lamaitre-Walker (FLRW) models, and similar others. Among all others, however, Bianchi and FLRW or popularly called as FRW models have been extensively studied. The FRW Model has successfully explained and define many physical parameters like the red shift, Hubble constant, deceleration parameter, expansion parameter, shear scalar, etc. But there had been certain issues, which have still to be addressed. Particle horizon, problem of flatness, entropy and monopole, accelerated expansion, inflation, etc. could not be solved by Friedmann cosmology. Cosmic microwave background (CMB) data, galaxy observations, Supernova explosions, redshift surveys, cosmic microwave background explorer (COBE) satellite data, Wilkinson microwave anisotropy probe (WMAP) data, have been the major experiments. All these have been landmarks of the field of observational cosmology. Among these observations WMAP has used to estimate the age of the universe of about 13.6 Gyrs, which is quite astonishing. Major breakthrough has occurred due to the supernova Ia experiments (1998-99) of Perlmutter and Reiss [3, 4]. The analysis of the data concluded the accelerated expansion of universe. The accelerated expansion, however, cannot be explained theoretically using FRW or any other models.

I.3

COSMOLOGICAL CONSTANT AND ITS SIGNIFICANCE Extensive studies have been undertaken in the past considering a variation of

the cosmological scale factor to address the acceleration of the universe. Researchers then veered to Einstein’s model and de Sitter’s model. Einstein’s model contained a constant () described as a cosmological constant. Interestingly, Einstein after Hubble’s universe expansion theory had conceded that there was no need for the

cosmological constant (). But Eddington and Lemaitre preferred to retain Einstein’s cosmological constant () to be able to introduce attractive features into cosmology. According to literature, the cosmological constant () represents vacuum energy density and it is factored into cosmological equations as dark energy, though its value is still unknown. The FRW models with cosmological constant, however, could conclude with accelerated expansion. The value of  has been calculated with the help of experimental data; but a huge discrepancy has been found between observed value and the value predicted by standard model (SM). This discrepancy is known as cosmological constant problem (CCP) [5]. Apart from cosmological constant problem, cosmological coincidence problem, large scale structure of the universe, mystery of dark energy and dark matter have been some of the puzzles went unattended till date. In 1989, Chen and Wu, in USA [6] found the solution for CCP. They concluded that the cosmological constant has not been really a constant. The issue has related to time variations of the cosmological constant, which led to the formulation of a new cosmology, called the lambda decay cosmology due to Chen and Wu. According to them cosmological constant varies as 1/R2, where R(t) is a scale factor. Thereafter, cosmological models with   H2,   qH2 (H - Hubble parameter, q - deceleration) and similar other variations have time dependent cosmological constant. As a result though worked upon by many cosmologists, but the CCP problem has not yet been solved. Carvalho and Lima [7] in 1990s described the cosmological model with generalized time dependent =H2 +R-2. This model has an advantage over other models, because accelerated expansion, isotropy, age of the universe can be explained. Although small positive value of cosmological constant had been predicted, but exact numerical value has yet to be confirmed. In addition to accelerated expansion, presence of some anisotropy in isotropic universe represented by WMAP data, early universe phenomenology and similar such criteria have motivated many cosmologists to look for dimensions more than the present four dimensional models.

I.4

HIGHER DIMENSION AND MODERN COSMOLOGY Dimensions are the foundations of theoretical physics wherein mathematical

equations express the physical phenomena. Historical review of dimensions dates back to Pythagoras era when his theorem propounded a revolution in the field of 5

science. Development of three-dimensional geometry from one dimension has taken more than half century. The Minkowsky’s work has instigated the new concept of non-Euclidean geometry, where time has been identified as fourth dimensions along with three normal space dimensions. The concept of space-time dimensions actually originated due to Mobius in 1827. Minkowsky space-time dimensions form the basis of Einstein special and general theory of relativity. Einstein general theory of relativity further got the recognition due to the use of Riemannian geometry, admitting the relation between matter and geometry, which has been the pioneer of modern cosmology. The mathematical models with four dimensions have continued to attract rigorously and work upon even today. Inadequacy of 4D models and rise of n-dimensional theory have attracted great attention due to the need of unifying all forces leading to the modern cosmology. The need of unifying forces has also been felt in sustainable with the advances in particle physics. Modern theory of particles necessarily deals exclusively with particle interactions. Quantum physics has greatly influenced the particle physics by defining the exchange quanta for the various types of interactions between the particles. Four types of forces have been defined to address interactions.

Besides basic elementary

particles, various types of sub-atomic particles discovered have led to a particle zoo. A mathematical model involving particle zoo, interactive forces and different kinds of particle phenomena all together has been designated as standard model (SM). The interaction phenomenon between particles, represented by Feynman diagrams has helped to understand the origin of different types of forces. Quantum mechanics has been extensively used to describe particle interaction phenomena of particle physics. Quantum electrodynamics (QED), quantum chromo dynamics (QCD), quantum field theory (QFT) have been the major topics of current research activity. Quantum field theory and Feynman diagrams have been the origin of string theory and n-dimensional theory. Strings, the topological defects produced during first phase transition at the early universe have played a key role in describing supersymmetry, and supergravity phenomena. All these had actually been unbelievable, but have supposed visibly happenings, essential for describing origin of the universe.

Exploration of n-

dimensional theory with the concept of quantum gravity, a latest research area, has led to Brane theory, multiverses, etc. String theory first propounded in a popular book 6

‘Theory of Everything’. It has motivated the researchers to study higher dimensional theory to address evolution of the universe from early to its late time stages. Actually, the foundation of higher dimension physics has been laid in 1920s during the attempts made to unify all four types of forces.

I.5

KALUZA-KLEIN COSMOLOGY AND EXTRA DIMENSION Gravitational force, the weakest force among all four forces whose origin, is

still not known., An indigenous work unifying for electromagnetism and gravitation of Kaluza [8] in 1921 had been actually initiated by Nordstorm which went unnoticed, has set the five-dimensional theory as a milestone for higher dimensional cosmology. Later on Klein [9] has combined gravitation with gauge field theory involving particle physics in five dimensional physics, and reached to the similar results of Kaluza. In five-dimensional theory besides four space and a time dimensions, an extra dimension has been considered similar to gauge potential of electrodynamics or any other parameter, which is space-like. It is possible to choose fifth dimension as a parameterization of mass. Fifth dimension of Kaluza-Klein theory has been independent of other from four dimensions. Kaluza explained five-dimensional theory assuming ‘hosepipe’ geometry. According to him, four dimensions are along the length of pipe and an extra dimension is along the circumference of hosepipe. Hosepipe geometry resembles cylindrical co-ordinate system. Five-dimensional hosepipe can be explained in analogy with its two-dimensional equivalent. Hosepipe that appeared one-dimensional when observed from long distance, but if observed closely, then appeared two-dimensional. These observations showed that an extra dimension is small. The universe may be having (4+d) dimensions, but for large distances extra dimension appears small. The smallness of scale corresponding to extra dimension may be such that it cannot be determined experimentally. However, it has been concluded that size of extra dimension should be of the order of Planck distance, which is about 10-35 m [10].

Extra dimension cannot be observed

experimentally but its effects can be detected experimentally. Importance of extra dimensions is not only pertinent to early universe phenomenology but also useful for the explanation of dark energy, dark matter, CCP, etc. An elegant and simple Kaluza-Klein (K-K) cosmology soon became popular to study the universe from the early stage phenomenology to its present features. Compactification of extra dimension further led to its implications to the present 7

universe. Usually, Kaluza-Klein (K-K) five-dimensional model has been considered as an extension of four-dimensional FRW model. The model however, can also be derived from Schwartzchild metric, which explains spherical symmetric system. It has been quite different from Bianchi type models. Bianchi models mostly convey anisotropic universe and constraint to particular scale. Higher dimensional Bianchi models had been investigated by a few cosmologists, and have been less popular due to their mathematical complexity. K-K cosmological models with varying cosmological constant have provided a detailed and clear picture of late time universe with accelerated expansion. Studying it with perfect fluid having flat as well as non flat curvature has not only demonstrated accelerated expansion but also evolution of extra-dimension, cosmological constant problem, dark energy, dark matter can be dealt with. Kaluza-Klein cosmology has been applied for Space-Time-Matter (STM) concept, which has been first put forth by Wesson [11] in late nineteenth century. In Wesson’s theory of gravitation, the mass has been parameterized so as to postulate STM theory. Fifth dimension believed to be embedded in four-dimensional matter so as to have induced mass concept. Also, geometrical concept of mass can also be probed with the help of STM theory. It has also been possible to understand the role of extra dimension and its effect on physical parameters. This theory has significantly helped to investigate K-K model with different forms of matter.

I.6

KALUZA-KLEIN COSMOLGY AND COSMOLOGICAL CONSTANT K-K cosmological model with variable cosmological constant has been

extensively studied [12]. Amongst them, models with cosmological constant Λ  R-2,

  H2 and   qH2 have often been studied and found to be equivalent. In this regard, Kaluza-Klein model with generalized  = H2 +R-2 has found to be quite expressive and can explain accelerated and isotropic universe. First term in the generalized lambda can explain accelerated expansion and solve age related issues. Further the second term in it has introduced to cope up with minimal anisotropic presence in the present universe. Other aspects of the universe can also be discussed using Kaluza-Klein model. In addition to this, the role of extra dimension and its implications on physical parameters can be described which encourages investigating the model with generalized cosmological constant. Kaluza-Klein model with 8

generalized lambda has found to be free from singularity, and can explain anisotropic universe at its early stage.

I.6.1

Kaluza-Klein Cosmological model with lambda and quarks The early universe has been often discussed with spherical symmetric or

Bianchi type models. Kaluza-Klein cosmology, based on higher dimensional theory, can explore it; and recently has been successfully attempted by many cosmologists. Early universe as well as present universe can be explained using K-K model using time dependent lambda provided investigated with the model of quarks. Quarks have been subatomic particles and they participate in strong interaction. Quarks have been generated at about 10-22 sec just after the Big-Bang. At early stage during first transition when the universe had been filled with radiation, thought to be in quark- gluon-plasma (QGP) state. Quarks paired resulted into hadron formation during the evolution of the early universe. Birth of compact star or quark star, which has been actually hypothetical, has been supposed during this time. Olinto has first explained importance of quarks in early universe. According to him, strange quark matters can exist at zero pressure. A team of scientists of Massachusetts Institute of Technology (MIT) has explained the internal structure of hadron. The hadron supposed to be composed of quarks, and has been like a ‘bag’ of quarks. Hadron has been thus, represented by MIT bag model. Using bag model, quarks have been modeled with equation of state relating its pressure and density as pq = 1/3(ρq – BC) which can be applied to Einstein field equations so as to obtain cosmological model with quarks. The validity of equation of state has also been confirmed after experimental results of Brookhaven’s relativistic heavy ion collider (BNL-RHIC) laboratory. BNL-RHIC has demonstrated that quarks-gluon act as perfect liquid. Since quarks play an important role at early universe, the models with quarks

gained importance. The model with quarks has been explored for solving

CCP, dark matter, phantom or dark energy, etc. As discussed previously, since extradimension play a key role for early universe happenings, study on cosmological models with extra dimension and quarks can be a milestone so as to have the solutions of the present problems. Four-dimensional cosmological model with quarks have normally been studied for astrophysical objects. Having seen the importance of quarks in cosmology, investigation on the models with it may have significant impact on solutions for the present problems. A few workers have already studied five9

dimensional physics with quarks, and research has been going on [13]. In this regard, study of Kaluza-Klein model with quarks can be useful due to its ability of depicting present as well as past universe. The role of extra dimension and its implications have been the important issues for better understanding of secretes of the universe. According to literature, quark matter can act as dark matter, but needs extensive investigations [14, 15]. Time varying lambda and equation of state for quarks can be used to solve dark energy and dark matter problems. Nonsingular behavior, eternal inflation have been some of the features of the universe, those can also be explained from the analysis of physical parameters of K-K model with quarks.

I.6.2

Kaluza-Klein model and strings Importance of strings is discussed previously. Early universe phenomena have

also been studied with strings. Inflation has been one of the phenomena among many conjectures observed at early universe. Consequently, inhomogeneities induced during this time, and this has also been one of the reasons for evolution of galaxies and other cosmic structures. Strings have been believed to produce density perturbation at the time of galaxy formation [16]. Strings have been actually massless objects but can attain mass if their stress energy and gravitational field are coupled. Consequently, gravitational effects of strings have been analyzed by deriving Einstein field equations (EFE). The strings attached to quarks have been considered for the analysis of string cosmological models. At present time, strings have been not observed. Thus, suggested that its density must have been decayed with time. The cosmological model with strings and time dependent lambda depicts the early universe epoch, presence of dark energy, accelerated expansion, etc. Investigation of Kaluza-Klein cosmological model with strings and generalized time dependent lambda can be carried out so as to understand the role of strings during the evolution. Derivation of physical parameters such as string density, quark density, pressure and cosmological constant has demonstrated the nature of cosmological model. Anisotropic nature of the universe can also be explained using string cosmological models by finding shear scalar to expansion scalar ratio. Extra dimension, also, affects the string density, and analysis of other physical parameters led to the accelerated nonsingular model.

I.6.3

Kaluza-Klein model with viscous medium and time varying constants Apart from dimensions, the natural constants have also played a major role in

cosmology, as they have been the basic contents of various physical quantities. Actually, Eddington has first worked upon ‘constants of nature’ in 1946. Later on, Planck and Wheeler have investigated these constants extensively. As a matter of fact, the charge (q) has its quantized value. The issue of how a particle gets its mass remained unsolved, however. In cosmology, as per cosmic principle, ћ, c, and G have been assumed equal to one. In early nineteenth century, investigations on varying constants have begun after Lord Kelvin. Later studies continued further due to Milne and Dirac independently in 1930s [17, 18]. They investigated time varying gravitational constant. Milne’s work suggested increase in G with time. Dirac, whereas, has suggested decrease in G with time. Milne’s work had not recaptured much importance. Dirac has postulated large number hypothesis, which has gained attention of many as it can be related to the age of the universe. In cosmology, Einstein field equations contain G so as to address both the geometry and matter. Realizing the importance of time varying G in cosmology, attempts have been made for studying cosmological models with time varying G. Since  is also varying as predicted recently, studies on cosmological models with varying G and  have been attempted in different contexts, recently [18]. One of the reasons to study models with varying G and  has been to address stress-energy conservation. The cosmological models with time varying G and  in viscous fluid are investigated in Kaluza-Klein metric during present work. The universe is believed to be filled with viscous fluid at it early stages.

Dissipative mechanisms at early

universe have been studied by considering it to be filled with viscous fluid. In this regard, casual theory proposed by Israel and Stewart and non-casual theory of thermodynamics by Eckart explained dissipative mechanisms independently [19]. According to these theories the equation of state have been modified due to modified pressure that depended upon viscosity coefficient. Time varying bulk coefficient has been then introduced to explain the present universe, which has been assumed filled with perfect fluid [20]. The models with time varying G and  have been studied for perfect fluid. These models have explained accelerated expansion but viscous model with G and  brought out much information about the evolution of the universe. The model in Kaluza-Klein metric has investigated the effect of extra-dimension on 11

viscosity coefficient as well as on other parameters. So the role of extra dimension in the evolution of the universe also analyzed along with effects of varying G and .

I.7

KALUZA-KLEIN COSMOLOGICAL MODEL AND ITS FUTURE ASPECTS The extra dimension, which actually has origin at early universe, can enlighten

some important phenomena at that stage. A brief review on thermal history depicts the different phases of the early universe. Phase transitions and time-temperature relation have significant impact on the evolution of the early universe. Four-dimensional FRW model has been able to explain them theoretically but Kaluza-Klein model express them so as to reconcile with observational data. Presence of dark energy and dark matter at early universe can also be dealt with K-K model with cosmological constant [21]. Present work includes a brief discussion on dark energy, dark matter and matter creation so as to give an idea of future scope of the Kaluza-Klein model of the universe. The Kaluza-Klein model of the universe can also be analyzed for dark energy and dark matter. Both are recent major problems and challenges to reconcile them with present observational data [22]. A few cosmologists have investigated matter creation, an important phenomenon in cosmology. Studying matter creation with Kaluza-Klein cosmological model can illuminate the evolution of the universe further.

I.8

STATEMENT OF PROBLEM The thesis entitled Kaluza-Klein type cosmological model is organized in the

following way. 1. First chapter includes historical background of cosmology and introduction to the topic so as to get a brief glimpse of the aims of present work. 2. Second chapter explains Friedmann-Robertson-Walker cosmological model which is a four dimensional standard model in cosmology. Other physical parameters are also defined and briefly described. Later derived then for the Kaluca-Klein model. Shortcomings of the model are also discussed in this chapter so as to set up a ground for the choice of the present work. 3. A theoretical background on Kaluza-Klein cosmological model is presented in third chapter. Kaluza mechanism for five-dimensional theory instigates the

importance of extra dimension and also explains its compactification. Thus, it was taken so as to carry further investigation on Kaluza-Klein theory. Wesson theory is also explained in third chapter, which infuses the study of KaluzaKlein model with matter. Kaluza-Klein model with constant lambda is also discussed so to know its inadequacies and, hence go further for time dependent lambda. 4. Fourth chapter takes care of study on Kaluza-Klein model with different kinds of time dependent lambda and their comparison. The model with generalized lambda is found to be significant and can be reconciled with the present observational data. Physical parameters and neoclassical tests are derived for the model to provide further in comparison with other four dimensional as well as other type of models. The age of the universe is also calculated using this model and hence with generalized lambda, it can be studied. 5. Fifth chapter deals with the investigation on Kaluza-Klein model with strange quark matter in presence of time varying lambda. The model can be studied for inflation era. Also expected to provide a solution to dark energy and dark matter problems. The model with generalized lambda is new one. It can also explain non-singular behavior of the model. 6. Sixth chapter encompasses the K-K model with strings and strange quark matter in presence of time varying lambda. The physical parameters derived for it concluded with accelerated model even with string density. It is also observed that string density decreases with time, which suggests that probably the early universe consists of strings and it decayed with time evolution. 7. Kaluza-Klein model in viscous medium with time dependent G and Λ is recently investigated so as to understand the evolution of anisotropic early universe to present isotropic universe. The cosmological model with time dependent G and Λ is studied to meet with stress energy conservation. The dissipative mechanisms along with G of the cosmological model explain theory of matter creation. The models with G  H-1 and G  H for KaluzaKlein viscous model with generalized lambda are investigated in seventh chapter. These models lead to dark energy models and can be useful to investigate the evolution of the universe.

8. A brief discussion on the early universe phenomena has been carried out in eighth chapter. Thermal history, phase transitions and MIT bag model are crisply described understand the evolution of the universe at its early stages. Time-temperature relation is discussed in context with FRW as well as with Kaluza-Klein models. A comparison between them depicts importance of extra dimension at early universe. 9. Ninth chapter provides a glimpse of future scope of K-K model and also summarizes the present work. The present work can be further carried out for interactive dark energy and dark matter, which are the current issues. K-K model for matter creation can also be taken up in future.

REFRENCES

[1]

Narlikar J.V., Elements of Cosmology, Chapter 1, first edition, Cambridge University Press, Cambridge (1993)

[2]

Friedmann H.P., Astrophysical Journal, 82, 284 (1935)

[3]

Perlmutter, S. Gabi S., Goldhaber G., Goobar A., Groom D. E., Hook I. M., Kim G., Kim M. Y., Lee J. C., Pain R., Pennypacker C. R., Small I. A., Ellis R. S., McMahon R. G., Boyle B. J., Bunclark P. S., Carter D., Irwin M. J., Glazebrook K., Newberg H. J. M., Filippenko A. V., Matheson T., Dopita M., Couch W. J., Astrophys. J. 483, 565 (1995)

[4]

Riess A. G., Filippenko A. V., Challis P., Clocchiatti A., Diercks A., Garnavich P. M., Gilliland R. L., Hogan C.J., Jha S., Kirshner R. P., Leibundgut B., Phillips M. M., Reiss D., Schmidt B. P., Schommer R. A., Smith R.C., Spyromilio J., Stubbs C., Suntzeff N. B, Tonry J., Astronom. J. 116, 1009 (1999)

[5]

Weinberg S., Rev. Mod. Phys. 61, 1 (1989)

[6]

Chen W. Wu Y. S., Phys. Rev. D 41, 695 (1990).

[7]

Carvalho J. C., Lima J. A. S., Waga I., Phys. Rev. D 46, 2404. (1992)

[8]

Kaluza T., Sitzungsber. Preuss. Akad. Wiss. k1, 966 (1921)

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Klein O., Zeit. Phys. 37, 895 (1926).

[10]

Chodos A. ,Detweiler S, Physics Review D 21, 2167 (1980)

[11]

Wesson P. ‘Space Time Matter : Modern Higher Dimensional Cosmology’ Singapore World Scientific Publishing Co. Pvt. Ltd. (1999)

[12]

Sahni V., Starobinsky A., Int. J. Mod. Phys. D 9, 373 (2000)

[13]

Witten E., Phys. Rev. D 30, 272 (1984)

[14]

Sagert I., Fischer T., Hempel M., Pagliara G., Schaffner- Bielich J., Thielemann F.-K., Liebendorfer M., J. Phys. G, Nucl. Particle Phys. 37, 094064 (2010).

[15]

Trimble V., Neutrinos and Explosive Events in the Universe, 181, Springer, Netherland (2005).

[16]

Polchinsky J., String theory, Cambridge University Press , Cambridge (1998)

[17]

Milne E A, Relativity, Gravitation and world Structure, Oxford University Press, Oxford (1935)

[18]

Dirac P A M , Proc. R. Soc. A165, 119 (1938) 15

[19]

Maartens R, Class. Quant. Grav., 12, 1455 (1995)

[20]

Arbab A. I, Gen. Relativ. Gravit., 29, 61 (1997)

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Gleiser M., Preprint, arXiv : hep-ph/9803291/v1 (1998)

[22]

Goodstein D., Adventures of Cosmology, chapters 9, pg 258, world scientific Co. Ltd. Singapore (2012)

CHAPTER II THE FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL: A STUDY

II.1

INTRODUCTION

II.2

FRIEDMAN-ROBERTSON-WALKER

COSMOLOGICAL MODEL II.2.1 The assumptions for the FRW model

II.3

II.4

II.5

(a) Weyl’s Postulate

(b) Cosmological principle

II.2.2 The line element

II.2.3 FRW model

(a) Matter dominated phase

(b) Radiation dominated phase

COSMOLOGICAL PARAMETERS

II.3.1 Hubble parameter

II.3.2 Deceleration parameter

II.3.3 Density parameter

FRW MODEL WITH COSMOLOGICAL CONSTANT II.4.1 Horizon problem

II.4.2 Flatness problem

II.4.3 Field equations and the model

AGE OF THE UNIVERSE FOR FRW MODEL WITH

COSMOLOGICAL CONSTANT II.6

LIMITATIONS OF FOUR DIMENSIONAL FRW MODEL WITH COSMOLOGICAL CONSTANT II.6.1 Cosmological constant problem (CCP)

II.6.2 Cosmological coincidence problem

II.7

LAMBDA DECAY COSMOLOGY

II.8

CONCLUSIONS

REFERENCES

II.1

INTRODUCTION Cosmology is the study of the universe. The universe, its origin, evolution and

its fate are still a curiosity for mankind and the scientific community extensively studies it since ancient times. Observing heavenly objects and their dynamics through sky gazing is a natural trend involved in the study of the universe, and this way of study is called Astronomy. It is the astronomical observations of planets, stars and galaxies that have created inquisitiveness to look beyond our Milky Way galaxy to distant objects from which light may take billion years to reach us. Now-a-day, modern techniques of observing the sky to study stars and galaxies has enabled us to understand dynamical and physical behavior of billions of galaxies spread across vast distances. Studying the extragalactic world and large-scale structure of the universe has gained popularity way back in the nineteenth century, when Einstein’s theory of relativity emerged. In the late twenties, tensor calculus revolutionized the geometry of space by involving it in the study. This led to the theoretical study on the universe and, thus, a cosmological model set up using tensor calculus and metric algebra. The cosmological model of the universe has been a mathematical model evolved by solving mathematical equations with the help of tensor calculus. The application of tensor calculus with Riemannian geometry brought about the concept of space-time curvature and geodesics in general relativity. A geodesic has been understood as the minimum stationary distance by any body on the space time curve similar to a curve on a 3D sphere. In tensor calculus, Christofflel tensor and Ricci tensor have been the main tensors to derive equations of geodesics. With the help of tensor calculus, Einstein published equations in general relativity, which relates space and geometry in 1915. With those equations Schwarzchild found internal and external solutions for space-time geometry of the spherical distribution of matter thereby developing the Schwarzchild metric. Einstein, De Sitter and many others implemented Schwarzchid metric successfully to study the universe. It was also successfully applied for the explanation of experimental measurements of solar system, which involves advance perihelion of a planet, bending of light, gravitational red shift, radar echo delay, etc. Einstein and de Sitter set up the cosmological models by line element as Schwarzchild metric. Both models can explain the universe successfully, but at some issues they are 17

contradictory to each other. Einstein universe is a steady state model and filled with matter without motion, but de Sitter universe is an expanding model filled with empty space. Both the models were inadequate to explain the expansion of the universe as observed experimentally by Hubble and Humason in 1929. Hubble proposed that universe is expanding after actually calculating moving galaxies with finite velocity, which is proportional to its distance from Earth. The proportionality constant is called as the Hubble’s constant H. Expansion of the universe has remained a question until the work carried out by Friedmann in 1922 and Lamaitre, in 1927 independently. Hubble recognizes it during his discovery in 1929, and later by the scientific community, which unfortunately went unnoticed. The expanding model can be explained by generalizing the line element so that nonstatic line element can be used to study various aspects of the universe including Hubble’s law. The nonstatic line element derived rigorously by Robertson and Walker (independently) in 1930s is called as the Robertson-Walker line element. The model, hence, developed is called as Friedmann-Lamaitre-Robertson-Walker (FLRW or FRW) model. This model is worked upon and explored so as to understand the universe since its evolution. Since two decades FRW model of the universe remained the area of research for many. Despite the successful explanation of expansion and age of the universe, FRW model suffers from some infirmities. These are related to horizon, flatness problems, etc. of the universe. Recent research activity to overcome the shortcomings is to look for higher dimension cosmology. Kaluza in 1921 and Klein in 1926 laid foundation of higher dimensional cosmology, which is actually the consequence of their attempts for unification of all four types of forces i.e., gravitation with electromagnetism, gravitation with particle interaction, etc. Higher dimension can play an important role to analyze the dynamics of early universe. The detailed study of FRW model will be taken up in the next section followed by kinematics and neoclassical studies. The shortcomings of FRW model will be addressed thereafter. The Kaluza-Klein cosmology is analyzed further so as to enable to develop a ground for the present work. Concluding remarks on the present chapter are given in final section.

II.2

FRIEDMANN-ROBERTSON-WALKER COSMOLOGICAL MODEL In 1930s the universe is observed as non-static by Hubble and Humason,

which required theoretical explanation. Friedman-Robertson-Walker (FRW) model is 18

the model of the universe representing Friedman universe in Robertson-Walker metric. The present work discussed in this section, is four-dimensional model. In order to study FRW model it is necessary to note the assumptions, which are accepted for generalized model for nonstatic conditions.

II.2.1 The assumptions for the FRW model (a) Weyl’s postulate. It states that in a space-time diagram, world lines representing history of galaxies never intersect each other and form a funnel like structure, increase steadily and these three-bundle of nonintersecting geodesics are orthogonal to a series of space like hyper surfaces. Weyl, a mathematician has obtained this postulate in order to understand the regularities in the world lines, which are space-time curve for the locations of galaxies. A particle moving in space can be represented by a curve in space-time plane. Thus, for the particles moving in space-time plane can be represented by several curves as shown in Fig.II.1 and 2. Fig.II.1 shows the curves intersecting each other while other one (Fig.II.2) represents the non-intersecting curves in space-time plane. In the subsequent figure, lines intersecting each other imply collision of particles. If this had been a situation in the universe then Einstein field equations would have been difficult to solve. The real universe does not appear so messy, therefore, the world lines representing galaxies are not intersecting and have funnel like structure as

t1 time

time t0 space space

e a b

Fig.II.1: Bundle of intersecting world lines

e a b

Fig.II.2: Bundle of non-intersecting world lines 19

depicted in Fig.II.2. Physical significance of Weyl postulate can be explained by considering co-ordinate and metric of space-time. For a typical world line in (1+3) dimensions there are three space co-ordinates and one time co-ordinate. For space like hyper surfaces the co-ordinates represented by xμ is considered to be constant. The galaxies lying on different lines are assumed to be like smooth fluid and so xμ represents both time as well as space co-ordinates together (0,1,2,3). The metric tensor for such a system is given by gij. From the orthogonality condition g0μ = 0 where suffix ‘0’ represents time co-ordinate provided μ is other than zero. The geodesic equation is given by:

d 2 xi

dx k dxi i  kl  0. ds ds ds 2

(II.1)

The above equation is satisfied for constant xμ and i =1, 2, 3, Christoffel Tensor of second kind is ikl which is defined as:

ikl 

1 ij  g jl gkj gkl  g    , 2  x k xl x j 

Гμ00 = 0 for μ = 1, 2, 3. This would give

(II.2)

g00 x

 0 . Without loss of generality consider

g00 = 1, The line element, therefore, becomes:

 

ds 2  dx0

 g dx dx  c 2dt 2  g dx dx .

(II.3)

The time co-ordinate in above equation is called as cosmic time, which is an important parameter for studying dynamics of the universe.

(b)

The Cosmological Principle. Another very important assumption that is used / employed for developing

cosmological model is the cosmological principle. It states that at any given cosmic time, the universe is homogeneous and isotropic. This suggests that the surface of hyper sphere of Einstein universe to be smooth; and surfaces with constant t exhibits

the properties of Einstein universe. This principle is the consequence of observations using Hubble’s telescope and recent observations by cosmic background explorer. The cosmic microwave background (CMB) observations allow inferring that the universe appears to be isotropic. In other words, the principle can be stated that at large scale the universe appears to be same from everywhere and there is neither any centre nor any boundary. Weyl postulate itself explains homogeneity and isotropic condition of the universe since, as seen earlier, the lines are not intersecting but parallel to each other, and space-like hyper-surfaces are orthogonal to time lines. With this principle, it is also possible to transform the space-time co-ordinates into cosmic standard co-ordinates representing density, pressure and temperature.

II.2.2 The line element With above assumptions it is possible to have three types of surfaces viz. flat surface, closed surface and open surface. Plain surface has zero curvature. An open surface has negative curvature and closed surface has positive curvature. These three types of curvatures have some constant value, which ensures to hold the properties of homogeneity and isotropy. If the curvature of space differs, it is possible to get another homogeneous and isotropic spaces by appropriate transformations. To get this, let us consider space co-ordinates which are given as xμ, where μ = 1,2,3,4. Here four instead of usual three dimensions are used so as to meet the criteria of cosmological principle [1]. Let us consider surface with negative curvature:

x12 + x 22 + x32 - x 42 = - R 2.

(II.4)

The substitution of x1 = R Sinh χ cos θ, x2 = R Sinh χ sin θ cos , x3 = R Sinh χ sin θ sin , x4 = R Cosh χ gives:

dx12 +dx22 +dx32 - dx42 = R2 [dχ 2 + Sinh 2 χ(dθ 2 + sin 2θdφ2 )] .

(II.5)

Negative sign for x4 suggests that the surface is embedded in four-dimensional pseudo - Euclidean spaces instead of normal 3-d Euclidean space. If we consider r = Sinh χ in above equation then Eq. (II.5) becomes:

 dr 2  d 2  R 2   r 2 (d 2  sin 2 d 2 )  , 1  kr 2 

(II.6)

where k = 0 for flat surface, k = -1 for open surface (negative curvature) and k = 1 for closed surface. The right hand side of above Eq. is simply Euclidean line element scaled by the factor R for flat surface. The scale factor R can depend upon cosmic time. The most general line element satisfying Weyl postulate and cosmological principle is represented by,  dr 2  2 2 2 2 ds  c dt  R (t )   r 2 d 2  sin 2 d 2  , 2  1  kr





(II.7)

where R(t) is the expansion factor or scale factor. Eq. (II.7) is called as RobertsonWalker line element or metric, which is due to the work of Robertson and Walker in 1930s, independently. The next subsection deals with the model obtained using R-W metric, which is often called as FRW model.

II.2.3 The FRW model Friedmann developed the cosmological model in 1922 with the help of relativistic cosmology, which is very much similar to that obtained by Newtonian cosmology. The model, which will be derived using Robertson-Walker metric, would be same as that obtained by Friedmann and so it is called as Friedmann-RobertsonWalker cosmological model. Here, let us have a glance at FRW model, which will enable us to set up a new model with different metric. Consider,

 dr 2 ds 2  c2dt 2  R 2 (t )   r 2 d 2  sin 2 d 2 2 1  kr





 .

(II.8)

Let x0 = ct, x1= r, x2 = θ, x3 = . In a curved space-time, the line element is given as ds 2  gijdxi dx j , where gij is a 44 metric tensor. Therefore, metric tensor for above

element is given by g00 = 1, g11 = -R2(t)/(1-kr2), g22 = -R2(t)r2, g22 = -R2(t)sin2θ. k is a curvature constant, k = 0 for flat universe, k =1 for closed universe and k = -1 for open

universe. As discussed earlier it is necessary to obtain Einstein field equations to obtain the model. The Einstein field equations can be obtained from following Eq. 1 Ri  R i   i  8 GT i , j 2 j j j

(II.9)

where ij = gilglj and gil is metric tensor, Rij - Ricci tensor, ij - Kronecker delta tensor, R - Ricci scalar, Tij - energy-momentum tensor,  - Cosmological constant, G gravitational constant. Ricci Tensor is written as Rij=gil Rlj and can be obtained as follows: a ij

a Rij    ia  b a   b a , ij ba ia bj a x x j

(II.10)

where Γs are Christoffel tensors of second kind and is given by:

1 ab   gbj  gib  gij  , g    i j b 2  x  x  x  

a ij 

(II.11)

generally, ћ = c = 1 is taken at the cosmic scale. R  R11  R22  R33  R44 , a ia 

 log  g xi

(II.12)

(II.13) ,

g =  gij . Field Equations can also be written as:

Gij 

8 G i T   g ij , 2 j c

(II.14)

and Tij is the energy momentum tensor for perfect fluid, and represented as  p  Tij      uiu j  gij p  c2 

(II.15)

ui 

d xi dt

is the 4-vector velocity component such that uiuj = 1 when i = 0,1,2,3

(space -time coordinates); p and ρ are pressure and density of matter distribution of the universe, respectively. Hence, from above equation, the energy momentum tensor Tij = (ρ, -p, -p, -p). This assumption, also, shows that the energy momentum tensor of the universe takes the same form as that of perfect fluid. As per cosmological principle, the universe is homogeneous and isotropic, so it is possible to assume ћ = c= 1 for deriving field equations. Here, let us consider  (the cosmological constant) is zero. In fact Λ is first suggested by Einstein in order to have relation between geometry and matter but later on discarded by him due to its inability to explain expansion of the universe. Derivation of field equations using Eq. (II.9 - II.15) for FRW model without Λ, following equations are obtained as:

(II.16)

and

(II.17)

G11  G22  G33 .

In addition, the conservation of energy requires Tij;j =0 that implies

(II.18)

In equivalent form d d (  R3 )  p ( R 3 )  0 . dt dt

(II.19)

The above relation is similar to energy conservation equation given as dE + p dV = 0 It should be noted that the field equations satisfy energy conservation equation. The above field equations are solved for (i) dust universe for which p = 0 and can be studied for open, closed and flat curvature of the universe, (ii) equation of state (EOS) for perfect fluid as given by p =  ,where,  is constant of EOS. The  determines the relation between pressure and density. If  = 0 then the universe is dust filled or in 24

matter dominated and if  

1 then it is in radiation dominated phase. Substituting 3

EOS in field Eq., the Eq. (II.17) and (II.18) can be rewritten as:

(II.20)

Also, have

(II.21)

General solution is obtained as, ,

(II.22)

where A is a constant of integration, which can be determined from initial conditions. From above Eq. it is possible to obtain the solution for matter-dominated and radiation dominated phases of the universe.

(a)

Matter dominated phase If the universe is matter dominated then EOS for it is given as  = 0,

consequently, Eq. (II.22) takes the form as: .

(II.23)

Scale factors for flat, close and open universe can be determined by substituting k = 0, k = 1 and k = -1, respectively in Eq. (II.23) as given below. 1. Flat universe, k = 0,

2 R 3



2  At  B  3 .



(II.24)

Here, B is constant of integration. If it is assumed to be zero, then R t2/3

2. Close universe, k = 1 R

t

A (1  cos  ) and 2

(II.25)

A (  sin  ) . 2

(II.25a)

Assuming at t = 0, R = 0 in above equations. 3. Open universe , k = -1 R

A (Cosh  1) 2

(II.26)

t

A (Sinh   ) 2

(II.26a)

Again here also R = 0 at t = 0. Plots of all above mentioned three types of models are shown in Fig.II.2.

Open model Cosmic time

Flat model

R  Q  P 

Closed model

Scale factor R Fig.II.3 Types of universe-open, flat and close models of the universe P, Q and R corresponds to present epochs [1].

As seen, open and flat models represent expanding universe, while close model has expansion for some time and thereafter it may contract. Recent observations led to a conclusion that the universe is flat. Radiation dominated phase 26

of the universe is analyzed in the subsequent subsection, which demonstrates the behavior of the universe at its early stages.

(b)

Radiation dominated phase. In Radiation dominated phase, EOS is given by p =  = /3 so,  = 1/3.

Substituting it in Eq. (II.22), gives: .

(II.27)

The solutions of above Eq. are obtained for flat, open and closed universe as given below.

1. Flat universe (k = 0). Eq. (II.27) is rewritten by considering k = 0, and the obtained solution is given by



R 2

At  B



1 2.

(II.28)

If constant of integration is zero, then R t1/2

2. Closed universe (k =1). Consider k = 1 in Eq. (II.27), the solution is of the following form as 1 2 2 . R  A  2 C  t 





(II.29)

If C = t0 then above Eq. can be rewritten as: 1 2 2 . R  A  2  t0  t 





(II.30)

For close universe as t → t0, scale factor R(t) reaches to a constant value, which suggests that the universe tends to become de Sitter universe for the closed model of the universe.

Open universe (k = -1).

For k = -1, the solution of Eq. (II.27) is as given below:



R  2 t  C 

1 A 2.



(II.31)

The constant C can be determined from initial conditions. If C = -t0 then as t → t0 the scale factor becomes imaginary; and if C = t0, R tends to reach steady value for t→t0. From Eq. (II.28 - II.30), it can be observed that the universe at early stages is in radiation-dominated era and in steady state condition. A comparison of matter dominated phase and radiation phase suggests that the present universe is matter dominated rather than radiation dominated. FRW models suffer from flatness problem as it can be seen from equations for scale factor where it was not possible to determine the constant A from the initial conditions. The problems encountered by FRW models in absence of cosmological constant are addressed later in this chapter. Prior to it, cosmological parameters are briefly discussed in following sections.

II.3

COSMOLOGICAL PARAMETERS In order to understand the geometry of the universe, several important

parameters are defined. They play key role for observing and understanding the universe i.e. in the observational cosmology. These parameters are i) Hubble parameter ii) Deceleration parameter and iii) Cosmological density parameter. A short introduction of these parameters is given below.

II.3.1 Hubble parameter Hubble parameter is derived from Hubble’s law, which is identified for velocity-distance relation. In 1929, Hubble established linearity between the velocity of galaxy and red shift of the light received on the Earth, using which distance of a galaxy from the Earth is determined. Hubble’s law states that velocity of a galaxy is directly proportional to the distance between it and earth. It is given by v = H0 D, where v-velocity of galaxy, D - arial distance between galaxy and earth, H0-Hubble’s constant at present epoch. H0 has dimensions of km/s/mpc. The mpc (mega parsec) is the unit for distance on the cosmic scale. Hence, Hubble constant has dimension of 28

inverse of time. It can be, therefore, used to determine age of the universe. This constant is now called as Hubble parameter. For present universe, it is given/represented by

, R(t) is a scale factor. A dot over scale factor R (t)

is the derivative of R(t) with respect to time. The Hubble parameter not only determines age of the universe but also an important parameter as a measure of observable size of the universe. The inverse of H0 is called Hubble’s time.

II.3.2 Deceleration Parameter Deceleration parameter q0 is also required for the explanation of expansion of the universe and it is useful for the expression of density of the universe. q0 is formulated as

and it is the deceleration parameter at present time.

According to recent observations by COBE, WMAP, BAO and several other experiments, the expansion of the universe is accelerating. Hence, deceleration parameter is defined for the measurement of accelerated expansion. The value of q0 should be negative for the acceleration of the universe. Since it is found to be greater than 1/2 for matter dominated FRW close model, the model is decelerating. It is, thus, thought that the present universe is not the closed one. The q0 being negative, the universe is accelerating forever for open and flat models. For flat universe q0  -1, this shows that the universe is accelerating with a constant rate. The acceleration equation is written using Eqs. (II.16) and (II.17) as;

(II.32)

In above EOS, p =  is substituted and q0 is obtained as 1 k   q  (1  3 ) 1  , 2  R2 H 2 

(II.33)

where

(II.34)

(II.35)

at any given time. For flat universe  > -1/3 i.e., universe filled with any fluid is accelerated forever. The q can also be represented in terms of Hubble parameter as, above Eq. can be used for closed or open model also.

II.3.3 Density parameter For defining the expansion of the universe, density parameter is formulated in terms of proper density and Hubble parameter and is given as:



8 G  3H 2

For present epoch, W0 =

(II.36)

8pr0 3H 02

, provided G = 1 is assumed.

Density parameter also determines the geometry of the universe. The density of the universe is same as critical density for flat universe. A critical density is defined as a watershed point between expansion and contraction of the universe [2]. Its expression is derived from field equations of Friedman model for flat universe, and it is given by:

C 

3H 2 . 8 G

Its calculated value is equal to 1.8810-29h02 g/cm3. So the present cosmological density parameter  is expressed as W =

r . rC

If universe is flat then  = 1. If  >1, it is close and stop expanding, while if

 < 1 then it is open and expands forever. Recent observations of WMAP concluded that  is nearly equal to one, which implies that the present universe is flat. If the cosmological constant  is introduced, then the more general expression for  can be expressed that includes  corresponding to  and k, i.e.  and k.  represents

vacuum density parameter, and,  is identified as vacuum energy i.e. (/3H2) [role of

 in cosmology will be discussed later in detail in this chapter]. Further k is expressed as (–k/R02 H02) that corresponds to density of curvature. The total density for the universe is now given by 0 + +k, which should be equal to 1. At early stages of evolution of the universe called visible matter had been dominated by radiation that is radiation-dominated stage. The visible matter in the universe is contained in the galaxies provided the density is approximately equal to 10-31 g/cm3 over the largest scales. This is equivalent to one proton per cubic meter. At this stage, the universe additionally contains neutrinos and gravitational waves, which are called as primordial radiations. Apart from visible matter now-a-day researchers are looking for the dark matter also. Hence, the total density parameter is not only related to visible matter and dark energy but also includes dark matter density, which is yet to be cracked. Dark matter and dark energy had been not the parts of the model during the time of Friedmann. Hence, the field equations did not have expression related to these matter. Although FRW model has become the standard model in cosmology but some observations figure out the inadequacies of FRW model without cosmological constant. Thus, FRW model with cosmological constant is taken as a task for many in the cosmology. The FRW model with cosmological constant is dealt in next section.

II.4

FRW MODEL WITH COSMOLOGICAL CONSTANT FRW model without cosmological constant had suffered from some of the

drawbacks and are discussed below. Two basic problems associated with FRW models, namely, horizon problem and flatness problem are described below.

II.4.1 Horizon problem In cosmology, two kinds of horizons are often discussed. Of these, particle horizon is related to the communication in past; and event horizon is related to the limits on communication in the future. Horizon problem deals with the problem related to largest distance traveled by light to reach an observer since the time of bigbang. The largest radial distance for such a case is given as t

rmax   0

cdt ' . R(t ')

rmax enables to obtain information contained in those particles which can be detected; and the three surfaces in space-time having this radius is called as particle horizon. Physical distance to the horizon at the time t is Dhorizon (t) = R(t) rmax (t). Dhorizon (t) is found to be equal to 2ct for radiation dominated phase for which p = ρ/3 and Dhorizon(t) = 3ct for matter dominated phase when p = 0. The different values of Dhorizon(t) for different phases is called as horizon problem. The problem raises the question on homogeneity of universe, since the cosmic microwave radiation data clearly indicates that we live in a nearly homogeneous universe.

II.4.2 Flatness problem Flatness problem is due to the total density calculated for open, flat and closed universe. Essentially, the total density is considered as  

 where ρc is the critical c

density of universe. Ω at t0 is calculated for present epoch. According to FRW model, Ω should deviate further and further from unity when k = ±1. However, even though it deviates continuously from unity for k = ±1 (retaining unity for k = 0) it reaches unity. Present day observations suggest values of Ω in the range 0.01 - 10. The discrepancy between theoretical and observed values is due to the flatness problem. Inflationary model can provide solutions for the two basic problems. But it can only give information about the early universe. It cannot explain the present universe. However, several efforts directed in past to solve these problems have gone in vain. This has led to the conclusion that the finite cosmological constant does exist and it has some physical significance in the study of universe. The importance of cosmological constant is discussed appropriately later in this Thesis. Since FRW model is understood as the basic standard model, this model containing cosmological constant

 is discussed at length here. II.4.3 Field equations and the model Field equations with Λ are obtained from FRW metric, and they are as given below:

(II.37)

and

(II.38)

G11  G22  G33 .

Above Eqs. can be solved with the help of EOS i.e., by considering p = . The first order differential Equation is obtained as:

(II.39)

The above Eq., first order solution to the field equations, is analyzed for dust dominated or matter dominated universe and radiation dominated universe. For dust dominated model of universe, let us consider  = 0. Thus, Eq.(II.34) now can be written as,

(II.40)

The real solution of above Eq. can be obtained if A = 0. In such case the solutions are     t for open model (k = -1), and R  C exp  t  for flat model, R  C Cosh 3  3 

R  C Sinh

 3

t for closed model (k = 1), where C is a constant.

Above solutions are quite different from these obtained when  = 0. The models for positive value of Λ represent steady state models, which resemble Einstein static models. Model with  < 0 will not give the real solution for closed or open model. Because  = 1/3 for the radiation dominated universe, Eq. (II.39) is simplified as:

(II.41)

If A ≠ 0, then, the above Eq. is rewritten as:

(II.42)

If k = 0, i.e. flat universe, the model is given by, 12

 3A    R Sinh  2 t  C   3  2   

(II.43)

If k = 1, for close universe,  3 A 9 R    2   4

    3  t  C     Sinh  2 3    2 

(II.44)

(II.45)

k = -1, for open universe,  3 A 9 R      42

    3  t  C     Sinh  2 3    2 

In conclusion, for simplicity it is possible to choose C = 0, which implies that the behavior of all models are similar in radiation dominated phase. All are expanding in radiation dominated phase. Flat model appears to have singularity while others are nonsingular models. It can also be observed that the positive Λ plays an important role in the study of the universe, which was initially discarded by Einstein. It is found that the results of FRW with lambda models can be reconciled with observational data [3]. expanding Universe k = 0,  > 0 R(t)

kK==1, 1, >0 >0

k = -1,  > 0 Loitering Universe

t k = 0, 1 or -1,  < 0 Collapsing Universe Fig.II.4: Dynamics of the universe for k = -1, 0, 1 and positive and negative lambda [1] 34

There are several reporting showing the importance of Λ in the study of the universe. The Fig.II.4 reveals the types of universes in presence of lambda. A close look at Fig.II.4, reveals expanding models when k = 0 and k = -1 with positive lambda; and at k = 1 FRW model is the bouncing universe.

II.5

AGE OF THE UNIVERSE FOR FRW MODEL WITH COSMOLOGICAL CONSTANT Age of the universe in cosmology, is estimated from H0-1 i.e. Hubble

parameter at present time. In fact, present universe being matter dominated, age t0  H0-1 and it is calculated as 14.4 Gyr. As per the observational data the calculated age is 13.350.35 Gyr. A small discrepancy in the age of the universe can be removed by considering density corresponding to Λ along with matter density. The relation between age and density parameter is explained here. Two important cosmological parameters (Hubble parameter and deceleration parameter) at present time are given by

and ,

, respectively. Considering R = R0 at t = t0. Eq.

(II.40) in presence of Λ is rewritten as: H 02 

k R02

8 G 0 1 .   3 3

The density parameter m0 + 0 + k0 = 1 is obtained provided m0 

(II.46)

8 G  3H 02

0 .

Other density parameters for Λ and curvature constant are defined in a similar way. Values of matter density and vacuum density (density corresponding to

) are

obtained respectively as m0 = 0.3 and 0= 0.7 approximately as per observational data. Hence, Eq. (II.41) is rewritten as:

1 2q02  H02  Rk2    8 G 0 .

(II.47)

Above Eq. is further simplified as:

1 q 0  (1  3 )m   . 2

(II.48)

1 ω = 0 i.e. for dust filled universe q 0  m   . 2 Current observational data estimates the values of Ωm and ΩΛ as 0.3 and 0.7, respectively. Considering these values for dust filled universe, the value of deceleration parameter is calculated as q0 = -0.55, which confirms that universe is accelerating. Age of the universe can be determined if the present universe is dust filled flat universe. In such case, scale factor R (t) ~ t2/3, and t0 = 2/3H0, which is approximately calculated as 9Gyr. The age of the oldest star is found to be 12Gyr, however. The discrepancy between the observed age of star and estimated age through Hubble parameter implies the existence of finite positive cosmological constant. In presence of cosmological constant, age of flat universe is determined by calculating t0. Age of the universe is determined by calculating

(II.49)

In terms of red shift, consider 1+z=R0/R, where z is red-shift magnitude. The age is given by:

(II.50)

Hence, from Friedmann equations, it is possible to represent Hubble parameter as a function of z. For matter dominated universe H (z) is derived from Eqs. (II.18) and (II.37) which is given as

(II.51)

Thus, H(z) = H0 f(z) , and R/R0 = H0 f(z)/(1+z). Further dR/R0 = -dz/(1+z)2 . This implies that Eq. (II.49) can be rewritten as: 36

   dz . t  H 01     0 [(1  z ) f ( z )] 

(II.52)

Hence,

 Ω 2 t0   Ω 1/2 Sinh 1   3H 0  Ω m

1/2 

  

. 

(II.53)

Substituting the values of m and  as 0.3 and 0.7, respectively the age of the universe for present model is calculated as 12.2Gyrs. From experimental observations by WMAP and COBE, age of the universe is 13.798 ± 0.037 Gyrs. Size of the universe is also an important parameter, which is related to age of the universe. Volume of the universe is normally calculated using Eq.(II.34):   2 2

V  R3  



r drsin d d (1  kr 2 )1/2

00 0

(II.54)

It will be infinitely large flat universe. Although this is a theoretical estimate, but we need to find size of the observable universe so as to facilitate the determination of particle horizon, which can eventually give past information of the universe. In cosmology there are two kinds of horizons namely, event horizon and particle horizon. Latter horizon relates the communication in the past and former horizon is related to the limits on communications in future. In spite of limitations for observable universe, determining Hubble radius rH defined by ctH can find still size of the universe for FRW model. Cosmologically, c = 1 so rH = tH. It is estimated roughly as 2998 h-1mpc [4]. More accurately it can be determined from an expression as discussed below. Let the light ray start from r = r1 at time t1 and it reach us at r = 0 at t0. Therefore, for received light we have 0

- ò dr = r1 = r1

ò R (t ) .

(II.55)

For dust filled universe r1  exp(a(t1-t0)) where, a depends upon Λ. Hence, horizon problem is solved by inclusion of Λ in the field equation. R(t) represents, here, an inflationary era. In such situation the particle at one point will still be casual in contact with another point for very large distance between them. This is possible because expansion rate is an exponential function of time. In a similar manner flatness of the universe is also solved [4]. From Eq. (II.38 - II.40) time evolution of an accelerating universe is described as given below. The area of the universe in accelerated phase for which the universe is flat and has some finite cosmological constant; whereas decelerating phase of the universe may be due to k = 0 and zero cosmological constant as, illustrated in Fig. II.5. It is also observed that there is bulging outward for the accelerating phase so as to have more age as compared to the universe with zero cosmological constant. Further, the accelerating expansion due to vacuum energy causing negative pressure tends to have flat universe. Hence, the model with finite positive cosmological constant solves age problem and flatness-problem. R(t)

Accelerating Universe k = 0,   0 Current size of the Universe k=0,=0 Decelerating Universe

Age -13.5 Gyrs. Age-9.3Gyr -15

-10

-5

Now t0 0.0

t 5

Fig.II-5: Time evolution of the universe-accelerating and decelerating Universe

Although FRW model with finite cosmological constant solves horizon problem, age problem and flatness problem but faces some of the other problems. In next section these problems are addressed

II.6

LIMITATIONS OF FOUR DIMENSIONAL FRW MODEL WITH COSMOLOGICAL CONSTANT Finite cosmological constant, representing vacuum energy density, plays an

important role in understanding dynamics of the universe and it can explain accelerated expansion of the universe. Cosmological constant, also, enables us to predict the age of the universe and relate it to observational data. However, recently some key problems related to cosmological constant i.e. lambda cosmology have been pointed out. These are given below.

II.6.1 Cosmological constant problem (CCP) The value of the vacuum energy density  is given by ~

, which is

related to the momentum of the zero mode of vacuum oscillations. Using the quantum field theory, the energy density ε related to cosmological constant is found to be ~ (Mpl)4 ~ (8πG)-2 i.e., ~ (1018 GeV)4 = 210110 erg/cm3[5]. While from observational calculations, it is found that  ≤ (10-12 GeV)4 ~ 210g-10 erg/cm3 [5]. The significant discrepancy between the two values of total density  is around10120

, which is called as cosmological constant problem (CCP)[5]. Several scientists put

forth a good number of proposals based on string theory, super symmetry, scalar field theory, etc. But most promising solution of CCP is suggested as lambda decay cosmology indicating time dependent cosmological constant that decays with time.

II.6.2 Cosmological coincidence problem. Cosmological coincidence problem has been the problem pertinent to the values of Λ and vacuum density. It is found that the vacuum density is comparable to the matter density at some cosmic time; and reduces drastically with the expansion of the universe. The value of Λ is not reduced accordingly, however. It seems that there may not necessarily be any coincidence between lambda and vacuum density although latter can be expressed in terms of former. Many researchers have suggested quintessence cosmological model [6] with self-interaction, Gaussian potential, etc. in order to solve this problem. While solving both cosmological constant value and coincidence problems, time varying lambda has become the most satisfactory answer. Anthropic solutions and existence of dark energy have also been the solutions proposed by cosmologists, anthropic solution means disturbing structure of the 39

universe, however. While existence of dark energy is still a puzzle and the dark energy content is unknown till date. Cosmologists are still working on it, and this area is also the topic of current research. It is also pointed out that the solution to cosmological constant problem may allow drawing some definite conclusions on the presence of dark energy in the universe, which fills almost 70% of the universe. As discussed previously here it is necessary to review lambda decay cosmology, which is motivated us to choose time dependent lambda for further research work.

II.7

LAMBDA DECAY COSMOLOGY Although cosmological model with constant  explains the expansion of

universe to some extent, but as discussed in the previous sections, a huge discrepancy has been found between observed value and predicted value based on standard model (SM) described in particle physics. It is known as cosmological constant problem (CCP).

To solve this problem, a natural suggestion given by many scientists,

cosmological constant may have large value, and with evolution of time its value decreased, because, since Big-Bang, the universe has undergone several phases. The early phase of the universe is also called the inflationary phase. In this phase, value of cosmological constant is assumed to be very high. Nevertheless, in present phase, value of Λ is calculated to be very small. Bondi had first discussed cosmological constant problem in 1960 [7]. There have been several articles published by quantum field theorists and others (Overduin and Cooperstock [8], and ref. there in) wherein, potential sources like scalar fields, tensor fields, non-local effects, worm holes, etc. are considered as various sources. These articles eventually concluded that cosmological constant is not really a constant but is varying. Overduin and Cooperstock [8] have established the relation between scale factor and cosmological constant. Sahni and others [9] have also stated that vacuum energy density as calculated by Zeldovich [10] is given by εvac.= ρvac . c2 = (Gm2/λ)/ λ3= Gm6c4/ ћ4 where λ=ћ/mc. When density calculated for pion mass compared with density related to Planck mass it is given as 6

1 m    4    pl  1.45ρpl ×10-123. 2  mpl 

Here ρpl is the Planck density. The above expression gives an idea about a huge difference between energy due to Λ and that of Planck energy. Here value of ρpl is calculated as ρpl = c5/G2 ћ ≈ 5×1093 g/cc. With this, during Planck epoch tpl ~ 10-43 sec, it would involve a fine-tuning of one part in 10-123! In some papers this time is calculated as 10-35 sec [11]. This shows that vacuum effects also play important role if the universe is expanding. The huge difference between the values of Λ for the early universe and present universe has been a key factor in the study of lambda decay cosmology. Many projects [12, 13, 14, 15] in which four dimensional model with time varying cosmological constant have been studied extensively. Overduin and Cooperstock have explained time variations of Λ as Λ ~ a-m, Λ ~ t-t, Λ ~ qr, Λ~H2 and Λ ~ H. In these models, they have considered specific values of m, t, and r, which are constants and discussed several oscillating and non-oscillating models. Berman and others [16] have concluded Λ ~ t-2. Arbab [17] has also reviewed cosmological model with Λ ~ H2, Λ ~ q and Λ ~ R2 and studied cosmic acceleration for positive cosmological constant and its implications on cosmological model. A list of time varying cosmological constant is available in the articles due to Sahni [9]. Table II.1 – List of Time varying Cosmological Constant [9].

t-2

(8,15, 16) (8)

A + B exp(- t)

(Beesham and others) (18)

a-2

(Chen &Wu) (29)

exp(- a)

(Rajeev S.G.)(19)

(Canuto and others)(20)

(17, 21) Lima and others

H2 + Aa-

Λ Λ Λ

(8)

f (H) g( , H)

Carvalho, Waga (22) Lima and Maia, (23) Lima and Trodden (24) Hiscock (25) Reuter(26)

The phenomenological models with variations of Λ with scale factor a (t), cosmic time t, cosmic temperature T, Hubble constant H as shown in above table II.1, have been discussed by many researchers [9] and references therein.

II.8

CONCLUSIONS It is observed that FRW model so called ‘standard model of the universe’ is

quite successful. Although it provided positive answers to age problem, horizon problem and expansion of the universe, but still there are several puzzles, which are yet to be solved. Lambda decay cosmology is gained attention recently for solving puzzle of dark energy, dark matter, large structure of the universe, the phases of the early universe etc. In the mean time existence of higher dimensions is also appeared as the subject of interest for carrying out study in unifying all types of forces. During the development of the study of higher dimensions, pioneer work of Kaluza in 1921 and Klein in 1926 in five dimensional physics is of great importance and has broader coverage area in Cosmology. Although Kaluza-Klein cosmology consistently gaining response from research community, but FRW cosmological model based on it, has been first discussed by Chodos and Detweilar [27] in 1984 and later on investigated by many authors in different context. However, Wesson developed a new theory called space-time–matter theory (STM) and discussed 5D models [28]. In the next chapter a review on Kaluza -Klein cosmology is presented in detail, which is actually the topic of interest for the present thesis.

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[25]

Hiscock, W.A. Phys. Lett., 166B, 285 (1986)

[26]

Reuter, M. and Wetterich, C. Phys. Lett., 188B, 38 (1987)

[27]

Chodos A, Detweilar S.: Phys. Rev. D 21, 2167 (1980)

[28]

Wesson P.S., “Space- Time-Matter Modern higher dimension cosmology” 2nd edition, World scientific, Singapore (2007)

[29]

Chen W., Wu Y.S., Physics Review D, 41, 695 (1990)

CHAPTER III KALUZA-KLEIN COSMOLOGY

III.1

INTRODUCTION

III.2

HIGHER DIMENSION

III.3

THEORETICAL BACKGROUND OF

KALUZA-KLEIN COSMOLOGY III.4

KALUZA-KLEIN MECHANISM

III.5

KALUZA-KLEIN COSMOLOGY AND MATTER

III.6

KALUZA-KLEIN COSMOLOGICAL MODEL

WITH COSMOLOGICAL CONSTANT

III.7

III.6.1 Dust dominated model

III.6.2 Radiation dominated model

CONCLUSIONS

REFERENCES

III.1

INTRODUCTION FRW model with and without cosmological model were reviewed in the

previous chapter. To some extent, FRW model with cosmological constant although able to explain some aspects of universe and solve age problem as well as expansion of the universe, but it encountered difficulties while addressing the cosmological constant problem, accelerated expansion, etc. Lambda decay cosmology found quite useful to solve these problems and can be incorporated to unfold the universe. This has been a quite popular among cosmologists and it is successful, also. Several attempts involving four dimensions with variable lambda have been directed to find mysteries of it, and still intensive research is going on. Some of the past attempts have been made towards unification of the forces. In this regard, Nordstorm and Kaluza, inspired by Minkowski’s work related to extension of 3D to 4D geometry, independently have attempted for unifying gravitation and electromagnetic forces. This has made them to think about involvement of fifth dimension. It has paved the way to unite electromagnetic forces and nuclear forces except gravitational force. According to particle physics interaction between particles are due to exchange of field quanta that gives rise to different types of forces. In this regard, Klein worked upon particle interaction with gravitation by involving extra dimension. In his work, he has also argued that higher dimension will reduce with time thus concept of ‘Compactification’ arose. The independent works of Kaluza and Klein have revolutionized the study on the universe; and became very much popular among scientific community, which is termed as ‘Kaluza-Klein’ Theory. Using this theory, a comprehensive study of the universe, especially, at its early times has been carried by many researchers. Five-dimensional analysis has not only changed the geometry of the universe in terms of space-time-matter (STM) co-ordinates but also provided the new type of particles called ‘Kaluza-Klein particles’ thought produced during second phase transition [1]. STM theory, the consequence of Kaluza-Klein theory, has been first put forth by Wesson [2]. Recently, this theory called as Wesson theory of gravitation and 45

will be introduced latter in this chapter. Knowing importance of higher dimensions in cosmology, in the subsequent section the brief information on it is first discussed, later Kaluza-Klein theory and the theory of cosmological model are addressed.

III.2

HIGHER DIMENSION In fact, dimensions play a key role in the study of universe. The standard, hot

and big-bang models of the universe have shown a fairly comprehensive picture of how the universe probably evolved to its present state. Apart from dimensions, the natural constants play the major role in cosmology since; they are basic contents of various physical quantities. In fact, ‘constants of nature’ had first worked upon by Sir Eddington in 1946. Later on, Max Plank and John Wheeler investigated them extensively [3]. Though the constants G, c, ћ, k have been assumed to be equal to one astronomically, even charge (q) has its quantized value, but the issue of how a particle gets its mass has remained unsolved. Other than natural constants, the cosmological constant has, recently, become important to understand the universe. Past evolution of the universe is qualitatively independent of the nature of the cosmological constant (Λ). Later on, it became important to obtain an answer for isotropic and homogeneous universe, its acceleration, existence of dark matter, dark energy, etc. Variations of either natural constants or cosmological constant alone can not provide any answer to the various questions regarding the features of the universe. Geometry of the universe does play a very important role in its study. Since there are many reasons for including higher dimension, it is necessary to discuss the higher dimension to solve the mysteries of the universe. Nordstrom has attempted this for the first time in 1914 and later by Kaluza in 1921 independently. But the work done by Nordstorm went unnoticed [1]. With the discovery of several particles or so-called particle zoo in 20th century, a standard model has been set up in particle physics. This model, a mathematical concept, has been in remarkable accord with observation facts over a broad range of phenomena involving different particles. The interaction phenomena between particles, involve Feynman diagrams and 46

also Dirac’s introduction of antiparticles in relativistic quantum theory have been forced into the framework of quantum theory of fields. Hence, quantum field theory (QFT) has considerable importance, which has led to the String Theory. The development in the QFT aims at studying field theories of more than 4 dimensions so that geometric interpretation of internal quantum numbers of elementary particles along with their energy and momentum can be obtained. With the evolution of QFT and string theory, the idea of higher dimension gained attention. The string theory involves manifold physics with at least 11 dimensions. Some researchers considered n-dimension physics and discussed super symmetry and super gravity and super geometry [2]. Higher dimensional physics is also useful in treating inflationary cosmology as well as cosmology for the early universe. These extra dimensions have been considered small for the early universe, so that they will merge into four dimensions as time increases. This is known as ‘Compactification’. Higher dimension is also useful to explain the black hole theory. Hawking first propounded string theory through his popular book, Theory of Everything [4]. New ideas from string theory, i.e. higher dimension theory, Brane theory, etc. are inculcated during this period. Being higher dimension theory, the Kaluza-Klein theory has been widely applied to solve the mysteries and it is enlightened in the following sub-section.

III.3

THEORETICAL BACKGROUND OF KALUZA- KLEIN COSMOLOGY As discussed earlier, the need of higher dimension is originated from string

theory that led to the idea of manifold physics. With a brilliant insight, Kaluza in 1921 had put forth a theory for unification of electromagnetism and gravitation in five dimensions. Although Einstein endorsed the five-dimensional physics for uniting Maxwell’s equations with gravitation but major credit has gone to Kaluza. In fact, Nordstrom in 1914, in the context of his theory of gravity, has explained five dimension space-time. According to him, 5D space-time equations can split into Einstein equations and Maxwell equations, but unfortunately his work went unnoticed. 47

In the mean time Kaluza published his derivation in 1921 in an attempt to unify electromagnetism with Einstein's general relativity. Both Nordstrom and Kaluza avoided the question of verification, with a thought that extra dimensions do not appear in physics, because they are compactified and unobservable on an experimentally accessible energy scale. Both have assumed that all of the derivatives with respect to the fifth dimension should vanish. Kaluza derived field equations by assuming five-dimensional physics to take place on a four-dimensional hyper surface in a five-dimensional universe (this is a cylindrical condition). In 1926 Klein has introduced quantum theory for unifying particle interaction and gravitation wherein the extra dimensions, which assumed to be microscopically small to be detected experimentally as it may be of the order of Planck scale. It has also been pointed out that extra dimension get embedded itself into four dimensions at late time of the universe. In fact, Klein tried to unite gauge theory of particle physics with general theory of relativity with the introduction of extra dimension, to explain the short-range interaction and long-range interactions together. The idea of uniting gauge theory with gravitation has been due to the calculations of mass of particles and its implications on quantum field theory (QFT). The masses of several particles are still not determined appropriately. Some particles like neutrinos have only upper limits on masses, but proper values of neutrino mass is still unknown. Smallness of mass or electric charge through Feynman graphs has been the consequence of some renormalization process for tiny charges, which acquires bare value due to shielding, resulting from vacuum polarization. Further, these small values of mass can be considered due to bare values just like charges; and these bare values might be some mathematical number i.e. 1 or 4π. The small values could also result by replacing divergent calculations of QFT to something convergent. Divergent sums and integrals in QFT involve more and more momentum, thus, referring to tinier and tinier distances. These conditions can be overcome by including gravitation with QFT calculations. Therefore, Klein has included extra-dimension in Einstein’s general 48

theory of relativity, to bring maturity in the field of the universe and particle physics. The theory proved to be quite elegant and also laid foundations for the study of n-dimensional theories such as 10D superstrings and 11D super-gravity [5]. Among several theories in n-dimension physics, five-dimensional theories seem to be more realistic; and among these Kaluza-Klein theory is most popular. As per Kaluza and Klein, the extra-dimensions are considered to be small. Smallness of dimensions are analogous with a ‘hosepipe’ as shown in Fig.III.1. As discussed earlier, if one looks at a hosepipe from a great distance, it appears to be one-dimensional, but when examined closely, it appears to be a 2-dimensional surface. Similarly, the universe looks like (4+d) dimensional hosepipe where d-extra dimensions are very small and cannot be experimentally determined. Degree of smallness or the scale of the tiny extra dimension (circumference of (4+d) hosepipe) predicted by Klein is of the order of the Planck distance (10-35 m). Although extra-dimensions are small enough not to be detected, their effect can be observed since the birth of the universe. Normal Space-time

Extra-dimension s Fig. III.1 Hosepipe structure of Kaluza Klein type Space –time [3].

In above Fig.III.1 normal space - time dimensions are along the length of hosepipe. The dimensions around the hosepipe are extra dimensions. A section of center is the present time at which the extra-dimensions are extended at both sides of hosepipe, which are too small to be detected experimentally. The idea of hosepipe structure arises from gauge invariance theory in which extra fifth dimension can be understood to be in circle group U (1) [U (1) is abelian gauge group of one dimension]. In particle physics, it is possible to unify all types of interactions with the 49

help of gauge theory [6]. The laws of electrodynamics are derived by imposing local U(1) gauge invariance with that of free particle Lagrangian [6]. Electroweak interactions can be represented by gauge symmetry SU(2) and strong interactions by SU(3). Except gravity, all of the symmetry groups can be brought under single group, such a group is called as grand unified theory (GUT), which has been put forth for the first time in Glashow-Salam-Weinberg (GSW) model, now called as Standard model [7]. Considering SU(3)  SU(2)  U(1) symmetry group combination which led to unification of forces and is also an origin of multi-dimensional theories. Isomeric group SU(3)  SU(2)  U(1) is also one of the important parameter, for studying super-symmetry, super-gravity, etc. This kind of symmetry involves both strong and weak interactions so as to account for bosonic and fermionic matter fields. Super-symmetry and super-gravity are yet to be understood completely. In fact, the idea of extra dimension actually has originated from string theory that pursued mainly for the study of early universe, super-symmetry and super-gravity; and this has been the chief reason for the popularity of Kaluza-Klein theory. It has been possible to unify gauge theories with that of gravitation on the basis of multidimensional theories. One of the ways for unification has been to use five-dimensional tensor rather than four-dimensional tensor, a gauge group in 4-dimensional space. Electromagnetic interactions being consist of gauge group of U(1) allows to form an isomorphic structure with one dimensional S1 over the Riemannian base-space of R4, over which the gravitation has been defined. This way of unifying electromagnetism and gravitation has been discussed by Trautman [8]. In other words, the space R4  S1 is a space of five dimensions. When such space is compactified then effective four-dimensional field is observed [9]. When extra dimension is curled in the form of circle of so small radius such that a particle travel a very short distance along the axis and returned to the same point, the distance travelled by the particle has normal space-time dimensions. Such conditions for compacting extra dimension are possible with the help of Yang-Mills theory wherein generalized gauge theory is discussed. 50

Apart from geometrical interpretation, it is very much necessary to understand physical interpretation of the theory. Although consideration of extra dimension in four dimensional metric and deriving Einstein field equations (EFE) have mathematical elegance, but physical explanation of the solutions obtained for field equations have much importance in the study of cosmology. Such ingenious way adopted in Kaluza-Klein cosmology helped in geometrizing electromagnetism and gravity. Solution of Einstein field equations by assuming vacuum conditions i.e. Rij =0 (Rij is the Ricci Tensor) has useful implications on the study related to cosmological behavior in absence of all physical fields in four-dimensional theory. However, in five-dimensional theory such as in present case, obtaining solutions for five-dimensional EFEs is to understand and study the cosmological behavior in the presence of gravity and electromagnetism thereby encapsulating all known physical fields of the present time. The next section Kaluza-Klein Cosmological model pioneered by Kaluza mechanism to bring together Einstein theory of gravitation and Maxwell theory of electromagnetism using extra dimension is discussed [10].

III.4

KALUZA-KLEIN MECHANISM Kaluza-Klein theory is essentially five-dimensional theory in general relativity

but it has two main constraints related to compactification and geometry of space-time. The condition of compactification introduced by Klein, has been explained in the previous section, whereas regarding geometry, Kaluza has introduced ‘cylinder’ condition. In this case, for reducing complexity of algebra smallness of fifth dimension considered and partial derivatives with respect to fifth co-ordinate are set to zero. Inclusion of the 5  5 metric tensor gAB in the line element has led to 0 - 3 as space-time co-ordinates and fifth one can be either gauge potential [‘internal’ co-ordinate in applications to particle physics] or any other quantity which can be space like, time like or both. In this regard, an extensive research carried by a number of researches has widened the scope of Kaluza Klein theory. At present let us have a 51

look at the case in which space-time co-ordinates gij = gij(x), g44 = -2(x) where x is the five-dimensional co-ordinate system. As per the constraints, space-time co-ordinates should be independent of fifth coordinate, so that the differentiation of these co-ordinates with respect to fifth one should be zero. Consequently, the transformation of fifth co-ordinate is of the form x5  x5’ = x5+ f(x), where f denotes any arbitrary function. Analogous to four vector of general relativity, here fivedimensional Ricci Tensor Rij can be calculated from 5D line element. The Einstein tensor and field equations can be derived subsequently and many researchers have obtained the solutions for it. A number of researchers have obtained the solutions for vacuum or empty form of field equations Gij =0 since Kaluza and Klein time. In gravitation problems, assumption of metric element persistent to gravitation aspects depends on choice of coordinates; whereas, it is a choice of gauge in particle physics. Fifth element may be included as either fifth coordinate or a gauge in the metric tensor, which is written in the form as given below:  g ij ĝ ij   g 5j 

g i5  . g 55  

(III.1)

As discussed earlier, fifth dimension should be compactified in the form of a circle having such a small radius that fulfills unobservable conditions at present time. Consequently, R4 S1 topology is accepted wherein each space dimension can be consisted of circles of extra-dimension radius. If 5D line element is given by: dS2 = gij dxi  dxj where, i, j = 0 - 4 and i  j.

(III.2)

i = j = 0 - 3 represents normal space-time, whereas, fifth one can be related to gauge field or scalar field or vector potential. The idea of selecting scalar field  as fifth coordinate has been originally due to Kaluza. He has combined it with vector potential A in electrodynamics and explained importance of extra dimension [11]. 52

Therefore, Eq. (III.1) can be rewritten as:  g  k 2 A Aj k 2 A  i i , g AB   ij  2 2     k Aj

(III.3)

where k is a coupling constant. Consequently, line element can be represented as, dS2 = ds2 + k2(dx4+Aidx) 2,

(III.4)

where ds2 is the line element for four-dimensional space-time. Although in the Eq. (III.3) k = -1 but it can have +1. The line element will be a particle like at k = -1 and, it will be a wave like at k =+1. Particle like behavior being discussed a number of times; negative sign has been introduced in Eq. (III.3) In fact, Einstein field equations can be derived from Einstein Tensor i.e. GAB is given by GAB= RAB - ½ R gAB , where RAB is the Ricci Tensor for 5D line element and R is the Ricci Scalar. It is also known that GAB = k TAB, where TAB is the 5D energy momentum tensor, related to density, pressure and electro-magnetic tensor (known as Faraday’s tensor). Faraday’s tensor is given by: F  4  A A  J , F   , c x x x

electromagnetic tensor also follows the identity as:

F F F     0, x  x x which is similar to Bianchi identity. Imposition of the constraints in which cylindrical conditions are assumed, allows derivatives of co-ordinates with respect to fifth co-ordinate to zero. Hence, the field equations reduced to only 15 simplified equations. Field equations are derived assuming ћ = c = 8πG =1 as per cosmic principle. Consideration of vacuum conditions GAB = 0 means equivalently RAB = 0. Using 53

Eq.(III.3) and RAB = 0 following relation is obtained [12].

(III.5)

where D’ Alembert operator, defined as □ = gij ∂i ∂j, is also called as wave operator, and  is differential operator in space. Tij is electromagnetic energy-momentum tensor [energy-momentum tensor related to density and pressure is zero in the present case], which is given as below:

Tij 

1 g F F   F α Fβj . i 4 ij

(III.6)

Other field equations are derived as follows:

i Fij  3

i



Fij

and

(III.7)

(III.8)

Eq. (III.7) represents the modification of electromagnetic field, whereas, Eq. (III.8) can be thought of as wave dependent nature. The Eq. (III.5) gives Einstein equations in 4D general relativity but with the energy momentum tensor derived from fifth dimension on right hand side. Eqs (III.5), (III.7) and (8) represent field equations, which can together account for gravity, electromagnetism, and a scalar field. In this way Kaluza explained the mechanism of unification of gravity and Electromagnetism. Here scalar field is also considered to include matter field [11]. If -2 = -1 together with k = 8πG/c2, then field equations reduce to equations which are read as:

Gij 

k 2 2 8 G Tij  Tij and 2 2 c

(III.9) 54

i Fij  0 .

(III.10)

These equations are similar to Einstein and Maxwell’s equations in four dimensions, which are 5D Einstein equations in five dimensions. Two consequences of above equations are (i) Maxwell equations are part of Einstein Equations, which geometrize electromagnetism and (ii) It is possible to induce matter relating to geometry. The latter facilitated matter to be observed in the geometry of the universe provided scalar field is not equal to -1. It is also observed that if A= 0, then field equations obtained are similar to that of Brans-Dicke scalar tensor theory. Although Kaluza’s ingenious way to unite gravity and electromagnetism looks to be very attractive but it has some shortcomings. The smallness of extra dimension has to be understood yet, and to be accepted if its effects can be found experimentally while cylindrical conditions seems to be quite unnatural. Thus, initially not much work has been progressed on this until Klein’s attempt. According to recent observational data, the universe is isotropic and homogeneous; concurrently, assumptions of spherical conditions in five dimensions are more feasible. Among many theories 5D FRW model is the most popular among cosmologists due to its quite simplicity. Interestingly, 4D FRW model is quite successful in explaining some of the key features of the universe to some extent. Therefore, it can be extended to higher dimensions, which can fulfill the requirement for compactification without the need of Kaluza constraints. The Kaluza-Klein cosmological model is first reviewed and then discussed its various aspects in the subsequent section.

III.5

KALUZA KLEIN COSMOLOGY AND MATTER Let us have a glance at the induced matter theory and Kaluza-Klein cosmology

before discussing the cosmological model. Wesson with the help of Kaluza-Klein theory has first put the idea of induced matter theory forth in 1997 [13]. This theory acted as a bridge between geometry and matter. Although gravity has been related to matter but in this case gravity and matter have been interrelated with geometry. 55

Further, it has been argued that the mass is supposed to be length, as fundamental constants c, G and ћ are used for transforming the physical dimensions so that they can be expressed either space-like or time-like. In this regard, density and pressure are represented in the form of geometrical quantities, as the case in 4D FRW model. The choice of mass as a fifth dimension, facilitates two possibilities namely (i) to know the properties of matter from solutions of the Einstein field equations derived from 5D Riemannian geometry and (ii) it is not necessary to have any assumptions regarding topology of fifth dimension [11]. Wesson’s theory uses 5D metric tensor with the signature of the diagonal as (+ - - - ) i.e., density and pressure have their usual signs but fifth dimension can have either + or – sign. The 5D field equations are derive from vacuum conditions i.e. RAB = 0 so also 5D Einstein field equations (EFE) are obtained as GAB = 0 although 4D EFE with matter is not zero and it is embedded in five-dimensional tensor given by Gij =8πTij, where I and j represent indices for four basic dimensions. It can be observed here that the effective four-dimensional Einstein tensor is an induced energy-momentum tensor Tij, which contains the properties of matter. The 4D tensor embedded into 5D tensor has been the direct consequence of Campbell’s theorem. This theorem states that any analytic N-dimensional Riemannian manifold can be locally embedded in an (N+1) dimensional Riemannian manifold Ricci flat (RAB = 0) [14]. Generalization of 4D field equations can be obtained in n-manifold physics using the above theorem. The field equations of Wesson’s theory are obtained with the help of Schwartzchild metric since, it has not much been worked upon in higher dimensional physics. Mostly five-dimensional FRW model considers with time like extra dimension but in some cases, space like extra-dimension is also considered. Here the method of Wesson is revisited to understand the way of involving fifth dimension in the metric, which will enable us to think in the direction of 5D FRW model. Consider 5D Schwartzchild metric as given below:





dS 2  e dt 2  e dr 2  r 2d  2  e dl 2 ,

(III.11) 56

The time co-ordinate x0 = t, space-co-ordinates x1, x2, x3 = r, ,  and d2 = d2 + sin2d2 in above metric element, which are similar to 4D metric. The new co-ordinate has been introduced as x4 = l. The metric coefficients,  and  depend upon l and t. Four independent Einstein field equations given below are Einstein tensor [12].

(III.12)

(III.13)

(III.14)

G11  G22  G33 and

(III.15)

Here a dot over coefficient represents the time differentiation and a dash over it is the differentiation with respect to length l. If it is related to four-dimensional field equations, as argued previously then first of all energy momentum tensor corresponding to 4D metric for perfect fluid can be the usual one represented as follows,

Tij   p    uiu j  pgij ,

(III.16)

where ui = dxi/ds , and T00 = , T11= -p for the density and pressure. It should also be noted here that the derivatives with respect to l and the terms depending coefficient  57

have been considered to know the effect of extra dimension. Hence, field equations in 5D with energy momentum tensor have been given by:

(III.17)

(III.18)

Eq. (III.17), (III.18) and Eqs.(III.12) - (III.15) put together, the following Eqs. obtained as:

(III.19)

(III.20)

(III.21)

(III.22)

If  = 0,  = log t and  = -log t then obtained density and pressure will be dependent on fifth dimension and time. The equations for density and pressure from Eq. (III.19) and (III.21) have been obtained as 8 = 3/4t2, and 8p = 1/4t2, respectively. These solutions enabled Wesson to induce matter for which mass can be obtained provided proper radial distance R = exp.(-/2r) has been introduced. Mass of the fluid in such case has been given by 4/3(+3p) R3, where ( + 3p) has been 58

the gravitational density. The latter has been also the consequence of the acceleration of the perfect fluid filling the universe. It should also be noted here that ( + 3p) > 0. Since p = /3 (represents radiation phase in the universe), the obtained solutions represent radiation dominated phase. The solutions have been modified to switch over to induce matter theory. As per Wesson theory (assuming e = l2 and considering Ponce de Leon (1988) [15] solutions), it has been concluded that the said model is identical to 4D Einstein-De Sitter dust model. Wesson, also, has carried studies pertinent to solitons (massive objects which have no gravitational effects), of 5D electro-magnetism and astrophysics applications, etc. Chatterjee et al [16] have also carried similar work and explained homogeneous model. The extensive research efforts have been focused on space-time-matter theory, which forms the future research since, it can provide an explanation for not only accelerated expansion of the present universe but also can explain the effect of extra dimension at early stages. Although Wesson 5D theory is quite attractive and it has several implications, but a much-simplified theory with 5D FRW metric is proposed so as to understand the behavior of the universe. Wesson theory has been implemented with the assumption of RAB=0. It is, however, necessary to consider stress-energy conservation i.e. Tij;j =0 in cosmology. Without considering this fact, Wesson theory explains impact of extra dimension that resulted in the form of induced matter. Wesson theory (STM theory) needs to be generalized and divergence of Einstein tensor has to be considered before implementing STM theory to cosmology. Past attempts made in this regard has opened some latest issues, which attracted the attention of researcher. Since the universe is isotropic and homogeneous as per cosmic principle, Kaluza-Klein theory can be discussed with the help of FRW metric. Such attempt could help to set up the model with different kinds of matter. Using 5D FRW (normally identified as Kaluza-Klein metric) it is possible to analyze the model with strings, quark matter apart from perfect fluid filled in the universe. By this time it is understood that cosmological constant is an important parameter in cosmology and it represents the dark energy density. Hence, a study on Kaluza-Klein cosmological 59

model mitigated the retrieval of some information about dark energy, which is addressed later in this thesis. The set up of Kaluza-Klein cosmological model for 5D FRW model is discussed in the next section.

III.6

KALUZA KLEIN COSMOLOGICAL MODEL WITH COSMOLOGICAL CONSTANT The Kaluza-Klein mechanisms discussed in previous section emphasize the

wide scope of the theory. With such a motivation let us consider 5D FRW metric with an extra dimension as given below:

éì 2 2 2 2 êï dr ds = -dt + ( R (t )) êí + r 2 d q 2 + sin 2 q d j 2 êï 1- kr 2 ëî

(

)

(

)

üù ïú 2 ýú + A (t ) d Y 2 , ïú þû

(

)

(III.23)

where, k is a curvature constant equal to 0, 1 , -1 for flat , closed and open universe, respectively.

R(t) and A(t) are fourth and fifth dimensional scale factors, and  is

fifth dimension which is similar to dl of previous section. The four-dimensional coordinates in above equation are given by x0 = t, and x1, x2, x3, x4 = r, , , , respectively. Metric elements are given by g00 = -1, g11= (R(t))2/(1-kr2), g22 = R2 r2, g33 = r2 sin2 , g44 = A2(t). Consider g= g00 g11 g22 g33 g44, to obtain field equation,

 r4  2 2 g   R6   sin  A ,  1  kr 2      2   r  sin  A .  g  R3  1  2 2  1  kr   





(III.24)

(III.25)

Christoffel symbol of second kind is written as,

a ij 

1 ab   gbj  gib  gij  g    . i 2  x j  xb   x

For the present metric it is calculated as,

kr , G1 11 = æ ö 2 ç1- kr ÷ è ø

2 = 1 = G 2 = G3 = G3 , , G12 21 13 31 r

G122 = -r (1- kr 2 ) ,

, G133 = r sin 2 q (1- kr 2 ) ,

2 = -sin q cosq , G3 = G3 = cot q , , G33 23 32

other Christoffel symbols are zero. Ricci tensor for the above line element is calculated as follows. It is known to us from previous chapter that,

¶Gaij ¶Ga b a b a Rij = + ia - Gij G ba + Gia G bj , a j ¶x ¶x

(III.26)

(III.27)

(III.28)

(III.29) R33  sin 2 R22 ,

(III.30) 61

(III.31)

It is possible to get the solutions assuming RAB =0. From Eqs. (III.27) and (III.31) inflationary solutions are obtained. Divergence of Einstein field equations is the necessary condition and it should be satisfied in cosmology, however. To derive Einstein field equations, consider Einstein field tensor as given below: 1 Gij  Rij  Rg ij . 2

(III.32)

In above Eq. R is called as Ricci scalar and it is given by: R  R11  R22  R33  R44  R55 ,

where Rii  g iσ Rσi . Using Eq. (III.27) to Eq. (III.32) field equations are obtained as,

(III.33)

(III.34)

G11  G22  G33 ,

(III.35)

Above field equations can be solved for different matters, which are analyzed later in the present thesis. 62

III.6.1 Dust dominated model Present work is aimed at dust dominated study of the universe for which p = 0. i.e. the pressure less universe. According to Einstein tensor, we have Gij   8 GTji   g ij ,

(III.36)

where T i     p  uiu j  pg i j j

is the stress-energy tensor; and ui and gij are defined

in usual way. The uiuj =-1 in the present case. For dust universe or matter dominated phase one have Tij = (-, 0, 0, 0, 0), so Eq. (III.33) - (III.35) will be written as:

(III.37)

(III.38)

(III.39)

From Eq. (III.38) and (III.39) following simplified Equation is obtainable.

(III.40)

Since Eq. (III.39) is the equation of one variable, it can be solved easily. Simplified version of Eq. (III.39) can be represented as:

(III.41)

Solution of above Eq. is obtained as

R2 =

æ 2L ö æ 2L ö + A1 exp.çç t ÷÷ + A2 exp çç t ÷, L 3 ÷ø è 3 ø è

(III.42)

where A1 and A2 are constants of integration. The above Eq. is the general solution of Eq.(III.41). The solution is similar to that obtained by Gron [17] for inflationary cosmology. If k = 0 (flat universe), A1 =1/2 and A2 = -1/2 are assumed then Eq. (III.42) takes the form,

1 2

 2 R   Sinh t . 3  

(III.43)

Similar equations can be obtained for closed and open universe. It is necessary to solve Eq. (III.40) so as to obtain an expression for A. The Eq. (III.40) is the non-homogeneous second order differential equation however. Therefore, to solve it power law relation between A and R is assumed, the consequence of the cosmic principle where the universe is assumed to be isotropic. In such a case A = Rn , where n is a constant. For isotropic nature of the universe, Shear scalar 2 = 0, where:

Hence, A(t) for flat universe is expressed using Eq. (III.44). n  2  2 A(t )   Sinh t , 3  

(III.44)

Similar expression can be obtained for closed as well as for open universe. Density parameter for dust-dominated phase is obtained from stress energy conservation 64

relation as described below:

 

So    0 AR3

1

 2   (t )   Sinh t 3  

(III.45)

  n  3 2

(III.46)

(III.47)

It can be observed from Eq. (III.45) - (III.47) that density decreases faster with time than scale factors. The value of n is still unknown, however. It could be a criterion for further research in higher dimensional cosmology.

III.6.2 Radiation Dominated model Since p = /3 for radiation dominated phase, field equations are rewritten by considering energy-momentum Tensor Tij = (-, p, p, p, p) for i = j and zero for i j. Assume 8G = 1 in the back ground of cosmic principle, so field equations are given as:

(III.48)

(III.49)

(III.50)

Divergence of Einstein field Eq. implies:





(III.51)

 i 1 i i i  R j  Rg j   Tj   g j  0 2 ;j  ; j

Conservation of Energy-Momentum tensor gives us following relation:

   0 R

4

(III.52)

4 A 3. 

(III.53)

The field Eq. (III.49) and (III.50) give,

(III.54)

Above Eq. can be solved provided we know the relation between A and R. As discussed in previous sub section, let us assume A = Rn. Consequently, above Eq. can be modified to have following general Eq.

(III.55)

Solution of above Eq. is obtained as: 1

 2kt 2  2 R(t )  ct  d   . (1  n)  

(III.56)

The above solution does not involve cosmological constant. However, if we consider Eq. (III.52) and simplify it with the help of field equations, one can obtain: 66

(III.57)

(III.58)

(III.59)

Substitution of Eq. (III.49) in above equation resulted,

(III.60)

Let,

m

2k  2(n  3)  3 2(n  3)  3 2(n  3) , k1  , k2  3 9(n  1) 9(n  1)

Eq. (III.60) can be modified as:

(III.61)

First order differential equation given below, is obtained when m = 1 as,

(III.62) And an expression for R (t) is given below, 2

2C0  k1  k 4(n  3) R     Sinh (t  c)  1 k2  k2  9(n  1) k2

(III.63)

It can be rewritten as,

R 2  a Sinh

k 4(n  3) (t  t0 )  1 , 9(n  1) k2

(III.64)

here, 2

2C0  k1  a    , c = -t0. k2  k2  Above Eq. represents non-oscillatory expanding model of the universe. As t  , Sinh(t-t0)  0. Hence, at late time it reaches to a steady state model. Other parameters can be determined from field equations with the assumptions, which are necessary for getting an expression for extra dimension. The previous as well as this section reveals that the model with constant lambda () is steady state model at late times. As per observations the universe has accelerating expansion however, since Eq. (III.56) does not show the effect of cosmological constant, stress-energy conservation equation is solved to know the role of .

III.7

CONCLUSIONS Kaluza-Klein (K-K) cosmology is reviewed extensively. Interestingly, K-K

cosmology has versatile feature. It cannot only be applied to higher dimensional studies but it can also be implicated for the study of the universe at early as well as late times. Kaluza theory, with the help of an extra dimension, facilitates unification of electromagnetism and gravitation. Although the idea of multi-dimension originated from string theory but Kaluza-Klein theory (essentially five-dimensional theory) is none-the-less has wide scope. An important application of it is to include matter as an extra-dimension, which involves gravitation. An important consequence is the STM theory, ingeniously attempted by Wesson. Wesson theory, provides an insight to the STM theory, is an example where 68

Kaluza-Klein theory is used to include matter. The latter theory has

identified as

four-dimensional with an extra dimension. Since Wesson has brought forward the concept of induced matter, it can be applied to the model with strings and strange quark matter. Although Wesson theory has elegant explanation in describing STM concept, but it is necessary to consider stress energy-conservation equation as divergence of Einstein tensor should be equal to zero so as to comply with cosmology. Moreover, it uses modified form of Schwartzchild metric, which complicates the derivation of field equations and their solutions. Assumptions of the simpler form of solutions have enabled studies on cosmological parameters, which have been adopted in Wesson Theory. Nevertheless still there is immense scope for STM theory. Since the universe is isotropic and homogeneous as per cosmic principle, the five-dimensional FRW metric can be considered to derive field equations and find solutions along with other parameters. Five-dimensional FRW metric, often called as Kaluza-Klein metric, has been quite popular for understanding the different aspects of the universe. Kaluza-Klein cosmological model studied with constant lambda for both dust dominated and radiation dominated phases has led to the expanding model, in accordance with observational data. The model with constant lambda becomes steady state model at late times, however. According to observational data, the universe is expanding as well as accelerating. Recent observations also indicate the existence of dark energy, which is not possible with the present model with constant lambda. The present model cannot solve the cosmological constant problem (discrepancy about the unexpected small value which is not solved yet). It is, hence, necessary to consider model with time varying lambda. A commanding work done in this direction provided a track for further studies on time dependent lambda and its implications on Kaluza-Klein model. The model with time varying lambda is studied in detail in the next Chapter.

REFERENCES [1]

Doru Sticlet , Master program of Theoretical Physics Student Seminar in Cosmology Titled ‘ Phase Transitions in the Early Universe’ Drd. Maarten VAN DE MEENT

[2]

(www.physics.umd.edu/courses/)

Wesson P. S. ‘Five dimensional Physics: A Classical and Quantum consequences of Kaluza Klein Cosmology’ World Scientific Publishing Co. Pvt. Ltd. (2006)

[3]

Penrose R. ‘The Road to Reality’ Vintage Publishers (2005)

[4]

Hawking S.W. ‘Theory of Everything: The origin and Fate of the universe’ Jaico Publishing House (2007)

[5]

Wesson P. ‘Space Time Matter: Modern Higher Dimensional Cosmology’ Singapore World Scientific Publishing Co. Pvt. Ltd. (1999)

[6]

Halzen F., Martin A.D., ‘Quarks and Leptons’ John Wiley & Sons Inc. (1984)

[7]

Salam A., Rev. Mod. Phys. 52, 515, (1980)

[8]

Trautman A., Rep. Math. Phys. 1, 29 (1970)

[9]

Wikipedia, the free encyclopedia ‘Kaluza Klein Theory’ (2000)

[10]

Pope C., ‘Lectures on Kaluza Klein theory’, http://faculty.physics.tamu.edu/ pope/ihplec.pdf)

[11]

Overduin J.M., Wesson P.S. Preprint arXiv: gr-qc/9805018v1 (1998)

[12]

Lessener G., Phy. Rev. D 25, 3202 (1982)

[13]

Overduin J., Wesson P.S. Phy. Rep., 283, 303 (1997)

[14]

Romero C., Tavakol R., Zalaletdinov R., Gen. Rel. Grav. 12, 2411 (1996)

[15]

Ponce de Leon J., Gen. Rel. Grav., 20, 539 (1988)

[16]

Chatterjee S. Banerjee A., Bhui B, Phys. Lett. A, 149, 91 (1990)

[17]

Gron O., Astron. and Astrophy., 193, 1, (1988)

CHAPTER IV KALUZA KLEIN COSMOLOGICAL MODEL AND IMPLICATIONS OF TIME VARYING COSMOLOGICAL CONSTANT ON THE MODEL

IV.1

INTRODUCTION

IV.2

THE FIELD EQUATIONS

IV.3

COSMOLOGICAL MODEL WITH TIME VARYNG COSMOLOGICAL CONSTANT IV.3.1 Cosmological model with   

IV.5

IV.3.2 Cosmological model with    1

R R

IV.3.3 Cosmological model with   

IV.3.4 Cosmological model with   

IV.4

R2 1  2 2 R R

81 81

NEO-CLASSICAL TESTS IV.4.1 Proper distance D(z)

IV.4.2 Luminosity distance

IV.4.3 Angular diameter distance

IV.4.3 Look back time

KALUZA-KLEIN COSMOLOGICAL MODEL IN MATTER DOMINATED AND RADIATION DOMINATED PHASE

IV.6

IV.5.1 Matter dominated phase

IV.5.2 Radiation dominated phase

DISCUSSION AND CONCLUSIONS

REFERENCES

IV.1

INTRODUCTION Consequent to the general theory of relativity having being tested

experimentally and found successful; Einstein and De-Sitter put forth models for the universe. These models, however, had unsatisfactory features and could not explain the expansion of the universe detected by Hubble. Einstein, initially, introduced the Λ; but later discarded it, describing it as “a big blunder”. Since the past three decades the cosmological constant Λ has received considerable attention of researchers attempting to resolve some of the most important unsolved problems of the universe. The cosmological constant represents vacuum energy density, actually originates from the concept of zero-point energy in quantum mechanics. The status and consequences of the cosmological constant reviewed by Weinberg [1] have evoked much interest among scientists to look for its physical significance. The value of the Λ has undergone a crisis due to a discrepancy, of about ∼10120, with respect to observed [2, 3] and predicted value using quantum field theory in particle physics. Barrow and Shaw as well as Caroll and Press [4, 5] have pointed out this discrepancy. This has been termed as the cosmological constant problem (CCP). The solution to this problem has been searched assuming different types of time-varying cosmological constants including the investigations by Ratra and Peebles [6] Dolgov [7], Padmanabhan [8], Sahni and Starobinsky [9]. A number of 4D cosmological models with similar variations of cosmological constant have also been investigated during the last two decades by Pavon [10], Carvalho et al [11], Lima and Trodden [12], Carneiro and Lima [13]. The study on CCP, thus, revealed that the cosmological constant has not really been constant but varying. Many researchers suggested that CCP might be due to the distribution of matter in the universe. A number of researchers have investigated the CCP and its consequences on cosmology in various contexts [14 - 23]. The relevant literature has mainly emphasized on dependence of cosmological constant on scale factor, which explains the dynamic behavior of the universe. However, it proved to be inadequate to explain features of early universe. Carvalho et al [11] and other researchers [24 - 26] have discussed the four dimensional cosmological models with

Further, their models have depicted observational constraints of some physical parameters (age parameters, density parameters, etc.) for certain ranges. Current 71

observations about distribution of matter in the universe have suggested that it consists of 4% visible matter, 22 % of dark matter and 74 % of dark energy. The existences of dark energy, dark matter and its constitution have been still a mystery, which could be resolved by knowing the nature of Λ as it represents vacuum energy density. The following pie diagram shows distribution of matter in the universe [27].

Fig.IV.1 Pie diagram for distribution of matter in the universe [27]. Surprisingly 4% of the universe is visible. Dark energy and dark matter, fills 96% of the universe, have been the major areas of research. They have been thought to be responsible for accelerated expansion and large structure expansion. Importance of dark energy and dark matter along with the current research on it will be discussed later in this thesis.

Since the cosmological constant represents vacuum energy

density, it is also identified as dark energy. Suggestively, decay of dark energy may be responsible for accelerated expansion [28]. According to previous discussion in this section, the cosmological constant is varying, but in 1990 Chen and Wu [29] endowed with a new conception. According to them at the cosmological constant varies as 1/R(t)2, where R(t) is the four dimensional scale factor of FRW line element. Four-dimensional cosmological constants with time varying Λ had been found inadequate to explain the behavior of the universe at its early stages. Concurrently, the theory of higher dimension, first put forth by Kaluza [30] and Klein [31] independently, evolved for unifying gravitation with electromagnetism and gauge theories in particle physics. Kaluza-Klein cosmology has been the focused area of interest to many researchers for solving mysteries of the universe; especially, for the study of early universe. Some aspects of Kaluza-Klein cosmology have been highlighted by Srivastav [32], Overduin and Wesson [33], Servant and Tait [34]. Chodos and Detweiler [35] have investigated the early phases of the universe by invoking extra dimensions. Recently, 5D field equations and their solutions have been 72

turning into an area of interest in the study of particle interactions. Five-dimensional cosmological models with modification of matter have been set up by Chatterjee [36], Fukui [37] and Chatterjee et al [38] to know the impact of Kaluza-Klein theory. Chatterjee et al [36] have demonstrated the importance of extra dimension. The extra dimension has a marked effect on time-temperature relation of the universe. Therefore, our universe appears to cool relatively slower in higher dimensional spacetime. The observations of COBE have concluded isotropic nature of the universe. In order to explain observed isotropy of the universe, Gron [39] has investigated the vacuum, radiation-dominated, matter-dominated models with the help of Wesson’s gravitation theory. Subsequently, explained observed constancy of the rest mass of isolated system. Gron [39] and Banerjee et al [40] have obtained fourth and fifth dimension scale factor leading to expressions for variable mass. They have also showed that the rest mass tends to have constant value with the time evolution. Importance of Kaluza-Klein cosmology, extra dimension, cosmological constant and Wesson theory of gravitation, already discussed in previous chapter, have been the motivation for us to take up further study in this direction. Khadekar et al [41] have investigated the solutions to Kaluza-Klein cosmological model (flat universe) with different forms of time varying cosmological constant i.e.

, ~,

where a is scale factor. They have concluded that these models are dynamically equivalent for a spatially flat universe. The solutions obtained for the model are for flat universe. Singh et al [42] have explained five-dimensional model with in higher dimensional space-time for matter–dominated phase as well as radiation-dominated phase of the universe. Most of the models have inferred age and acceleration of expanding universe with certain constraints. Nevertheless, investigation on the role of extra dimension and lambda decay in its generalized form and their implications on cosmological model can explain the behavior of universe and its time evolution, and also can be important criteria for in depth study in cosmology. Implications of the decay law for cosmological constant and higher dimensional cosmology have been investigated by El-Nebulsi [43]. He related the decay law with extra dimensions to explain cosmological wormholes. 73

From above discussions, it can be concluded that the lambda decay cosmology with Kaluza-Klein theory may provide ground for the study of those features of the universe, which are not yet understood properly. Motivated by these facts, a fivedimensional Kaluza-Klein cosmological model with different forms of time varying cosmological constant, as a test case is developed to investigate the universe for matter dominated and radiation dominated phases. This would enable us to know the variation of extra-dimension in different phases of the universe for closed, flat and open models of universe. Implications of time varying cosmological constant on the model is worked out later so as to address some of the cosmological parameters and to reconcile with observational data. In next section field equations are obtained with the help of Kaluza-Klein metric with generalized cosmological constant. Subsequently, the models with different types of time dependent cosmological constant are investigated and determined the cosmological parameters. In the last section the discussion on implications of time varying lambda is followed by conclusions on the present topic.

IV.2. THE FIELD EQUATIONS Consider 5D FRW line element as given below:

 2   dr ds  dt  R  t     r 2 d 2  sin 2  d 2 2 1  kr    2









   2 2   A  t  d  ,   

(IV.1)

here k is curvature parameter and equal to 0, 1 and -1 for flat, closed and open universe, respectively. R (t) and A (t) are scale factors. Ψ is fifth dimension. The universe, filled with perfect fluid, satisfies the equation p = (γ -1) ρ, and the energy momentum tensor of perfect fluid is represented by,

Tji     p  uiu j  pg ij

(IV.2) 74

where ρ and p are the energy density and pressure of the cosmic matter, respectively; and ui is the five dimensional velocity vector in a co-moving co-ordinates and can be defined as uiuj = -1 for i = j , and uiuj = 0 for i ≠ j. Einstein field equation is given by,

1 Gij  Rij  Rg ij   8 GTji   g ij . 2

(VI.3)

While calculating Einstein field equations, ћ, c, and 8G = 1 are assumed. R ij is Ricci tensor, R is Ricci scalar, g ij is metric element, stress-energy tensor is T ji = diag (-ρ, p, p, p, p), Following tensor calculus in Riemannian geometry and from Eqs (IV.1) and (IV.3), Einstein field equations are obtained as,

(IV.4)

(IV.5)

(IV.6)

Divergence of Einstein’s tensor implies,





 i 1 i i i  Rj  Rg j   Tj   g j  0 . 2 ;j  ; j

(IV.7)

From Eq. (IV.7) continuity Eq. can be written as:

(IV.8)

Actually, divergence of Einstein field tensor for normal matter has not been strictly satisfied due to additional lambda term, but effective energy-momentum tensor should be conserved. Note that if Bianchi identities have to be satisfied then effect of decay of the cosmological term has to transfer energy from  to the perfect fluid, to have cosmological constant as a source term. In order to keep the discussion as general as possible, it has been assumed usually that the non-vacuum component obeys the γ-law equation of state [44]. On the other hand, to satisfy zero divergence of Einstein field tensor R.H.S. of Eq. (IV.7) is taken as zero.

i (T j;j )eff = (-T ji + L g ji );j = 0 ,



(IV.9)

In order to get exact solutions, ansatz power law equation A = Rn is followed. Substituting it in Eq. (IV.3) it yields:

(IV.10)

Differentiating Eq.(IV.18),

(IV.11)

(IV.12)

Using equation of state (EOS) i.e. p = (γ - 1)ρ in above Eq., it is rewritten as:

(IV.13)

Using Eq. (IV.6) in above Eq. (IV.13), the latter can be rewritten in the following form:

(IV.14)

In the next section, Eq. (IV.14) is modified for time dependent cosmological constant as: (1)

(2)

(3)   

1 R

(4)

, here α, β, γ1 are free parameters and

; and hereafter solutions are obtained. Other physical parameters

are also determined.

IV.3

COSMOLOGICAL MODEL WITH TIME VARYING COSMOLOGICAL CONSTANTS Eq. (IV.14) after simplification and can be rewritten as,

(IV.15)

In above Eq. substituting different types of time dependent cosmological constant, Cosmological models are analyzed in following subsections.

IV.3.1 Cosmological model with

, so as obtaining the solutions for the Eq. (IV.15).

Consider

Substituting  in it, following Eq. is obtained:

(IV.16)

Let,

m

k1 

 (n  3)  2 3(n  1)   (n  3) , 6(n  1)

k   (n  3)  2 2(n  1)

(IV.17)

(IV.18)

The Eq. (IV.16) takes the form as:

(IV.19)

Simple mathematical manipulation results in the first integral equation, which is given by,

, where m > 0 .

(IV.20)

Above Eq. is analyzed for flat and non-flat universe, as discussed here. 78

(a) If k = 0 i.e. for flat universe, the solution is obtained as: 1 m R(t )  {(m  1)( A1t  C} 1 ,

(IV.21)

where, C is constant of integration. Consider initial condition now R (0) = 0 at t = 0, from which constant of integration is obtained as zero. Similarly,

n A(t )  R(t )  (m  1)( A1t  C ) m+1 . n





(IV.22)

This shows that the fifth dimension decreases as time increases. Consequently, five dimensional model gets embedded into a four dimensional model if and only if n<m or for negative n. It should be noted that n ≠ -1 in order to get the solution. It is also seen that extra dimension has a significant role at early universe. It decreases considerably as the universe transits from matter dominated phase to radiationdominated phase.

(b) If k 0 then General solution of Eq. (IV.20) is hyper- geometric function. It is, thus, difficult to find a solution for it. However, for simplicity, let us consider m = 1. Expression for scale factor is obtained as,

(IV.23)

Initial conditions are assumed as per present epoch so as to get constants. To find A1, the terms R0 and H0 are considered as the scale factor and Hubble factor, respectively at present epoch. Substituting initial conditions, A1 is obtained as:

A1  H 02 R04 – k1R02 .

(IV.24) 79

To get R (t), integration of Eq. (IV.22) leads to,

R2 

- A1  k1(t  c)2 , k1

(IV.25)

where, c is constant of integration. To find c and constant A1, consider R = R0 at t=t0 (t0 is the present epoch) in Eq. (IV.25). Thus,

c

H 02 R04

1 A1  k1R02 - t0 ,  c  k1

k12

- t0 

H 0 R02 - t0 , k1

 H R2  -A R 2  1  k1  (t - t0 )  0 0  .  k1 k1  

(IV.26)

The fifth dimension A is represented by:

22   H 0 R02   n  - A1 . A R   k1  (t - t0 )    k1    k 1    

(IV.27)

H 0 R02 Universe passes through minimum at t  t0  when k1 > H02 R02. The solutions k1 are nonsingular only for certain range of parameters; but they will be singular for

k1 

R02  2t0 H 0 -1 t02

IV.3.2 Cosmological model with Eq. (IV.15) can be modified for above lambda value as,

(IV.28)

The constants m and k1 are defined as:

m

3(n  1)  2  3 (n  3) 

 1(n  3)  6(n  1)

, k1 

  (n  3)  2 3k  1(n  3)  6(n  1)

The solution of the Eq. (IV.28) can be obtained in the same fashion as that obtained in previous sub-section, i.e. following steps from Eq.(IV.19 - IV.26). However, the values of constants are quite different in both the models.

IV.3.3 Cosmological model with   

1 R2

Rewriting Eq. (IV.15) for above  resulted Eq. given below;

(IV.29)

Assume,

m

 (n  3)  2 , and 2

k1 

  (n  3)  2 3k   (n  3) 6(n  1)

Eq. (IV.29) is modified, which is similar to that obtained for the models (IV.3.1), (IV.3.2), and (IV.3.3). Although constants are different in Eq. (IV.16), (IV.28) and (IV.29), but the expressions for R(t) and other physical parameters of the models



with

1 R2

are similar to that obtained for

i.e. Eq. (IV.21),

(IV.22) for flat model, and Eq. (IV.25) and (IV.26) for non-flat models. The solutions for models with variations in (1)

(2)

and (3)   

1 R2

are

analyzed and compared in matter dominated and radiation dominated phases in next section.

IV.3.4 Cosmological model when Substitution of above value modifies the differential equation Eq. (IV.15)  and it can be rewritten as,

. (IV.30)

Let

m

 (n  3)  2 3(n  1)   (n  3) , 6(n  1)

k1 

3k  (n  3)  2   (n  3) 6(n  1)

Eq. (IV.30) is simplified as:

(IV.31)

With the help of simple mathematical manipulation, the first integral Eq. is derived as,

, here m> 0,

(IV.32)

General solution of above Eq. is hyper-geometric function, which is quite complicated and difficult to solve further so as to obtain final expression for scale factor. For simplicity, m = 1 is considered to find an expression for scale-factor. So Eq. (IV.32) is further simplified in the following form,

(IV.33)

Initial conditions as per present epoch are assumed to determine constants. To find A1, the terms R0 and H0 are taken as scale factor and Hubble factor for the present epoch, respectively. Substituting initial conditions in Eq. (IV.33), the expression for constant is obtained as,

A1  R04 – k1R02 .

(IV.34)

To get R(t), scale factor R(t) can be arrived at by integrating the Eq. (IV.33) as:

R2 

 A1  k1 (t  c) 2 , k1

(IV.35)

where, c is constant of integration. 83

To find c, substitute constant A1 and, R = R0 at t = t0, where t0 is the present epoch gives

c

1 k1

A1  k1R02  t0  c 

H 02 R04 H 0 R02  t   t0 , 0 k12 k1

 H R2   A1 R   k1  (t  t0 )  0 0  . k1 k1   2

(IV.36)

The fifth dimension A can be written as,

n 2  2  H 0 R02   n   A1 . A R   k1  (t  t0 )    k1  k1   



(IV.37)



The universe passes through minimum at t  t0 

H 0 R02 when k1 > H02 R02. The k1

solutions are nonsingular only for certain range of parameter but they will be singular for k1 

R02 2t0 H 0  1 . From Eq. (IV.37), it can be observed that if n < 1 then fifth t02

dimension becomes very small as time lapses. Eq. (IV.36) is further analyzed for flat and non-flat models as given below.

(a)

k = 0 for flat universe Consider k = 0 i.e. for flat universe, k1 

 (n  2)

. Hence, Eq. (IV.36) is

rewritten by substituting k1 in the following way.

R  2

 A1 (n  2)



 H 0 R02 (n  2)    .  k1  (t  t0 )    

(IV.38)

Here, it should be noted that scale factors for flat models (1)

(3)   

1 R2

and that for

, (2)

are quite different which can be seen

through Eq. (IV.21) and Eq. (IV.37). On the other hand, Eq. (IV.26) or (IV.36) is the generalized form of scale factor for open or closed model. By substituting m and k1 in equations (IV.20) and (IV.37), all physical parameters related to scale factors can be determined for the models under study. Since model with

is the

most general model, other physical quantities are calculated for it. Expression for pressure also can be obtained from equation of state i.e. p = (γ - 1) ρ.

(b)

k = ± 1, Open /Closed model If m = 1,  and  are obtained as follows;



3k  (n  3)  2  6(n  1)

,

 (n  3)

 (n  3)  4 3(n  1) .  (n  3)

In this case Eq. (IV.36) in a simplified form



R 2   A1  (t  t0 )  A1  R02



, 2

(IV.39)



22  A  R n    A1  (t  t0 )  H 0 R02  .  

(IV.40)

Other physical parameters are obtained as:

(t ) 

    (t  t0 )  H 0 R02 



 2   t  t0   H 0 R0 



  A1

  A1  

(IV.41)

As per the field equation, density is determined using Eq. given below:

2   3(n  1)     (3k   )(t  t0 )  H 0 R02   (3k   ) A1  (t ) 

(t  t )  H R   A  0

2 2 0

(IV.42)

Pressure is related to density through EOS, so it is given by:

p = ( -1),

Volume of the universe can be calculated as V = R3 A:





3 n

2 2  V    A1  (t  t0 )  H 0 R02  .  

(IV.43)

Hubble parameter and deceleration parameter are obtained as follows:

(IV.44)

It is observed that H (t)  t-1. Deceleration parameter is determined as given below. :

(IV.45)

If k1 =1 then q (t) → t-2, which shows that the acceleration is decreasing. Sahni and Starobinski [45] have first predicted slowing down of the acceleration of the expansion. The variation of density and deceleration parameter with respect to cosmic time are shown in Fig.IV.2

and

Fig.IV.3,

Fig.IV.2: Graph of Density vs. Cosmic Time

respectively for arbitrary values of α, β and n i.e. α = 0.5, β = -4, n=0.5, A1 = 0.5, t0 = H0 R02, k1=1. Fig.IV.2 reveals that density is quite high during the early stages of the universe. It decreases as t   for all types of universe. Hence, present model seems to represent inflationary model.

Fig.IV.3: Graph of q(t) vs. Cosmic Time

Further, Fig.IV.3 suggests that the acceleration is high at early stages of the universe but it decreased rapidly with laps of time. This shows the transition of universe from inflationary phase to steady state phase. Age of the universe can be calculated by assuming q = q0 and H = H0. Using Eq.(IV.31) and (IV.32), the expression for age of the universe is obtained as:

t = ( H 0 R0 )-1 

dR q 0  R0 2m 1+ m m  R 

(IV.46)

Above Eq. reduces to a simpler form, which has been explained by Carvalho et al [11] in four-dimension cosmology. If q0 = m then,

H 0t 

1 (m  1) R0m 1

R m 1 ,

(IV.47)

assuming constant of integration c = 0. Suggestively, age of the universe is affected by the values of n and α. Although it seems to be independent of β values; but under certain conditions age parameter is free from β. In the subsequent section some of the physical parameters under Neo-classical tests are carried out, which would help while dealing with the horizon problem.

IV.4

NEO-CLASSICAL TESTS

IV.4.1 Proper distance D (z): It is necessary to establish causality connection between observer and source at any time. Let an observer at r at time t receiving signal from source at a distance r = r1 at t = t1 then, the proper distance between the source and observer is given by:

d ( z )  R0  R

dR . RR

(IV.48)

First integral solution for k1 = 0 and if m ≠ 1 leads by

, substituting it in

Eq. (IV.48) and carrying out integration, results into,

d ( z) 

1 m A1

R0[ R0m  R m ] .

(IV.49)

Simplification further yields:

d ( z) 

1 m A1

R0m 1[1  1  z 

m

]

(IV.50)

here, R t  1 z  0   0  R  t 

m 1

(IV.51)

For m =1,

d ( z) 

1 2 1 R0 [1  1  z  ] , A1

(IV.52)

which is independent of γ.

IV.4.2 Luminosity Distance In theoretical cosmology luminosity distance manifests distribution of light to an observer who is at a distance from the source, obeying inverse square law for its 89

intensity. In other words, it is defined as the amount of light received from a distant object. If dL is the luminosity distance for the object, then

 L 2 dL    ,  4l 

(IV.53)

where L is the total energy emitted by the source per unit time, l is the apparent luminosity of the object, Therefore, we arrive at:

dL = (1 + z)d(z).

(IV.54)

Using Eq. (IV.49) luminosity distance can be calculated. It can be seen that d(z) depends on red shift. This is also an increasing function of z.

IV.4.3 Angular Diameter Distance It is a measure of how large an object appears to be. The angular diameter dA of a light source at proper distance d is given by,

dA = d(z)(1 + z) −1 = dL(1 + z) −2

(IV.55)

Using Eq. (IV.49) the expression for angular diameter is obtained as,

d A  d ( z )(1  z ) 1 

1 m A1

R0m 1[(1  z ) 1  1  z 

 m 1

(IV.56)

Evidently, dA is decreasing function of z. Further, the luminosity distance and angular diameter have different dependency on red shift. Since luminosity distance is directly 90

proportional to z; angular diameter is a decreasing function of red shift. Hence, the objects appear smaller as distance of object increases, which is consistent with the experimental results.

IV.4.3 Look back time The time in the past when the light that now received from a distant object was emitted and is called the look back time. How long ago the light was emitted (look back time) depends on the dynamics of the universe. The radiation travel time or look back time (t − t0) for photon emitted by a source at instant t that received at t0 is given by:

t  t0 

 R

t  t0 

dR , R

(IV.57)

1  R m 1  R m 1  ,  A1 (m  1)  0

above Eq, is written in a simplified form as:

t  t0 

R0m 1

1  z (m 1)  .  A1 (m  1) 

(IV.58)

Using binomial expansion, we find red shift, which can be a measure of look back time. For z < 1, luminosity distance and angular diameter are proportional to z. These results are similar to those obtained by Carmeli [46]. After neoclassical tests of the model, matter dominated phase and radiation dominated phase of the present model are investigated in the following section to understand the behavior of the universe at different phases.

IV.5

KALUZA KLEIN COSMOLOGICAL MODEL IN MATTER DOMINATED PHASE AND RADIATION DOMINATED PHASE The universe had passed through several phases since Big-Bang. The different

phases of universe are listed in table IV.1 based on the relation between pressure and density. The equation of state, ω, describes the relationship between pressure and density in a material according to ω = p/ρ. Here are some examples of the EOS for common fluids. When matter is at rest (pressure less dust) it has ω=0, but as it picks up velocity (v), its EOS increases until ω → 1/3. Table IV.1: Equation of state (EOS) and phase of the universe Phase of the Universe

Equation of State (ω)

Matter (pressure less)

Radiation

1/3

Curvature

-1/3

Cosmological Constant

-1

Matter (in general)

0< ω<1/3

Quintessence

-1< ω< -1/3

In this section, as considered previously ω = (γ - 1) is taken to study matter dominated phase as well as radiation dominated phase; and analyzed the constants for different values of  so as to understand the behavior of the universe in different phases of the model with different kinds of lambda.

IV.5.1 Matter dominated phase

(a)

Model with For matter dominated phase  =1. Thus, pressure p = 0 and constants are

determined as follows: Consider,

m





3 n 2  2n  1   (n  3) 6(n  1)

, and k1 

k . 2

If, m = 1, then,  

3(n  1)(n  1) , for k =0 and 1. n  -3, 1 for the model to be (n  3)

nonsingular. For n = -2,  = 9.  is obtained here, is similar to that considered by Lima [47]. It is also noticed that R (t)  t1/2, whereas, A(t)  t-1. Extra dimension decreases more rapidly as compared to four-dimensions, which is in accordance with observational data. If n = 0, then cosmological constant can be negative that results into contraction of universe rather than its expansion. (b)

Model with For matter dominated phase, as per last subsection  = 1, constants from Eq.

(IV.28) can now be rewritten as:

m

3(n  1)  2  (n  3)

 1 (n  3)  6(n  1)

, k1 

3 n  1 k

3(1  n ) , if m = 1 then  1  . 1 6(n  1)   (n  3) (n  3)

For n = -2, lambda is negative and this leads to a contraction of model, which should be avoided in order to relate the model with present data. Hence, if n = -4 then lambda is positive. Extra dimension reduces more rapidly as compared to the previous model however. It can be proved that models with

and

are

dynamically equivalent as  = -γ1, in accordance with literature [43].

(c)

Model with   

1 R2

The constants for the model with above lambda are obtained as follows.

m

  n  1 3k   (n  3) n 1 , and k1  . 6(n  1) 2

Here it can be observed that if m = 1, then, n = 1. This model, therefore, is quite different from above two models as k1 depends upon . For open model k = -1, and, if k1=1,  

 3  n  1

 n  3

, this leads to a negative cosmological constant. But, if -2 ≤ n <

-1 then there will be a small positive , responsible for expansion of the universe. However, for closed universe  

-9  n  1

 n  3

, so, the universe expands when n ≤ -2.

Even for k = 0 it can be observed that cosmological constant becomes positive when n > -3/5. The literature [12, 41], pertinent to the cosmological models with varying  discussed the matter density and vacuum density in matter dominated phase suggests that m + v = 1 in both phases of the universe. In the present models

m 

(d)

 3(n  1) H

and  v 

 3(n  1) H 2

Model with

The constants  and β are determined when m = 1 and k1 = 1 as explained in the subsection (IV.3.4) and they are as follows:



 (n  3)  4 3(n  1) ,  (n  3)



3k  (n  3)  2  6(n  1)

 (n  3)

Substitution of  = 1 in the above constants gives

 = 3n (n+1)/(n+3) and  = 3(k+2)(n+1)/ (n+3).

To have positive values of  and  it is necessary that n  1, -1or -3. The following subsection discusses the behavior of the models in radiation-dominated phase.

IV.5.2 Radiation dominated phase The models when 1)

and 3)   

investigated for the above desired phase and the model when

1 R2

are initially

discussed later in this subsection so as to compare it with other models. For this phase

=4/3 and p = ρ/3

(1)

,m 

(2)

m

(3)   

1 R2

9(n  1)

3(n  1)  2n  3 9(n  1)  2 (n  3)

m

For m = 1,  

 2n  3 (n  1)  2 (n  3) , k

 2n  3 , k



, k1 



k  2n  3 3(n  1)

3 2n  3 k 9(n  1)  2 1 (n  3)

  2n  3 3k  2 (n  3) 9(n  1)

 3n(n  1) (n  3)(n  1) and,  1  , however, value of  is found from n3 (n  3)

k1 i.e. taking it equal to 1 and calculated as:



 2n  3 3k  9(n  1) 2(n  3)

In cases (1) and (2) it is observed that when n  -1,  takes positive value. Since α and γ1 do not depend on k, the value of β can be calculated for different values of k. In cases of flat model:

For k = 0,  

9(n  1) . 2(n  3)

For k = 1,  

3(5n  6) and 2(n  3)

for k = -1,  

3n . 2(n  3)

The lambda of this model is found to be positive but effect of extra dimension cannot be observed if n = 0.

(4) For Radiation dominated phase, since  = 4/3, so,

 = 3n (n+1)/(n+3) and  = 3[k(2n+3)+2(n+1)]/2(n+3).

In both the cases n  -3. Since  is independent of k, finding  for different k values give the values of  for flat, open and closed models.

IV. 6 DISCUSSION AND CONCLUSIONS The models with  R-2, H2,

and R-2+ H2 in Kaluza-Klein space-time

metric are investigated and compared with each other. Exact solutions of Einstein field equations are obtained with the help conservation of energy-momentum tensor. Cosmological models for   H2 and

are found to be equivalent when k = 0,

which is in accordance with the literatures [41, 48]. Models when  R-2, and when

R-2+ H2 is quite different for flat universe. It is also observed that the models are quite different from 4D models in principle. Significant difference is observed in the 96

solutions of the model with R-2+ H2 and the models with variation of  as H2,

etc. Since m and k1 depend on γ, physical parameters - density and pressure will be different for matter as well as radiation- dominated phase. The cosmological model with R-2+ H2 is nonsingular expanding model. The solutions for all models are found to be dependent upon index factor n of power law equation and constants free parameters. cosmological constants vary as H2 and hand, the models for  R-2 and

Cosmological models with

depend upon constant m. On the other

depend on curvature factor, also.

It is observed that the latter model mostly satisfies the conditions for expansion of the universe due to small positive value of Λ and time decay of extra dimension. It is found that when ρ > 0 and p > 0 for the same model then R < 0 that meets the criteria of expansion. Eq. (IV.27) and (IV.37), reveals that the five-dimensional model reduces to four-dimensional model as t  t0; further Eq. (IV.40), suggests that t-2. The model with  t-2 has been widely accepted [19, 21]. During the present study on models, nature of extra dimension is investigated and inferred that it decreases with time; but the effect of it is prominently observed in a model with R-2 + H2 compared to other models. Also, the present model becomes steady state model as time lapses. One can, also, conclude that all physical parameters are dependent on

 and . Free parameters  and  are found to be different for matter and radiation dominated phases, because their values are determined by value of n. For n = 0.5, behavior of density for flat or non-flat universe is similar. The present model seems to represent inflationary phase of the universe. The q reaches to constant value in Eq. (IV.45). Consequently, it may be able to explain the slowing down of cosmic acceleration as explained by Sahani [45]. The Eq. (IV.41) and graphs suggest that normal density decreases more rapidly than effective density for flat as well as nonflat models. Since effective density is equal to normal matter and vacuum-density, if free parameters α and β> 3(n+1) then we get early universe situation. Also, enabled us to determine volume and age parameters depending on values of H0 and R0. Matter dominated and radiation dominated phases for flat, closed and open curvature 97

constants i.e. k = 0, ±1 are also studied during present work. Values of free parameters α, β, and γ1 are determined for all types of models of matter and radiation dominated phases. The results are quite different and require several assumptions, however. It is, also, concluded that for n = -2, all models gave positive cosmological constant, except when  

, which is a different result. For open and closed

models with different cosmological constants, it is possible to obtain non-singular solutions. This is an important result, which is not discussed so far. Neoclassical tests are also carried out for all models for the sake of completeness. These tests can be carried out for open and close universe; and the values of free parameters can be adjusted accordingly. Neoclassical tests conducted for flat universe so as to relate it with observational data. Also, for z < 1 luminosity distance and angular diameter are directly proportional to z. Results of present study are similar to those of Carmeli [46].

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Khadekar G.S., Vaishali Kambdi, “Int. Journal of Theo. Phy.” 48, 3147, (2009)

100

CHAPTER V KALUZA-KLEIN COSMOLOGICAL MODEL IN PRESENCE OF STRANGE QUARK MATTER WITH DECAYING LAMBDA

V.1

QUARKS AND COSMOLOGY

101

V.2

INTRODUCTION TO KALUZA-KLEIN COSMOLOGICAL

103

MODEL WITH QUARKS V.3

METRIC AND FIELD EQUATIONS

106

V.4

SOLUTIONS OF FIELD EQUATIONS WITH TIME

108

VARYING OSMOLOGICAL CONSTANT V.4.1 The model for different types of the universe

V.5

(i) Flat universe for k = 0

111

(ii) Closed universe for k=1

112

(iii) Open universe for k = -1

113

COSMIC IMPLICATION OF THE MODEL V.5.1 Physical features of the model

114

V.5.2 Present model and eternal inflation

115

V.5.3 The model and phase divide crossing

115

V.5.4 Comparison of present models and

116

gravity models V.6

CONCLUSIONS

116

REFERWNCES

118

V.1

QUARKS AND COSMOLOGY In nineteenth century, only a few elementary particles namely, electrons,

protons and neutrons along with electromagnetic field quantum (photon) had been known. It had been thought that every matter have been consisted of these particles. Discovery of cosmic rays and experiments on them at higher energies have led to the discovery of a large number of unstable states, particles with a life time 10-23 sec. These states, called as hadrons, have been found to group themselves into families or multiples. Their classification follows from properties they share. By now, these hadrons have been understood to be made up of fundamental point like spin ½ fermions, which carry fractional electric charge (2/3e-, 1/3 e-, where e- is charge of an electron). These fundamental particles have also been called as quarks. Other classes of fundamental particles, carry integral electric charge have been called as Leptons. It has also been established from the experiments that protons, neutrons have now no longer regarded as elementary particles, but made up of quarks. As a matter of fact, six quarks and their counterpart antiquarks exist in nature. Recent CERN experiment (2000) has proved the existence of massive quark namely tau quark. It should also be noted that quark matter has also been related to Baryon number. With the discovery of strange quark matter (SQM), Baryon number can be more than 10 3. This has suggested possibility of very heavy atomic nuclei. The classification of fermions into quarks and leptons has mainly due to difference in strength of interaction in which they participate. Quarks have been participating in all the interactions viz. strong, electromagnetic, and weak interactions. On the other hand, charged leptons exhibit only last two interactions. The fundamental interactions, which had been widely believed to describe by quantum field theory, have created new excitements in particle physics. The interactions have a major role in defining the kind of interactive forces. Strong forces and weak forces have been identified by strong and weak interactions, respectively. These interactions have been classified depending upon exchange of quanta between the interactive particles. With the evolution of the particle physics a new field called quantum chromo dynamics (QCD) has come up, Essentially, QCD deals with physics of quarks and gauge theory, which has brought revolution in physics and led to a standard model (SM) in particle physics. The SM model not only has an important role in the study of particle physics but also has a key role in cosmology. 101

According to cosmology, journey of quark began at 10-12 sec just after BigBang, which has been called as quark epoch. The quark epoch has an importance in the evolution of the universe. Particularly, the early universe when phase transitions had taken place. As a belief, the present universe has been the consequence of these phase transitions. Quarks have played a major role in the evolution of the universe. At very early stages of the universe evolution, it had been in the soup of quarks and gluons, which together have formed quark-gluon-plasma. During this stage, the density of the universe had very high value and so also the temperature nearly about 1010 ºC. After the first transition, due to the lowering of temperature, the quarks paired in such a way so as to form the core of compact stars. These compacted stars may be hypothetical quark stars, which remained undiscovered. Quark stars have been the objects with ranging densities between neutron stars and black holes. Also, the possibility of existence of strange quark stars in present time cannot be ruled out. Strange quark particles have been considered as stable (metastable) particles. Strange quark matters have been stable even at zero pressure, and Olinto [1] have explained physics behind it. Several experimental attempts in search of SQM have also been made in past. Bodmer [2] has explained the conversion of hadrons under high density and high temperature into droplets of quarks consisting of not only up u and down d quarks, but also into strange quarks, and since then, it has been studied thoroughly [3]. These droplets of quarks have been called as strangelets. These strangelets have been quite important, because they act as tool to study very old stars and hypothetical strange stars. These strangelets could have been one of the constituents of the matter at the early universe. In recent past, strange quarks matters (SQM) have attracted attention in the study on the early universe phenomenology. Investigations on neutron and strange stars can prove the existence of SQM and, hence, depict the formation of large scale structure of the universe. Cosmological studies with quarks have gained momentum since the inception of MIT Bag model in late 60s [4]. The latter model treats the quarks as free particles forming quark gas by considering it as perfect fluid. This has not only facilitated to find normal fluid pressure - quark pressure and normal fluid density - quark density relations, but also opened the doors to look at radiationdominated era of the universe. Besides astrophysical importance of quarks, and SQM, from particle physics point of view, it has been possible to deal with several problems encountered in 102

cosmology. Among all, cosmological constant problem, dark energy, dark matter, etc. have been the major problems to deal with. In fact, since SQMs’ have been stable or metastable particles, they can be an answer to the long-standing cosmological dark matter problem, which is a recent topic of research. Considering the significant role of quarks, SQM, next section introduces Kaluza-Klein cosmological model in presence of SQM. An important parameter in cosmology called cosmological constant was considered along with the model so as to have detailed understanding and broaden the view of the topic of the thesis.

V.2

INTRODUCTION TO THE KALUZA-KLEIN MODEL WITH SQM The physical situation prevailed during the early stages of the formation of the

universe has been still a challenge and an area so far of frontier research in cosmology. Extensive efforts have been directed in this area, which prompted us to look for dimensions more than space-time (3+1) for describing the early universe. The necessity for higher dimensions has been thought for the study of early universe, due to its very smallness at early stages. The extra dimensions become compactified and got embedded into four dimensions due to expansion of the universe. Hence, experimental detection of an extra dimension has been not possible today. Though certain practical limitations exist, but its effects can be observed. Contemporarily, the research community to find its origin has directed efforts for unification of forces. Kaluza [5] in 1921 and Klein [6] in 1926, independently, proposed the theories of higher dimensions for the unification of all forces of nature and particle interaction, respectively. A voluminous literature on the Kaluza-Klein (KK) theory of gravitation and its cosmic implications along with astrophysical consequences has been available now. All these have been widely used to address issues like accelerated expansion, mystery of dark energy, dark matter, etc. As a matter of fact, the universe has been expanding and also accelerating. The driving force for accelerated expansion has supposed to be dark energy. Gu [7] has enlightened the relation between extra dimensions with dark energy. Interacting dark energy models with K-K cosmology have been discussed by many authors [8 - 11]. El-Nabulsi [12] has touched the extra dimension topology and accelerated expansion of the universe. Thus, the K-K cosmology and its models have gained importance among the scientific community to learn the secrets of the universe, its behavior at early times, etc. 103

In this chapter, the K-K cosmological model in presence of strange quark matter (SQM) with decaying lambda is examined. The importance of quark matter lies in the structural formation of the universe and, subsequently, its evolution. Further, since it is being part of dark matter, its importance also lies in the interaction with dark energy. Both these have been seriously debated as evident from published literature [13 - 15]. Quarks can also be studied using domain walls and strings. Currently, models with quark matter with domain walls, strings, etc. have been studied by Adhav and Nimkar [16], Ozel et al. [17], Bali and Pradhan [18], and Yilmaz and Yavuz [19] in different contexts. Hence, it is worth to take a look at the role played by strange quark matter in the early stages of the evolution of the universe, since the big bang. In fact the quark-gluon-plasmas (QGPs) exist since the beginning of the universe. After the big bang, the universe has undergone two phase transitions: the first transition occurred at a critical temperature that resulted in stable topological defects, while the second transition occurred at the cosmic temperature of the universe TC ~ 200 Mev. At this temperature, a QGP has been converted to a hadron gas. Witten [2] in 1984 has pointed out the astrophysical consequences of phase transitions. The existence of quark matter had been first discussed by Itoh [20] and Bodmer [3] in the 1970s independently and later, Witten [2] have proposed models for the quark-gluonhadron gas transition and the conversion of neutron stars into strange stars at ultrahigh densities, two ways for the formation of quark matter. Sagert et al [21] discussed experimental analysis of explosive astrophysical systems and enlightened quarkhadron phase transitions. Farhi and Jaffe [22], Xu [23], and Lipkin [24], independently, reviewed the physical nature of SQM. They concluded that it is stable. Properties of quark matter [u (up), d (down), s (strange), etc. quarks] have been well explained in particle physics where quark matter participates in strong interactions and forms the basic constituent of baryons. Thus, study on SQM and quark matter can provide an idea on the structure and the geometry of the universe [25]. A study of quark matter and SQM has been a major research area of interest among scientists as it would not only provide information about the early universe, but can also solve the mystery of dark matter. In this context, Trimble [26] has given an excellent review. In a typical cosmological model with SQM, the quark matter has been modeled using phenomenological Bag model, where the quark confinement has been described as an energy term proportional to the volume [20]. In this model, 104

quarks have been assumed to be degenerated Fermi gases, which exist only in a region of space endowed with a vacuum energy density BC (called the Bag constant). Quark matter consists of massless u, d, and massive s quarks and electrons. In the simplified version of Bag model, the quark density and quark pressure have been related to each other and given as pq = ρq/3, and total pressure p = pq – BC, total density ρ= ρq + BC, where BC is Bag’s constant. Under this condition, it has been possible to obtain equation of state (EOS) for strange quark matter as p = 1/3(ρ -4BC). Experimental results obtained in Brookhaven’s relativistic heavy ion collider (BNLRHIC) laboratory concluded that quark-gluon plasma has been the perfect fluid, having quark matter as a basic constituent. Yilmaz and Yavuz [19] and others have inferred that the presence of extra dimensions and SQM tended to exert negative pressure with constant density in the early universe. This has been concluded to be the dark energy. In fact, the cosmological constant  represents dark energy, and it plays an important role. According to recent development of cosmology, the acceleration of the universe has been due to negative pressure, which has been proportional to the related vacuum density. Present day astronomical observations [27] have yielded the value of  ≤ 10-56 cm2. But the huge difference between the present small observed value of cosmological constant and that calculated using Glashow-Weinberg-Salam [28] model of particle interaction has been found to be of the order of 1050, and this has been known as cosmological constant problem (CCP). It is also an attractive area of research for many contemporary cosmologists. Bambi [29] has discussed the CCP and SQM, and remarked in his article that strange stars, thought consist of SQM (if these exist), can be a good laboratory to bring information about the early universe and their physical conditions during those early stages. It can be considered to investigate the CCP and to test the nature of dark energy. The time varying cosmological constant  has been suggested to solve the CCP, because it has been thought that perhaps  might have a large value in early universe and decayed with time so as to have the present small value. Chen and Wu [30] have first explained a decaying lambda. They have suggested that Λ  R-2. Later, Sahni and Starobinsky [31], Padmanabhan [32], and Overduin and Cooperstock [33] have reviewed the cosmological models with time varying  in four dimensions in different contexts. Recently, Khadekar et al [34 – 36] have dealt with the solutions of the K-K cosmological model with different 105

, 

forms of time varying cosmological constants, i.e.,

1 a2

and    , here a is the scale factor. According to them, these models have been dynamically equivalent for spatially flat universe. El-Nebulsi [37] has discussed higher dimensional nonsingular cosmology dominated by varying cosmological constant. One of the motivations for introducing  has been to reconcile the age parameters and density parameters with recent observational data. The variation of lambda has also been discussed in different context [38 - 45]. Encouraged by aforementioned facts, the KK cosmological model in the presence of quark matter with the variation of

(which was first

suggested by Carvalho and Lima [46]) has been investigated. Particularly, the generalized form of the model in this chapter is studied. Here α and β are the dimensionless free parameters. The exact solutions of Einstein field equations, so obtained, are applied to study variation of quark density and lambda for different types of universe. The next section deals with field equations. The solutions are found in section V.4. The discussion on types of the model is given in section V.5. Cosmic implications of the present model are discussed in section V.6 and finally the conclusions on the topic are drawn. V.3

METRIC AND FIELD EQUATIONS Einstein field equations are obtained with the help of Kaluza-Klein metric, as

given below:

éì êï dr 2 2 2 2 ds = -dt + R t êí + r 2 d q 2 + sin 2 q d j 2 êï 1- kr 2 ëî

()

(

)

(

)

üù ïú ýú + A 2 t d Y 2 ïú , þû

()

(V.1)

ћ = c = 8G = 1 are assumed in accordance with cosmic principle. R(t) and A(t) are fourth and fifth dimension scale factor, and k is curvature constant. k = 0, 1 for flat, and open/closed models of universe. The universe is assumed to filled with perfect fluid and represented by quark matter. Its energy-momentum tensor is given by: Tij = (p+)ui uj + pgij,

(V.2) 106

where ui is the five velocity vector, which satisfies the relation uiuj = -1. Here, p and

 are quark pressure and quark density, respectively; and p and  for quark matter are related by EOS, as per Bag model, as follows: p

1    4BC  . 3

(V.3)

Einstein field equations with time dependent cosmological constant, (t), is given by: 1 Rij  g ij R  Tji  (t ) g ij , 2

(V.4)

Divergence of Einstein’s tensor implies:





 i 1 i i i  R j  Rg j   Tj   g j  0 . 2 ;j  ; j

(V.5)

With the help of Eq. (V.1) and (V.2) the field equations are derived as:

and

(V.6)

(V.7)

Conservation of energy-momentum tensor gives the following relations:

and

(V.8)

(V.9)

To find exact solutions of field equations, using ansatz, the power law equation A(t)=Rn(t), which is assumed by many researchers [47,and references therein]. The ansatz power law equation is used in view of anisotropy of the universe, despite our assumption of isotropic and homogeneous universe. Expansion scalar  is 107

proportional to the shear scalar  useful for measurement of anisotropy [48]. Consequently, the relation between metric potentials R(t) and A(t) as A(t) = Rn(t). There are four unknowns i.e. R, A, ρ, and , but, three independent equations, Using A(t) = Rn(t) ansatz in equations (V.3), (V.5) (V.6), and solving them with (V.9), following Eq. is obtained:

(V.10)

Above Eq. can further be simplified on substituting  and  so as to get the solution and other physical parameters. This is explained in the next subsection.

V.4

SOLUTIONS OF EINSTEIN FIELD EQUATIONS WITH TIME VARYING COSMOLOGICAL CONSTANT To determine solutions of Einstein field equations, assume

as first suggested by Carvalho and Lima [46]. In this expression, first term is taken to deal with age and low-density problem, while second term is taken to satisfy the assumption of isotropic universe. Simplifying Eq. (V.10) by substituting  from Eq. (V.7) and Λ, the following equation is obtained.

(V.11) In above Eq., consider

m

6n2  15n  9  2 (n  3) , 9(n  1)

(V.12)

k1 

3k (2n  3)  2 (n  3) , and 9(n  1)

(V.13)

k2 

2(n  3) BC . 9(n  1)

(V.14) 108

Eq. (V.10) now can be simplified as:

(V.15)

After some mathematical manipulations, the general solution for above differential equation is obtained as

(V.16)

The solution of above equation is hyper geometric function but for analytic purpose, we assume m = 1, so as to get simplified solution. First integral equation of above differential equation is given by

(V.17)

where C0 is the constant of integration. Above equation can further be solved to obtain the solution as:

2 2C 0 æ k1 ö k R = - çç ÷÷ Sinh2 k 2 (t + c ) + 1 . k2 è k2 ø k2 2

(V.18)

2C0  k1     , f = 2 k 2 (t + c ) Let a  k2  k2 

and k 3 

k1 , k2

Choose c = - t0 as per present epoch, and,   2k2 (t  t0 ) Thus, Eq. (V.14) can be . rewritten as: 1 2

R (t ) = ( a Sinh f + k3 ) .

(V.19)

Other physical parameters are calculated as follows: 109

n 2

A(t )   aSinh  k3  ,

(V.20)

k2 1 aCosh  aSinh  k3  , 2

H (t ) 

(V.21)

(V.22)

  k2a 2Cosh 2  2 (aSinh  k3 )  (t ) 

(V.23)

2(aSinh  k3 )2

 (t ) 

[3(n  1)   ]k2a 2Cosh 2  2(3k   )(aSinh  k3 )

p(t ) 

[3(n  1)   ]k2a 2Cosh 2  2(3k   )(aSinh  k3 )

2(aSinh  k3 )2

6(aSinh  k3 )

, and



(V.24)

4BC 3

(V.25)

Expression for quark pressure and that for quark density can be found out as per Bag model. We know that p= pq – BC and ρ= ρq + BC.

 q (t ) 

pq (t ) 

3(n  1)    k2a 2Cosh 2  2(3k   )(aSinh  k3 )  B

C and

2(aSinh  k3 )2

3(n  1)    k2a2Cosh 2  2(3k   )(aSinh  k3 )  BC 6(aSinh  k3 )2

(V.26)

(V.27)

The model is investigated for flat, closed and open universes in the following subsection.

V.4.1 The model for different types of the universe It can be observed from Eq. (V.11) that depending on the values of constants m and k the behavior of the model can be studied for different kinds of the universe.

110

Considering m = 1 and k1 = 1 in Eq. (V.11), the expressions for free parameters  and  are obtained as:



3(n  1) , (n  3)

(V.28)



3k (2n  3)  9(n  1) . 2(n  3)

(V.29)

Density and pressure for different universes are determined as follows:

(i)

Flat universe, k = 0 It is observed that  is independent of k, but  depends on k. Substitution of k

= 0 in Eq. (V.28) and Eq.(V.29), yields:



3(n  1) , and (n  3)

(V.30)



-9(n  1) . 2(n  3)

(V.31)

It is observed that k2 is independent of free parameters and depends only on BC. Thus, Eq. (V.13) is modified as follows, while Eq. (V.14) remains unchanged.

k1 

 2 (n  3)  1 and 9(n  1)

(V.32)

k2 

2(n  3) BC . 9(n  1)

(V.33)

k Since k3  1 , it is obtained from Eq.(V.32) and Eq.(V.33) can be represented as: k2 k3  

 BC

(V.34)

111

9C0 (n  1)    1 a    2(n  3) BC  BC  BC

9C0 BC (n  1) 2 2(n  3)

and

2(n  3) BC (t  t0 ) . 9(n  1)



(V.35)

(V.36)

So density for flat model is obtained as:

 (t ) 

3(n  1)(n  2) 9(n  1) k2 a 2Cosh 2  (a Sinh   k3 ) (n  3) 2(n  3) (a Sinh   k3 )2

(V.37)

This can be rewritten in the following form:

ρ(t)=

3(n+1 )(n+ 2 )BC2 k2a 2Cosh 2 + 9(n+1 )BC(BCa Sinh  - β) 2(n+ 3 )(BCa Sinh  - β)2

(V.38)

Since β is quite small as compared to Sinh, it can be neglected as compared to Sinh. Hence, as  → 0 i.e. for present epoch, density, (t) → BC, and quark density, q, approaches to zero. From Eq. (V.3) it is seen that the pressure becomes negative by which accelerated expansion of the universe can be explained for present epoch.

(ii)

Close universe, for k = 1 Applying conditions for close universe i.e. k = 1, α and β are obtained from

Eq. (V.28 -V.29) as:



 3n 3(n  1) , and   2(n  3) (n  3)

k1 

3(2n  3)  2 (n  3) , 9(n  1)

k2 

2(n  3) BC , 9(n  1)

112

k3 is given by k3 

1 . On substitution of β and k3 in Eq. (V.24), expression of k2

density is simplified as:

 (t ) 

3(n  1)(n  2) 3 2 9(n  2) k2 a Cosh 2  ( ) k 2 (k 2 a Sinh   1) (n  3) (n  3) 2(k 2 a Sinh   1) 2

3(n  1)(n  2)k23a 2Cosh 2  9(n  2) k 2 (k 2 a Sinh   1) 2(n  3)(k 2 a Sinh   1)2

(V.39)

(V.40)

As compared to the flat model, it is found that as  → 0 the density approaches to a constant value.

(iii)



For open universe, k = -1  15n  18 3(n  1) and   2(n  3) (n  3)

 (t ) 

3(n  1)(n  2) 9n k2 a 2Cosh 2  ( )(a Sinh   k3 ) (n  3) (n  3) 2(a Sinh   k3 )2

(V.41)

Simplifying above Eq. further in the following form as:

 (t ) 

3(n  1)(n  2)k2a 2Cosh 2  9n(a Sinh   k3 ) 2(n  3)(a Sinh   k3 ) 2

(V.42)

Comparison of densities for flat, closed and open models, suggests that they approach to a constant value, but, the density for flat model appears to be more realistic than open or closed model. Flat model density reaches to Bag’s constant (BC) while other models tend to have constant value of density and independent of BC. In next section, cosmic implications of the model are discussed in which dependence of deceleration parameter, Hubble parameter, and other parameters on free parameters  and  are 113

analyzed. It is observed from above equations that the free parameters depend on n, which is index of power law equation.

V.5

COSMIC IMPLICATION OF THE MODEL With the evolution of cosmology along with particle physics, the importance

of quark matter in cosmology was explained in previous sections. Up till now a few models on quark matter and time varying lambda are investigated. These models emphasize only on the behavior of the universe, but cosmic implications remained unattended. The present model brings out several features i.e., the nature of the model, eternal inflation, phantom divide crossing and comparison with gravity models. The above features are briefly discussed in next sub-section.

V.5.1 Physical features of the model Eq. (V.19) and (V. 20) indicate that fifth dimension decreases more rapidly than fourth dimension for n < 2. It is also found that the fifth dimension scale factor is more dominant for small t. In the expression (V.19), the scale factor R (t) → constant value, but does not attend zero in either case, that is for t → 0 or  → 0. This concludes that the present model is a nonsingular model. Further, Eq. (V.21) reveals that as t  t0, H (t) tends to reach constant value. Eq. (V.23) manifests that Λ approaches to small positive value as t  . This is in accordance with recent observational data [59, 60]. The Eq. (V.24) - (V.26) show that both the density and pressure are dependent on nature of lambda. It is observed, that for present epoch, density and pressure are reduced due to the effect of time varying lambda. From Eq. (V.37) and Eq. (V.38), it can be observed that, for n = -2 and  (t) = 0 p = -BC. This suggests that universe is expanding. Presence of negative pressure indicates the existence of dark energy (DE). Further study on the model of DE is beyond the scope of present study. From Eq. (V.34) it is clear that at n = -2 densities are different for close universe as compared to that of flat or open universe. It is also seen that at n = 1, Eq. (V.10) is similar to that obtained by Ozel [17] for flat universe with β = 0. The constant integer n here is very important, because it provides information on the nature of extra dimension. Evidently, Eq. (V.37) and Eq. (V.38) suggest that quark density and quark pressure are dependent on Bag’s constant. Also, Eq. (V.22) indicates that the deceleration parameter does not depend on time varying lambda. 114

The nature of it is found to explain certain important features of the universe. With the help of it an important phase of the universe called ‘Eternal inflation’ is discussed in the following subsection.

V.5.2 Present model and eternal inflation Eternal inflation is an inflationary universe model, which is itself an outgrowth or extension of Big Bang theory. In this situation, inflationary phase of the universe lasts forever in at least some regions of the universe. Exponential expansion of the regions along with quite rapid growth has resulted into multiverses. According to Bousso et al [52] the eternal inflation predicts about end of the time. Douglas and Kacharu [57] explained distribution of vacuum, which depends on inflation. Eternality of inflation means whether inflation has a beginning and/or an end. Eternality of inflation can be related cosmologically to have some parts of universe to be inflated while others to exit inflation [55]. The present model, also, can have eternal inflation. From Eq. (V.22) it is observed that q = -1 when t  t0. This also implies that universe is accelerating. This condition, also, is related to eternal inflation of universe. Eternal inflation is unending inflation due to expansion of universe [49] and its consequences have been pointed out and much discussed in literature [50 - 58].

V.5.3 The model and phantom divide crossing In near recent past, intensive study is focused on phantom divide crossing. Phantom with ω ≤ -1 is dubbed as phantom energy [61]; ω = p/ρ is a constant in equation of state (EOS). ω = -1 is the phantom divide; ω > -1 is the quintessence era while ω < -1 is Phantom era. Phantom energy is the dark energy in the ω < -1 region. In this scenario dark energy density is ever increasing with time. Since phantom field requires negative kinetic energy, its particle excitations have negative energy. In this situation, empty space can decay into positive energy gravitons and negative energy scalar particles. El Nebulsi has [70] explained it. Higher dimensional cosmology with phantom energy is quite important for solving some of the major problems in physics; and it should also be noted that extra dimension plays a crucial role in solving key problems, which are dealt with initial inevitable singularities in cosmological past and hierarchy problem in particle physics. Physics of wormhole, big rip also can be studied in phantom era. El Nebulsi [71] has studied extensively in this regard. It 115

should also be noted that FRW model filled with phantom matter could escape big rip singularity through formation of wormholes. There is ample scope for research in this direction, and can be worked upon in future. In the present model, for phantom divide crossing ω = -1, results into ρ= BC. This is obtained for spatially flat universe. From Eq. (V.37), it is observed that as t→t0, ρ = BC is provided k1 = k2 and C0 =

9(n+1) 8(n+3)BC

. It is, also, noticed here that if ω < -1

then ρ < BC ; and that when ω > -1, ρ >BC. Thus, it is possible to have phantom divide crossing in present model. It is also known from various published literature [62, 64] that the phantom dark energy model can also explain the accelerated expansion apart from lambda decaying models.

V.5.4 Comparison of the present model with gravity models The models in which the scalar field either varies with R (Ricci scalar) or varies with T (trace of stress energy tensor) are called gravity models. Various models with f(R), and f(T) (gravity models) have been explained by Jamil et al [65], Jamil and Momeni [66], Momeni and Ajadi [67]. Because these models are mostly discussed in Bianchi metric, they explain the anisotropic universe. Recent observational data reveals isotropic nature of the present universe. They could explain accelerated expansion of universe, however, these models led to a singularity at late times. In this regard, model with quark matter in f(R) gravity could be useful for the research interest since, quark matter (QM) behaves like phantom type dark matter. The solutions for such model can provide information on inflation, and QM can be considered as source of dark energy at early universe. Yilmatz et al [68] have enlightened it. Klinkhamer [69] has discussed modified gravity model at QCD scale. They have claimed that their model is advantageous over lambda cold dark matter (ΛCDM) model.

V. 6

CONCLUSIONS An introduction of quarks and its importance in cosmology infused an interest

to take up the study on the quarks with Kaluza-Klein model of universe. The model with strange quark matter is investigated, and a discussion on cosmic implication brought out several features of the model along with an improvement over other models. Exact solutions in generalized form for Kaluza-Klein cosmological model in 116

presence of quark matter for flat, closed and open universe are derived. The obtained model is nonsingular, expanding and it generalizes the work of Ozel [17]. With dependence on n (index factor), it concludes that physical parameters depend on extra dimension. The present model also demonstrates importance of Bag’s constant on which quark density and quark pressure are dependent. The expressions for density and pressure are observed to be similar. It is also inferred that the model is accelerating at late times. Transition of universe from radiation-dominated phase to matter dominated phase, the present era, occurs as density as well as pressure decreases exponentially. It is, also possible to have phantom divide crossing in the present model; at the phantom divide crossing, density is equal to Bag’s constant, which is in line with the reports of Yilmatz [19]. Phantom divide crossing is more prominent for flat model than open or closed model; however, the present model can explain eternal inflation. Nonsingularity behavior of proposed model is advantageous over f(R) and f(T) models. Apart from eternal inflation, phantom divide crossing etc. models along with strings also are studied by many researchers. Recently, considerable amount of work has been carried out by invoking the model with strings attached to quarks that enabled the present work to study physics of the early universe. Study of string theory with quark matter is not only useful for understanding the early universe phenomena but also it acts as a source of information about dark matter and dark energy. The next chapter aims at the study of Kaluza-Klein cosmological model with string and strange quark matter, which is definitely a new aspect of Kaluza-Klein model.

117

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CHAPTER VI KALUZA-KLEIN COSMOLOGICAL MODEL WITH STRING, SQM AND TIME VARYING LAMBDA

VI.1

INTRODUCTION TO STRING COSMOLOGY VI.1.1 Strings

121

VI.1.2 Strings and cosmology

122

VI.1.3 Strings and higher dimensional cosmology

124

VI.2

KALUZA-KLEIN COSMOLOGICAL MODEL WITH STRINGS

127

VI.3

EINSTEIN FIELD EQUATIONS

129

VI.4

SOLUTIONS OF FIELD EQUATIONS FOR   

R2 1  2 2 R R

132

VI.4.1 Physical parameters

133

VI.4.2 String density and other parameters

134

VI.5

DISCUSSION

135

VI.6

CONCLUSIONS

136

REFERENCES

137

VI.1

INTRODUCTION TO STRING COSMOLOGY

VI.1.1 Strings Recently string cosmology has gained considerable attention due to its key role in obtaining information about the early universe. Since the time of Big-Bang, the early universe phenomenology has been a major curiosity among many cosmologists. The idea of string theory has come up to describe events of early stage of evolution of the universe. The origin of strings has lied in the phase transition through which the universe has passed during its early stages. Phase transition is a phenomenon of the nature where a reduction in the ambient temperature induces an abrupt gross overall change in nature of the material. Phase transition of gases to liquid, liquid to solid, unmagnetized state to magnetized state have been some of the well-known phenomena of nature. A phase transition occurs on lowering temperature, and it has often been accompanied by symmetry breaking, but it is not essential. At the very early stages of evolution of the universe, it had been assumed that when the universe pass through phase transition at critical temperature, symmetry of the universe has broken spontaneously. Phase transition in the early universe is discussed in detail later in the chapter VIII. As a consequence of first phase transition that took place at 10-12 sec after the Big-Bang, electroweak symmetry has broken. Consequently, topological defects appeared, such as magnetic monopoles, strings and domain walls, etc. The strings are actually the consequences of quantum field theory. They have been used to explain hadron scattering through Feynman diagrams. The concept of strings in Feynman diagrams has been first explained by Veneziano [1] in 1968 and Japanese /American physicist Nambu in 1970 [2]. The concepts of strings have been put forth to explain the conversion of two hadron particles A and B into another two type of particles C and D as shown in Fig.VI-1 and Fig.VI-2. The problem has arisen when particles A and B combined to form third particle, which decayed into two different particles. There could be many possibilities but most promising explanation has been to exchange a point like particle during the interaction between two particles A and B. These points like particles have been suggested as strings so that Feynman diagrams (Fig. VI.1 and Fig.VI.2) look similar. Eventually, solves the puzzling process of hadron physics. It has also been understood that strings behave like elastic bands with a string tension increased to the amount of 121

stretch of the string. Similar concepts of the strings will be discussed in context with quantum field theory in next subsection.

X C

Fig.VI.1: A & B combined to form X particle which decayed to C & D.

Fig.VI-2: Exchange of Y particle during interaction of A & B.

Kibble in 1970s has described strings from cosmological point of view. Cosmic strings have been topologically stable defects, which might be observed during a phase transition of the early universe [3]. It has also been believed that strings are density perturbations that led to the formation of galaxies [4]. These strings have, also, been useful in the study of gravitational effects as they have stress energy, and can be coupled to gravitational effects [5]. The gravitational effects of the strings can be studied, if they are assumed to consist of string cloud with particles. Stachel [6] also explained string dust model where null strings have been taken as massive. Vilenkin [7] has studied the gravitational effects of domain walls with strings. Strings have also been considered to be important while discussing quantum gravity, which is useful in obtaining information about graviton. Recent advances in observational cosmology depict many other features of the universe, such as, the accelerated expansion, discovery of dark energy, large structure of the universe, etc. With the upsurge of string theory, it has been possible to explain these features. A brief discussion on string cosmology is carried out in the following subsection.

VI.1.2 Strings and cosmology String theory in cosmology has important physical significance. As discussed in previous section, observed accelerated expansion, large structure, inflation at early universe and several other observations, dealt by general relativity, are still require more clarity and need to have proper explanation theoretically. There have been many cosmological models, which have provided explanations for these features.

The 122

developments in quantum field theory (QFT) have facilitated existence of standard model of particles. Attempts for unifying all types of forces have been made in past. Among them, higher dimension theory has gained importance. Particularly, the concept of higher dimension has arisen from QFT. QFT has originated when standard model of particle physics required study of quantum theory for particle physics, which should be consistent with Einstein special theory of relativity. Particle - antiparticle interaction, also, led to the beginning of new era in the field of QFT. Particle-antiparticle interactions have been represented by Feynman graphs. The interactions between particles or antiparticles or particleantiparticle are thought to be due to exchange of quanta, which is represented by strings as shown in Fig.VI.2. Exchange quantum is a virtual particle, which transferred so as to change internal states of the interacting particles.

Time

Fig.VI.3: Exchange of photon during the interaction between two electrons.

The Fig.VI.3 represents Moller scattering in which the interaction between electrons is represented by exchange of electromagnetic quantum or photon. Based on different types of exchange quanta, the interactions have been classified as strong, weak, electromagnetic, etc. It has been possible to represent different types of interactions in various ways in the form of Feynman graphs, and has been referred to as topology [8]. Feynman graphs have also been used for representing particle-antiparticle creation by involving charge renormalization. Renormalization, an essential part of QFT, allows different Hilbert spaces   and 0  (two different vacuum states) to be operated on the particle so as to have physical sense of   (all negative states), and to obtain zero charge even for infinite total charge value of the sea of electrons. Similarly, negative mass also needed to be renormalized. In fact, QFT outlines the background for 123

standard model of particle physics. Spontaneous symmetry breaking, an important phenomenon during inflation, can be explained with the help of QFT that involves different vacuum states. Also, phase transition in QFT is described as quantum mechanical processes and involves forming new vacuum states due to it. Symmetry breaking, the major phenomenon at early universe, has the origin of cosmic strings. Strings with cosmology can be dealt at Planck scales. Cosmic microwave background (CMB) observations are at high-energy scales and revealed successfully an inflationary scenario. However, inflation at its early stage can be understood by incorporating strings with cosmological model [9]. Inflation can solve horizon and flatness problems but failed to deal with dark energy problems, space like singularity, eternal inflation, etc. Phase transitions at early universe have led to origin of strings and that facilitated to obtain information about pre Big-Bang scenario. Gasperini [9] reviewed the string cosmology for pre Big-Bang scenario and related it with the primordial magnetic fields.

Another

important consequence of string theory has been the higher dimensional cosmology, which has been one of the attractive topics of research. In fact, string theory has been instrumental for higher dimension. Next subsection provides brief information on it.

VI.1.3 Strings and Higher dimensional cosmology Journey of cosmology revealed that there are many parameters, which are yet to be understood so as to visualize a complete universe. Although this is a difficult task but some past attempts have been partly successful. In order to explain these cosmological parameters, various proposed ideas have been based on extradimensions, strings, higher-dimension structures called membranes or simply branes, etc. These ideas have basically put forth for explaining a puzzling concept called supersymmetry, which is an extension to idea of symmetry. As a matter of fact, nature followed the symmetry, but the idea of supersymmetry has come from QFT models developed for explaining symmetry inter-relationship between fermions and bosons based on ‘Lie algebra’ of group theory. Supersymmetry theories have been proved to be quite useful in particle physics. Essentially, new particles such as 0-spin electron or s-electron, as partner to electron, s-quark or 0-spin quark, etc. have been discovered. As per this theory, each particle in nature has a partner with a spin differing from original particle by (1/2)ħ. In this way fermions and bosons are interrelated and a new 124

generation of particles i.e., W and Z bosons, gluons, graviton and most important particle called Higgs boson (God particle) have been recognized. The idea of manifold or dimensions more than space-time has been put forward to explain the partner of graviton called gravitino. Graviton, the gravitation quanta, is a boson particle. Hence, a particle is postulated as a partner of graviton having spin (3/2)ħ [10]. In short, supersymmetry, supergravity, etc. are the reasons for generating higher dimensional theories. Feynman graphs and string theory, cannot only explain supersymmetry and supergravity but also, it is noticed that if dimensionality is increased then matters look much more promising [11]. Although Kaluza and Klein have laid foundations for five dimensional theories for uniting electromagnetism, gravitation and particle physics, but Kaluza-Klein theory with strings can be pursued so as to employ both supersymmetry and higher dimensions; and understand the nature in more comprehensive way. String theory motivations for extra space time dimensions have been revealed from quantum gravity, spacetime singularities of Big-Bang and black holes, hot topics of cosmology, which are considered to be anomalies. These anomalies could not be understood by normal symmetry theory in quantum mechanics. Involving string theory as parameterization invariance in the description of string and considering dimensions more than four, it seems to have solutions for such anomalies. In this regard, a number of dimensions can be increased from four to twenty six [12, 13]. Theory based on increasing number of dimensions more than four could not gain popularity due to certain reasons. Out of many reasons, explanations based on Twistor theory, dealing with tachyonic behavior (faster than light propagation), has led to the conclusion of reducing number of dimensions from 26 to 10 [14]. Moreover, growing success of standard model, puzzling features of hadronic physics (tachyonic behavior, etc.) and simplicity of string theory, Veneziano, Nambu and others have started work on the road of strings that explained quantum chromodynamics (QCD) in terms of gluon-quark picture. Above discussions suggests that the string theory plays promising role in cosmology; and if observational evidence found then it can be a milestone in the study of universe. Gravitational lensing has been one of the tools that can provide evidence of cosmic strings. In this regard, in 2003 a group led by Sazhin [15] accidently discovered two seemingly identical galaxies very close to each other. It has been 125

believed that gravitational lensing using straight section of cosmic string may produce two identical undistorted images of the galaxy. However, observations through Hubble telescope showed that identical pair of galaxies has actually been present in the universe instead of their images through it lack evidences it. Observations of double quasar called Q0957+561 A, B have been originally due to Walsh Dennis, Carswell, and Weymann in 1979. Satisfactory explanation of double image of quasar can be given by gravitation lens effect. Double image of quasar have been understood due to the presence of galaxy between it and earth. Gravitational lens effect causes bending of the quasar's light so that it follows two paths of different lengths to earth. Consequently, two images of the same quasar one arrive at a short time after the other (about 417.1 days later). However, a team of astronomers at Harvard Smithsonian center of astrophysics led by Schild [16] found that during July 1994 to Sept.1995 double images appeared simultaneously having same brightness and occurred at four different occasions. According to them it is possible if cosmic strings are assumed to pass with great speed between quasar and earth during the same time with period of oscillation of 100 days. Thus, evidences of cosmic strings can be searched with the help of gravitational lens effects. The earthbound laser interferometer gravitational wave observatory (LIGO) and especially the space-based gravitational wave detector laser interferometer space antenna (LISA) are the experiments, which aim at searching for gravitational waves. They are quite sensitive for signals for cosmic strings. Presence of anisotropy in cosmic microwave background (CMB) can also provide the information about cosmic strings. In this context, Brandenberger and Turok in 1986 [17] have explained energy density spectrum and anisotropies in microwave background radiations by using string cosmological model. In a similar way Moessener et al [18] and Perivolaropoulos et al [19] have also discussed signatures of strings on CMB. Looking at various aspects of cosmic strings and possibility of getting solutions of the problems of the universe, an attempt is made to set up a cosmological model with string in Kaluza-Klein metric. The next section introduces Kaluza-Klein cosmological model with string, which is followed by setting up of the model, and thereafter a discussion on the results derived from the proposed model.

126

VI.2

KALUZA-KLEIN COSMOLOGICAL MODEL WITH STRINGS Higher dimensional string cosmological model in presence of quark matter

and time varying cosmological constant in Kaluza- Klein metric have been investigated. After the Big-Bang, universe had undergone another phase transition i.e. quark phase to hadron phase when TC ~200MeV. Such state has also called as quarkgluon-plasma (QGP) state. Witten [20], Fahri and Jaffe [21] have discussed its importance in describing early universe. They pointed out that the critical role of quark matter at early universe. Gerlach [22], Ivanecka et. al [23], Bodmer [24] , Itoh [25] have also discussed the importance of quarks during phase transition. The concept of quark matter has emerged from the quantum chromodynamics (QCD). The possibility of so called quark star or compact star, smaller than neutron star has also been thought and supported by degenerated quark pressure. It is plausible to attach quark matter to strings because strings are free to vibrate in different modes. The different modes represent different particles. Also, different modes of vibrations have been assigned to different masses or spins. Charged strange quark matter, attached to string cloud in cylindrical space time admitting conformal motion, has been studied by Mak and Harko [26]. On the other hand, Oli [27], Pradhan et al [28], Bali and Pradhan [29] , Mahanta et al [30], Singh and Sharma [31], Banerjee et al [32], Yavuz and Tarhan [33] have discussed the string cosmological model attached to quark matter in Bianchi space time. Tripathi [34], Reddy [35], Glovanini [36], Jotania [37], Saha et al [38], Singh and Singh [39] have studied Bianchi type string cosmological model in presence of electro-magnetic or magnetic field in different context. Pawar [40], Bali [41] have obtained the solutions of Einstein field equations for the string cosmological model with viscous fluid and bulk viscous fluid respectively, while Xing-Xang Wang [42] has discussed BI string model with bulk viscosity and magnetic field. Since the concept of higher dimension has originated from string cosmology, so it plays an important role in the study of early universe. Higher dimensions put forth independently by Kaluza [43] and Klein [44] have also gained attention while unifying gravitation and particle interaction, electromagnetism, gauge theories, etc. There are many reportings on the studies of higher dimension for exploring the idea of Brane cosmology, scalar-tensor field theory, parameterization of mass, etc. Wesson [45] has enlightened Kaluza-Klein theory so as to develop the new idea of Space -Time - Matter theory and applied to 127

many phenomenon i.e. particle interactions, gravitation, electromagnetism, etc. Study of higher dimension cosmological model with the string cloud and strange quark matter will be fruitful for obtaining information at early stage of the universe. As discussed earlier that the cosmological constant  has played the key role in the study of universe. Observational data of SN Ia by Perlmutter [46] and Reiss [47] in 1998 when combined with CMB measurements implied that universe is accelerating, which indicates towards finite and small positive value of cosmological constant. However, this has different value than the value predicted by standard model of particle physics. This problem is now known as cosmological constant problem (CCP). Chen and Wu [48] suggested that time varying cosmological constant can solve this problem by assuming ansatz   R-2, where R is the scale factor. There are several reporting on studies of cosmological models with time varying cosmological constant in different context. Overduin and Cooperstock [49], Sahani [50] and Padmanabhan [51], Arbab [52] have reviewed cosmological models with various time varying lambda. The literature suggests that the value of cosmological constant  might be large at the time of evolution of universe, when strings had also assumed to be dominating. In this regard, Pradhan et al [53], Tiwari [54], Abbasi and Rajmi [55], Anil Yadav [56] have discussed string cosmological model with time varying cosmological constant. Ray, Rahaman and Mukhopadhay [57] have explained scenarios of cosmic strings with variable cosmological constant assuming  = 3r-2,  = 8πρ, ρs = 3r-2, here r has its usual meaning. Later on, in order to reconcile age parameter and to deal with low-density problem with observational data, lambda has been further generalized by Carvalho and Lima [58] as =αH2+βR-2. Yilmatz [59], Katore [60] have explained string cosmological model in presence of quark matter in Kaluza-Klein metric without cosmological constant. Above discussion prompted to obtain exact solutions of Einstein field equations of Kaluza-Klein cosmological model with string in presence of quark matter and time varying cosmological constant. For simplicity this chapter is organized as follows. In section VI.3, Einstein field equations of string cosmological model in Kaluza-Klein metric are obtained. In section VI.4, solutions for string cosmological model in presence of quark matter and time varying cosmological constant are derived. Expressions for densities, pressure and other physical parameters 128

are also obtained with the help of solutions of EFE. Finally, the conclusions are drawn on the cosmological model and its behavior based on obtained expressions of physical parameter, which seems to describe physical situations at early stage of the universe.

VI.3

EINSTEIN FIELD EQUATIONS The line element for five-dimensional Kaluza-Klein model is represented as:

   dr 2 ds 2  dt 2  R 2  t     r 2 d 2  sin 2  d 2 2   1  kr 



 



  2 2   A  t  d  ,  

(VI.1)

where, k is curvature parameter, which is equal to 0, 1, -1 for flat, closed and open universe, respectively. R(t) and A(t) are scale factors. Ψ is fifth dimension. According to cosmic principle, ћ = c = 8G = 1. Five-dimensional energy-momentum tensor for string cloud attached to quark matter [59] is given as:

Tji     p  uiu j  pg ij   xi x j ,

(VI.2)

where p is isotropic pressure;  is the proper energy density for a cloud of strings with particles attached to them;  is the string tension density; ui = (0,0,0,0,1) is fivevelocity and time like vector; xi is a unit space like vector such that xixi = 1 in the direction of xi   4i which represents the directions of cloud i.e. directions of anisotropy. uiui = -1 and uixj = 0 are also considered here. Proper energy density for strings attached to particles can also be written as =p +. According to Bag model, quark density p = q+ BC and quark pressure pq = ρq/3, pp = pq – Bc, where Bc is the Bag constant. In accordance with Bag model, it is the difference between the energy density of the perturbative and non- perturbative QCD vacuum. Thus, proper density is represented by  = q + BC +. To obtain Einstein field equations, consider the following Eq.(VI.3): 1 G ij  Rij  Rg ij   g ij , 2

(VI.3)

129

where, R ij is Ricci tensor, R is Ricci scalar, g ij is metric element. Einstein field tensor is given by Gij  Tji   g ij , where T ji = diag (-ρ, p, p, p, p-), T ji is the stress-energy tensor. Einstein field equations are obtained From Eq.(VI.1) as follows:

(VI.4)

and

(VI.5)

(VI.6)

Conservation of energy momentum tensor is given by:

 Tji   gij  ; j  Tji eff ; j  0 . Hence, conservation equation followed from field equations:

 3R A   A        0 . A  R A

  ( p   )

(VI.7)

Above Eq. can be obtained from field equations also. The six variables R, A, p, , , and  are unknowns. To solve above Eqs. explicitly, three more equations are needed. Generalized  as a function of time is given by

. Since

quarks are considered to be massless particles, and quark fluid is supposed to be perfect fluid as per Bag model, the equation of state (EOS) for quark matter is given by, p

1    4BC  . 3

(VI.8)

130

To obtain the scale factors R(t) and A(t), field equations are solved using Eq. (VI.4) and (VI.5) as given below.

and

(VI.9)

(VI.10)

Physical variables, expansion factor and shear scalar in five dimensional metric are defined as

and

(VI.11)

(VI.12)

Use of EOS for quark matter in Eq. (VI.9) results into;

(VI.13)

Substituting Eq. (VI.6) in Eq. (VI.13) and simplifying it, following equation is obtained:

(VI.14)

In order to solve above Eq. ansatz A = Rn is assumed. This power law equation is considered to be due to the fact that there is still anisotropy for the flat and homogeneous universe and   ij (shear tensor), polynomial relation between metric coefficients. Thus, Eq. (VI.14) is simplified as:

(VI.15) 131

(VI.16)

In next section variable cosmological constant, generalized, first put forth by Carvalho and Lima [58], is considered; and the solution of above field equation is obtained. Physical parameters i.e. density, pressure, string tension density, quark density and quark pressure are determined subsequently.

VI.4 SOLUTIONS OF EINSTEIN FIELD EQUATIONS FOR To find the exact solution of field equation for flat universe, consider k = 0. The substitution of

and k = 0 in the Eq. (VI.16) yields:

Assuming m 

(VI.17)

3n 2  6n  6  4 4 4 BC , k1  , and k 2  , 3(n  2) 3(n  2) 3(n  2)

Eq. (VI.17) now can be simplified as:

(VI.18)

First order integral solution of above equation is:

(VI.19)

For simplicity assume m = 1, the solution of above Eq. is

(VI.20) 132

where, C0 is a constant of integration. Concurrently, the scale factor R(t) for KaluzaKlein metric is obtained as:

R2 

2C0  k1  k    Sinh 2k2 (t  c)  1 . k2  k2  k2

(VI.21)

Let,

k1 2C0  k1  3(n  2)C0  BC  , a      ,   2k2 (t  c) and k 3  k2 k2  k2  2    Eq. (VI.21) is modified in the following form, R2  aS inh  - k3 ,

 R(t ) =

(VI.22)

 a Sinh  - k3  .

(VI.23)

VI.4.1 Physical parameters Knowing R (t), other physical parameters are obtained as follows :

A(t )   a Sinh   k3 

H (t ) 

(t ) 

n 2

(VI.24)

k2 1 aCosh  aSinh  k3  , 2

(VI.25)

[ ]k2a 2Cosh 2  2 (a Sinh   k3 ) 2(a Sinh   k3 )2

3(n  1)    k2 a 2Cosh 2  2(  )(a Sinh   k3 )   (t )  2(a Sinh   k3 ) 2

(VI.26)

(VI.27)

133

1 p(t )  (   4 BC ) and 3

(VI.28)

(VI.29)

Above Eqs. are similar to those obtained for the cosmological model with quarks. The similarity is due to strings attached to quarks assuming same EOS as that of quarks. However, the constants in above equations are quite different. Hence, analysis for string density and the model with it, are discussed in the following subsection.

VI.4.2 String density and other parameters To determine string tension and density, consider Eq. (VI.9). Since k = 0, Eq. (VI.8) is rewritten as,

(VI.30)

substitution of A(t) = Rn (t) results into:

 k (n  1)a 2Cosh 2  ak Sinh  (a Sinh   k )  2 2 3  . 2   (a Sinh   k3 )  

  (1  n) 

(VI.31)

(VI.32)

From above Eq. the condition n  1 has to be satisfied. To find quark density, consider proper density as  = q + BC + . , -  = q + BC.

 k (n2  3n  2   )a 2Cosh 2  [   (1  n)ak Sinh  ](a Sinh   k )  2 2 3 B 2   C (a Sinh   k3 )   (VI.33)

q  

Quark pressure can be obtained as follows: 134

æ ö k 2 (n 2 + 3n + 2 - a )a 2Cosh 2j -[b + (1- n )ak 2Sinh j ](a Sinh j - k3 ) ÷ BC ç pq = = 2 ÷ 3 3 ç 3( a Sinh j k ) è ø 3

(VI.34) Expression for expansion factor is found out with the help of Eq. (VI.11) as follows:

(VI.35)

Similarly, expression for shear scalar can be found using Eq. (VI.12),

(VI.36)

VI. 5. DISCUSSION The Eq. (VI.27) reveals decrease in the total energy density with time, and it is always positive. Further, Eq. (VI.28) suggests that in absence of BC, radiation dominated phase of the model can be studied. Eq. (VI.26) manifests decrease in the cosmological constant decreases with time and approaches to a small positive value at late time, which is in good agreement with the recent astrophysical observations [46, 47]. According to Eq. (VI.29) as t → 0, q (t) = -1 suggesting that the model is deSitter in the early stage of the evolution of the universe; and this situation also leads to eternal inflation as discussed in fifth chapter. It is, also, observed that when t → ∞, q(t) = 0. Consequently, the model reaches to a steady state at late time, so universe expansion rate falls down if time increases. It can also be inferred from Eq. (VI.32) that string density λ decreases with time, and it is always positive. Eq. (VI.35) and

  constant , predicting the anisotropic nature of present t  

(VI.36) suggest that lim model.

135

VI.6

CONCLUSIONS Exact solutions for the higher dimensional cosmological model with strange

quark matter and a variable cosmological term  are derived. The model is also a nonsingular and represents inflationary phase, if k1 is assumed to be zero. It is the generalized model in which dimensionless parameters α and β define the behavior of the model. For real solution it is necessary to have

3C0 (n  2)    2 and n  -2 .   i.e. C0  BC 3(n  2) BC  BC 

With  = 0, the present work generalizes the work of Ozel et al [61] in absence of string cloud. The model is expanding and it disappears at late time. The derived model, also generalizes the work of Yilmatz [59] in absence of any pressure and density.

136

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139

CHAPTER VII KALUZA-KLEIN BULK VISCOUS COSMOLOGICAL MODEL WITH TIME DEPENDENT GRAVITATIONAL CONSTANT AND COSMOLOGICAL CONSTANT

VII.1 INTRODUCTION

140

VII.2 TIME VARYING GRAVITATIONAL CONSTANT AND BULK COEFFICIENT VII.2.1 Time varying gravitational constant

143

VII.2.2 Time varying bulk viscosity coefficient

145

VII.3 EINSTEIN FIELD EQUATIONS

148

VII.4 SOLUTIONS OF FIELD EQUATIONS

R2 1 VII,4.1 Case (i) G = G0/H , ξ = ξ ρ and    2   2 R R

152

R2 1  2 2 R R

155

VII.4.2 Case (ii) G = G0H , ξ = ξ ρd and    VII.5 DISCUSSION

157

VII.6 CONCLUSIONS

161

REFRENCES

162

VII.1 INTRODUCTION Kaluza-Klein cosmology, with its venerable history and several interesting features, has become popular since; it brings about a revolution in the study of the universe. It made possible to unfold the universe at its early stages and study its evolution and behavior by including an extra dimension in the Einstein field equations. Although, it has been difficult to prove the existence of an extra dimension experimentally due to unavoidable practical limits, but its effect can be observed. Extensive work on multi-dimensional physics and the Kaluza-Klein Theory [1 - 4], brought forth the concept of the universe comprising more than four dimensions, which had been large at the early stage of universe, but later became too small to be measured experimentally with available data. Recent studies using SNIa [5, 6], BAO [7], and WMAP [8] experiments have concluded accelerated expansion of the universe. The common interpretation of the accelerated expansion has suggested the presence of dark energy and dark matter in the universe. Both have been yet not understood properly. Several published literature, explaining implications of dark energy to some extent, have been now available on dark energy models. Interactive dark energy models, dark matter models and viscous dark energy models [9 - 12] have discussed dark energy models in the context of time varying cosmological constant . Two parameters, Newtonian universal gravitational constant G and cosmological constant  (Lambda), introduced by Einstein in his general theory of relativity are crucial. These two constants have been the major influential quantities in cosmology to understand the relation between geometry and matter. Recent studies on the universe have indicated time varying nature of both cosmological and gravitational constants [13, 14]. The discrepancy of about 10120 in the value of , determined using cosmologically and Standard model of particle physics [15], has been described as the cosmological constant puzzle (CCP). A time varying cosmological constant has been proposed to overcome this puzzle has led to a new area of research for many cosmologists. The time varying cosmological constant, first put forth by Chen and Wu [16], has been later adopted in various forms by many scientists studying cosmological models with   R-2,   H2 and  

[17-20].

140

The cosmological constant   H2 + R-2, and the generalized lambda, first suggested by Carvalho and Lima [21] applied by Singh et al [22] and Jain et al [23] has explained the non-singular, accelerated and expanding universe in higher dimension. Cosmological models with generalized cosmological constant with strange quark matter (SQM) or strings can be utilized for the study of early universe [24, 25]. Strange quark matter/strings play a significant role in the study of early universe, as they came into existence just after the Big-Bang. Some authors also have put forth the idea that SQM may be the part of dark matter [26]. Interaction between dark energy and dark matter has also been discussed in the literature that explains the present behavior of the universe to some extent [27, 28]. The pioneering work of Dirac [29] on large number hypothesis (LNH) has led to the idea of time dependent G. He pointed out that G  t-1. According to Milne [30] G  tm, where m is any positive number. Although his work has indicated importance of time varying G in the explanation of accelerated expansion, early universe phenomenology and large-scale structure of the universe, but it could not clarify whether G is increasing or decreasing with time. Models with time varying G of Barrow [31], particle creation in the universe according to the model with time varying G due to Harko and Mak [32], and dark energy models suggested by Ray et al [33], have considered decrease in G with time. In contrast, some published work [34, 35] has pointed out that the gravitational constant increases with time. Several research teams through astronomical observations within the range of solar and binary systems investigated a possible variation in G [36]. In other words, many researchers have explored cosmological models with time varying G and  [37 - 40]. Occhionero and Vagnefti [41] have calculated the value of G and co-related it with experimental one. A detailed literature survey has suggested that G  tm; where m is a constant but its sign (positive or negative) is not known [37 - 41]. The present acceleration of the universe can be very well explained by considering it to be filled up with perfect fluid. To understand the evolution of the universe, however, it is necessary to deal with the universe filled with viscous fluid. The current isotropic nature of the universe has been thought to evolve due to the dissipative effects of the viscosity of fluid [42 - 44].

141

The material distribution in the universe has also been one of the criteria for assumption that the universe is filled with viscous fluid. The phenomenon at early stages of the universe has indicated that when radiation decouples from matter in the form of photons and neutrinos, then it behaves as viscous fluid. In fact, viscosity decreases with time during evolution. A number of authors have investigated viscous cosmological models in different contexts [45, 46]. In this regard, Weinberg [47] has explained the role of a viscous medium during the evolution of the universe. Also, he suggested that the general form of energy-momentum tensor, which contains bulk viscous coefficient factor , is proportional to scalar expansion factor . Consequently, Both explain the decrease in pressure and accelerated expansion. The concept of evolution of the universe filled with viscous fluid whose bulk co-efficient

  ρd, investigated thoroughly by many researchers, has led to the generalized study of the universe [47 - 50]. It is possible to understand the presence of dark energy at early stages of the universe through the study of viscous cosmological models. Dark energy models with viscous fluid have been studied by Arbab [51, 52] and Ren and Meng [53] in different contexts. It has also been noted that the viscous cosmological model not only explained accelerated expansion, but, also, indicated the existence of dark energy [54]. The models with only variable G, variable lambda or variable bulk coefficient alone can explain accelerated expansion with certain constraints. The model with variable G as well as lambda considered on energy conservation ground, can explain late time acceleration; whereas, the viscous models deals with early universe. Arbab [55] has explained nonstandard cosmology with both the variables G and Λ with bulk viscous fluid. The unified description of the model with time dependent G and time dependent  with viscous fluid has brought out several interesting features related to cosmic evolution. Singh et al [56] studied the viscous model with time varying G, and Λ for early universe. Paul and Debnath [57] studied modified gravity model with viscous effect variables G and Λ. Further, the model can be extended to five dimensions for generalized lambda that can allow us to investigate transformation of anisotropic early universe to isotropic accelerated model and bring useful information on dark energy.

142

Aforementioned facts inspired as to set up the cosmological models based on time varying G and  with viscous Kaluza-Klein metric with a view to understand the accelerating universe. Additionally, the present model was set to reconcile with existing observational data. Also, in order to understand the universe at its early stages and its late time acceleration; models with gravitational constants as a function of H-1 and H are investigated. Further, an attempt is made so as to obtain information on dark energy, along with time varying G and . The Bulk coefficient of viscous fluid as  = 0 ρd is used in the present models. Concurrently, its effect on the models are observed. Essentially, time varying  modifies Einstein field equations that also illustrated the behavior of the universe at its early as well as its present state. Section VII.2 of this chapter discusses the role of both time dependent gravitational constant and time variation of bulk coefficient on the universe; both of them have been already introduced earlier. Section VII.3 sets Einstein field equations (EFE) with  = 0 ρd and  = H2 +R-2. Section VII.4 offers solutions of EFE for G

 H-1. In Section VII.5 model for G  H is derived, followed by discussion in Section VII.6. Conclusions are drawn in the subsequent section.

VII.2. TIME VARYING GRAVITATIONAL CONSTANT AND TIME VARYING BULK COEFFICIENT

VII.2.1Time varying gravitational constant The revolutionary rise or advancement in the physical science began in sixteenth century after the discovery of gravitational force due to great and renowned scientist Sir Issac Newton. Gravitational constant has been introduced as proportionality constant in the expression of gravitational force between the two masses. The value of gravitational constant is 6.7 × 10-11 m3/kgs2 and it is also called as universal constant as it is applied to all bodies including heavenly bodies and also unifying astronomical phenomena. Introduction of fundamental constants become essential so as to understand and establish the spectacular physical theories from macroscopic world to microscopic world. Fundamental constants, also, characterizes physical theories (Newton’s constant is called as classical theory while Planck constant as quantum theory). 143

Among many constants, speed of light c is the only fundamental constant, which is consistent in all fields of physics. The fundamental constants, which are 25 in number at present, have been regarded as constants till nineteenth century. Lord Kelvin has initialized investigation on varying constants in nineteenth century. In 1935 Milne [30] has proposed ‘bimetric’ theory that relates time in atomic phenomena with time in gravitational phenomena and requiring mass of the universe to be constant. Thus, it is proposed that G is directly proportional to time. Dirac [29] in 1937, however, has pointed out that gravitational constant is varying inversely with time by postulating large number hypothesis (LNH), which affected the dynamics of the universe. The large number, figured out by Dirac, has been the ratio of electrical force and gravitational force between two electrons and noted as 1040. This huge magnitude has been demonstrated as approximately equal to age of the universe in atomic units. Suggestively, it can be explained if G varies with time so that large value of electric force and gravitational force can be considered, as universe is old. Actually, large number hypothesis first proposed by Eddington. Later Dirac carried forward to time varying G [63 reference therein]. The proposed hypothesis of G  t-1 could not survive for longer time, because Teller [64] in 1948 has notified that high value of G in early universe would have resulted into elimination of life due to very high temperature of the earth during pre Cambrian era. A discussion between Dirac and Gamow in 1967 [65] with the support of Dicke [66], however, provided a boost to Dirac hypothesis for time varying cosmological constant. Thereafter the Dirac hypothesis got recognition and later acted as a stimulus to many scientists working in the field of cosmology. In this regard, Jordan, Brans, Dicke [67], have used time variation of G with scalar field theory and developed Brans - Dicke theory, which has also been a new research area. Involving variable G in Newton’s law of gravitation has been much easier than that in Einstein field equations; but Einstein field equations have been more versatile as they dealt with both matter and geometry, and can involve variations of other constants, also. Higher dimensions, spontaneous symmetry breaking process, eternal inflationary scenario, and quantum gravity at the beginning of the universe are some of the parameters forced to consider the time variation of G and obtain the apt model. According to Barrow, cosmologically the value for the

144

≤10-2H0 where, H0 is the Hubble constant at present universe. Barrow [68], Demarque et al [69] and Copi et al [70] have found

is nearly equal to 2×10-11

through astronomical calculations. Interestingly, cosmologists carried theoretical analysis assuming G  H. However, as discussed in previous section, some authors remarked that G increases with time. Science community being divided on the variation of G, it is interesting to first investigate the models with G = G0 H and G = G0 /H and then compare them on cosmological ground. A number of attempts have been directed in past for explaining accelerated expansion using cosmological models with time varying G. the models, however, modified by involving time varying cosmological constant can explain various aspects of the universe. In the present work, hence, Kaluza-Klein bulk cosmological model with varying G and Λ are considered. Presence of bulk viscosity in the proposed model is already highlighted in previous section. The next subsection gives a brief account of it so as to know its effects and motivation for the present work.

VII.2.2 Time varying bulk viscosity co-efficient There is inquisitiveness about the early universe since long ago. The phenomena at early universe suggest that the universe has been filled with viscous fluid at its early epoch as it has undergone several transitions and particle creation due to nucleosynthesis. Temperature degradation and its large-scale structure took place at this time. It needed to be identified with proper physical explanation. Various attempts have been made in this regard. Currently, it has been quite possible to understand various phenomena of early universe. As discussed in previous section, matter distribution in the universe at its early stages, universe had been filled with viscous fluid due to very high temperature. Consequently, particle creation rate has been less dominating. The early universe has been proposed as highly anisotropic. Thus, studies on the viscous model with variable viscosity coefficient demonstrated its anisotropic nature and explained present isotropic universe with time evolution. A quantum effect at early universe has played an important role in describing vacuum polarizations and particle productions due to quantum nature of matter. 145

Interactions between matter-radiation, quarks-gluons and particle-particle produced dissipative stresses have implicated the viscous universe. Viscous stresses, has also been due to phase transitions, decay mode of massive superstring mode, etc. [71 and ref. therein]. Several authors have studied viscous cosmological models to address several interesting effects due to bulk viscosity. Particularly, related to avoidance of singularity [72], primordial anisotropy [73] along with inflation and deflation [74]. A number of attempts made in past to study bulk viscosity with grand unified theory (GUT) have led to the inflationary universe [74 - 75]. Some of the models with perfect fluid usually suffered from singularity at some time even with time varying G and lambda, but viscous model can overcome this difficulty [48]. The dissipative mechanisms, also, helped in understanding the presence of dark energy, its evolution, dark matter, etc. Bannerjee et al [48] have pointed out that time varying viscous coefficient can also account for large entropy per baryon of the present universe. Most of the cosmological models have been considered with the dependence of bulk coefficient on density. Maartenes, Gron, Zimdahl

[50], [4], [44] have

reviewed non-casual thermodynamic theory of Eckart [76] and casual theory of Israel and Stewart [77]. According to them, the bulk coefficient   m (m is a constant). Non-casual theory admits the dissipative signals with superluminative velocities and instability of equilibrium states at all times [78]. Maartenes have pointed out the drawbacks of non-casual theory of thermodynamic from the observed deviation of first order from equilibrium for thermodynamical system. Currently Israel-StewartHiscock (ISH) full casual theory has widely accepted due to its capability to explain relativistic viscous fluid by considering higher terms of deviation from equilibrium in the transport equation. Energy-momentum tensor for relativistic viscous fluid is given [15] as:

T 'ij  Tij  Tij ,

(VII.1)

where ∆Tij is the correction term that can be related to viscous fluid. The general form of it that includes viscosity coefficient factor [79] is given by: k Tij   H ik H jlWjl   H iju;k ,

(VII.2) 146

2 k where, W ij  ui; j  u j;i  giju;k , is the shear tensor; and Hij is the projection tensor, 3

which is written as Hij = gij – ui uj on hyper plane normal to ui . The ζ and ξ are shear and bulk viscosities, respectively. Shear viscosity for FRW type model is zero. Hence,

k k k . Tij   H iju;k   giju;k   uiu ju;k

(VII.3)

Using Eq.(VII.3) in Eq. (VII.1) following Eqs. are obtained: k , T 'ij  Tij   H iju;k

(VII.4)

k k T 'ij  (   p   u;k )uiu j  ( p   u;k ) gij

(VII.5)

k Hence, effective pressure for viscous fluid peff is given by peff  p   u;k , of which k p is pressure for perfect fluid and  u;k is viscous term. Isotropy and homogeneity of

the universe are not affected by the viscous term due to its decay with time evolution k of the universe. It is well known that expansion factor   u;k , so peff= p – ξθ, which

has been used in Einstein field equations to study the viscous model. The bulk coefficient ξ is usually of the form ξ = ξ0ρd, where ξ0 and d are constants [76]. Some of the workers used bulk viscosity coefficient ξ  H [79, 80]. These models explained certain aspects of the universe but cosmological constant problem (CCP), problem of dark matter, dark energy, supersymmetry, supergravity and similar problems are remained unattended. Role of extra dimension is already discussed in previous chapters. Fourdimensional viscous models, proposed by Arbab, can be extended to five dimensions. Inspired by Arbab [51,55], and Singh et al [56] the model with variable G and Λ in viscous fluid is explained in Kaluza-Klein metric, and subsequently Einstein field equations are derived in next section.

147

VII.3 EINSTEIN FIELD EQUATIONS Kaluza-Klein metric given below is considered, so as to obtain Einstein field equations of present cosmological model:  1  2 2  ds 2  -dt 2  R 2 (t )  dr  r (d 2  sin 2  d 2 )   A2 (t )d 2 ,  2  1- kr  

(VII.6)

where k is curvature parameter, which is equal to 0, 1, and -1 for flat, closed and open universe, respectively; R(t) and A(t) are fourth and fifth dimensional scale factors, respectively; Ψ is the fifth dimension and r, , and  are the usual spherical coordinates. The comic principle allow us to consider ћ = c =1 and assume G (t) is the time dependent gravitational constant. Einstein field equations for time dependent G and  can be read as: Gi  8 G(t )T i  (t ) g i , j j j

(VII.7)

1 Gij  Rij  Rg ij . 2

(VII.8)

Here R ij , R and g ij have their usual meaning. The general form of energy momentum tensor for viscous fluid is taken as: Tji     peff  uiu j  peff g ij ,

(VII.9)

where  is the density for cosmic matter and peff is the effective pressure of the fluid of the universe. Five-dimensional effective pressure of viscous fluid is related to normal pressure as [47]: peff = p -  ,

(VII.10)

where,  is the coefficient of bulk viscosity, and scalar expansion factor is defined as

 = ui;i where semicolon represents differentiation of co-moving velocity ui, uiui = 1, ui is five dimensional velocity vector, and ui;i is the co-variant differentiation of 148

velocity vector. The  term consisting of viscosity co-efficient and expansion factor, introduced by Arbab [51] for viscous effects, has acted as a source term for matter creation and relates to expanding universe with dissipations. It has been argued by many authors [51-55, 58] that effective pressure reduces due to decay of the coefficient of bulk viscosity and this might be the reason for accelerated expansion. Energy momentum tensor looks similar to that of perfect fluid except for the extra term , which is very small as compared to normal pressure and density.

ij Hence, energy - momentum conservation for perfect fluid T; j  0 can still be considered here. For the metric Eq. (VII.6), using Eq. (VII.9) along with ansatz A(t)=R (t)n, Einstein field equations obtained and are given below,

(VII.11)

(VII.12) .

(VII.13)

Divergence of Einstein’s tensor implies,





 i 1 i i i  Rj  Rg j   8 GTj   g j  0 . 2 ;j  ; j

(VII.14)

From field Eqs. (VII.11) - (VII.13) and with the help of Eq. (VII.14), energy conservation is obtained as:

(VII.15)

Substituting peff from Eq. (VII.10), above Eq. can be rewritten as :

(VII.16) 149

ij The usual energy conservation relation T; j  0 implies that continuity Eq. should have an extra term linking it to the bulk coefficient term, i.e.,

. The work

of Singh et al [45], Arbab [51, 52] and similar literature revealed that applying a dominating energy condition as could still satisfy the usual conservation law given below

. Since the proposed model is a non-causal and dissipative, the dissipative

effects can be linked to time dependent G and , and can arrive at:

(VII.17)

From Eq. (VII.16), it follows that,

(VII.18)

Expansion of scalar factor  can mathematically be written as:

(VII.19)

Substituting  in Eq. (VII.18), leads to:

(VII.20)

To arrive at the field equations satisfying energy conservation Eq. (VII.21), a second order differential equation is obtained by differentiating Eq. (VII.13) with respect to time and dividing by 8G.

(VII.21)

150

For simplification of the above equation, let us apply “gamma law” EOS for pressure and density as p = ( - 1)ρ, where  is the adiabatic parameter. In cosmology, the value of  is 0    2, which can take the values 0, 1, 2/3, etc. for the universe filled with dark energy, matter, radiation, etc., respectively. Substitution of “gamma law” EOS in Eq. (VII.17), results in:

(VII.22)

r = r0 R -g (n +3) ,

(VII.23)

where ρ0 is constant density. Using Eq. (VII.20) and (VII.22), Eq. (VII.21) is simplified to:

(VII.24)

Applying Eq. (VII.13) in the above Eq., and after some mathematical manipulations, following differential equation is obtained:

(VII.25) The above Eq. may be further simplified by considering time dependent , coefficient of bulk viscosity  and gravitational constant G. The models for G that increases with time have been investigated by a few authors [34, 35] for the perfect fluid. On the other hand, the model for G  H has been discussed by a number of cosmologists [33], [38], [39]. The model with G  1/H is discussed so as to study the effect of increasing G with time. An another model with decreasing G, inspired by the earlier investigations [33, 38] is taken up to and study the models for generalized cosmological constant [21 - 23]. The solutions to the above differential Eq.(VII.25) 151

are obtained for different G in the following section, and later evaluate them for the universe with viscous fluid.

VII.4 SOLUTIONS OF FIELD EQUATIONS A solution to differential Eq.(VII.25) is obtained using

, =

0 ρd and for two different gravitational constants viz., (i) G = G0/H and (ii) G = G0H as discussed below.

VII.4.1 Case (i) : G=G0/H, =0 ρd and Assuming G is inversely proportional to Hubble’s constant, and

. It

suggests that the G increases with time as H  t-1. Increase in G with time has, also, been discussed in literature [38, 39]. Substituting G,  and  in Eq. (VII.25), it takes the form as:

(VII.26) Let 3 (n  3)(n  1)  6(n  1)   (n  3) , 6(n  1)

m1  k1 

 (n  3)(3k   )  6 k

m2 

6(n  1)

(n  3)2 8 G00 0d 6(n  1)

Using Eq. (VII.23) in m2, Eq. (VII.26) is rewritten as: 152

(VII.27)

The solution to the above differential Eq. is:

(VII.28)

where C is constant of integration. A solution to the above Eq. is, however, complicated. For a realistic solution, the above Eq. is simplified by assuming m1 = 1 and d (n+3) = 2. In such a case, the solution to the above Eq. is obtained as:

(VII.29)

m Assuming m2'  2 , the solution obtained as: 2

R2 





 (m'  k )(t  C )2  C  . 2 1 1  ' (m2  k1)  1

(VII.30)

The above Eq. can be rewritten by considering the initial condition, at t = t0, R = R0 and H =

2   2 H 0 R0  C  R 2  (m2'  k1)  (t  t0 )   .  ' (m2  k1) m2'  k1    





(VII.31)

To arrive at a realistic solution, one should have m2 > 2k1. Other physical parameters are obtained as follows:

153

(VII.32)

(VII.33)

2 (m'  k )(t  t )  H R 2   C 0 0 0  G  2 1 G (t )  0   H (m'  k ) (m'  k )(t  t )  H R 2 2 1 2 1 0 0 0



  0



1 2 2  H R C 0 0  (m2'  k1)  (t  t0 )    (m2'  k1  (m2'  k1)  

2    H 0 R02   '  C     0 (m2  k1)  (t  t0 )    k1  (m2'  k1)      

(VII.34)

(VII.35)

(VII.36)

 (n 3) 2

(VII.37)

The above physical parameters will be further analysed in the succeeding sections. In the following section the solutions to the field equations at G = G0H are determined.

154

VII.4.2 Case (ii): Model with G=G0H, =0 ρd and Substitution of G,  and  in Eq. (VII.20), leads to following Eq.:

(VII.38)

Consider 3 (n  3)(n  1)  6(n  1)   (n  3) , 6(n  1)

m1  k1 

 (n  3)(3k   )  6k

m2 

and

6(n  1)

(n  3)2 8 G00 0d 6(n  1)

Eq. (VII.38) is rewritten as:

(VII.39)

The solution to the above differential equation can be found under the conditions that m1 = 1 and d (n+3) = -2. Consequently, these conditions, Eq. (VII.39) is modified as: , If

y

(VII.40)

then, the obtained solution is:

1 k1 Coth k1m2 (t  t0 ) 2 m2

(VII.41) 155

In the above equation, constant of integration c is assumed to be c = -m2t0 . Hence, obtained R(t) is : 1  1 2 R(t )   logeSinh k1m2 (t  t0 )  . m  2 

(VII.42)

Other physical parameters are presented as below,

H (t ) 

k1m2 Coth( k1m2 (t  t0 )) , 2loge Sinh( k1m2 (t  t0 ))

(VII.43)

logeSinh( k1m2 (t  t0 )) Coth 2 ( k1m2 (t  t0 ))  4    1 , q(t )   Coth 2 ( k1m2 (t  t0 ))

(VII.44)



  k1m2 Coth( k1m2 (t  t0 )   4 log eSinh( k1m2 (t  t0 )  2logeSinh( k1m2 (t  t0 )   

 1    0  logeSinh k1m2 (t  t0 )   m2 

 (n 3) 2

2   (n 3)  1    0 0 logeSinh k1m2 (t  t0 )  .   m2 

, (VII.45)

(VII.46)

(VII.47)

Using m2 in the above Eq., it is further simplified to: 156

 6(n  1)  logeSinh k1m2 (t  t0 )  , 2  (n  3) 

 

G  G0

k1m2 Coth( k1m2 (t  t0 )) . 2logeSinh( k1m2 (t  t0 ))

(VII.48)

(VII.49)

In order to study the physical significance of the parameters derived above, let us, initially, consider approximation of Eq. (VII.40). Series expansion of y is given as

y

1 k1 2 m2

  -2 k1m2 (t-t 0 ) 4 k1m2 (t-t 0 )  2e    . 1  2e  

(VII.50)

Neglect higher terms of exponential terms in above Eq. because they reduces to zero as t  . The Eq. (VII.50) is then modified to:

y

1 k1  2 k1m2 (t-t 0 )  1  2e  . 2 m2  

(VII.51)

Eq. (VII.41) can now be rewritten as,

R2 

k1 m2

 2 k1m2 (t-t 0 )  t  4 k1m2 e .  

(VII.52)

For the sake of convenience, let us assume constant of integration as zero. Other parameters can also be derived in a similar manner. It can be observed that as t  t0, the scale factor R (t) reaches constant suggestively, to a steady state condition. Next section deals with discussion on the physical significance of the parameters derived for both the models (G = G0/H and G = G0H)

VII.5 DISCUSSION Both the models predict a small positive value of the cosmological constant, in accordance with observational data [5, 6]. The constants  and  can be obtained for both the models as they depend upon m1 and k1. Since k1  0 for flat model, it 157

becomes possible to analyse the flat as well as non-flat models. Nevertheless, for flat model,  and  are determined as,



3(n  1)( (n  3)  4) 6(n  1) ,  ,  (n  3)  (n  3)

provided k1 = 1 is assumed. From these expressions for  and, it can be observed that for a small, positive value of ,  should be greater than ; the index factor n should be positive and its value should be 0 < n < 1 so as to have compactification of extra dimension. For γ =1 i.e., for a matter dominated phase  >  so  is positive but, its value is small. For radiation dominated phase, also,  is smaller than . Eq. (VII.32), (VII.33) and (VII.34), (VII.35) represent peculiar behaviour of H(t) and q(t). This can be illustrated through the Fig. (VII.1 - VII.4). 1] G = G0/H , ξ = ξ0ρd and  = H2 +R-2 {Assumed initial conditions : C = 0.25, H0 R02 = 0.5 and other constants equal to 1}

Fig VII.1 Plot of q(t) v/s t [for G = G0/H ].

Fig VII.2 Plot of H(t) v/s t (not as per the scale) [for G = G0/H ]. 158

2] G = G0H , ξ = ξ0ρd ,  = H2 +R-2 Assuming constants k1 and m2 equal to 1

Fig VII.3 Plot of q (t) v/s t (G = G0H).

Fig VII.4 H (t) v/s t (not as per the scale) (G = G0H). Although in the Fig.VII.4, it appears that H (t) = 0 at some time but actually it is extremely small which cannot be plotted in the graph. It can also be evident from Figs.VII. 2 and VII.4 that H (t)  1/t for present universe, and has a smaller value than that of perfect fluid when it is filled with viscous medium. In earlier stages of the universe, the expansion of time dependent viscous medium enhances leading to acceleration at later stages. This could be due to the decrease of the coefficient of viscosity at a faster rate than G. Decrease of H (t) is faster for G H-1 as compared to that of G  H as also evident from Fig. (VII.2) and Fig.(VII.4). Such behaviour implies that the universe reaches a steady state condition earlier in the first case as compared to the latter. A little bump, however, observed in the Fig.VII.4 for a very short time, which is not observed practically; and an unusual behaviour is attributed 159

to the approximations made for plotting graphs, and is beyond the scope of present work. The Eq. (VII.33) and (VII.44) reveal that for the model with G = G0/H, q(t) → - 0.5 when t > t0 and for G = G0H, q (t) becomes negative after some time. This implies that there is an eternal inflation. According to Guth [59], eternal inflation has resulted in the rise of multiverses, which has been the latest area of research in cosmology. In this regard, Borde [60], and Taotao et al [61] have enlightened eternal inflation in different contexts. Fig.VII.1 shows the behaviour of the universe at later stages of the universe, while Fig.VII.2 indicates the behaviour at early stages of the universe. For dark energy models, if we consider  = 0, then differential equations Eqs. (VII.22) and (VII.33) can be reformed. However, the case (ii) G = G0H, model with dark energy explains accelerated expansion of the present universe. The case (i) G = G0/H model with dark energy explains early universe phenomena. If   -1, then the model with G = G0H (case ii) is able to explain accelerated expansion, while the model with G = G0 /H (case (i)) explains the present universe phenomenology for 1. Nevertheless, phantom divide crossing, the transition from phantom era ( < -1) to quintessence era ( > -1) [ = 0 is the phantom divide] is more prominent in the case (ii) model, as compared to the case (i) model as evident from Fig.VII.1 and Fig.VII.3 The model with G = G0H, filled with perfect fluid, has been explained in literature relating to early universe phenomena. The co-efficient of viscosity plays an important role; the present isotropic condition of the universe can be very well explained with the help of the universe filled with perfect fluid, but the universe filled with viscous medium with time dependent bulk co-efficient leads from the anisotropic early universe to present isotropic universe. This can be understood from the values of 2/ . Shear scalar  is 3 obtained from the relation  2  (1  n)2 H 2 [22] and scalar expansion  is defined 8

as  = (n+3) H [previously written as  from Eq. (VII.14)]; (2/) can be easily found from Eq. (VII.27) and (VII.37) for both models. Since (2/)  H, it is observed that in case (i) model with G = G0 /H, / → 0, while in case (ii), G=G0 H,

2/ → k in the present scenario. 160

VII.6 CONCLUSIONS The models with gravitational constant G, which is proportional to H-1 as well as proportional to H are analysed. G increases with time for G = G0/H, and it decreases with time for G = G0H. The latter model with G = G0H is similar to that of discussed by Ray et al [62], provided the universe is filled with perfect fluid, while the model with G = G0/H is similar to that of discussed by Singh [22] for the universe filled with perfect fluid. The cosmological constant is also found to have a small positive value as per observational data. Bulk viscous Kaluza-Klein model with time dependent cosmological constant and varying G is observed to be expanding and accelerating. The model with G = G0/H explains the early universe if it is filled with perfect fluid. Although existing literature explain universe with G and  both decreasing with time to reconcile it with present observational data; the model with G increasing with time is found to be forever accelerating. This is possible because the co-efficient of viscosity () decreases with time. Hence, these models can also be studied for dark energy models. A close scrutiny of the models with G = G0/H and G = G0H, suggests that both the models explain the universe, which are physically viable in different ways but the model with G = G0H can be reconciled with the present observation data. It is also found that the model with G = G0/H is a non-singular, accelerating universe; while the model where G = G0H represents an inflationary universe as observed from Eq. (VII.42) at early stages of the universe but later will tend to de Sitter universe. It is also concluded that effect of time dependent viscous medium along with variables G and  are responsible for acceleration of the universe.

161

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165

CHAPTER VIII THE EARLY UNIVERSE AND KALUZA KLEIN COSMOLOGICAL MODEL

VIII.1 INTRODUCTION

166

VIII.2 PHASE TRANSITIONS IN THE EARLY UNIVERSE

167

VIII.2.1 Thermal history of the universe

168

VIII.2.2 Phase transition and effective potential

171

VIII.2.3 MIT Bag model and critical temperature

174

VIII.3 TIME-TEMPERATURE RELATION IN FRW AND

178

KALUZA-KLEIN COSMOLOGICAL MODEL VIII.4 CRITICAL TEMPERATURE AND PHASE TRANSITION

186

IN KALUZA-KLEIN MODEL VIII.5 CONCLUSIONS REFERENCES

190 191

VIII.1 INTRODUCTION Man is always curious about secretes of the universe. Birth of the universe, origin, its behavior at early stages and other long-standing issues, are still not explored to the complete satisfaction. In this regard, major revolution is due to Big-Bang theory. This theory is the most successful, though yet incomplete owing to inability to explain certain features of the universe such as presence of dark matter, dark energy, large-scale structure, accelerated expansion, etc. Emergence of the universe from nothing has been an amazing situation. In this regard, Lamaitre has proposed ‘hypothesis of primeval atom’ in 1927. The theory has also predicted that just after 10-37 second when the temperature and pressure had been very high enough to cause cosmic inflation [1]. The universe had been in the sea of matter and radiation in thermal equilibrium with each other. After the inflation, another phase transition occurred that resulted in quark-gluon -plasma (QGP) state. Around 10-6 sec after the Big-Bang, in spite of low temperature so high, pair production and annihilation had taken place. Both have resulted into the production of quarks and leptons. Today quarks and leptons are considered as basic building blocks of elementary particles. Baryogenesis, i.e. generation of Baryons continued with the evolution of the universe. It is, also, seen that Baryogenesis even violated the conservation of baryon number in the process of matter creation so as to have some structure for the universe. Although matter creation at early stages of the universe sounds to be appealing, but it has been certain that the nature preferred matter generation than antimatter creation. The ratio of matter to antimatter is unequal for the universe. This problem, however, has yet not solved. With the time evolution of the universe, due to pair annihilation it has been filled with photons, neutrinos, electrons and protons. The inflation has caused the universe to expand continuously. As a result, the temperature of the universe had fallen to several Kelvin. A number of workers [2, 3, 4] predicted that the universe at very early stages initially had been anisotropic. Later become isotropic due to its expansion. Big Bang model had been challenged by steady state model [5], which has 166

explained the universe and its behavior with the help of cosmological principle, specifically termed as perfect cosmological principle. Expansion of the universe, its isotropic nature and matter creation have been well explained using the steady state model. Certain observations however, created a major setback to steady state model. In this regard, Big-Bang model has proved to be more successful than steady state model. The observations of Deep space radio telescope indicated that the universe at its early stages was quite different than the present universe [6, 7]. Cosmic microwave background for the universe, as per COBE satellite [8, 9], had been inferred. Consequently, the universe appeared to be permeated uniformly by cosmic radiation in the background, and thus, it appeared to be isotropic in present time. Apart from Big-Bang model, higher dimension theory in the field of cosmology has brought a revolution, which can be mainly useful in the study of the early universe. One of the most important milestone in the higher dimension theory has been the Kaluza-Klein theory that came into existence for unifying electrodynamics, particle physics with that of gravitation. Many researchers have studied Kaluza-Klein theory [10] by setting up various models so as to understand the early universe.

The discussions of previous chapters of this thesis revealed that the

K-K model can explain certain features of early universe. In the next sections phase transitions followed by implications of K-K model and its importance in the study of early universe are discussed.

VIII.2 PHASE TRANSITIONS IN THE EARLY UNIVERSE It had been and it will be obvious curiosity for the mankind that how matter has been created from nothing. In fact, the Big-Bang theory is a violation of energy conservation law. This idea, however, got accepted worldwide. In fact, probably the matter had been created when the universe passed through transitions while cooling i.e., during its evolution. These transitions have been termed as phase transitions. Generally, phase transitions have been correlated with the change of phase of matter i.e., similar to solid changes to liquid called melting, liquid changes into gas (boiling) and vice-versa. 167

There can be several effects of the phase transitions in the normal matter. Symmetry breaking has been the major effect of the phase transition. Normally, it is observed that water has been more symmetric than ice; and steam is more symmetric than water. The symmetry of the matter is thus related to the temperature. Any phase exhibits higher symmetry at high temperature. Thus, for a fraction of second after the birth of the universe it can be thought that it was highly symmetric due to its very high temperature. Phase transition can be well understood in the light of thermal history of the universe as explained in the next subsection.

VIII.2.1 Thermal history of the universe The thermal history of the universe [Fig (VIII.1)] has suggested that before 10-43 sec, there had been Planck epoch and temperature must have been around 1010 GeV as per the Standard Model (SM). Also, there might have existed quantum gravity barrier. The onset of inflation resulted into expansion of the universe. The Planck era ended at about 10-35 sec. At this time, a transition might have occurred which unified strong, electromagnetic and weak forces, except gravity as per grand unified theory (GUT) (GSM Model [11]), and the temperature at that stage would have been about 1015GeV. The topological defects introduced due to this transition have resulted into magnetic monopoles, cosmic strings, etc. Inflation of both the universe and Baryogenesis has taken place as of this a consequence transition. At this transition, the unified force might have been due to interactive quanta, called super heavy Boson of mass m~1015GeV. Concurrently, strong and electro-weak forces emerged. Quarks and leptons have supposed to come into the existence at this stage. The inflation followed by Baryogenesis has been responsible for generation of baryons (protons, neutrons, etc.). It suffered, however, from Baryon asymmetry. Concept of Higg’s Boson has also been brought in at this stage so as to correlated the mass of the particle. Principle of gauge invariance [13] (PGI) has been related to Higg’s mechanism. Gauge invariance is violated for gauge field associated with mass. This idea itself has revealed the breaking of symmetry spontaneously. The major breakthrough includes the evolution of the universe that took place at 10-12 sec, 168

when electro-magnetic forces and weak forces became independent by the process of symmetry breaking. As a result bosons, photons and electrons got their mass. The time gap between GUT transition and electroweak transition, however, has been thought of as a particle desert; and the universe has been filled with hypothetical particles called as axions and supposed to be dominated by radiations. At this phase rather than symmetry, super symmetry of the universe is broken, which had been assumed to be present before GUT transition. Occurrence of QCD transition at 10-6sec has led to the generation of quarks, and the importance of quarks eventually gained attention for the study of internal structure of hadrons. Interestingly, quarks and leptons have been conceptualized from this transition and formed the basis of standard model.

Fig. VIII. 1 Schematic diagram of Thermal History of the Universe [12]

The process of nucleosynthesis had began at t = 10-5 sec, T  1GeV≈1013K and ended at t  5s, and T  0.5 MeV ≈109K. In this era, light elements like Deuterium, Helium, Lithium, etc. have been created. The universe started growing at a faster rate 169

than previous rates at different transitions. This era, is also, called as lepton era. Similar to the electrons and neutrinos, created during electroweak transition in large density, have started playing substantial role in the construction of large-scale structure of the universe. A major event that took place at t =1sec during the evolution. At this time, neutrino decoupled from radiations through weak force. Neutrinos have been held in equilibrium with electrons due to the interactive reactions

as    e  e ;

and

photons

equilibrium

with

electrons

as     e  e . The neutrinos being participated in weak reactions, its interaction with the electron has been relatively slower than the photon-electron interaction. Hence, they decoupled from photons during cooling of the universe. Neutrinos decoupling or freezing out early has resulted into its number density comparable to that of photons. Since neutrino has been relativistic particle exhibiting similar properties to that of photons, they have been also called as hot relics. Although its detection has been very difficult due to weak interaction with matter, but this nature has helped us to understand cosmic microwave background (CMB) as it cools very slowly. About t = 10-5 sec, due to electron-positron annihilation, the entropy and the temperature of the universe have increased. Since the universe has been filled with the radiations, the number of free electrons has reduced in nature. First three minutes [14] had decisive phase for defining the universe. Its further development in the light of the present scenario of the universe with the stars, galaxies, planet and life on the planet would be interesting.

The universe underwent nucleosynthesis when time

changed from t ~ 1 sec to 100 sec and temperature from T ~ 1 to 0.1 MeV, which has also been called as Big-Bang Nucleosynthesis (BBN). Elementary particles (protons, neutrons) produced in a large number due to Baryogenesis, have interacted with each other at high temperature and created light elements like helium, lithium, etc. The abundance of He4 in the universe has been the manifestation of nucleosynthesis, which has been actually the consequence of stellar nucleosynthesis. The matter-antimatter asymmetry has also been noticed during this process which is still a 170

puzzle in cosmology. The question raised regarding expansion of the universe and reduction in its temperature to present temperature T ~ -269 C has been traced by science community intensively. Later it has been solved with the help of experiments conducted using Hubble’s telescope. Having discussed about several phase transitions, the cause of phase transition that occurs due to the peculiar nature of effective potential is studied in the following subsection.

VIII.2.2 Phase transition and effective potential After the Big - Bang the temperature of the universe at 10-12 sec, might have been of the order of several hundreds of GeV. Electro-weak symmetry had broken [15] due to an expansion of the universe. Electromagnetic force and weak nuclear forces have got separated resulting in identity of electrons, massive Bosons and photons. W and Z0 Bosons have been responsible for weak forces as they have been massive field quanta. Photons have been the electromagnetic quanta. Toy models [16] have been introduced using simple Lagrangian, to understand the phase transitions. The phase transitions have been of two types viz. first order phase transition and the second order or crossover phase transition. Electroweak transition has been an example of first order phase transition. The QCD (quantum chromo dynamics), the second order phase transition has dealt with interaction of quarks with matter that influenced the universe. Thermal history also revealed that the GUT transition, Electro-Weak transition, and QCD transitions have been the major transitions during evolution of the universe and its proposed shape and structure. Other transitions at various times are shown in the Fig. (VIII.1). The theory of phase transition has been conceptualized by deriving effective potential from Lagrangian [15]. Let us revisit a simple Lagrangian to understand the phase transition. The Lagrangian as,

1 L    V , 2

(VIII.1)

where  is scalar field and V = V() is the potential. Also, consider 171

V ( ) 

 4

( 2  v 2 )2 ,

(VIII.2)

where  and v are constants. For the minimum potential, ∂V/∂ = 0. This will result into two roots of . It shows that the quantum fluctuations occurred at zero state also and spontaneously global symmetry of Lagrangian is broken at vacuum state. If the potential is temperature dependent, then above Eq. can be modified and can be written as [16];  2 2  T 2  . 2 Veff ( , T )      v     4 2  4  

(VIII.3)

The above Eq. is useful to determine critical temperature below which symmetry is broken. It can be written in terms of mass of the particle [16] as given below: 4

Veff ( , T )      (T 2  m2 ) ,

(VIII.4)

where, ,  are constants and m is mass of the particle. The modified Lagrangian for the first order phase transition by considering field strength tensor Fμ and vector field function Aμ (gauge potential) is given as, L

1  1 D  D  (   v 2 )2  F F  , 2 4 4

(VIII.5)

Here Dμ = ∂μ + i g Aμ and g is the charge. Consequently, potential is modified as

 g 3T 3 1   g 2  2 Veff ( , T )         T  v2   2 . 4 4 2  3 4   



(VIII.6)

Expectation value of φ is  in above Eq. and due to 3term; it is observed that besides critical temperature TC, there are three minima of potentials where the temperature drops. The plots of effective potential with that of  are as shown in Fig. VIII.2 [16]

172

Fig.VIII.2 First order phase transition: V() v/s  [16]

Tn in Fig.VIII.2 represents nucleation temperature and TC is the critical temperature. At nucleation temperature Tn, the possibility of tunneling the potential barrier to the true vacuum is more than expansion rate of the universe. At this stage symmetry breaks so as to have nonzero expectation value of the field. The gauge field attains mass by this process. Since Veff has different roots at ground energy level, symmetry breaks spontaneously. It is also observed that the mass can be assigned to the field by considering m (φ) = g ∙ φ where, g is the charge. Gauge invariance is, also, violated due to assignment of mass to the field, however. The first order phase transition is similar to a process of thermal equilibrium at bubble walls. If pressure difference across the bubble wall is different then bubble wall can be broken and releases the energy, which can be in other forms. The formation of the domain walls, generation of gravitational waves and other certain topological defects are the consequences of first order transition. Cosmic strings are also topological defects that came into existence before the electroweak transition. The second order phase transitions are also called crossover transitions that must have taken place at time 10-6 sec at the temperature probably about 1GeV. As universe expanded, during cooling the quarks-hadron interaction resulted into its appearance in bound form in both baryons and mesons. Prior to this transition, quarks have been free to move in space, called as quark-gluon plasma state. The transition of quarks facilitated the interaction leading to the formations of baryons and mesons. In this way, they became the building blocks for hadrons or baryons. In other words, 173

second order transition led to the confinement of quarks, so also the formation of hadrons. Since the Lagrangian for crossover Transition does not include the term in 3, of three minima at ground level of the potential evident in Eq. (VIII.2) are understood. Since universe has cooled down further, the process of generation of hadrons remained continued. Quarks played a very important role in the time evolution of the universe. Also, they are important to understand large-scale structure of the universe. Minimum Standard Model [17] has explained first order transition but failed to elucidate QCD transition. There are, however questions regarding order of QCD transition, critical temperature at the transition, quantitative analysis through phase diagram, etc. These problems can be solved with the help of MIT Bag model or lattice QCD simulations. The MIT Bag model that provides most promising theoretical explanations is discussed briefly in next subsection. Further, it includes how to obtain critical temperature for hadron formation, which is an essential phenomenon for large structure of the universe.

VIII.2.3 MIT Bag model and critical temperature The MIT Bag model, the model of hadrons, has been proposed by scientific community from Massachusetts institute of technology (MIT) [18]. This model explains the structure of well-accepted concept of hadron. According to this model, all particles have been supposed to be composite systems including associated quark and gluon field variables. Further, the hadrons have been treated as bags containing quarks confined as free particles. The bag term has been chosen for hadrons because its internal structure cannot be considered to be made up of cluster of particles. The internal structure of hadrons can be thought of as association of particle and field variables. The quarks being quantized fields, and so internal structure of hadron must account for field variables and their interactions. Unlike ordinary field that vary over space, here it has been assumed that the field varies over some points inside the extended object confined to some region. This set of points has been called as ‘Bag’ for the hadron internal structure. It has also been pointed out that the quarks have been bound together by interactive field quanta called gluon, which can be understood as 174

exchange particle responsible for strange reactions among hadrons. Hadron formation can be understood by determining critical temperature TC at which QCD transition took place. Consequently, quarks come together to form hadrons. Bag model has been developed in such a way so as to treat the quarks to be free particles inside hadron beg but, cannot come out of it. To determine critical temperature TC, according to Bag model, hadron has been assumed to be a spherical body of radius R and mass M. Bag model assumed a constant called Bag’s constant B, which can be associated with the volume of hadron and it has dimensions of mass. Bag’s constant has an important significance in the Bag model. It has been equivalent to surface pressure (Dirac pressure) due to quark-gluon field inside the hadron [19]. Surface energy of hadron in association with Bag’s constant Es 

4 3 R BC , and kinetic energy of quarks has 3

been assumed to be ~ 1/R as per uncertainty principle. So total energy of the quarks inside the hadron can be given as:

EH 

4 3 C , R BC  3 R

(VIII.7)

where, C is the constant determined by minimizing EH with respect to R. The constant C depends upon total number of quarks and antiquarks occupying the space inside the hadron. Let

dEH  0 . Thus, C = 4BC R4. The C can also be determined quantum dR

mechanically from stress- energy conservation. It is also given by C = 2.04n, where n is number of quarks and anti-quarks at ground state. By minimizing EH, it can easily be understood that

1 4

 C R  .  4 BC 

(VIII.8)

It obviously predicts that the radius of hadron depends upon number of quarks and anti-quarks inside the hadron. Substituting C = 4BCR4 in EH, so, 175

EH 

16 3 R BC , 3

(VIII.9)

which is equivalent to mass of hadron M. So,

M

16 3 R BC . 3

Pressure inside the hadron is given by P = -∂VEH, where, ∂VEH is

P  

(VIII.10)

EH , V

EH C ,   BC  V 4 R 4

(VIII.11)

If mass of hadron is approximately equal to 1GeV then (BC)1/4 = 200MeV. Several assumptions considered to obtain critical temperature, have been given below. 1.

u and s quarks are lightest particles.

Chemical potentials for quarks inside hadrons are taken to be zero

Near critical temperature mesons are lightest hadrons, which are assumed to be present.

Hadron fluid pressure is related to temperature and is given by [17, 18],

 2  4 PH   3   T  BC .  90   

(VIII.12)

For three types of pion (, ) hadron pressure, which depends upon temperature, is given by PQGP. The calculated PQGP for eight types of gluons with two helicity states and for 2 quarks, 2 anti-quarks with each two helicity states having Fermi statistic factor as 7/8 and three colors. It can be written as 2 4 7   T PQGP   2  2  2  3   2  8  . 8   90

(VIII.13)

176

Calculating PH using (BC)1/4 ~200MeV and equalizing with PQGP critical temperature can be obtained as: 1

 45B  4 TC    ~ 144MeV .  17 2 

(VIII.14)

It is also seen from an expression for R that if EH is equivalent to MH then it is possible to get a relation between MH and R given below. 1

4 M H  (4 BC ) 4 C 4 . 3

(VIII.15)

From above Eq. it is observed that size of hadron depends upon R of the bag in which quarks are confined. Hadron formation and QCD transition are the milestones and responsible for the structure of present universe. Although MIT Bag model is inadequate to explain certain properties of particle (chiral symmetry, color confinement) but it predicts that hadron is made up of quarks. Quark matter played an important role in the formation of stars and galaxies. Using the bag model, the equation of state for quark matter has been obtained as pq = 1/3 q [4]. Pressure and density, of quarks are related to that of normal matter and can be written as p = pq - BC and  = q + BC. Recently, the universe has been observed with accelerated expansion. Further, it consists of only 4% visible matter, while 22% dark matter and rest is filled with dark energy. Several workers argued that one of the constituents of dark matter can be thought of as quark matter. So equation of state of quark matter is proved quite useful in the study of several aspects of the early as well as present universe. With the initiation of nucleosynthesis, the universe has started building itself leading to its expansion. The expansion of the universe has been first observed with the help of Hubble’s telescope. Expansion rate, as observed by Hubble, has been called as Hubble

177

parameter, and it is given by

. The light elements created at high temperature

during nucleosynthesis led to the matter-dominated universe. Before nucleosynthesis, since the universe consisted of radiations, that phase has been termed as radiation dominated universe. Radiation dominated universe suffered from mainly two types of transitions, namely, GUT and QCD transitions. Several theories developed to explain origin, time evolution and fate of the universe. These theories laid down the cosmological models with the help of metric tensor algebra. Among several cosmological models, Friedmann-Robertson-Walker (FRW) model came with appealing explanation of the present universe. The early universe phenomena have remained unfolded however. In late forties, many models have been proposed after several arguments, with the help of Einstein field equations in higher dimension metric. These models could throw light on early universe scenario to some extent. In these regard, Kaluza-Klein theory has gained popularity due to its ability to adopt multidimensional physics. Five dimensional physics actually, initially evolved from Kaluza-Klein theory and later, as an extension to it, higher dimension theory has been proposed by many which has depicted the early universe scenario. The following section is focused to study time-temperature relations of the early universe of FRW and Kaluza-Klein models.

VIII.3 TIME-TEMPERATURE RELATION IN FRW AND KALUZA-KLEIN COSMOLOGICAL MODELS A new theory, put forth by Kaluza in 1921 [20] and Klein in 1926 [21] for unifying gravitation, electro-magnetism, and particle interaction, have certainly gained an attention to look beyond the boundary of the present universe. A close look at thermal history of the universe has revealed occurrence of an important transition at 10-37 sec. This transition is called GUT transition. At this stage, all major four types of forces strong, weak, electromagnetic, and gravitation forces have been thought together. The standard model (SM) in particle physics can explain unification of three of the above four forces. The gravitational forces being very weak during the early 178

stage of the universe it could not be brought under the same roof as that for other three forces, until attempts of Kaluza in 1921, Einstein in 1926, and grand unification theory by Guth [22]. Out of these attempts, Kaluza-Klein (K-K) theory gained popularity because of its wide scope and vulnerability. According to K-K theory, the size of extra dimensions might be large at very early universe, but it curled to very small value and get embedded into four dimensions, which is known as compactification. Hence, the K-K theory gained attention of the scientific community. The role of extra dimensions can be thought to become effective only after GUT transition as photon and neutrino decoupling occurred after it. Big Bang nucleosynthesis provided the way for the formation of both the star and galaxy formation, as the basic elements, and their constituents, which are produced during that time. Hubble telescope has been instrumental for observing the universe, which helped mankind to understand the universe at various stages. Although the observational cosmology with spectacular results has enabled to visualize the present universe to some extent, but theoretical explanation of the universe has gained importance in Einstein era during which field equations are developed with the help of Riemannian geometry. Schwarzchild, Einstein, De Sitter and many others have established a proper cosmological model but, their models had been inadequate to explain the universe completely however. Friedmann-Walker model, the simplest model, proved to be milestone in the study of the universe, and it can define various parameters like red-shift, Hubble parameter H, density, pressure, etc. It has been possible to find critical temperature cosmologically with the help of Hubble parameter in Kaluza-Klein metric, a five dimensional metric, which is an extension of 4D FRW metric. Hubble parameter -temperature (H -T) relation for 4D FRW model has is revisited in the following subsection. Further, it is investigated in five dimensional K-K model. Let us first consider four-dimensional FRW field equations, which are as follows:

179

(VIII.16)

(VIII.17)

In above equations R(t) is the scale factor defined in FRW metric, k is curvature constant,  is the normal density, p is the normal pressure,  is the cosmological constant. A dot above R represents the time derivative of it. G is the gravitational constant. The above equations are obtained by assuming that ћ = c =1, in accordance with cosmic principle. Let

then equations (VIII.16) and (VIII.17) are

modified as:

3H 2  3

k R2

 8 G    and

(1  2q) H 2 

k R2

(VIII.18)

 8 Gp  

In Eq. (VIII.19), q is deceleration parameter defined as

(VIII.19)

Here , cosmological constant, plays a significant role in the study of early universe as it represents vacuum energy. At the early stage of the universe k can be taken as zero, i.e. the universe at its early stage can be assumed to be flat. In order to find TC,  is assumed to be constant factor at the early universe, although it is not really a constant as inferred through the observations in late nineties. To determine TC, consider energy conservation Tj;i j  0 , so the following Eq. is obtained as,

(VIII.20)

180

Substituting k = 0 as well as constant  in Eq. (VIII.18), and solving it with Eq. (VIII.20) results in Eq. given below,

d (  R3 )  3 pR 2  0 dR

(VIII.21)

For radiation dominated universe, since p = (1/3)  , the   R-4 . Let us consider radiation density U for past epoch in the early era to determine

R4 critical temperature. The radiation density for past epoch R is given by U  U 0 0 , R4 where U0 and R0 are the initial radiation energy density and initial epoch at t = 0, respectively.

Consider the early universe as perfect black body, and hence, energy

density in the perfect black body is given by U = T4, where  is radiation constant and T is its temperature. This situation can have T 

A . To observe the effect of R

lambda on critical temperature, initially the field equations are obtained without lambda and later, modified by including lambda in them. Consider, now, field equations without lambda. The Eq. (VIII.18) can be modified by including  = U =

T4, as,

(VIII.22)

On substitution of T = A/R and assuming at t = 0, R = 0, solution of Eq. (VIII.22) is obtained as, 1 1

 3 4 2 R  A  t .  32 G 

(VIII.23)

Substituting R in above Eq. and T = A/R, time-temperature is given by,

181

1 1 4 2 t

 3 T    32 G 

(VIII.24)

Above Eq. gives direct relation between T and t. But to calculate transition temperature, as we know it that at early stage of the universe, the particles are assumed to behave like relativistic gas at high temperature. If particle interaction is slower than expansion rate H [15] then density and Hubble parameter are modified as

2 * 4  g T .

(VIII.25)

From Eq. (VIII.8) considering k = 0 and neglecting  , Eq. (VIII.8) is rewritten as

3H 2  8 G  (T ) .

(VIII.26)

Assuming Planck’s mass M Pl 

1 then 8 G

(VIII.27)

Since T 1/R, the expression obtained for time-temperature relation for FRW model is given below,

t

2

1  1   2 g  T 

(VIII.28)

here, g* is related to number of degrees of freedom of the particles. Direct information on the critical temperature could not be obtained from equations (VIII.24), (VIII.25) and (VIII.26) but, calculations of TC can be made simple by knowing g* at the phase 182

transitions. The GUT transition suggests, g* = gb + gf, where gb and gf are internal degrees of freedom for bosons and fermions, respectively. From the particle data group [23], g* is calculated as 106.75 at the time of GUT transition while g*=17.25 at QCD transition. The phase transition temperatures TC are calculated for different g*, which are different at GUT, electroweak and QCD transitions. The relation (VIII.24) is not obeyed strictly at the transitions as it represents continuous t - T variations. In order to get t - T relation and obtain transition temperature, the higher dimension Kaluza-Klein model involving extra dimension can be considered. Emel’Yanov and Dienes et al [24, 25] have explained the implications of extra dimension of t - T relation in higher dimension, which is revisited here. Field equations of Kaluza-Klein model are modified for five dimensions as follows: ,

(VIII.29)

(VIII.30)

G11  G22  G33 , and

(VIII.31)

here, A is the extra dimension. k can be taken as zero for early universe. Since the universe was almost flat at very early universe,  can be neglected as compared to pressure and density assuming that they are very large. Thus, Eqs. (VIII.29 - VIII.31) are simplified to get H-T relation; and are rewritten in the modified form as:

(VIII.32)

183

, and

(VIII.33)

(VIII.34)

Above field equations can be solved easily to obtain the relation between R and A. To determine density from energy conservation relation, consider,

(VIII.35)

According to cosmological principal, the universe being filled with perfect fluid, the equation of state (EOS) for it is given by p = ( -1). An expression for density is thus, obtained as:

  0 R3 A .

(VIII.36)

If ansatz A = Rn is assumed (n is constant), then anisotropic universe at its early stages, then 2  θ, where  is a shear scalar and θ is an expansion scalar. Hence, the metric potentials are related by power relations, as mentioned in previous chapters. Substituting A in Eq. (VIII.36), following Eq. is obtained:

  0 R (n 3) .

(VIII.37)

In this case, temperature dependence on scale is given by,

T  T0 R



 (n 3) 4

(VIII.38)

184

  [ T0   0   

,  is the Stefan’s constant]. Assuming 8G = 1 and solving field Eqs.

(VIII.32) and (VIII.33) the resultant Eq. is as given below: 2

  (n  3)   (n  3) . Rd t  C  2 

(VIII.39)

Assuming initial conditions at t = t0, R = R0 and H = H0

C

 (n  3) 2

t0 

1 , d   H0  H0

2  (n  3)

R0 .

Thus, time - temperature relation can be obtained as, 

   n  3   (n 3) T  T0  H 0  t  t0   1 . 2  

(VIII.40)

The temperatures at radiation dominated phase and matter dominated phase can be determined from above Eq. For radiation dominated phase,  = 4/3, therefore, above Eq. is rewritten as:



 3  n  3  2(n  3) T  T0  H 0  t  t0   1 .  2 

(VIII.41)

It is also observed that if n < 1, then the temperature decreases rapidly as per recent observational data. From Eqs. (VIII.24) and (VIII.40) it can be observed that temperature in five-dimensional universe is lower than that for four-dimensional universe. This can be due to the presence of extra dimension in the early universe. Eq. (VIII.10) also shows that T depends upon . Hence, different phase transition temperatures of the universe can be calculated. Although above expression explains 185

time - temperature relation in higher dimension but, it does not explain critical temperature at the phase transition. To determine critical temperature in five-dimensional universe, it is necessary to account number of degrees of freedom in five dimensions for both Fermions and Bosons, which are supposed to be major constituents at early stage of the universe. In the following section, critical temperature is calculated and analyzed to understand the nature of phase transition.

VIII.4 CRITICAL TEMPERATURE AND PHASE TRANSITION IN KALUZA -KLEIN MODEL In fact, first order and second order phase transitions have been explained using field theory. The theory considers effective potential as a function of the field and temperature. Also, it has been established that the phase transition took place for minimum or zero effective potential. Transition temperature, in previous section was calculated in four-dimensional model. Transition temperature in five-dimensional universe is glanced at in this section with the effect of extra dimension. Particularly, since the importance of extra dimension cannot be ignored at early stages of universe, an attempt is directed in this section to study critical temperature in Kaluza - Klein model. To get an idea about phase transitions and critical temperature in five dimensional physics, Dienes et al [25] have argued that mass of the particle can be expressed with the radius of extra dimension, and particle is thought to be in Kaluza Klein mode. The mass of the particle in Kaluza -Klein mode can be written as

n2 , m2  m02  r2

(VIII.42)

where m0 is mass in zero mode, n is the Kaluza-Klein excitation number and r is the radius of the circle in which extra dimension is compactified. According to the field theory, effects of extra dimensions are unobservable at low masses but it can be observed at respectively higher masses since, Kaluza-Klein excitations can be observed at energies ~ r-1. The radius a is also called as Compton wavelength for 186

Kaluza-Klein mode of excitations. It is expected that for T >> r-1, and the relevant number of Kaluza-Klein modes ~ rT. It is important to note that Kaluza-Klein mode of excitations represent light particles or rather massless particle because, heavy modes are suppressed owing to their large masses at T >> r-1. Using simple scalar field theory, the critical temperature can be determined by calculating the effective potential V(C) for Kaluza-Klein field [25], and it is written as,

V (C )  

2 2

4 2 C , C 

(VIII.43)

where μ is the bare mass, independent of any background field, C is <  > ( is the quantum field for K-K mode),  is coefficient of quadric term C4 of V(C). The expectation value of C, i.e., C 

 . In four dimensional cosmological model μ is 

related to critical temperature; and considered as a correction to mass i.e., as

m2(T)= T2/4. The critical temperature in four-dimensional physics is given by,

TC D  4  2

 . 

(VIII.44)

For T > TC, expectation value of scalar field vanishes, which is a signal of phase transition. In order to get an idea about critical temperature, let us suppose that μ < 1/ r-1. If temperature is high such that rT >> 1 then, the universe will have effectively five-dimension. Dienes et al [25] have shown that as temperature decreases, the correction to bare mass (μ2) term becomes negative, which is related to radius as О(1/r2). Consequently, phase transition occurs at TC = О(1/r), and it is independent of bare mass term. This shows that, in presence of extra dimension at TeV scale with Higgs field, electroweak transition is quite different. If plasma mass term (correction to bare mass term) m(T) is larger than -μ2 and rT ≥ 1 then phase transition is

187

expected under the condition that (1/r) < T < (r)-1. The critical temperature is then represented as: 1

 2  2  3  . TC D 5    3  (3)  r  

(VIII.45)

A comparison of TC in four-dimensional and five-dimensional physics suggests that extra dimension plays a significant role in phase transition. The idea about critical temperature has been explained by Emel’Yanov [23, 24] using the calculations of effective potential at the finite temperature in one loop approximation in K-K theory. Essentially, bosons along with massless fermions are considered to calculate effective potential. Such calculated effective potential had to be modified for massive fermions too, because these fermions play an important role during phase transition. The role of boson, however, does not play any significant role as ‘forces’ correspond to the effective potential that depends more upon fermion numbers and their degrees of freedom. Therefore, Candelas and Weinberg [26] have made an attempt, by considering massive fermions and massless bosons, for calculating effective potential for the field in extra dimension. They derived dependence of effective potential on the size of the extra dimension. It is given by

 3 (5)   f , L5 << 1/m << β and Veff    4 2  L45

f Veff

 m2      L , 1/m << L5 << β.  120 2  5  

(VIII.46)

(VIII.47)

In above expression L5 is effective circumference of extra dimensional circle. It is written as L5 = 2 r (C)1/2 , β =1/T ; and m is the mass of fermion. It should also be noted that effect of massive fermions could be considered through its contribution in the induced cosmological constant, which is given as, 188

 m5  ,  120 2   

i   Ni 8 G5 

(VIII.48)

where Ni is the number of fermions, G5 is the fifth dimensional gravitational constant which is assumed to be equal to G4L5. The cosmological constant, introduced here to eliminate the contribution of fermions, is responsible for repulsive forces. Contribution of bosons can further be considered by including in the Einstein-Action principle for cosmological constant i.e.,

Scosm =

b  L5 3 d x . 8 G5 

(VIII.49)

i and b together is supposed to be the total cosmological constant. This is an important consequence as it states the importance of cosmological constant at early stages of the universe and also at the phase transitions. Considering Eqs.VIII.46 VIII.47, the dependence of effective potential on the size of extra dimension is represented by Fig.VIII.3 [24]

Veff

β 1/m

Fig. VIII.3 Dependence of Effective potential on size of extra dimension in Kaluza-Klein model

Fig. VIII.3 reveals two minima. First one, observed due to the condition that

189

b << i is not absolute; and it is unstable state due to a large negative value. The negative value suggests that there can be finite probability for tunneling the barrier however. Most stable condition arises when L5 = β, which can be called as transition state in K-K theory. This determines the transition temperature in K-K theory. Nevertheless, the transition temperature can be calculated from Eq.(VIII.42).

VIII.5 CONCLUSIONS The universe at early stages had gone through several phases. Some problems at early stages are not addressed so far. Thermal history enlightens the phase transitions at several stages. Phase transition depends on the nature of effective potential by which critical temperature at different phases can be found out. They can also be studied with FRW and Kaluza-Klein models. It is seen that the calculated value calculated of critical temperatures and effective potential are close to the observational data. It is also observed that the temperature decreases relatively faster with time in Kaluza-Klein model. The extra dimension plays a very important role during phase transition. It is also observed that in Kaluza-Klein theory considers one loop approximation of effective potential depending upon size of extra dimension. Further, through the inclusion of cosmological constant in Kaluza-Klein model, it is possible to explain the transition at specific phase.

190

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Nakamura et al (Particle data group collaboration), J. Phys. G 37, 075021 (2010) ; FERMILAB-PUB-10-665-PRD (2010) 191

[24]

Emel’Yanov V.M., Physics Reports , Vol. 143, 01 (1986)

[25]

Dienes K.R., Dudas E., Gherghetta T., Riotto A., Preprint, arXiv: hep-ph/9809406 (1998)

[26]

Candelas P. Weinberg S., Nu. Phys. B, 237, 397C (1984)

192

CHAPTER IX KALUZA-KLEIN MODEL OF UNIVERSE: A FUTURE SCOPE

IX.1

INTRODUCTION

193

IX.2

KALUZA-KLEIN PARTICLES AND DARK MATTER

194

IX.2.1 Direct detection of Kaluza-Klein Dark Matter

197

IX.2.2 Indirect detection of KKDM

198

KALUZA-KLEIN COSMOLOGY AND DARK ENERGY

201

IX.3.1 Observational evidence of dark energy (DE)

201

IX.3.2 Cosmological model and dark energy

203

KALUZA-KLEIN COSMOLOGICAL MODEL

205

IX.3

IX.4

AND MATTER CREATION IX.5

CONCLUSIONS

208

REFERENCES

210

IX.1

INTRODUCTION Theoretical explanation for the experimental observations of the universe has

been the most challenging work. Also, it has a wide scope for research in cosmology. Since nineteenth century, observations in space have regularly been carried out. Theoretical analysis however, has started only after the work of genius Einstein. An extension of three space co-ordinates to four space-time dimensions has also been understood during the same period. Theoretical models of the universe in four dimensions, developed during 1930s, have partly been successful in explaining experimental observations of ‘Hubble’ telescope. This has increased the curiosity about the universe and consequently, accelerated the research in theoretical as well as experimental cosmology. With the growing research in cosmology and particle physics, a trial for uniting laws of physics, i.e. to bring them under one roof, has also taken place. In this regard, indigenous work of Kaluza and Klein has opened the doors to look for a new area of physics of dimensions more than four space-time coordinates. Theoretical concept of higher dimension physics had been quite appealing but needed to be verified experimentally. The latter seems to be a quite difficult task. Because extra dimensions have been of Planck scale and difficult to achieve due to human limits. Although extra dimension could not have direct evidence but its effects can be observed. Kaluza-Klein model, five-dimensional theory, has become popular due to its simplicity, elegance and can be treated with perspectives of modern cosmology. The Kaluza-Klein theory have facilitated cosmologists to study different matters and, also, enabled to explain many unsolved problems including problems related to early universe phenomena to certain extent. Researchers have consistently directed their efforts in this regard and optimistic for both the better results and answers for unsolved problems of dark energy, dark matters, etc..

Apart from studies on

accelerated expansion, cosmological constant problem and similar other problems have been treated with the help of Kaluza-Klein theory. A new theory called SpaceTime-Matter, put forth by Wesson, has been a modified form of K-K theory, which primarily deals with parameterization of mass, giving a new meaning to the concept of matter. In late nineties, the intense work has been carried on early universe phenomenology. These studies include Big-Bang, supersymmetry, phase transition, 193

thermal history of the universe, evolution of present universe, fate of universe, etc. All these phenomena at early universe can be explained with the help of string cosmology and higher dimension theory.

Hawking has hypothesized strings.

According to him, the pressure due to invisible matter in the universe has negative value, and thus universe has been expanding and accelerating. An important phenomenon called phase transition has been accompanied by process of symmetry breaking at early stages of the universe. Consequently, string theory has been evolved. Strings played a key role in higher dimension theory and search of quantum gravity, which might be due to Big-Bang. Matter content of early stages of the universe has to be properly identified to get an idea of evolution of the universe. It has, also, been believed that during the phase transition at early stages, the matter creation also took place. Concurrently, led to an idea of some new particles. Owing to Kaluza-Klein theory, Kaluza-Klein particles representing Kaluza-Klein mode of excitations have been thought to exist during one of the phase transitions. Kaluza-Klein particles have been proposed to behave as dark matter at early stage of the universe. Kaluza-Klein dark matter, dark energy has been some of the fields, which led to rigorous research activities and gained momentum recently. It should also be noted that experimental detection of Kaluza-Klein dark matter could help to understand the nature of dark matter. The next section throws a light on these issues.

IX.2

KALUZA-KLEIN PARTICLES AND DARK MATTER During first phase transition, took place at t~10-11 sec after Big-Bang,

topological defects actually energy perturbations had been occurred. Consequently, domain walls, magnetic monopoles, strings, etc. have been formed. As discussed in Chapter VIII, the phase transition has been the consequence of abrupt changes in potential accompanied by sudden fall in temperature. According to Kaluza-Klein theory, extra-dimension or fifth dimension represented radius of circular cross section of hosepipe having lengths equal to normal space and time co-ordinates. Kaluza-Klein particles have been thought to be quantization of the gauge field, which contribute to a extra dimension. Some of the 194

cosmologists have shown that inflation can be described with the help of KaluzaKlein theory [1]. The importance of Kaluza-Klein theory, also, lies in introduction of extra-dimension in the model of the universe.

Another concept regarding extra

dimension, called universal extra dimension (UED) has been postulated. The UED has shown the existence of extra dimension through the propagation of standard model fields freely [2]. The size of extra dimension has been predicted as R~TeV-1[3] and later on it came out as K-K state that can serve as dark matter [4]. It has been assumed that the potential depends upon mass and can be represented in the form of fourier series. Zeroth order of the fourier series has been thought to be lightest Kaluza-Klein mode of excitations, called lightest Kaluza-Klein particle or abbreviated as LKP. Another concept regarding K-K excitation states has been to express tree level masses representing tower of K-K states and treating extra dimension as heavy masses [5]. The tree tower masses are represented as:

m2X (n) 

n2 R

 m2X (0) .

(IX.1)

In above equation X(n) is the nth K-K excitation state of the standard model field X and X(0) is the zeroth mode of K-K excitation. In order to consider lowest mode of excitation as lightest Kaluza-Klein particle (LKP), which should be stable and viable candidate for dark matter, condition for conservation of K-K number (n) required to be satisfied. The stability of LKP depends upon K-K conservation number called as K-K parity, similar to R-Parity of the lightest supersymmetric particle. An open question, is it necessary to have some interface between particle physics and cosmology? An answer to this lies in the nature of dark matter. Fritz Zwicky has proposed first indication of unseen mass in 1933 in the halo of galaxies [6]. The existence of dark matter has been inferred from the observed discrepancy between rotational speed of galaxies and amount of visible stars within them. The rotational speed for visible matter can be calculated from the following equation;

v( r ) 

GN (r ) M N (r ) , r

(IX.2)

where MN (r) is mass interior to r. 195

The number of stars has not been enough to account for observed rotational speed, however. The orbital velocity of the object outside the central region of the galaxies does not have values as per above equation. The rotation curves of galaxies have revealed that for some regions M(r)  r and ρ(r)  1/r2. This led to the conclusion that along with visible matter, galaxies contain non-luminous matter also. The composition of this matter termed as dark matter that is still unknown. Sources like CMB and COBE observations, revealed that 22% of the universe consists of dark matter. Particularly, a large part of dark matter consists of non-baryons as evident from CMB anisotropy and Hubble telescope observations. The evidence of dark matter searched in last decade, based on gravitation lensing, has been the major achievement of many cosmologists. Hooper [5] has given a brief description on the efforts made by several scientists. Particularly, galactic rotation curves, supernova Ia observations, formation of large scale structure, weak gravitational lensing of distant galaxies by foreground structure and weak modulation of strong lensing around massive elliptical galaxies have been some of such attempts [[5] and references therein]. Gravitational lensing is the method in which an image of distant galaxy is to be located behind massive cluster of galaxies in the form of Einstein ring. The mass of the galaxy calculated in this way has suggested the presence of dark matter in the universe. Walsh, for the first time has observed two images of galaxies having similar red shift, spectrum and magnitudes but separated by only 5.6 arc while working at Kitt peak National Observatory [7]. Similarly, Lynde and Petrosian in 1988 have seen multiple arclets (part of Einstein ring) of clusters [8]. Garrett and Duda [9] have provided a detailed information about dark matter that can throw more light on its existence.

Through the observations of galaxies, it has been inferred that non-

luminous matter extends significantly to magellanic clouds.

Gravitational micro

lensing of the stars in the region of magellanic clouds in the gravitational field of massive compact halo objects (MACHOS) has revealed the presence of another type of non-luminous matter. But this non-luminous matter forms the small fraction of dark matter (DM) due to limits set up by Big-Bang nucleosynthesis on baryonic density [10]. The calculations of density and orbital speed of galaxy have indicated the presence of DM in the universe. The existence of DM though proved, its content has still been unknown to us. Neutrinos have been thought to be one of the constituents of DM. Since it has negligible mass with ambiguity about it due to its 196

peculiar nature, it forms a small fraction of DM. Hence, the puzzle of DM has remained unsolved. The concepts of cold dark matter, warm dark matter and hot dark matter have been proposed by the scientific community. Consequently, a zoo of new DM particles has been suggested beyond standard model (SM). Further, some of DM have been relativistic and others non-relativistic [11]. The content of DM had been searched at the early stage of the universe because most of the new particles have been postulated at this time only. Weakly interacting massive particles (WIMP), axions etc. have been the major constituents of DM. Among these, WIMP has been most suitable. Its interactions with matter and photons have been quite weak. Stable, electrically neutral and colorless particles have been the prominent signatures of the particles those constitute DM. Not only WIMP axion, but also, supersymmetric particles (SM) satisfying R - parity conservation R = (-1)3B+L+2S , [B-Baryon no., L-lepton no. and S-spin] have been supposed to be constituent of DM. The SM particles and their superpartners have R = ±1. Some lightest supersymmetric particles have been neutralinos and sneutrinos.

Super

partners have been one of the current and important research topics of astroparticle physics. The theory of universal extra dimension (UED) suggests that K-K particles or K-K dark matter (KKDM) have been the most viable candidate for DM as discussed previously in this section. Photon, neutrinos, Higgs boson, or graviton can have first mode of KK excitation. The next subsections takes care of the detection of KK and the information on DM, which could prove the existence of dark matter in the universe.

IX.2.1 Direct Detection of Kaluza-Klein Dark Matter (KKDM) Recently, experiments for direct detection of KKDM have been carried out by finding cross-sections of Higgs exchange. The Higgs exchange is proportional to 1/(m2B(1), mh4). The cross-sections of KK quark exchange is proportional to 1/(m2B(1),∆4) ,where ∆ = (m q(1)- m B(1))/ m B(1) is the fractional mass splitting of the KK quarks. B(1) cross-section calculations has shown small values. Consequently, experimental detection becomes difficult on such scale.

One of the first direct

detection experiments using earth-based detector has been directed to study interaction of LKP with matter. An interaction of DM with atoms may scatter off nuclei so as to produce nuclear coil, which can be detected. It had also been thought 197

that electrons can scatter off the atom by DM particles while interacting with normal matter. Due to low event rates and low cross section, this method encountered difficulty in detecting LKP or DM particles. First direct detection experiment for KKDM had been set up at Gran Sasso National Laboratory at Italy (DAMA/NaI), an underground experiment employing 100g sodium iodide as the detector material. Later, an improved version of above mentioned detector has been called as ‘DAMA/LIBRA’, has contained 250g NaI as detector material. They found 400 ≤ mB(1) ≤ 1200 with 5% ≤ ∆ ≤ 15% assuming Higgs mass 115 GeV ≤ mh ≤ 200 GeV. The predicted event rate 50 per year for extremely low allowed region has been found lesser than maximum allowed parameters [12]. There are other detectors namely, GENIUS, MAJORANA, etc. which are in used for searching the evidence for KKDM through detecting particles and study an interaction with detecting material.

IX.2.2 Indirect detection of KKDM Apart from direct detection, search for detection of KKDM using indirect method have also been attempted. The later involves LKP annihilation in distant region of the universe. Further, the annihilation radiations from high-density regions comprising galactic centers and dwarf spheroidal galaxies have been observed. Essentially,, they interact with matter and generate particle-antiparticle pairs. These particles have been normally neutrinos, positrons, electrons and antiprotons. Even gamma rays have been produced due to annihilation of WIMP particles. Since DM particles have been annihilated at a rate proportional to square of their density [13], highly dense regions can be more suitable for finding its evidence. Detection of gamma rays as a product of WIMPS (one of the constituents of DM) has been the most suitable way of searching evidence for DM. The highly energetic gamma rays at GeV scale cannot be significantly attenuated and retain their spectral information. Hence, DM can be detected over cosmological distances [14]. Gamma ray flux can be calculated using DM annihilations with the help formula given below:

  E ,  

dN  1 1  XX v  2dld .  2 2 dE 4 mX los

(IX.2)

198

In above equation  XX v is the WIMP’s annihilation cross section (actually it is cross section times relative velocity), ψ is the observed scattering angle, ρ is the DM density, dNγ/dEγ is the gamma ray spectrum produced per annihilation. The integration is obtained over line-of-sight. Detectors have been classified based on different types of telescopes to observe gamma ray flux. These telescopes include the satellite-based Fermi gamma ray space telescope (FGST), ground based atmospheric Cerenkov telescope namely HESS, MAGIC, and VERITAS. Ground based telescopes have extensively been used to study the emission from small angular fields but with far greater exposure. The energies of gamma rays greater than 100 GeV have been analyzed using FGST. Fermi studied the rays in the energy range from 100 MeV to 300 GeV. Based on these experiments, the limits on DM annihilations have been put forth in dwarf spheroidal galaxies, dark matter substructures, etc.[15,16]. Signatures of DM present in cosmic ray spectrum observed for galactic centers can be detected by studying the spectrum of positrons and antiprotons produced during DM annihilation. Though the charged particles produced in WIMP annihilations have been deflected by galactic magnetic field, but they can also be used to study annihilation rate of DM in our Milky Way galaxy. Recent experiments for detecting DM have three-year satellite mission PAMELA, and HEAT balloon experiment, AMS-01aboard international space station, etc. Surprisingly, the energetic electrons and positrons have been found to exist unexpected everywhere throughout our galaxy. The emission of excess microwave from the central region of the Milky Way, detected by WMAP experiment, has been interpreted as synchrotron emission from a population of electrons and/or positrons with hard spectral index [17]. Apart from detection of gamma rays, highly energetic electrons, and neutrino based telescopes have been the latest developments in the search of DM. Solar neutrinos, atmospheric neutrinos and cosmic neutrinos have been of prime importance as they are not affected by matter due to weak interaction. The WIMPS can potentially generate neutrinos through a variety of annihilations. Neutrinos can be the byproduct of the annihilations to heavy quarks, tau leptons, gauge bosons, Higgs bosons, etc. They can be directly annihilated to neutrino-antineutrino pair. Ice cube experiments at Antarctica, involving large 199

volume neutrino telescope and Super-Kamiokande detector, have been the latest developments in this regard. Observation and study of muon neutrinos from sun can provide useful information regarding existence of KKDM as muon neutrino detectors have observed muon tracks.

Ice cube detectors analysis have been particularly,

important in this regard because data regarding WIMP annihilation daughters depend upon mass of WIMP. The information about the identity of DM can prove existence of KKDM. All Indirect searches though quite promising for the detection of DM but, they have some inadequacies. Indirect searches have quite large uncertainties surrounding their observations. Occurrence of WIMP annihilation is quite uncertain and so also its location, i.e., centre of sun, earth or a galaxy etc. is not properly known. This has caused difficulty in the detection of DM. Moreover, considerable research efforts have been directed to understand new physical phenomenon such as supersymmetry, supergravity, etc. Hence, the evidence of KKDM has been thought due to large hadron collider (LHC) or linear collider, which enabled to search SUSY particles. The KK parity conservation has been responsible for decay of first mode of K-K particle to LKP state. The possibility to have evidence of DM is obvious from above discussion, and if detected experimentally then the structure of universe can be more visualized. To account for the existence of dark matter in the present universe, currently, LambdaCDM (cold dark matter) cosmological model has been studied more attentively due to its elegance and simplicity. For the present universe, cold dark matter (CDM) is needed to understand the evolution of the universe. In fact, at the beginning or at very early universe of Big-Bang the temperature of the universe had been very high, which has decreased rapidly with the time evolution of the universe. The constitutes of DM have been still unknown, hence, the classification of DM as hot, warm and cold DM has been done so as to know present structure formation. Also Lambda-CDM has included cosmological constant. Consequently, it has been possible to deal with nonrelativistic dark energy-dark matter (DE-DM) puzzle. Recently, research has also been carried on the cosmological model with an interaction between DE and DM that can explain large-scale structure, acceleration of the universe and can deal with cosmological constant problem. In next section another important constituent called 200

dark energy is highlighted, which can be useful for further research in the direction of present work.

IX. 3 KALUZA-KLEIN COSMOLOGY AND DARK ENERGY As a matter of fact, 74% of the universe consists of dark energy (DE). The DE has been the energy of the region devoid of matter and radiation, and it resembles vacuum energy. Vacuum has been a sort of emptiness having importance in the evolution of the universe and it has also been possible to have evolution of space-time in it. According to quantum mechanics, vacuum quantized energy can play a significant role in understanding the accelerating expansion of the universe.

IX.3.1 Observational evidence of dark energy (DE) Observational evidence of dark energy has been discovered first through the data on distance-red shift relation of supernova type Ia (SNIa) experimentation of Perlmutter et al [19] and Reiss et al [20]. The evidence has been inferred on the basis of difference between the luminous distance of dark matter dominated and dark energy dominated universe. A large luminous distance has been predicted for objects with high red shifts for dark energy dominated universe. As a consequence, the object of fixed intrinsic brightness has appeared fainter in dark universe. Since 1998, dark energy (DE) discovered from experimental evidence of SNIa research on DE has been progressing. It has, however, been still unknown what DE consist of. Type Ia Supernova (SN Ia) explosions have been seen while mass of rotating carbon-oxygen white dwarf, when its mass approached Chandrashekhar limit. Thermonuclear burning started at this moment and then exploded. Ia nomenclature for such supernova has been suggested due to presence of prominent silicon (Si II) lines in its spectrum. Experimental study on SN Ia has been normally carried out by finding apparent magnitude as follows.  d  m  M  5 log10  L   10 pc 

(IX.3)

where m, M, dL are apparent magnitude, absolute magnetic and luminosity distance, respectively. Luminosity distance (L) can facilitate determination of luminosity flux 201

of standard candles (a hypothetical object having fixed luminosity) using flux inverse square law (Eq. IX.4), f 

L 4 d L2

(IX.4)

For a standard candle, M (equivalently L) is approximately same for each object. Therefore, measurements of apparent magnitude of each object can provide information about luminosity distance. The experiments on supernova have been carried out since 1970s. The major breakthrough, however, came in 1990s when two teams of SN researchers supernova cosmology project (SCP) led by Perlmutter and High-z supernova search team led by Schmidt could obtain the results from distant SN. Observations of Phillips, an astronomer, have indicated the broadening of lines in the SN light curves and noticed as intrinsic brightness of SN Ia. Reiss in 1996 with his team improved the technique by introducing multicolor observations for measuring SN magnitudes. Most important technique, used to establish existence of DE, has been due to the development and application of charge-coupled devices (CCDS) for telescopes at Kitt Peak and Cerro Tololo [21]. High redshift observed for distant SN, and lower brightness of SN than expected one has indicted accelerated expansion of the universe and presence of DE in it. Acceleration of expanding universe cannot be possible with matter dominant universe with any amount of matter regardless of its curvature. In addition to Type Ia Supernovae, other probes of DE operate using different physics. Baryon acoustic oscillations (BAO) [22], weak gravitational lensing, and galaxy cluster abundance have been some of such probes. BAO probe measures length-scale characteristics of acoustic oscillations, which are calculated using present day correlations of galaxies by baryonic physics at the epoch of recombination and then calculating angular diameter distance for the distant galaxies. This method has also predicted accelerated expansion of the universe. Gravitational lensing has been actually the probe for DM but it has also been useful probe for DE. Distorted image of distant galaxies obtained, can allow understanding the distribution of DM and its evolution with time to be measured. Consequently, facilitates probing on the influence of DE on the growth of structure [23]. Galaxy clusters have been the largest objects 202

in the universe. Therefore, prediction of number density analytically followed by numerical simulation and then comparing with observed data may provide some clue about DE. The absolute number of clusters has dependency on minimal mass of clusters in the survey of solid angle survey with center at redshift z and thickness ∆z. Useful information about the rise of DE can be obtained from the geometry and growth of structure. Besides above-mentioned primary probes of DE, there have also been other secondary probes. Primary and secondary probes have been discussed and summarized by Goodstein in the book ‘Adventures in cosmology’ [[13], chap. 11]. Observations of gamma ray Bursts (GRB) has been suggested as another probe, other than supernova SNIa, to measure accelerated expansion with high precision because measured red shift z and it can be greater than 5. Hooper and Dodelson [24] have explored this idea and concluded that GRB observations can detect DE with high statistical significance. Not only detection of DE but it can also provide the quantitative information about presence of it at early universe. From above described various experiments, equation of state of DE can also be inferred. It has been found that p = -ρ for DE. This has indicated the presence of negative pressure, which explained the accelerated expansion. The experiments have also revealed the dominance of DE, and its smooth geometry. It affects geometry and growth of structure. The presence of DE has led to the present universe which is isotropic and spatially flat. Theoretically, its density can be related to critical density. The research in theoretical field is going on besides an experimental search. Both are briefly discussed in next subsection.

IX.3.2 Cosmological model and Dark Energy Previous subsection revealed that the cause of acceleration is DE, which is a form of vacuum energy. Acceleration of expanding universe can only be explained using the cosmological model provided cosmological constant Λ, is included in Einstein field equations. Cosmological constant Λ is already introduced in chapter II. It has, also, been predicted that currently, except Λ, no other quantity represents dark energy. The calculated energy density for matter is only 26% of the total density of the universe, whereas, energy density related to Λ is nearly equal to 74% of the total energy density. 203

There has also been inquisitiveness about the origin of DE. Since the development of lambda decay cosmology, the Λ has been predicted to have small positive value in the present universe. The universe with constant Λ has been, however, ruled out because of its inadequacy to explain accelerating universe. Hence, the model with varying Λ has been the area of current research, wherein the value of Λ thought to be quite high at the early universe. Further, due to decay of Λ, the present universe has small positive value of Λ. In a similar manner, it has also been thought that very early universe in radiation dominated phase has revealed the small value of DE. Present small value of DE has been assigned to the lambda decay, and also the cause of accelerated expansion. Knowing the fact that accelerated expansion is due to negative pressure, cosmological models with equation of state p = -ρ many cosmologists have explained. The study of cosmological models with DE has been the recent research area worked out in various contexts. Mostly, DE models have been studied with time varying lambda, but some of the models have used time varying gravitational constant while some have viscous models. Dark energy models have been referred to as quintessence models in which equation of state is p = -ρ and equation of state parameter w = -1. This condition has, also, been called as Phantom divide crossing. Phantom has been presumed to be dark energy quanta. Phantom dominated era has been supposed to be for w < -1, while w>-1 has been the condition to be dominated by normal matter. Higher dimensional theories developed recently have paved the way to the modern cosmology.

Involvement of extra dimension in the field equations has

provided some clue to early universe phenomenology and a new physics regarding brane world, multiverses, quantum gravity, etc. Accelerated expansion although can be explained through the presence of DE but, origin of DE itself is not well known yet.

Kaluza-Klein cosmology implementation in context with dark matter as

discussed in the previous subsection has provoked the idea of K-K cosmology with dark energy. In present thesis, also, K-K cosmological models with different matters discussed in previous chapters have concluded with the nature of DE. More clues can be obtained with the Kaluza-Klein theory with string cosmology. Since, strings have been the origin of higher dimensions. It is long way to go in search of DE and DM and there is lot of scope for studying Kaluza-Klein models with interactive and non204

interactive DE and DM. Presence of Dark Energy in the present universe may also be due to the decay of viscous coefficient with time as discussed in previous chapter.

IX.4

KALUZA-KLEIN COSMOLOGY AND MATTER CREATION According to previous sections K-K cosmology can deal with the problem of

DM and DE. Also it can provide the information on the universe since early to late stages. In present thesis cosmological models with perfect fluid, strange quark matter, strings, viscous fluid, etc. in Kaluza-Klein metric have already been discussed in previous chapters. These models can explain accelerated expansion and role of extra dimension in the universe, early universe phenomenology, cosmological constant problem and some physical features of the universe. One of the important phenomena in the universe is matter creation. Matter creation has also been an important issue that can be explored with KK cosmology, and it could be a future research topic of K-K Cosmology due to less studied. The creation of matter from nothing is still a mystery. Big-Bang theory, however, explains the creation of matter at very early universe. The model with BigBang theory, explains all the relevant physical processes, has some drawbacks. It, however, had singularity in past and may have one in future, which should be avoided. With time evolution, energy conservation has been violated in the Big-Bang model, and it has the small particle horizon. Consequently, horizon problem and flatness problem have arisen. Hence, FRW model set up with matter creation can explain the physical processes of the universe with energy conservation successfully. Theoretical explanation for creation of matter in context with cosmological model has been first dealt by Hoyle and Narlikar [25]. The detailed investigation on steady state theory (first explained by Gamow) and steady state model based on it has widened cosmological scene by past two decades. Although the model has been close to bigbang epoch, but has no answers to problems related to origin and fate of the universe. Large scale structure and expansion of the universe R(t)=e2H0t, and certain other problems have been remained unattended until the work of Hoyle in 1948 [26]. Hoyle with Bondi and Gold [27] have not only explained the steady state model but also able to deal with creation of matter. Increase in mass of the universe has been explained using simple theory and exponential scale factor. The explanation of large structure of the universe obtained, by assuming perfect cosmological principle has been based on 205

uniform and isotropic universe at any cosmic time. The metric element has been given by





ds 2  c2dt 2  e2H0 t dr 2  r 2 d 2  sin 2  d 2  .  

(IX.5)

The model has the line element with k = 0. In above equation H0 is the Hubble’s constant. Since, Hubble parameter H = (1/R) (dR/dt) = constant = H0, hence R2 (t) = e2H0t. The value of H0 plays an important role. If H0 = 0, then the universe is static and it would be infinitely old Euclidean universe. If H0 < 0 then the universe would be contracting, and H0 > 0 suggests expanding universe. Hence, H0 < 0 or H0 = 0 are not applicable to present universe. Volume of such universe can be given by V e3H0t. Therefore, (1/V)(dV/dt) = 3 H0. If density is constant then the mass, M, of the universe depends upon volume of the universe. Since M = Vρ0, dM/dt = 3 H0 Vρ0, The current density depends on variation of mass and represented as J = 3 H0 ρ0. The value of J is obtained as J = 2×10-40(ρ/ ρc) h03 g cm-3 s-1, where ρc and h0 are the critical density and Hubble’s constant at present times, respectively. Small value of J indicates the slow and continuous matter creation, which is in contrast to infinite and explosive approach to matter creation at t = 0 as per Big-Bang theory. The continuous creation of matter is very important from the large-scale structure point of view. Alternate to FRW metric, Hoyle [28] in 1948 has proposed field theory to understand the creation of matter, and eventually for studying cosmological model. Pryce in the early 1960s has suggested adopting scalar ‘creation field’ (C-field). The C-field has been massless and charge-less in the energy-momentum tensor without violating energy conditions. The matter creation has necessarily required a negative energy field that can act as source term. Hoyle and Narlikar [28] have extensively used stress-energy tensor for C-field in early 1960s thereby proposed Einstein field equations. Modified Einstein field equations with C-field have been written as:





1 8 G ij ij Rij  g ijR   T(m)  T(C) . 4 2 c

(IX.6)

206

ij ij where, T(m) is the stress-energy tensor for normal matter, while T(C ) has been given

by 1   ij T(C)   f  C iC j  g ijC k Ck  , 2  

(IX.7)

Negative sign represents negative energy density that produces repulsive gravitational effect as T(ijC ) < 0 for f > 0 where f is coupling constant. This, also, explains about the driving force of accelerated expansion. Considering energy conservation condition, it



ij ij is observed that T(m)  T(C)

This results to



ij ij = 0, which implies T(m); j  T(C); j .

ij i j T(C); j   f (C C ); j .

It is equivalent to

dxi mgij  C j  0 . This ds

explains that the four momentum of the created particle is compensated by four momentum C-field. The Equation (IX.6) has been solved for four-dimensional FRW cosmological model by assuming dC/dt = mc2 and solving field equations k  cf  1n , where n is number accordingly. The source term for C-field theory is C;k

density or in other words, it is the number of net creation events per unit proper 4volume. Density at present state is given by ρ = ρ0 =

3H 02 2 fm for flat universe with 4G

exponential scale factor. The equation is the relation between density and elementary creation process. Besides continuous matter creation through steady state model, A explosive matter creation can, also, be explained if C  3 , where A is a constant. CS

field FRW model with varying G has been discussed by Bali and Tikekar [29]. Many cosmologists have proposed similar models to explain them with barotropic fluid, time varying lambda, bulk viscosity, etc. [30-32]. Higher dimensional model in Bianchi space-time with creation field have been studied by Adhav et al [33]. The role of extra dimension for matter creation can be studied in future, which may help in better understanding the large scale structure. Chattergy and Banerjee [34] have investigated higher dimensional model with C-field theory. According to them, C- field theory has been advantageous over 4D FRW model proposed by Hoyle and Narlikar [35]. The density in dust filled 5D C-field 207

model decreases until it reaches to a constant value in contrast to 4D model in which it continuously decreases as t → ∞. The matter creation is slow and a continuous process without any need of singularity in 4D model. On the other hand, there is at least one singularity that can be obtained in 5D model, which may be in absence of Cfield.

It is also found that five dimensional model becomes effectively four-

dimensional model with time. It is possible to explain accelerated expansion along with matter creation, which is not yet been attempted in HN model. According to thermal history of the universe, the matter creation depends upon thermodynamics of the universe. The entropy of the universe also increases with matter creation, leading to disorderliness. Prigogine et al [36] suggested that matter creation and entropy production have taken place simultaneously in the early universe. Further, according to them particle production can be treated macroscopically instead of quantum mechanically. Since universe is supposed to be thermodynamically open system, adiabatic stress-energy relation is modified in order to account particle creation. Einstein field equations can be solved using the adiabatic relation between density and pressure of fluid that fills the universe. It is necessary to know evolution of number density for the matter creation that obtainable thermodynamically. In this process, creation of matter is irreversible process. Space-time can be transformed into matter but vice versa is not possible. It is, also, shown that transformation of gravitational energy to particle creation results into negative pressure. From above discussion, considering the importance of extra dimension, Kaluza-Klein cosmological model can be set up with the help of creation field to study matter creation. Garriga and Verdaguer [37] have discussed the role of extra dimension in particle creation. They have shown that rapid contraction of extra dimension results into particle creation. Kaluza-Klein cosmological model can be studied in future by involving C-field theory, which not only will be helpful for matter creation but also to understand the present thermodynamical conditions of the universe.

IX.5

CONCLUSIONS Presence of dark matter and dark energy is an important feature of the large-

scale structure of the universe. Kaluza-Klein cosmology can play a significant role as demonstrated in the present chapter. Kaluza-Klein particle can be one of the 208

constituents of dark matter. Using Kaluza-Klein cosmology, it is possible to explore dark matter. A detail study on direct and indirect detection of the K-K particle led to its existence and, hence, inspires further theoretical study on it. Dark energy and its existence are established experimentally. The cosmological constant is identified as dark energy; thus, the present work can be implicated to study it in future. Matter creation is one of the important phenomena in the universe, so a discussion on it, will be helpful in future for further research work.

209

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LIST OF PUBLICATIONS/PRESENTATIONS  Number of research papers published: (05) 1) ‘Kaluza-Klein Bulk viscous model with time dependent Gravitational Constant and Cosmological Constant’, Namrata I. Jain, Shyamsunder S. Bhoga, 2538-X (2015) published online 12/2/15 2) ‘Friedmann-Robertson-Walker Cosmological Models :A study’, by Namrata I. Jain. Shyamsunder S. Bhoga , Int. Jour. of Mathematical and Physical science Research, Vol.2 (2), 70, (2014) 3) ‘Kaluza-Klein Cosmological model with Strange Quark Matter and time varying lambda’ by Namrata I. Jain, S. S. Bhoga, G. S. Khadekar, Z. Naturoforsch A, 69(2), 90 February (2014) 4) ‘Implications of Cosmological Constant on Kaluza-Klein Cosmological model’Namrata I. Jain, S. S. Bhoga, G. S. Khadekar, International Journal of Theoretical Physics, 53(2), 4416, August(2013) 5) ‘Kaluza-Klein Cosmological model with string, SQM and time varying lambda’, Namrata I. Jain, S. S. Bhoga, G. S. Khadekar, ARPN Journal of Science and Technology, 6(3), 647 June (2013)  Number of papers presented (05) 1) Presented paper on “The Early Universe Scenario and Kaluza-Klein Cosmological Model” at UGC funded National conference organized by NM Mehta college, Bordi from 22-23 Jan. 2015 2) Presented paper on ‘Higher dimensional string Cosmological model with strange quark matter and a new cosmological constant’ at ICMS Conference 2012 organized by SSES Amt. Science College, Nagpur from 28th-31st Dec. in 2012. 3) Presented paper on ‘study of Higher dimensional Cosmological model with and varying Cosmological Constant’ at ICMS international conference (CONIAPS-XIV) organized by SVNIT, Surat in Dec. 2011. 4) Presented paper on ‘The Cosmological model with time varying Cosmological constant’ at two days National level Seminar organized by ICLES College Vashi in association with UGC in 2010 5) Presented a paper on ‘Higher dimension Cosmological model with Varying Cosmological Constant’ at 3- days state level seminar at C. K. Thakur college Panvel, in association with UGC held on date 6-7 Feb. 2010

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Kaluza-Klein Bulk Viscous Cosmological Model with Time Dependent Gravitational Constant and Cosmological Constant Namrata I. Jain & Shyamsunder S. Bhoga

International Journal of Theoretical Physics ISSN 0020-7748 Int J Theor Phys DOI 10.1007/s10773-015-2538-x

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Kaluza-Klein Bulk Viscous Cosmological Model with Time Dependent Gravitational Constant and Cosmological Constant Namrata I. Jain · Shyamsunder S. Bhoga

Abstract Cosmological models with time varying gravitational constant, G, and cosmological constant , in the presence of viscous fluid in Kaluza-Klein metric were investigated. The solutions to Einstein Field Equation were obtained for different types of G, with bulk coefficient ξ = ξ0 ρ d (where ρ is density of the Universe, d is some constant) and lambda = αH2 + βR−2 where H and R are Hubble parameter and scale factor respectively. Two possible models are suggested, one where G is proportional to H and, the other where G is inversely proportional to H. While the former leads to a non-singular model, the latter results in an inflationary model. Both Cosmological models show that the Universe is accelerating; but at the early stage of the Universe the behaviour of both models is quite different,which has been studied through the variation of decelerating parameter q with time. Keywords Cosmological model · Einstein field equations · Kaluza -Klein metric · Cosmological constant · Gravitational constant

1 Introduction Kaluza –Klein Cosmology, with its venerable history and several interesting features, has become popular since it brought about a revolution in the study of the Universe. It made it possible to unfold the Universe at its early stages and study its evolution and behaviour by including an extra dimension in the Einstein Field Equations. Although, it has been difficult to prove the existence of an extra dimension experimentally due to unavoidable practical limits, its effect is observed . Extensive work in multi-dimensional physics and the Kaluza-Klein Theory (Emel’Yanov et al. [1], Wesson [2], Chatterjee and Banerjee [3], N. I. Jain ( ) Department of Physics, M. D. College, Parel, Mumbai, 400012 India e-mail: namratajain.jain2@gmail.com S. S. Bhoga Department of Physics, RTM Nagpur University, Nagpur, 440033 India e-mail: msrl.physics1@gmail.com

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Gron [4]) brought forth the concept of the Universe comprising more than four dimensions, which were large at the early Universe stage, but later became too small to be measured experimentally with available data. Recent studies using SNIa (Perlmutter et al. [5], Reiss et al. [6]), BAO [7], and WMAP [8] experiments have concluded accelerated expansion of the Universe. The common interpretation of the accelerated expansion suggests the presence of Dark Energy and Dark Matter in the Universe. Both are yet not understood properly. Several published literature, explaining implications of Dark Energy to some extent, are now available on Dark Energy Models, Interactive Dark Energy Models, Dark Matter Models and Viscous Dark Energy Models (Pradhan et al. [9], Mukhopadhyay et al. [10], Rahaman et al. [11]. Ray and Mukhopadhyay [12] have discussed Dark Energy models in the context of time varying Gravitational Constant ( ). Two parameters, Newtonian Universal Gravitational Constant G and Cosmological Constant , (Lambda), introduced by Einstein in his General Theory of Relativity, have been major influential quantities in Cosmology to understand the relation between geometry and matter. Recent development in the study of the Universe by several researchers indicate that both Cosmological and Gravitational Constants have been time varying [14, 15]. The discrepancy of about 10120 in the value of found cosmologically and that determined using Standard model in Particle Physics [15], has been observed. It is described as the Cosmological Constant Puzzle (CCP). To overcome this puzzle, a time varying Cosmological Constant has been suggested leading to a new area of research for many cosmologists. The time varying Cosmological Constant, first suggested by Chen and Wu [16], has been later adopted in various forms by many scientists studying cosmological models with ∝ R−2 , ∝H2 , ∝ 1/R(d 2 R/dt 2 ) [17–20]. The Cosmological Constant ∝ αH2 + βR−2 is the generalized lambda first suggested by Carvalho and Lima [21] and applied by Singh et al. [22], Jain et al. [23] to explain the non-singular, accelerated and expanding Universe with generalized lambda in higher dimension. Cosmological models with generalized Cosmological Constant with Strange Quark Matter (SQM) or strings can be utilised for the study of early Universe, as argued in some published literature [24, 25]. Strange Quark matter/strings play an important role in the study of early Universe, as they came into existence just after the Big-Bang. Some authors also put forth the idea that SQM may be the part of Dark Matter [26]. Interaction between Dark Energy and Dark Matter has also been discussed in the literature that explains the present behaviour of the Universe to some extent [27, 28]. The pioneering work of Dirac [29] on ‘Large Number Hypothesis’ (LNH) led to the idea that G depends on time. He pointed out that G ∝ t−1 . However, from Milne’s work [30] G is found to be proportional to tm , where m is any positive number. Although their work indicated that time varying G can play an important role in the explanation of accelerated expansion, early Universe phenomenology and large scale structure of the Universe, but, it could not clarify whether G is increasing or decreasing with time. Model with time varying G by Barrow [31], particle creation in the Universe by the model with time varying G by Harko and Mak [32], and Dark Energy Models by Ray et al. [33] have considered G decreasing with time. But some published work [34, 35] points out that the gravitational constant increases with time. A possible variation of G has been investigated by several teams through astronomical observations within the range of solar and binary systems (Uzan [36]). Hence, cosmological models with time varying G and have been explored by many researchers (Berman [37], Beesham [38], Rahaman [39], Sistero [40]). Value of G has been calculated and

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co-related it with experimental value in the published literature by Occhionero & Vagnefti [41]. A detailed literature summary suggested that G∝tm ; where m is a constant but its sign (positive or negative) is not known [37–41]. The present acceleration of the Universe can be very well explained when considering it to be filled up with perfect fluid; but to understand the evolution of the Universe, it is necessary to deal with the Universe filled with viscous fluid. The current isotropic nature of the Universe is thought to have evolved due to the dissipative effects of the viscosity of fluid (Calzetta and Thibeault [42], Chimento and Jakubi [43], Zimdahl et al. [44]). The material distribution in the Universe is also one of the criteria for assuming that the Universe is filled with viscous fluid. The phenomenon at early stages of the Universe indicates that when radiation decouples from matter in the form of photons and neutrinos, then it behaves as viscous fluid. In evolution, viscosity decreases with time. A number of authors have investigated viscous cosmological models in different contexts (Singh and Singh [45], Bali and Tinker [46]). In this regard, Weinberg [47] have explained the role of a viscous medium in the evolution of the Universe and deduced that the general form of energy-momentum tensor which contains bulk viscous coefficient factor ξ is proportional to scalar expansion factor θ . Consequently, pressure decreases and accelerated expansion is explained. The concept of evolution of the Universe filled with viscous fluid whose bulk co-efficient ξ ∝ ρ d has been investigated thoroughly by many researchers, has led to the generalized study of the Universe [47], Banerjee et al. [48], Ghate and Mhaske [49], Maartens [50] It is possible to understand the presence of Dark Energy at early stages of the Universe through the study of viscous cosmological models. Dark Energy Models with viscous fluid have been studied by Arbab [51, 52] and Ren and Meng [53] with time dependent G and in different contexts. It is also to be noted that the Viscous Cosmological Model explains accelerated expansion and, hence, indicates the existence of Dark Energy [54]. The models with only variable G, Variable lambda or variable Bulk coefficient alone can explain accelerated expansion with certain constraints. The model with variable G and lambda are considered on energy conservation ground can explain late time acceleration whereas the viscous models deals with early Universe. Nonstandard Cosmology with variable G and with Bulk viscous fluid has been explained by Arbab [55]. The unified description of the model with time dependent G, time dependent with viscous fluid brings out several interesting pictures related to cosmic evolution. Singh et al. [56] studied the viscous model with time varying G, for early Universe. Paul and Debnath [57] studied modified gravity model with viscous effect variable G and . The model can be extended to five dimensions for generalized lambda which can allow us to investigate anisotropic early Universe to isotropic accelerated model and bring an useful information on Dark energy. Motivated by the above discussions, it was possible to set up the cosmological models based on time varying G and with viscous Kaluza-Klein metric so as to understand an accelerating Universe. Another motivation for the present model is to reconcile it with existing observational data. To understand the Universe at its early stages and its late time acceleration; models with gravitational constants varying with H−1 and H are investigated. The present study is also focussed on deriving information about Dark Energy. Along with time varying ‘G’ and ‘ ’, we assume bulk coefficient of viscous fluid as ξ = ξ0 ρ d to see its effect on the model, as time varying ξ modifies Einstein Field Equations, to illustrate the behaviour of the Universe at its early, as well as, its present state. In this paper, Section 2 sets Einstein Field Equations (EFE) with ξ = ξ0 ρ d and = αH2 + βR−2 .

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Section 3 offers solutions of EFE for G∝H−1 . In Section 4, G ∝H is first derived, then discussed and Conclusions are drawn in the next section.

2 Einstein Field Equations Kaluza-Klein metric given below, is considered to obtain Einstein field equations of present cosmological model, 1 2 2 2 2 2 ds 2 = −dt 2 + R 2 (t) dr + r (dθ + sin θ dϕ ) + A2 (t)dψ 2 (1) 1 − kr 2 where k is curvature parameter, which is equal to 0, 1 and −1 for flat, closed and open universe respectively, R(t) and A(t) are fourth & fifth dimensional scale factors respectively,

is fifth dimension and r, θ , ϕ are the usual spherical co-ordinates. As per the Cosmic Principle, we assume = c=1, ‘G’ is the time dependent gravitational constant. Einstein Field Equations for time dependent ‘G’ and ‘ ’ can be read as : Gij = −8πG(t)Tji + (t)gji 1 Gij = Rji − Rgji 2 Here Rji is Ricci tensor, R is Ricci scalar and gji is the metric element. The General Form of Energy Momentum Tensor for viscous fluid is taken as: Tji = ρ + peff ui uj − peff gji

(2) (3)

(4)

where ρ is the density for cosmic matter and peff is the effective pressure of the fluid of the Universe. Five dimensional effective pressure of viscous fluid is related to normal pressure as [47]: (5) peff = p − ξ

Where ξ is the coefficient of bulk viscosity and scalar expansion factor is defined as = ui;i where semicolon represents differentiation of co-moving velocity ui , ui ui = −1, ui is five dimensional velocity vector and ; is the co-variant differentiation of velocity vector. ξ term consisting of viscosity co-efficient and expansion factor, has been first introduced by Arbab [51] for viscous effects which act as a source term for matter creation and relates to expanding Universe with dissipations. It is argued by many authors [51–55, 58] that effective pressure is reduced due to decay of the coefficient of bulk viscosity and this may be the reason for accelerated expansion. Energy Momentum Tensor looks similar to that of perfect fluid except for the extra term ξ , which is very small as compared to normal pressure and density. Hence, energyij momentum conservation for perfect fluid, T;j = 0 can still be considered here. For the metric (1), using (3) along with ansatz A(t)= R(t)n, , Einstein Field Equations obtained are given below, R̈ Ṙ 2 k + (n2 + n + 1) 2 + 2 = −8πG(t)peff + (t) R R R Ṙ 2 R̈ k G44 = 3 + 3 2 + 3 2 = −8πG(t)peff + (t) R R R Ṙ 2 k G55 = 3(n + 1) 2 + 3 2 = 8πG(t)ρ + (t) R R

G11 = (n + 2)

(6) (7) (8)

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Divergence of Einstein’s Tensor implies 1 = −8πGTji + gji =0 Rji − Rgji ;j 2 ;j

(9)

From Field (6)–(4) and with the help of (9), we get: ρ̇ + (ρ + peff ) (3 + n)

˙ Ġ(t) (t) Ṙ + ρ+ =0 R G(t) 8πG(t)

(10)

Substituting peff from (5), we get : ρ̇ + (ρ + p − ξ ) (3 + n)

˙ Ġ Ṙ + ρ+ =0 R G 8πG

(11)

The usual energy conservation relation T;j = 0 implies that continuity equation should have an extra term linking it to the bulk coefficient term i.e. ξ (3 + n) Ṙ R. By reviewing the work by Singh et al. [45], Arbab [51, 52] and similar literature, it is found that the usual conservation law given below can still be satisfied by applying a dominating energy condition as R̈ > 0. Since our model is a non-causal, dissipative model the dissipative effects can be linked to time dependent ‘G’ and ‘ ’, arriving at: Ṙ =0 R

(12)

˙ Ġ Ṙ ρ+ = (3 + n)ξ

G 8πG R Expansion scalar factor is mathematically written as:

(13)

ρ̇ + (p + ρ) (3 + n) From (11), it follows that

Ȧ Ṙ Ṙ + = (3 + n) R A R

(14)

Substituting in (12), we get: ˙ Ġ ρ+ = (3 + n)2 ξ G 8πG

2 Ṙ R

(15)

To arrive at field equations satisfying energy conservation, a second order differential equation is obtained by differentiating (8) with respect to time and dividing by 8πG, leading to:

˙ Ġ 6 Ṙ Ṙ 2 k R̈ ρ̇ + ρ + = − 2 − 2 (16) (n + 1) G 8πG 8πG R R R R To simplify the above equation, we apply “gamma law” Equation Of State (EOS) for pressure and density as p = (γ − 1)ρ, where γ is the adiabatic parameter. In Cosmology, the value of γ is 0≤ γ ≤ 2. which can take the values 0,1, 2/3 etc. for the Universe filled with Dark Energy, matter, radiation respectively. Substitution of “gamma law” EOS in (12), results in: Ṙ ρ̇ = −γ (n + 3) (17) ρ R ρ = ρ0 R −γ (n+3) where ρ0 is constant density.

(18)

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With the help of (15) and (17), (16) is simplified to:

R̈ Ṙ 2 Ṙ k 2 − 2 − 2 (n + 3) 8πGξ − 8πGργ (n + 3) = 6 (n + 1) R R R R

(19)

Applying (8) in the above equation, and after some mathematical manipulation, we get: 3γ (n + 3)(n + 1) − 6(n + 1) Ṙ 2 (3γ (n + 3) − 6)k R̈ + − R 6(n + 1) R2 6(n + 1)R 2 +

γ (n + 3) (n + 3)2 8πGξ Ṙ − =0 6(n + 1) 6(n + 1) R

(20)

The above equation may be further simplified by considering time dependent Lambda , Co-efficient of bulk viscosity ξ and gravitational constant G. The models for G increases with time have been worked upon by a few authors [34, 35] for the perfect fluid while the model for G∝H has been discussed by many cosmologists [33, 38, 39]. The model with variable G ∝1/H is discussed so as to study the effect of increasing G whereas another model with decreasing G inspired by the investigations carried out by [33, 38] and investigate the models for generalized Cosmological Constant [21–23]. The Solutions to the above differential equation will be derived for different G in the following section and investigate them for the Universe with viscous fluid.

3 Solutions of Field Equations 2

A solution to differential (20) is obtained by using = α Ṙ + β R12 R2 d ξ = ξ0 ρ and for two different gravitational constants (i) G=G0 /H and (ii) G =G0 H which are discussed below. 3.1 Case (i) 2

G=G0 /H, ξ = ξ0 ρ d and = α Ṙ + β R12 R2 Assuming time dependent gravitational constant ‘G’ inversely proportional to Hubble’s −1 Constant whereH = Ṙ R . This variation of ‘G’ suggests G increases with time as H ∝ t . Increase in G with time has been discussed in literature [38, 39]. Substituting G, ξ and in (20), it takes the form as : R̈ 3γ (n + 3)(n + 1) − 6(n + 1) − αγ (n + 3) Ṙ 2 + R 6(n + 1) R2 + Let m1 =

γ (n + 3)(3k − β) − 6k (n + 3)2 8πG0 ξ0 d ρ =0 − 2 6(n + 1) 6(n + 1)R

3γ (n+3)(n+1)−6(n+1)−γ α(n+3) 6(n+1)

m2 =

, k1 =

(21)

γ (n+3)(3k−β)−6k 6(n+1)

(n + 3)2 8πG0 ξ0 ρ0d 6(n + 1)

Using (18) in m2 , (21) is rewritten as: k1 Ṙ 2 R̈ + m1 2 + 2 − m2 R −dγ (n+3) = 0 R R R

(22)

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The solution to the above differential equation is arrived at : Ṙ 2 =

C m2 k1 R −dγ (n+3)+2 − + 2(m1 + 1) − dγ (n + 3) m1 R 2m1

(23)

Where C is constant of integration. A Solution to the above equation is, however, complicated. For a realistic solution, the above equation is simplified by assuming m1 =1 and dγ (n+3) = 2. In such a case, the solution to the above equation is obtained as: Ṙ 2 = Assuming m 2 =

m2 2

C m2 − k1 + R2 2

(24)

, the solutions obtained as 1 R2 = m 2 − k1 (t + C1 )2 − C m2 − k 1

(25)

The above equation can be rewritten by considering the initial condition, at t=t0 ,R=R0 and Ṙ H= H0 = R , t=t0

2 H0 R02 C R = m2 − k1 (t − t0 ) + − m2 − k 1 m2 − k 1

(26)

To arrive at a realistic solution, one should have m2 >2k1 . Other Physical parameters are obtained as follows: m2 − k 1 m2 − k1 (t − t0 ) + H0 R02 Ṙ H (t) = = 2 R (m2 − k1 )(t − t0 ) + H0 R02 − C R R̈ C q(t) = − 2 = − m − k1 (t − t0 )2 + H0 R 2 Ṙ

(27)

α 1 Ṙ 2 =α 2 +β 2 = R R

(28)

2 m2 − k 1 m2 − k1 (t − t0 ) + H0 R02 +β 2 2 m2 − k1 (t − t0 ) + H0 R02 − C

2 2 − k ) + H R −C m (t − t 1 0 0 2 0 G0 G(t) = = H m2 − k 1 m2 − k1 (t − t0 ) + H0 R02 ξ = ξ0

⎡

m2 − k 1

1 (t − t0 ) +

H0 R02

H0 R02 ρ = ρ0 ⎣(m2 − k1 ) (t − t0 ) + k1

(30)

(m2 −k1

(29)

(31) −

C (m2 −k1 )

⎤ −γ (n+3)

2 −

(m2 − k1 )

⎦

(32)

The above physical parameters will be further analysed in the succeeding sections. In the following section the solutions to the field equations for G=G0 H are determined.

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3.2 Case (ii) 2

Model with G=G0 H, ξ = ξ0 ρ d and = α Ṙ + β R12 R2 Substituting G, ξ and in (20), it follows that, R̈ 3γ (n + 3)(n + 1) − 6(n + 1) − αγ (n + 3) Ṙ 2 + R 6(n + 1) R2 2 γ (n + 3)(3k − β) − 6k (n + 3)2 8πG0 ξ0 d Ṙ ρ + − =0 6(n + 1)R 2 6(n + 1) R Consider m1 =

3γ (n+3)(n+1)−6(n+1)−γ α(n+3) 6(n+1)

m2 =

, k1 =

(33)

γ (n+3)(3k−β)−6k 6(n+1)

(n + 3)2 8πG0 ξ0 ρ0d , 6(n + 1)

Equation (33) is rewritten as: R̈ Ṙ 2 k1 + m1 2 + 2 − m2 R −dγ (n+3) R R R

2 Ṙ =0 R

(34)

The solution to the above differential equation can be found under the conditions that m1 =1 and dγ (n + 3) = −2, Assuming these conditions, (34) gets modified as below : R R̈ + Ṙ 2 − m2 R 2 Ṙ 2 + k1 = 0 If y = R Ṙ then, the solution obtained is: 1 k1 y= Coth k1 m2 (t − t0 ) 2 m2 In the above equation, constant of integration ‘c’ is assumed to be c= -m2 t0 Hence R(t) obtained is : 1 2 1 R(t) = loge Sinh k1 m2 (t − t0 ) m2 Other Physical Parameters are obtained as follows, √ √ k1 m2 Coth( k1 m2 (t − t0 )) H (t) = √ 2 loge Sinh( k1 m2 (t − t0 )) log Sinh(√k m (t − t )) Coth2 (√k m (t − t )) − 4 1 2 0 1 2 0 e √ − 1 q(t) = − 2 Coth ( k1 m2 (t − t0 )) √

(35)

(36)

(37)

(38)

(39)

2 √ √ k1 m2 Coth( k1 m2 (t − t0 ) + 4β loge Sinh( k1 m2 (t − t0 ) 2 √ 2 loge Sinh( k1 m2 (t − t0 ) (40) −γ (n+3) 2 1 loge Sinh k1 m2 (t − t0 ) (41) ρ = ρ0 m2 −2 1 loge Sinh k1 m2 (t − t0 ) (42) ξ = ξ0 ρ0γ (n+3) m2

α Ṙ 2 1 =α 2 +β 2 = R R

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Using m2 in the above equation, it is further simplified to: 6(n + 1) loge Sinh k1 m2 (t − t0 ) (43) ξ= (n + 3)2 √ √ k1 m2 Coth( k1 m2 (t − t0 )) G = G0 √ (44) 2 loge Sinh( k1 m2 (t − t0 )) To study the physical significance of the physical parameters derived above, let us, initially, consider approximation of (35). Series Expansion of y is given as √ √ 1 k1 1 + 2e−2 k1 m2 (t−t0 ) + 2e−4 k1 m2 (t−t0 ) + −− (45) y= 2 m2 Neglecting higher terms in exponential terms in above equation as they will be reducing to zero as t→ ∞. Equation (45) is modified to: √ 1 k1 1 + 2e−2 k1 m2 (t−t0 ) (46) y= 2 m2 Equation (36) can now be rewritten as, √ k1 2 R = t − 4 k1 m2 e−2 k1 m2 (t−t0 ) m2

(47)

For the sake of convenience, we assume constant of integration as zero. Other parameters can also be derived in a similar manner. It is observed that as t→t0 , scale factor R(t) reaches constant value , leading to steady state condition. In the next section we discuss the physical significance of the parameters derived for both models (G=G0 /H, G=G0 H)

4 Discussion: Both the models predict a small positive value of the Cosmological Constant according to observational data [5, 6]. The constants α and β can be obtained for both the models as they depend upon m1 and k1 . Since k1 = 0 for flat model, it becomes possible to analyse the flat as well as non-flat models. For flat model, α and β is determined asα=

6(n + 1) 3(n + 1)(γ (n + 3) − 4) ,β = γ (n + 3) γ (n + 3)

(48)

if we assume k1 = 1. From the above expressions for α and β, we observe that for small, positive value of ‘ ’, the index factor ‘n’ should be positive and its value should be 0<n<1 to have compactification of extra dimension. Equations (27), (28) and (38), (39) represent peculiar behaviour of H(t) and q(t). This can be observed through the graphs given in Figs. 1–4. 1] G =G0 /H, ξ = ξ0 ρ d and = αH2 + βR−2 {Assumed initial conditions : C=0.25, H0 R20 =0.5 And other constants equal to 1} 2] G =G0 H, ξ = ξ0 ρ d , = αH2 + βR−2 Assuming constants k1 and m2 equal to 1 * Although in the Fig. 4, it seems that H(t) =0 at some time but actually it is very small which can not be plotted in the graph.

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t-to 0

0.5

1.5

-0.5 -1 q(t) -1.5 -2 -2.5 -3 -3.5 Fig. 1 q(t) v/s t

To explain peculiar behaviour of H(t) , it is observed from Figs. 2 and 4, that H(t) ∝ 1/t for present Universe, and has a smaller value than that of perfect fluid, when it is filled with viscous medium. In earlier stages of the Universe, the viscous medium enhances expansion that leads to acceleration at later stages. This could be due to the decrease of the coefficient of viscosity at a faster rate than G. Decrease of H(t) is faster for G∝ H−1 as compared to that of G∝ H as evident in Figs. 2 and 4. Such behavior shows that the Universe is reaching a steady state condition earlier in the first case as compared to the second case. A little bump however observed in the graph Fig. 4 for a very short time which is not observed practically and this unusual behaviour may be due to the approximations made for plotting graphs and require further research to be carried out in this direction. Fig. 2 H(t) v/s t (not as per the scale)

H(t)

t-to

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2 1 q (t) 0 0 -1 -2

0.5

1 t-to

1.5

2.5

-3 -4 Fig. 3 q(t) v/s t ∗

The (28) and (39), reveal that for the model with G=G0 /H q(t)→ −0.5 when t>t0 and for G=G0 H., q(t) becomes negative after some time. This shows that there is an eternal inflation. According to Guth [59], eternal inflation results in the rise of multiverses, which is the latest area of research in Cosmology. In this regard, Borde [60], and Taotao et al. [61] have enlightened eternal inflation in different contexts. Figure 1 shows the behaviour of the Universe at later stages of the Universe while Fig. 2 indicates the behaviour of the Universe at early stages of the Universe. For Dark Energy Models, if we consider γ=0, then differential equations (22) and (33) can be reformed. However, the case (ii) G=G0 H model with Dark Energy explains accelerated expansion of the present Universe while the case (i) G=G0 /H model with Dark Energy explains early Universe phenomena. If γ ≤ −1, then the model with G=G0 H (case ii) is able to explain accelerated expansion, while the model with G=G0 /H (case (i)) explains the present Universe phenomenology for γ ≥ −1. However Phantom Divide Crossing which is the transition from phantom era (γ < − 1) to quintessence era (γ > − 1) [ γ =0 is the phantom divide] is more prominent in the case(ii) model, as compared to the case(i) model which is evident from Figs. 1 and 3. The model with G=G0 H filled with perfect fluid has been explained in literature relating to early Universe phenomena. It is observed that the co-efficient of viscosity plays an important role; the present isotropic condition of the Universe can be very well explained with the help of the Universe filled with perfect fluid, but the Universe filled with viscous medium with time dependent

Fig. 4 H(t) v/s t (not as per the scale) H(t)

t-to

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bulk co-efficient leads from the anisotropic early Universe to present isotropic Universe. This can be explained from the values of σ 2 /θ . Shear scalar σ is obtained from the relation σ 2 = 38 (1−n)2 H 2 [22] and Scalar expansion θ is defined as θ =(n+3)H [which is previously written as from (14)]; (σ 2 /θ ) can be 2 easily found from (27) and (37) for both models. Since (σ /θ ) ∝ H, so it is observed that in case (i) model with G=G0 /H, σ /θ →0, while in case (ii), G=G0 H, σ 2 /θ →k in the present scenario.

5 Conclusion The models with gravitational constant G, which is proportional to H−1 as well as proportional to H are analysed. G increases with time for G=G0 /H and it decreases with time for G=G0 H. The model with G=G0 H is similar to that discussed by Ray et al. [62], if the Universe is filled with perfect fluid, while the model with G=G0 /H is similar to that discussed by Singh [22] for the Universe filled with perfect fluid. The Cosmological Constant is also found to have a small positive value as per observational data. Decay of Cosmological Constant along with time varying G for Bulk viscous KaluzaKlein model is observed to be expanding and accelerating. The model with G=G0 /H explains the early Universe if it is filled with perfect fluid. Although many papers explain Universe with G and decreasing with time to reconcile it with present observational data; the model with ‘G’ increasing with time is found to be forever accelerating. This is possible because the co-efficient of viscosity (ξ ) decreases with time. Hence these models can also be studied for Dark energy models. Comparing the models with G=G0 /H and G=G0 H, Both models explain the Universe which are physically viable in different ways but the model with G=G0 H can be reconciled with the present observation data. We also found that the model with G=G0 /H is a non-singular, accelerating Universe; while the model where G=G0 H represents an inflationary Universe as observed from (37) at early stages of the Universe but later will tend to de Sitter Universe. It is also concluded that effect of time dependent viscous medium along with variable G and ’ are responsible for acceleration of the Universe. Acknowledgments We are highly thankful to Dr. Farook Rahaman, Jadavpur University and Dr. Saibal Ray, Government College of Engineering & Ceramic Technology for providing their views and suggestions for substantial improvement of this paper.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Emel’yanov, V.M. et al.: Phys. Rep. 143(1), 1 (1986) Wesson, P.S.: Astrophys. J. 344, 19 (1992) Chatterjee, S., Banerjee, A.: Class. Quan. Grav. 10(1), 1 (1995) Gron, O.: Astro. Sp. Sci. 173, 191 (1990) Perlmutter, S. et al.: Astrophys. J. 517, 565 (1998) Reiss, A. et al.: Astrophys. J. 116, 1009 (1998) Eiesenstein, D.J. et al.: Astrophys. J. 633(2), 560 (2005) Bennett, C.L. et al.: arXiv:1212.5225 (2012) Pradhan, A. et al.: Preprint arXiv:gr-qc/0310023v2 (2005) Mukhopadhyay, U. et al.: Int. J. Theo. Phys. 49, 1622 (2010) Rahaman, F. et al.: Astro. Sp. Sc. 288, 379 (2003)

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56. 57. 58. 59. 60. 61. 62.

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International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 2, Issue 2, pp: (70-86), Month: October 2014 – March 2015, Available at: www.researchpublish.com

Friedman –Robertson –Walker Cosmological models: A study 1

Namrata I. Jain. 2Shyamsunder S. Bhoga

Department of Physics, M.D. College, Parel, Mumbai India -400012 Department of Physics, RTM Nagpur University, Nagpur India -4400033

Abstract: Friedman –Robertson-Walker model is the standard model which has been studied thoroughly. The FRW model is successful in explaining expansion of the Universe but it has certain limitations which are also discussed here. FRW model with and without Cosmological model have been compared which leads to the conclusion that Cosmological Constant plays an important role in the study of the Universe. It is also observed that Cosmological Constant is not really constant but it is varying. Although FRW models are inadequate in the study of the Universe but its study will be a milestone for the future research. Keywords: Cosmology, Cosmological Model, Einstein Field Equations.

INTRODUCTION

Cosmology is the study of the Universe. The Universe, its origin, evolution and its fate are still a curiosity for mankind and is intensively studied by the scientific community since ancient times. Observing heavenly objects and dynamics by sky watching has been a natural trend involved in the study of the Universe and this study is called Astronomy. It is the astronomical observations of planets, stars and galaxies that have created inquisitiveness to look beyond our Milky Way Galaxy to distant objects from which light may take billion years to reach us. Nowadays, modern techniques of observing the sky to study stars and galaxies have enabled us to understand dynamical and physical behavior of billions of galaxies spread across vast distances. Studying the extragalactic world, large scale structure of the Universe has gained popularity in the nineteenth century, when A. Einstein‟s Theory of Relativity emerged. In the late twenties, tensor calculus revolutionized the geometry of space by involving it in the study. This led to the theoretical study of the Universe and thus a cosmological model could be set up using tensor calculus and metric algebra. The Cosmological Model of the Universe is a Mathematical model which is developed by solving mathematical equations with the help of tensor calculus. The application of tensor calculus with Riemannian geometry brought out the concept of Space-Time curvature and geodesics in General Relativity. A Geodesic is understood as the minimum stationary distance of anybody on the space time curve which is similar to a curve on a 3D sphere. In tensor calculus, Christofflel tensor and Ricci tensor are the main tensors to derive equations of Geodesics. With the help of Tensor calculus, A. Einstein published equations in General Relativity which related space and geometry in 1915. With these equations K. Schwarzchild found internal and external solutions for space –time geometry of the spherical distribution of matter thereby developing t he Schwarzchild metric. [1] The Schwarzchid metric has been successfully implemented in the study of the Universe by A. Einstein, W. de Sitter and many others. It has also been successfully applied to explain experimental measurements of the solar system which involve Advance Perihelion of a planet, Bending of Light, Gravitational Red Shift, Radar Echo Delay. Einstein and de Sitter set up Cosmological models by line element as Schwarzchild metric. Both models could explain the Universe successfully but at some issues they were contradictory to each other. Einstein‟s Universe was a Steady State Model and filled with matter without motion; while de Sitter‟s Universe was an Expanding Model with empty space. Both

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International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 2, Issue 2, pp: (70-86), Month: October 2014 – March 2015, Available at: www.researchpublish.com models were inadequate to explain the expansion of the Universe which was observed experimentally by E. Hubble and M. Humason in 1931 [2, 3]. E. Hubble in 1929 found that the Universe is expanding by actually calculating moving galaxies with finite velocity which is proportional to its distance from Earth. The proportionality constant is called the Hubble‟s Constant H.[2] Expansion of the Universe was a puzzle until the work carried out by A. Friedmann in 1922 and A. Lamaitre, in 1927 independently which have been recognized later due to the discovery by E . Hubble in 1929 by science community which unfortunately had gone unnoticed. The expanding model could be explained by generalizing the line element so that the non static line element could be used to study various aspects of the Universe including Hubble‟s Law. The non static line element was derived rigorously by H.P. Robertson and A.G. Walker (independently) in the 1930s which is called the Robertson –Walker line element [4,5]. The model thus developed is called the Friedmann – Lamaitre- Robertson- Walker model which has been worked upon and explored to understand the Universe since its evolution. Since two decades, the FRW model of the Universe has become an area of research for many. Despite the successful explanation of expansion and age of the Universe, FRW model suffers from some infirmities. These are related to Horizon, Flatness etc. of the Universe. The Cosmological Constant () which is an important parameter in Cosmology, plays a key role in resolving these infirmities. Several researchers have pointed out the importance of () by setting up models. They succeeded in overcoming some of the problems faced by their models without. Among these researchers, some explained dust model with p=0 while others considered the relation between p and  .[see 6 and references therein] While the problems related to age, horizon, flatness are solved to a certain extent, FRW models with constant  - which is four dimensional model - is still unable to explain certain observational facts . With reference to accelerated expansion, Existence of Dark Energy, Dark Matter and Cosmological Constant Problem (CCP) there is a mismatch between observed and calculated value , which are yet to be resolved. One solution for CCP is to consider the model with time dependent Lambda which has been quite successful in explaining accelerated expansion and other features of the Universe, However CCP is not yet completely solved. Besides CCP, the mystery of Dark Energy and Dark Matter, there is a curiosity among Cosmologists about the early Universe phenomena. Recent researches attempting to overcome these shortcomings and unfolding the early Universe have suggested that one must look for higher dimension cosmology. Kaluza in 1921 [7] and Klein in 1926 [8] lay the foundation of higher dimensional cosmology which is actually the consequence of their attempt to unify all four types of forces i.e. gravitation with electro-magnetism and gravitation with particle interaction. Higher dimension plays an important role in the analysis and the dynamics of the early Universe. This paper has been organized as: The detailed study of FRW model will be taken up in the next section followed by Kinematics and neoclassical study. The shortcomings of FRW model will be studied thereafter. Kaluza Klein Cosmology have been analyzed further to enable us to develop a ground for the present work. Final section is the concluding remarks on the present work.

II.

FRIEDMAN –ROBERTSON – WALKER COSMOLOGICAL MODEL

In the 1930s the Universe was observed to be non static by E. Hubble and M. Humason which had to be explained theoretically. Friedman -Robertson –Walker (FRW) model is the model of the Universe representing Friedman universe in Robertson –Walker metric. The model which is going to be discussed in the present section, is a four dimensional model. In order to study FRW model it is necessary to note the assumptions that are accepted to generalize the model for non static conditions. 2.1 – The assumptions for the model A. Weyl’s postulate It states that in a space-time diagram, world lines representing history of galaxies never intersect each other and form a funnel like structure, increase steadily and these three–bundles of nonintersecting geodesics are orthogonal to a series of

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International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 2, Issue 2, pp: (70-86), Month: October 2014 – March 2015, Available at: www.researchpublish.com space-like hyper surfaces. This was postulated by a mathematician H. Weyl in 1923 to understand the regularities in the world lines which are space-time curve for the locations of galaxies.[9] A particle moving in space can be represented by curve in a Space –time plane. Thus, for particles moving in a Space-time plane can be shown by several curves as shown in Fig. 1 (i) and (ii) . Fig.1 (i) shows the curves intersecting each other while the other figure represents the non-intersecting curves in a space-time plane. In the following figures, lines intersecting each other denote collision of particles. If this would have been situation in the Universe then Einstein Field Equations would be difficult to solve. The real Universe does not appear so messy; therefore the world lines representing galaxies are not intersecting and have a funnellike structure as shown in Fig. 1 (ii).

The physical significance of Weyl‟s Postulate can be explained by considering co-ordinates and metrics of space time. For a typical world line in (1+3) dimensions there are three space co-ordinates and an one-time co-ordinate. For spacelike hyper surfaces the co-ordinates represented by xμ are considered to be constant. the galaxies lying on different lines are assumed to be like smooth fluid and so xμ represents both time as well as space co-ordinates together(0, 1,2,3). The metric tensor for such system is given by gij . From the orthogonality condition g0μ =0 where suffix „0‟ is for time co-ordinate if μ is other than zero. The geodesic equations is given by d 2 xi dx 2

 ikl

dx k dxi 0 ds ds

(1)

i The above equation is satisfied for constant xμ .We also have i=1, 2, 3, Christoffel Tensor of second kind is  kl which is

defined as ikl 

1 ij g 2

 g jl g kj g kl   k  l   x x j   x

Гμ00 = 0 for μ=1, 2, 3 .This will give

(2) g 00 x 

 0 . Without loss of generality we take g00 = 1

The line element therefore becomes

 

ds 2  dx 0

 g  dx  dx  c 2 dt 2  g  dx  dx

(3)

The time co-ordinate in the above equation is called the Cosmic time which is an important parameter for studying dynamics of the Universe.

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International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 2, Issue 2, pp: (70-86), Month: October 2014 – March 2015, Available at: www.researchpublish.com B. The Cosmological Principle. Another very important assumption that is adopted while developing a cosmological model is the Cosmological Principle. It states that at any given cosmic time, the Universe is homogeneous and isotropic. This gives the surface of hyper sphere of Einstein‟s Universe to be smooth and surfaces with constant„t‟ exhibit the properties of Einstein Universe. This principle is the result observations made by Hubble‟s Telescope. From Cosmic microwave background (CMB) observations it is inferred that the Universe appears to be isotropic. In other words, the principle can be stated that at large scale the Universe appears to be the same from everywhere and there is neither a centre nor a boundary. Weyl‟s postulate itself explains homogeneity and an isotropic condition of the Universe- as we have seen earlier that the lines are not intersecting but parallel to each other and space-like hyper-surfaces are orthogonal to time lines. With this principle, it is also possible to transform the space-time co-ordinates into Cosmic standard co-ordinates representing density, pressure and temperature. 2.2 The line element With the above assumptions it is possible to have three types of surfaces - the Universe can have a flat surface, or aclosed surface or an open surface. While a plain surface has zero curvature, an open surface has negative curvature and a closed surface has a positive curvature. These three types of curvatures have some constant value which ensures the properties of homogeneity and isotropy. If the curvature of space differs, than it is possible to get other homogeneous and isotropic spaces by appropriate transformations. To get this, we consider space co-ordinates which are given as xμ where μ = 1,2,3,4 . Here four instead of the usual three dimensions are used to meet the criteria of the Cosmological Principle [10] Let us consider the surface with negative curvature,

x12  x22  x32  x42  R2

(4)

where S is a constant. The substitution x1=R sinhχ cosθ , x2 = R sinhχ sinθ cos , x3 = R sinhχ sinθ sin , x4 = Rcoshχ gives

dx12  dx22  dx32  dx42  R2 [d  2  sinh2  (d 2  sin2  d 2 )]

(5)

The negative sign for x4 shows that the surface is embedded in four dimensional pseudo- Euclidean spaces instead of normal 3-d Euclidean space. If we consider r= sinhχ in above equation, then Eq. (5) becomes:

 dr 2  dσ 2 =R 2  +r 2 (dθ 2 +sin 2θdφ2 )  2 1-kr 

(6)

Where k=0 for flat surface, k=-1 for open surface (negative curvature) , k=1 for closed surface and right hand side of above Eq. is simply Euclidean line element scaled by the factor R for flat surface. The scale factor R can depend upon cosmic time. The most general line element satisfying Weyl‟s postulate and the Cosmological Principle is given by

 dr 2  ds 2 =c2 dt 2 -R 2 (t)  +r 2 dθ2 +sin 2θdφ2  (7) 2 1-kr  Where R(t) is the expansion factor or scale factor. Eq. (7) is known as the Robertson –Walker line element or metric which was obtained by H.P. Robertson and A.G. Walker in the 1930s independently. In the next subsection we will obtain the model using R-W metric which is often called the FRW model.





III.

THE FRW MODEL WITHOUT LAMBDA

A. Friedman developed a cosmological model in 1922 with the help of relativistic cosmology which is very similar to that obtained by Newtonian cosmology.

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International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 2, Issue 2, pp: (70-86), Month: October 2014 – March 2015, Available at: www.researchpublish.com The model derived using Robertson- Walker metric by A. Friedman and it is called as Friedman-Robertson-Walker Cosmological model. Here in this section let us glance at the FRW model which will enable us to set up a new model with different metric. Consider

 dr 2  ds 2  c 2 dt 2  R 2 (t )   r 2 d 2  sin 2  d 2  2  1  kr 0 1 2 3 Let x =ct , x =r , x = θ , x =  ,





(8)

In a curved space time, the line element is given as ds 2  gij dxi dx j where gij is a 44 metric tensor. Therefore metric tensor for above element is given by g00 =1, g11= -R2(t)/(1-kr2), g22= -R2(t)r2 , g22 = -R2(t)sin2θ . k is a curvature constant , k=0 for flat Universe, k=1 for closed Universe, k=-1 for open Universe. To derive the model it is necessary to obtain Einstein Field Equations. Einstein Field Equations can be obtained from the following equation: 1 (9) R ij - Rδij +Λδij =-8πGTji 2 Where ij = gil glj , gil is metric Tensor, Rij –Ricci Tensor, ij – Kronecker delta tensor, R-Ricci scalar, Tij-energymomentum tensor, -Cosmological constant, G- gravitational constant. Ricci Tensor is written as Rij=gil Rlj Ricci tensor can be obtained as follows: Rij  

a ij



a ia

b a b a  ij ba  ia bj

x x where Γ is Christoffel Tensor of second kind and is given by

ija 

1 ab   gbj  gib  gij  g     , 2   xi  x j  xb 

(10)

(11)

Generally ћ =c=1 is taken at the cosmic scale. R  R11  R22  R33  R44 a ia 

(12)

 log  g

xi g =  gij  . Field Equations can also be written as 8πG i G ij = Tj + Λ g ij 2 c and Tij is the energy momentum tensor for perfect fluid which is given as

 p  Tij =  2 +ρ  u i u j -gijp c 

(13)

(14)

(15)

d xi is the 4- vector velocity component such that uiuj =-1 for i= 0,1,2,3 (space -time coordinates) p and ρ are dt pressure and density of matter distribution of the Universe respectively. Thus, from the above equation Energy momentum tensor Tij = (ρ, -p, -p,-p). This assumption also shows that the energy momentum tensor of the Universe takes the same form as that of perfect fluid. ui =

According to the Cosmological Principle, the Universe is homogeneous and isotropic, so it is possible to assume ћ =c=1 for deriving field equations. At present we consider „‟ (the Cosmological Constant) to be zero. The Cosmological Constant was first suggested by A. Einstein to show a relation between geometry and matter but was later discarded by him as the Einstein model was inadequate to explain expansion of the Universe. Deriving field equations from (9-15) for FRW model without lambda, we get the following equations:

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R2 R

3

k R2

 8 G 

(16)

R R2 k    8 Gp 2 R R R2

(17)

G11  G22  G33 Additionally, the conservation of energy requires T ij;j =0 which implies : R 0 R In equivalent form d d (  R3 )  p ( R3 )  0 dt dt The above relation is similar to the energy conservation equation given by dE + pdV=0 in thermodynamics.

  (   p)3

(18)

(19)

It should be noted that the field equations satisfy the energy conservation equation. The above field equations are solved for (i) dust Universe for which p=0 and can be studied for open, closed and flat curvature of the Universe (ii) Equation Of State (EOS) for perfect fluid given by p=  where  is constant of EOS .  determines the relation between pressure and density . If  = 0 then the Universe is dust filled or matter-dominated and if = 1/3 then it is in a radiation-dominated phase. Including EOS in field equations , Eq. (17) and (18) are now rewritten as : R2 2

k 2

8πGρ 3

(20)

R R We also have

R R2 k + + =-8πGωρ R R2 R2 Solving the above equations, a general solution is obtained as , 2

(21)

R 2 =AR -(1+3ω) -k

(22)

A is a constant of integration, which can be determined from initial conditions. From the above equation it is possible to obtain the solution for a matter-dominated and radiation-dominated phase of the Universe However, the solution is analyzed for matter-dominated phase for simplicity as the present Universe is matterdominated. A radiation-dominated phase will be analyzed for the model with lambda in the later section. 3.1 Matter Dominated Phase If the Universe is matter-dominated then EOS for it is given by taking =0, Thus Eq. (22) takes the form as

R 2 =AR -1 -k

(23)

Scale factors for flat, closed and open Universe can be determined by substituting k=0 , k=1 and k=-1 respectively in Eq. (23) which are as follows. 1.

Flat Universe , k=0 ,





2 3 R=  At+B  3  Where B is constant of integration and is assumed to be zero, so R t2/3 2. Close Universe , k=1 A R= (1-cos ) 2

(24)

(25)

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International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 2, Issue 2, pp: (70-86), Month: October 2014 – March 2015, Available at: www.researchpublish.com A ( -sin ) 2 To arrive at above Eq. initial conditions t=o , R=0 are assumed. 3. Open Universe , k=-1 A R  (cosh   1) 2 A t  (sinh    ) 2 Here also we have R=0 at t=0 . Plot of all three models is shown in Fig.2 below.

(25a)

(26) (26a)

Fig.2 Types of Universe –open, flat and closed models of the Universe. P, Q and R correspond to present epochs.

From Fig. 1.2, it is observed that open and flat models represent an expanding Universe, while a closed model has expansion for some time and thereafter it may contract. Recent observations lead to the conclusion that the Universe is flat. But FRW models suffer from flatness problem as it can be seen from Equations for scale factor that constant A has not been determined from initial conditions. The problems faced by FRW models in absence of cosmological constant will be discussed in next section. Let us have an overview of the cosmological parameters in the following section.

IV.

COSMOLOGICAL PARAMETERS

To understand the geometry of the Universe, several important parameters are defined; which play a vital role in observing and understanding the Universe i.e. observational cosmology. These parameters are i) Hubble parameter ii) Deceleration parameter iii) Cosmological density parameter. A short introduction of these parameters is given below. A. Hubble Parameter Hubble parameter is derived from Hubble‟s Law which is defined for velocity –distance relation. In 1929, Hubble established linearity between the velocity of a galaxy and a red shift of the light received on Earth by which the distance of the galaxy from the Earth could be determined. Hubble‟s Law states that the velocity of a galaxy is directly proportional to the distance between it and the earth. It is given by v = H0 D.

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International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 2, Issue 2, pp: (70-86), Month: October 2014 – March 2015, Available at: www.researchpublish.com V-velocity of galaxy, D- distance between galaxy and earth, H0 - Hubble‟s constant at present epoch. H0 has dimensions of km/s/mpc , mpc (mega parsec ) is the unit for distance measurements on the cosmic scale , hence the Hubble Constant has a dimension inverse of time. Hence, it can be used to determine the age of the Universe. This constant is now called the Hubble Parameter which, for present Universe, is written as H 0 =

R(t) R(t) t

, [R(t) is a scale 0

factor. A dot over scale factor R (t) is the derivative of R (t) with respect to time]. The Hubble parameter not only determines age of the Universe but is also an important parameter in the measure of observable size of the Universe. The inverse of H0 is called Hubble‟s Time. B. Deceleration Parameter Deceleration Parameter „q0‟ is required for the explanation of expansion of the Universe and is useful for the expression of density of the Universe. q0 is formulated as q 0 =-

RR R2 t 0

q0 is the deceleration parameter at present time. According to recent observations by COBE, WMAP, BAO and several other experiments, the expansion of the Universe is accelerating. Hence the deceleration parameter is defined to measure the accelerated expansion. The value of q0 should be negative for the acceleration of the Universe. For the matter-dominated FRW closed model, it is found to be greater than 1/2, so the model is decelerating; hence it is thought that the present Universe is not a closed one. For open and flat models, since q0 is negative the Universe is accelerating forever. For flat Universe q0  -1 shows that the Universe is accelerating at a constant rate. From field equations, the acceleration equation is derived from Eqs. (16) and (17) which is given by, R =-4πG(ρ+3p)=-4πGρ(1+3ω) R In the above EOS, p=  has been substituted. With Eq. (27) , q0 is obtained as 3

(27)

1 k   q= (1+3ω) 1+  2  R 2 H2 

(28)

Where q=-

(29)

at any given time. For a flat universe >-1/3 i.e. universe filled with any fluid accelerates forever. q can also be represented in terms of the Hubble Parameter as,

=-(1+q) H2 The above equation can also be applied for closed or open models.

(30)

C. Density Parameter To define the expansion of the Universe, a Density Parameter is formulated in terms of proper density and Hubble parameter, which is written as: 8πGρ Ω= (31) 3H 2 8πρ0 For present epoch we have Ω0 = when G= 1 is assumed. 3H02

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International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 2, Issue 2, pp: (70-86), Month: October 2014 – March 2015, Available at: www.researchpublish.com The Density Parameter also determines the geometry of the Universe. For a flat Universe, the density of the universe is same as critical density. Critical density is defined as a watershed point between expansion and contraction of the Universe [2] . Its expression is derived from the field equations of the Friedman model for flat Universe, and is given as

ρC = Ω=

3H 2 .Its value is equal to 1.88 10-29 h02 g/cm3. So the present cosmological density parameter  is expressed as 8πG

ρ .If the Universe is flat, then =1 . If the Universe is closed and has stopped expanding, then >1 , If the ρC

Universe is open and expands forever, then <1 then it. Recent observations by WMAP have concluded that  is nearly equal to one which implies that the present Universe is flat. When the Cosmological Constant „‟ is introduced then a more general expression for  c where  relates to  and k can be expressed i.e.   and k respectively.  represents vacuum density parameter defined as,  represents vacuum energy i.e. (/3H2 ) [ role of  in cosmology will be discussed later in detail in this chapter] and  k is expressed as (–k/R02 H02 ) which corresponds to density of curvature. The total density of the Universe is now given as 0 + +k which should be equal to 1. At early stages of evolution of the Universe, visible matter was dominated by radiation, that is to say that it was a radiation-dominated stage. If the visible matter in the Universe is contained in the galaxies the density is approximately equal to 10-31 gm/cm3 over the largest scales. This is equivalent to one proton per cubic meter. In this stage the Universe also contains neutrinos, gravitational waves which are called as primordial radiations. Apart from visible matter nowadays researchers are looking for dark matter too. Hence, the total density parameter is not only related to visible matter and dark energy but also includes dark matter density which is yet to be cracked. Dark matter and Dark energy were not parts of the model during the time of Friedman, Hence his field equations do not have any expression related to these. Although the FRW model has become the standard model in cosmology; some observations inferred the inadequacies of the model without the Cosmological constant. Thus developing a FRW model with the Cosmological constant is being undertaken by many in cosmology. In the next section a FRW model with the Cosmological Constant will be explained.

V. FRW MODEL WITH COSMOLOGICAL CONSTANT The FRW model without the Cosmological constant suffered from some drawbacks. Basically these related to the Horizon and Flatness. A. The Horizon Drawback In Cosmology, two kinds of Horizons are often discussed. These are the Particle Horizon - which relates to the communication in past and the Event Horizon – which relates to limits on communication in the future. The Horizon drawback deals with the problem of the largest distance traveled by light to reach an observer since the time of the Big-Bang. t

cdt ' . R(t ') 0

The largest radial distance for such a case is written as rmax = 

From r max we get information that has been obtained from those particles which could be detected and the three surfaces in space-time having this radius is called as particle horizon. Physical distance to the Horizon at the time of t is D horizon (t) = R (t) r max (t). D horizon (t) is found to be equal to 2ct for radiation-dominated phase for which p = ρ/3 and D horizon (t) = 3ct for matter dominated phase for which p=0.

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International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 2, Issue 2, pp: (70-86), Month: October 2014 – March 2015, Available at: www.researchpublish.com The different value of D horizon (t) for different phases is called the Horizon Problem. This problem raises the question of homogeneity of the Universe, as cosmic microwave radiation data clearly indicates that we live in a nearly homogeneous Universe. B. Flatness Drawback The Flatness Drawback is due to total density calculated for open, flat and closed Universe, where we define total density ρ given by Ω= where ρc is the critical density of Universe. Ω at t0 is calculated for present epoch. ρc According to the FRW model, Ω should deviate further and further from unity for k = ±1. However, even when deviating continuously from unity, it reaches unity for k = ±1 (even though retaining unity for k=0). From present day observations, Ω has values between 0.01 and 10. The discrepancy between Theoretical and Observed values is the Flatness Problem. An inflationary model can provide solutions for the two basic problems. But it can only give information about the early Universe. It cannot explain the present Universe. There have been several unsuccessful attempts by researchers to solve these problems, leading to the conclusion that the finite Cosmological Constant existed and it has some physical significance in the study of the Universe. The importance of the Cosmological Constant will be discussed subsequently in the present thesis. Since we know that the FRW model is a basic, standard model the FRW model with Cosmological Constant „‟ will be discussed here.

VI.

THE FIELD EQUATIONS AND THE MODEL

Field equations with Λ are obtained from FRW metric and these are given below:G 00 =

R2 R

G11 =2

1 8πGρ - Λ= 3 R 3

(32)

R R2 k + + -Λ=-8πGp 2 R R R2

(33)

G11 =G 22 =G33

The above equations can be solved with the help of EOS where p=  . Solving the above equations, the first order differential Equation is given by

ΛR 2 k 3 1+3ω For dust model Universe, we have =0, Thus the equation (34) will now be given as: R 2 =AR -(1+3ω) +

ΛR 2 -k 3 The real solution of the above equation can be obtained if A=0. In such a case we have: R 2 =AR -1 +

For flat model R=Ce

Λ t 3

(34)

(35)

For open model i.e. k=-1, R=Ccosh For closed model k=1 R=Csinh

Λ t 3

Λ t, 3 Where C is a constant.

The above solutions are quite different from the results obtained for =0. The models for positive lambda represent steady state models which resemble Einstein‟s static models. A Model with <0 will not give a real solution for closed or open models.

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ΛR 2 k 3 2 If A≠0 ,then, we have the following equation R 2 =AR -2 +

(36)

ΛR 4 kR 2 3 2 If k=0 ,i.e. Flat Universe , the model is given by , R 2 R 2 =A+

(37)

 3A  Λ  R=  sinh  2 t+C   3     2 If k=1, Closed Universe, we have,

(38)

 3A  Λ  3  9  R=  sinh 2 t+C     +  2  3   4Λ   Λ 16Λ  For k=-1, Open Universe,

(39)

 3A  Λ  3  9  R=  t+C  (40)   sinh  2 2 3  Λ 16Λ    4Λ  From the above, and for simplicity we choose C=0 which will imply that the behavior of all the models are similar in the radiation-dominated phase. All are expanding in radiation-dominated phase. A Flat model appears to have singularity while others are nonsingular models. It can be observed that the positive cosmological constant can play an important role in the study of the Universe which was initially discarded by Einstein, but since then it has been found that results of FRW with lambda models can be reconciled with observational data [3] . There are several reportings which have shown the importance of lambda in the study of the Universe. The following Fig. (4) represents the types of Universes in the presence of lambda.

Fig.3 – Dynamics of the Universe when k= -1, 0, 1 and positive and negative lambda [1] Error! No text of specified style in document. From Figure it can be easily understood that when k=0 and k=-1 with positive lambda represents an expanding model; whereas when k=1 there is a bouncing Universe model.

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AGE OF THE UNIVERSE FOR FRW MODEL WITH COSMOLOGICAL CONSTANT.

In Cosmology, the age of the Universe is estimated by calculating H0-1 i.e. Hubble parameter at present time. Since it was known that the present Universe is matter-dominated, hence by age t0  H0-1 it is calculated as 14.4Gyr. According to observational data, Age=13.350.35 Gyrs . is calculated. A small discrepancy in the age of the Universe can be resolved by considering density corresponding to Lambda along with matter density. The relation between Age and density parameter is explained here. Two important Cosmological parameters (Hubble parameter and deceleration parameter) at present time are given by

H0 =

R RR , q 0 =Rt R2 0

, considering R=R0 at t=t0 Eq.(32) in presence of lambda is rewritten as H02 + t

k R 02

8πGρ0 1 - Λ= 3 3

(41) =1 , where we consider density parameter for

Considering the definition density parameter, we have  m0 +0 +k0 8πGρ 0 matter as Ω m0 = . Other density parameters for lambda and curvature constant are defined in a similar way. 3H 02

Values of matter density and vacuum density (density corresponding to ) are obtained as  m0 =0.3 and  0 =0.7 approximately as per observational data. Eq. (33) is rewritten as k 1-2q02 H02 + -Λ=-8πGωρ 0 R 02

(42)

The above equation is further simplified to : 1 q 0 = (1+3ω)Ωm -ΩΛ 2 For a dust-filled Universe ω=0 Hence we have q0= Ωm/2- ΩΛ .

(43)





Current observations estimate Ωm=0.3 and ΩΛ =0.7 values. So, for a dust-filled Universe, the value of deceleration parameter is determined as q0 = -0.55 which confirms that Universe is accelerating. The Age of the Universe can be determined in the following way: If we assume the present Universe is in a matter-dominated phase i.e. dust-filled, flat Universe, then, Scale factor R(t) ~ t2/3 and t0 = 2/3H0 - which is approximately calculated as 9Gyr. But the age of the oldest star is found to be 12Gyr. The discrepancy between the age of a star and that estimated through Hubble parameter forces us to accept the existence of a finite, positive Cosmological constant. In the presence of the Cosmological constant, the age of a flat Universe is determined by calculating t0. R(t)

t=H-1 0 

dR' R'

(44)

In terms of red shift, Consider 1+z=R0/R, where z is red-shift magnitude, we can find age as given by formula =H-1 0

(1+z)-1



dR' R'

(45)

Hence from Friedman‟s equations, it is possible to represent the Hubble parameter as a function of z.

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R H(z)= =H0 Ωm (1+z)3 +Ωk (1+z)2 +ΩΛ  2   R Thus we write H(z) = H0 f(z) , now we have R/R0 = H0 f(z)/(1+z).

(46)

We have further dR/R0 = -dz/(1+z)2 . This implies that Eq. (44) can be rewritten as   dz t=H-1  0 -   0 [(1+z)f(z)]  Hence we have

t0 =

(47)

1/2   2 1/2 -1  ΩΛ   Ω sinh  Λ    3H0  Ωm  

(48)

Substituting the values of  m and  i.e. 0.3 and 0.7 respectively, the age of the Universe for the present model is calculated as 12.2 Gyrs. From experimental observations by WMAP, COBE, the age of the Universe is found to be 13.798±0.037 Gyrs. The size of the Universe is also an important parameter which is related to the age of the Universe Volume of the Universe is normally calculated as:  π 2π r 2 drsinθdθdφ

V=R 3   

00 0

(49)

(1-kr 2 )1/2

It will be an infinitely large, flat Universe. Although this is a theoretical estimate, we need to find the size of the observable Universe so that the particle horizon can be found which can give past information of the Universe. In Cosmology there are two kinds of horizons - Event Horizon and Particle Horizon. Particle Horizon relates to the communication in the past and Event Horizon relates to limits on communications in future. Since we have limitations for observable Universe but still size of the Universe can be found out for FRW model by determining Hubble radius rH defined by ctH . Cosmologically c=1 so rH= tH .It is roughly estimated to be 2998h-1mpc [4] It can be determined more accurately by the following expression: Let the light ray start from r=r1 at time t1 and it reaches us at r=0 at t0. Therefore for light received, we have: 0

-  dr=r1 = 

dt R(t)

(50)

For a dust-filled Universe r1ea(t1-t0) - where a depends upon Lambda. The Horizon problem is solved if the Cosmological constant is included in the field equation. R(t) represents an inflationary era . In such a situation the particle at one point will still be in casual contact with another point for a very large distance between them. This is possible because the expansion rate is an exponential function of time. In a similar manner, flatness of the Universe is solved [4]. From Eqs. (38-40) time evolution of an accelerating Universe is as shown below. Fig.4 illustrates the area of the Universe in an accelerated phase for which the Universe is flat and has some finite Cosmological Constant, whereas decelerating phase of the Universe may be due to k=0 and zero Cosmological Constant. It is also observed from the fig.(4) that there is an outward bulging during the accelerating phase so as to have age more as compared to the Universe with zero Cosmological Constant. It should also be noted that the accelerating expansion due to vacuum energy causing negative pressure tends to have flat Universe. Hence the model with finite, positive Cosmological Constant solves the Age problem and the Flatness problem.

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International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 2, Issue 2, pp: (70-86), Month: October 2014 – March 2015, Available at: www.researchpublish.com

Fig. 4 Time evolution of the Universe – Accelerating and Decelerating Universe

Although FRW model with finite Cosmological Constant has solved the Horizon problem, Age Problem and Flatness Problem it still t faces some other problems. In next section these problems are going to be discussed.

VIII.

LIMITATIONS OF FOUR DIMENSIONAL FRW MODEL WITH COSMOLOGICAL CONSTANT.

Finite Cosmological Constant representing vacuum energy density plays an important role in explaining dynamics of the Universe and it can explain accelerated expansion of the Universe. With Cosmological Constant it is also possible to predict the age of the Universe and relate it to observational data. However recent researches in cosmology have pointed out some key problems related to Cosmological constant i.e. Lambda Cosmology. These are given below, A. The Cosmological constant problem The value of the vacuum energy density  is given by as ~ħkmax4 which relates to momentum of the zero mode of vacuum oscillations. By quantum field theory, energy density ε related to cosmological constant is found to be ~ (Mpl)4 ~ (8πG)-2 which is calculated as ~(1018GeV)4= 2 10110 erg/cm3. From observational calculations, it is found that  ≤ (10-12 GeV)4 ~ 210-10 erg/cm3 . Discrepancy between the two values of total density  is around 10-120 which is called the Cosmological problem.[5] . Several proposals have been put up by several scientists which are based on string theory, super symmetry, scalar field theory etc. But the most promising solution of CCP is suggested as Lambda Decay Cosmology indicating time dependent Cosmological Constant which decays with time. B. Cosmological Coincidence Problem. The Cosmological Coincidence Problem is regarding the values of the Cosmological constant and vacuum density. It is found that Vacuum density is comparable to matter density at some cosmic time and reduces drastically with the expansion of the Universe. However the value of lambda does not reduced correspondingly. Seemingly, there may not necessarily be any coincidence between lambda and Vacuum density; although it can be expressed in terms of lambda. To solve this problem, Quintessence Cosmological model [6] with self interaction, Guassian potential etc have been suggested by many researchers. In order to solve both Cosmological constant problems, coincidence problems, time varying lambda becomes the most satisfactory answer.

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International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 2, Issue 2, pp: (70-86), Month: October 2014 – March 2015, Available at: www.researchpublish.com An Anthropic solution and existence of Dark Energy are also solutions suggested by cosmologists, but an anthropic solution means disturbing the structure of the Universe. Since the existence of Dark Energy is still a puzzle and the Dark Energy content is still unknown, cosmologists are still working on it and this area is also the current topic of research. Dark Energy and Dark Matter is still a mystery which is yet to be solved and it is the latest research topic. It is also pointed out that the solution to Cosmological Constant problem may draw some definite conclusion on Dark Energy in the Universe which fills almost 70% of the Universe. As discussed previously here it is necessary to review Lambda Decay Cosmology which motivates us to choose time dependent lambda for further research work.

IX.

LAMBDA DECAY COSMOLOGY

Although a cosmological model with constant  explains the expansion of the Universe to some extent as seen in preceding sections; a huge discrepancy exists between the observed value and the predicted value by the Standard Model described in Particle Physics. This discrepancy is also called the „Cosmological Constant Problem‟ (CCP). To solve this problem, a popular suggestion given by many scientists was that the Cosmological Constant may have a large value, and with the evolution of time its value decreased. Since the Big-Bang the Universe has undergone several phases. The early phase of the Universe is also called the inflationary phase. In this phase, the value of the cosmological constant is assumed to be very high. However, in present phase, value of Λ is calculated to be very small. The Cosmological constant problem was first discussed by H. Bondi in 1960 [7]. There are several articles by quantum field theorists and others ( [8], and ref. therein ) in which sources of vacuum energy have been considered as potential sources like scalar fields, tensor fields , non local effects, worm holes etc. These articles concluded that the Cosmological Constant is not really constant but is varying. The relation between scale factor and cosmological constant was also established by [8] in their paper. V. Sahni and others [9] have stated that vacuum energy density as calculated by Zeldovich [10] is given by εvac.= ρvac . c2 = (Gm2/λ)/ λ3= Gm6c4/ ћ4 where λ=ћ/mc. When density calculated for pion mass compared with density 1  mπ related to Planck mass it is given as ρ Λ =  2π 4  m pl

  ρ pl =1.45 ρpl ×10-123 gm/cc  

Here ρpl is the Planck density. The above expression shows that there is a huge difference between energy due to Λ and that of Planck energy. Here value of ρpl is calculated as = c5/G2 ћ ≈5 × 1093 gm/cc. With this, during Planck epoch t pl ~ 10-43 sec, it would involve a fine-tuning of one part in 10-123 3! In some papers this time is calculated as 10-35 sec. This shows that Vacuum effects also play an important role, if the Universe is expanding. The huge difference between the values of Λ for the early universe and present universe has been a key factor in the study of lambda decay cosmology. There are many projects [11, 12, 13,14] in which a Four dimensional model with time varying cosmological constant have been studied extensively. Overduin and Cooperstock explained time variations of Λ as Λ~a-m , Λ~t-t Λ ~ qr ,Λ~H2 , Λ~H. In these models, they considered specific values of m, t, r which are constants and discussed several oscillating and non-oscillating models. Berman and others [15] had concluded Λ~t-2 . Arbab I. Arbab [16] has also reviewed cosmological models with Λ~H2, Λ~ q and Λ~ R2 and studied cosmic acceleration for positive cosmological constant and its implications. A list of time varying cosmological constant is available in the articles by Sahni V. [9].

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International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 2, Issue 2, pp: (70-86), Month: October 2014 – March 2015, Available at: www.researchpublish.com

Table I. List Of Time Varying Cosmological Constant

t-2

[8,14, 15]

[8]

A + B exp(- t)

(Beesham & others) [17]

-2

[Chen &Wu] [28]

[8]

exp(- a)

[Rajeev S.G.][18]

[Canuto & others][19] 2

H 2

[16, 20] Lima and others -

H + Aa

Carvalho, Waga [21] Lima and Maia, [22] Lima & Trodden [23]

f (H)

Hiscock [24] Reuter [25]

g( , H)

The Phenomenological models with variations of Λ as scale factor a (t) , cosmic time t, Cosmic temperature T , Hubble constant H as shown in the above table, have been discussed by many researchers[9] and references therein)

CONCLUSION

In this chapter we conclude that the FRW model, the so-called „standard model of the Universe‟ is quite successful in providing positive answers to the age of the Universe problem, Horizon problem, and expansion of the Universe problem;, but there are several puzzles still to be solved. Lambda decay Cosmology has gained attention recently for solving puzzle of Dark Energy, Dark matter, Large structure of the Universe, the phases of the Early Universe etc. In the meantime, existence of higher dimensions has also become a matter of interest to study unifying all types of forces. Pioneering work, in the study of higher dimensions, by Kaluza T. in 1921 and Klein O. in 1926 in five dimensional physics is of great importance and has a wider coverage area in cosmology. Five dimensional FRW Cosmological model was first discussed by Chodos and Detweilar [26] in 1984 and later on investigated by many authors in different contexts . Five dimensional FRW cosmology is alternately called as KaluzaKlein Cosmology which is consistently gaining response from research community . Recently, using Kaluza Klein cosmology, Paul Wesson developed a new theory, called it Space-Time–Matter theory [27] and explained 5D models in detail in his books. ACKNOWLEDGEMENT It is my pleasure to give my sincere gratitude to my Co-Guide Prof G.S. Khadekar for providing his valuable guidance. REFERENCES [1]

Narlikar J.V., An Introduction to Cosmology , Chapter 3, 3rd edition, Cambridge University Press, Cambridge (2002)

[2]

Wikipedia ,(The free encyclopedia) „Big Bang‟ en.wikipedia.org/wiki/Big_Bang

[3]

Eddington A, „Space, Time and Gravitation. An Outline of the General Relativity Theory‟ Cambridge University Press, Cambridge (1920)

[4]

Banerji S., Banerjee A. „General Relativity and Cosmology‟, Chapter 11, Reed Elsevier Pvt. ltd., Elsevier (2007)

[5]

Caroll H. Press T. „Ann. Rev. Astron. Astrophysics. 30, 499 (1992)

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International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 2, Issue 2, pp: (70-86), Month: October 2014 – March 2015, Available at: www.researchpublish.com [6]

El Nabulsi A. Rami , Gen. Relat. Gravit ,43, 261 (2010)

[7]

Bondi H. „Cosmology‟ , Cambridge University Press, Cambridge.(1960)

[8]

Overduin J.M. & Cooperstock F.I. , Preprint Arxiv astro-ph 9805260v1 (1998)

[9]

Sahni V. , Starobinsky A., Int. J. Mod. Phys. D 9, 373 (2000)

[10]

Zeldovich Y.B. et. al Sov. Phys.-JETP 40, 1 (1975)

[11]

Peebles, P.J.E. , Ratra, B. ApJ, 325, L17 (1988).

[12]

Dolgov, A.D. , Phys. Rev. D, 55, 5881 (1997).

[13]

Padmanabhan T., Preprint Arxiv : hep-th/ 0212290 (2002)

[14]

Maia, M.D. , Silva, G.S. Phys. Rev. D, 50, 7233 (1994).

[15]

Berman, M.S. , Som, M.M. Int. J. Theor. Phys. 29, 1411 (1990).

[16]

Arbab I.A. Preprint Arxiv : gr-qc/9909044 (1999)

[17]

Beesham, A. Phys. Rev. D, 48, 3539 (1993).

[18]

Rajeev, S.G. Phys. Lett., 125B, 144 (1983)

[19]

Canuto V., Hsieh S.H. , Adams P.J. Phys. Rev. Lett. 39,429 (1977)

[20]

Lima, J.A.S. , Carvalho, J.C. Gen. Rel. Grav. 26, 909 (1994).

[21]

Carvalho, S. C., Lima, J.A.S. , Waga, I, : Physics Review D 46, 2404 (1992).

[22]

Lima, J.A.S. , Maia, J.M.F. , Phys. Rev. D, 49, 5597 (1994).

[23]

Lima and Trodden, M: Physics Review D 53, 4280 (1996)

[24]

Hiscock, W.A. Phys. Lett., 166B, 285 (1986).

[25]

Reuter, M. and Wetterich, C. Phys. Lett., 188B, 38 (1987).

[26]

Chodos A, Detweilar S. : Phys. Rev. D 21, 2167 (1980)

[27]

Wesson Paul S. “Space- Time-Matter Modern higher dimension cosmology ” 2nd edition, World scientific, Singapore (2007)

[28]

Chen W., Wu Y.S. , Physics Review D, 41, 695 (1990).

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Kaluza–Klein Cosmological Model, Strange Quark Matter, and Time-Varying Lambda Namrata Jaina , Shyamsunder S. Bhogab , and Gowardhan S. Khadekarc a b c

Dept. of Physics, Mahrshi Dayanand College, Parel, Mumbai-400012,Maharashtra, India Dept. of Physics, RTM Nagpur University, Nagpur-440033, Maharashtra, India Dept. of Mathematics, RTM Nagpur University, Nagpur-440033, Maharashtra, India

Reprint requests to N. J.; E-mail: nam_jain@rediffmail.com Z. Naturforsch. 69a, 90 – 96 (2014) / DOI: 10.5560/ZNA.2013-0079 Received May 28, 2013 / revised October 2, 2013 / published online December 18, 2013 In this paper, exact solutions of the Einstein field equations of the Kaluza–Klein cosmological model have been obtained in the presence of strange quark matter. We have considered the timevarying cosmological constant Λ as Λ = αH2 + β R−2 , where α and β are free parameters. The solutions are obtained with the help of the equation of state for strange quark matter as per the Bag model, i.e. quark pressure p = 1/3(ρ − 4BC ), where BC is Bag’s constant. We also discussed the physical implications of the solutions obtained for the model for different types of universes. Key words: Kaluza–Klein Cosmological Model; Bag Model; Cosmological Constant. PACS numbers: 98.80-k; 98.80,Hw; 98.80 Vc

1. Introduction The physical situation prevailing during the early stages of the formation of the universe is still a challenge and an area of major research in cosmology. A lot of efforts are directed in this area, which prompted us to look for dimensions more than space-time (3 + 1) for the early universe. The necessity for higher dimensions is thought for the early universe, as it was very small in its early stages. The extra dimensions become compactified and get embedded into four dimensions due to expansion of the universe. Hence, experimental detection of an extra dimension is not possible today due to certain practical limitations, however, its effects can be observed. Contemporarily, the unification of forces had been worked upon by the research community to find its origin. Kaluza [1] in 1921 and Klein [2] in 1926 independently put forward theories of higher dimensions for the unification of all forces of nature and particle interaction, respectively. A voluminous literature on the Kaluza–Klein (KK) theory of gravitation, its cosmic implications and astrophysical consequences are available now, which has been widely referred to solve issues like acceler-

ated expansion, mystery of dark energy, dark matter, etc. It is well known that the universe is expanding and also accelerating. The driving force for accelerated expansion is supposed to be dark energy. The relation between extra dimensions with dark energy is enlightened by Gu [3]. Interacting dark energy models with KK cosmology are discussed by Ranjit et al. [4], Chakraborty et al. [5], Sharif and Jawad [6], and Sharif and Khanum [7]. The extra dimension topology and accelerated expansion of the universe have been touched upon by El-Nabulsi [8]. Thus, the KK cosmology and its models have gained importance among the scientific community to learn the secrets of the universe, its behaviour at early times, etc. In this paper, we examine the KK cosmological model in presence of strange quark matter (SQM) with decaying Lambda. The importance of quark matter lies not only in the structural formation of the universe and, subsequently, its evolution; but since it is a part of dark matter, its importance lies also in the interaction with dark energy. These are hotly debated issues discussed in published literature recently [9 – 11]. Quarks can also be studied with domain walls and strings. Currently, models with quark matter with do-

N. Jain et al. · Kaluza–Klein Cosmological Model, Strange Quark Matter, and Time-Varying Lambda

main walls, strings, etc. have been studied by Adhav and Nimkar [12], Ozel et al. [13], Bali and Pradhan [14], and Yilmaz and Yavuz [15] in different contexts. Hence, it is necessary to take a look at the role played by strange quark matter in the early stages of the evolution of the universe, since the big bang. It is well known that quark-gluon-plasmas (QGPs) exist since the beginning of the universe. Since the big bang, the universe has transited through two phases: the first transition occurred at a critical temperature resulting in stable topological defects, while the second transition occurred at the cosmic temperature of the universe T ∼ 200 Mev. At this temperature, a QGP converted to a hadron gas. The astrophysical consequences of phase transitions have been pointed out by Witten [16] in 1984. The existence of quark matter was first discussed by Itoh [17] and Bodmer [18] in the 1970s and later, Witten [16] proposed that the quark-gluon-hadron gas transition and the conversion of neutron stars into strange stars at ultra-high densities were the two ways for the formation of quark matter. Sagert et al. [19] discussed experimental analysis of explosive astrophysical systems and enlightened quark-hadron phase transitions. Farhi and Jaffe [20], Xu [21], and Lipkin [22] independently reviewed the physical nature of SQM, concluding that it is stable. Properties of quark matter [u (up), d (down), s (strange) etc. quarks] have been well explained in particle physics where quark matter participates in strong interactions and forms the basic constituent of baryons. Thus, undertaking a study of SQM and quark matter can provide an idea on the structure and the geometry of the universe [23]. A study of quark matter and SQM has been a major area of interest among scientists as it could not only provide information about the early universe, but can also solve the mystery of dark matter. In this context, Virginia Trimble has given an excellent review [24]. In a typical cosmological model with SQM, the quark matter is modelled with the help of a phenomenological Bag model, where the quark confinement has been described as an energy term proportional to the volume [17]. In this model, quarks have been assumed to be degenerate Fermi gases, which exist only in a region of space endowed with a vacuum energy density BC (called the Bag constant). Quark matter consists of

massless u, d, and massive s quarks and electrons. In a simplified version of the Bag model, quark density and quark pressure have been related with each other as pq = ρq /3, while the total pressure is p = pq − BC and the total density ρ = ρq + BC . Under these conditions, one obtains the equation of state (EOS) for SQM described as p = 1/3(ρ − 4BC ). Experimental results obtained in Brookhaven’s relativistic heavy ion collider (BNL-RHIC) laboratory conclude that a quark-gluonplasma is the perfect fluid, of which quark matter is a basic constituent. Yilmaz and Yavuz [15] and others inferred that the presence of extra dimensions and SQM tended to exert negative pressure with constant density in the early universe. This was concluded to be dark energy. In fact, the cosmological constant Λ represents dark energy, and it plays an important role. In a recent development in cosmology, it was found that the acceleration of the universe is due to negative pressure which is proportional to the related vacuum density. Presentday astronomical observations [25] indicate that the value is ≤ 10−56 cm2 . But the huge difference between the present small observed value of the cosmological constant and the one calculated by the Glashow– Weinberg–Salam model [26] from particle interaction has been of the order of 1050 . This is known as the cosmological constant problem (CCP) and is also a major area of research for many contemporary cosmologists. Bambi [27] discussed the CCP and SQM and remarked in his article that strange stars, which are thought to consist of SQM, if these exist, can be a good laboratory to bring information about the early universe and their physical conditions during those early stages. It can be considered to investigate the CCP and to test the nature of dark energy. The time varying cosmological constant Λ was suggested for solving the CCP, as it has been thought that perhaps Λ might had a large value in early universe and decayed with time so as to have the present small value. A decaying Λ has been first explained by Chen and Wu [28] who have suggested that Λ ∝ R−2 . Thereafter Sahni and Starobinsky [29], Padmanabhan [30], and Overduin and Cooperstock [31] reviewed cosmological models with time varying Λ in four dimensions in different contexts. Recently, the papers by Khadekar et al. [32 – 34] dealt with the solutions of the KK cosmological model with different forms of time varying cosmological constants, i.e., 2 Λ ∝ aȧ2 , Λ ∝ a12 , Λ ∝ äa , Λ ∝ ρ; here a is the scale fac-

N. Jain et al. · Kaluza–Klein Cosmological Model, Strange Quark Matter, and Time-Varying Lambda

tor. It has been shown by the authors that these models are dynamically equivalent for a spatially flat universe. El-Nebulsi [35] has discussed a higher-dimensional nonsingular cosmology dominated by a varying cosmological constant. One of the motivations for introducing Λ is to reconcile the age parameters and density parameters with recent observational data. The variation of Λ has also been discussed in literature in different contexts [36 – 43]. Encouraged by aforementioned facts, we investigated the KK cosmological model in the presence of quark matter with the variation of the cosmological 2 constant Λ = α ṘR2 + β R12 which has been first suggested by Carvalho et al. [44]. Particularly, the present paper is focused on the study of a generalized form of the Kaluza–Klein cosmological model. Here α and β are considered as the dimensionless free parameters. The exact solutions of the Einstein field equations so obtained are applied to study the variation of quark density and Λ for different types of universes. Cosmic implications of the model under consideration are also discussed. The paper is organized in four sections. In Section 2, we first set up field equations and later, in Section 3, found solutions followed by discussion and conclusions in Section 4 and Section 5. 2. Metric and Field Equations

quark density, respectively, which are related by EOS per the Bag model given as 1 (ρ − 4BC ) , (3) 3 where BC is Bag’s constant. The Einstein field equations with time dependent cosmological constant Λ (t) are given by p=

1 Rij − gij R = −T ji + Λ (t)gij . 2 The divergence of Einstein’s tensor implies 1 i i = −T ji + Λ gij ; j = 0 . R j − Rg j 2 ;j

(4)

(5)

With the help of (1) and (2), we obtain the field equations as k Ä R̈ Ṙ Ȧ Ṙ2 + + = −p + Λ , 2 +2 + (6) R R A R2 R2 A 2 Ṙ Ȧ k Ṙ +3 2 = ρ +Λ . (7) 3 2 +3 R RA R Conservation of the energy-momentum tensor gives us the following relation: 3Ṙ Ȧ ρ̇ + (p + ρ) + + Λ̇ = 0 , (8) R A 3Ṙ Ȧ ρ̇ + Λ̇ = −(p + ρ) + . (9) R A

To obtain the solutions of Einstein Field equations let us consider the Kaluza–Klein metric, which is given as follows:

According to published literature, exact solutions are obtained using as ansatz the power law equation A(t) = Rn (t) which is also assumed by many researchers [45, and references therein]. The ansatz power law equads2 = − dt 2 + R2 (t) (1) tion is used in view of anisotropy in the universe, de spite our assumption of isotropic and homogeneous dr2 2 2 2 2 · + r dθ + sin θ dϕ + A2 (t) dΨ 2 . universe. The expansion scalar θ is proportional to the (1 − kr2 ) shear scalar σ , which can be used for a measurement of Also, assume that h̄ = c = 8πG = 1 in accordance with the anisotropy [46]. This leads to the relation between cosmic principle. R(t) and A(t) are the fourth and fifth the metric potentials R(t) and A(t) as A(t) = Rn (t). It is seen from (6), (7), and (8) that there are three dimension scale factors and k is the curvature constant: k = 0, ±1 for flat, open, and closed model of the uni- independent equations, and four unknowns R, A, ρ, and verse, respectively. The universe is assumed to be filled Λ . Hence, to solve the field equations, we substitute with a perfect fluid represented by quark matter. The A(t) = Rn (t) as ansatz in (3), (5), (6), and solve them with (9), getting the following equation: energy-momentum tensor is given by Ti j = (p + ρ)ui u j − pgi j ,

(2)

where ui is the five velocity vector, which satisfies the relation ui u j = 1. Here p and ρ are quark pressure and

Ṙ3 kṘ R̈Ṙ 6(n + 1) 2 − 6(n + 1) 3 − 6 2 = R R R 4ρ 4 Ṙ − − BC (3 + n) . 3 3 R

(10)

N. Jain et al. · Kaluza–Klein Cosmological Model, Strange Quark Matter, and Time-Varying Lambda

The above equation can be simplified further by substituting ρ and Λ so as to arrive at a solution and derive other physical parameters. This is explained in the next section. 3. Solutions of Einstein Field Equations with Time-Varying Cosmological Constant

where C0 is the constant of integration. The solution of the above equation is arrived by s 2 p 2C0 k1 k1 2 R = − sinh 2k2 (t + c) + . (15) k2 k2 k2 r Let a =

To determine solutions of the Einstein field equa 2 tions, the assumed form of Λ is Λ = α ṘR2 + β R12 , suggested by Carvalho et al. [44]. In this expression, the first term was taken to deal with age and lowdensity problems while the second term was taken to satisfy the assumption of an isotropic universe. By substituting ρ from (7), thereafter using Λ , and simplifying (10), the following equation emerges: 2 R̈ 6n + 15n + 9 − 2α(n + 3) Ṙ2 (11) + R 9(n + 1) R2 3k(2n + 3) − 2β (n + 3) 1 2(n + 3) + − BC = 0 . 2 9(n + 1) R 9(n + 1) Following assumptions have been made in the above equation: 6n2 + 15n + 9 − 2α(n + 3) , 9(n + 1) 3k(2n + 3) − 2β (n + 3) k1 = , 9(n + 1) 2(n + 3) k2 = BC . 9(n + 1) m=

k1 k2 C0 R2 + 2m . Ṙ = − + m (m + 1) R

k2 2 C0 R + 2, 2 R

2 k1 k2

,ϕ=

√ 2k2 (t + c), and k3 =

k1 k2 ,

substituting c = −t0 as per present epoch, we can √ write ϕ = 2k2 (t − t0 ). Consequently, (14) takes the form 1

R(t) = (a sinh ϕ + k3 ) 2 .

(16)

Other physical parameters are calculated as follows: n

A(t) = (a sinh ϕ + k3 ) 2 , p H(t) = k2 a cosh ϕ (a sinh ϕ + k3 )−1 ,

a + k3 sinh ϕ

RR̈

, q(t) = − 2 = −

Ṙ a cosh2 ϕ

[α] k2 a2 cosh2 ϕ + β (a sinh ϕ + k3 ) , Λ (t) = (a sinh ϕ + k3 )2 ρ(t) =

(17) (18) (19) (20) (21) ,

[3(n + 1) − α]k2 a2 cosh2 ϕ + (3k − β )(a sinh ϕ + k3 ) 3(a sinh ϕ + k3 )2 4BC − . (22) 3 (12)

(13)

The above equation is a hyper geometric function but for analytic purpose, we assumed m = 1 to get a further simplified solution. The first integral equation of the above differential equation is given by Ṙ2 = −k1 +

−

p(t) =

After some mathematical manipulation the general solution of the above differential equation is 2

2C0 k2

[3(n + 1) − α]k2 a2 cosh2 ϕ + β (a sinh ϕ + k3 ) (a sinh ϕ + k3 )2

Equation (11) now gets simplified as R̈ Ṙ2 1 + m 2 + k1 2 − k2 = 0 . R R R

(14)

The expressions for quark pressure and quark density can be found out as per the Bag model. We know that p = pq − BC and ρ = ρq + BC , ρq (t) = [3(n + 1) − α] k2 a2 cosh2 ϕ + (3k − β )(a sinh ϕ + k3 ) (a sinh ϕ + k3 )2 − BC , (23) pq (t) = [3(n + 1) − α] k2 a2 cosh2 ϕ + (3k − β )(a sinh ϕ + k3 ) 3(a sinh ϕ + k3 )2 BC − . (24) 3

N. Jain et al. · Kaluza–Klein Cosmological Model, Strange Quark Matter, and Time-Varying Lambda

If m = 1 and k1 = 1, we obtain the expressions for α and β as as given below: α=

3k(2n + 3) − 9(n + 1) 3(n + 1) , β= . (n + 3) 2(n + 3)

Density and pressure for the different values of k are determined as follows: 3(n+1) (i) For flat universe: k = 0, β = −9(n+1) 2(n+3) , α = (n+3) , −2β (n + 3) 2(n + 3) k1 = , k2 = BC , 9(n + 1) 9(n + 1) β k1 = , as k3 = k2 BC s 2 9C0 (n + 1) β a= − 2(n + 3)BC BC s 1 9C0 BC (n + 1) −β2 , = BC 2(n + 3) s ϕ=

(25)

4(n + 3)BC (t − t0 ) , 9(n + 1)

(26)

ρ(t) =

(27)

9(n+1) 2(n+3) (a sinh ϕ + k3 )2

3(n+1)(n+2) k2 a2 cosh2 ϕ + (n+3)

(a sinh ϕ + k3 ) .

(ii) For closed universe: k = 1, so, α = −3n 2(n+3) ,

3(n+1) (n+3) ,

β =

3(2n + 3) − 2β (n + 3) 2(n + 3) , k2 = BC , 9(n + 1) 9(n + 1) ρ(t) = (28) 3(n+1)(n+2) k2 a2 cosh2 ϕ + 9(n+2) (n+3) 2(n+3) (a sinh ϕ + k3 ) . (a sinh ϕ + k3 )2 k1 =

(iii) For open universe: k = −1 then α = β = −15n−18 2(n+3) ,

3(n+1) (n+3)

ρ(t) =

and

(29)

3(n+1)(n+2) k2 a2 cosh2 ϕ + (n+3)

9n 2(n+3) (a sinh ϕ + k3 )2

(a sinh ϕ + k3 ) .

The following section focuses on the dependence on the deceleration parameter, the Hubble parameter, and on the free parameters α and β . From the above equations, it is observed that the free parameters depend upon n which is the index of the power law equation.

4. Discussion Equations (16) and (17) indicate that the fifth dimension decreases more rapidly than the fourth dimension for n < 2. It is also observed that the fifth dimension scale factor is more dominant for small t. Equation (18) suggests that as t → t0 , H(t) tends to reach a constant value. Further, (19) suggests that q = −1 if t → t0 , that the universe is accelerating. This condition is also related to eternal inflation of the universe (unending inflation due to expansion of the universe [47]), and its consequences have been pointed out and discussed in literature [48 – 56]. Eternality of inflation can be related to cosmology to have some parts of the universe to be inflated while others to exit inflation [53]. From (21) and (22), it is observed that for n = −2, ρ(t) = 0, and so p = −BC . This shows that the universe is expanding. From (20) it is evident that Λ approaches to a small positive value as t → ∞. This is in accordance with recent observed data [57, 58]. From (29) it is clear that for n = −2 the density is different for an open universe as compared to that of a flat or closed universe. Furthermore, for n = 1 in (10) is same as that obtained by Ozel et al. [13] for a flat universe with β = 0. The constant integer n here is very important, as it provides information on the nature of extra dimensions. It can be seen from (23) and (24) that quark density and quark pressure depend upon Bag’s constant. Currently, a lot of studies are going on phantom divide crossing. Phantom with ω ≤ −1 is dubbed as phantom energy [59]; ω = p/ρ is a constant in the EOS; ω = −1 is the phantom divide; ω > −1 is the quintessence era while ω < −1 is the phantom era. In present model, if ω = −1, we have ρ = BC , and it is possible to have the phantom divide. For the spatially flat universe from (27), we found that as t → t0 , ρ = BC 9(n+1) if C0 = 8(n+3)B . C Thus, it is possible to have a phantom divide crossing in the present model. It is also known from various published literature [60 – 62] that phantom dark energy models can also explain the accelerated expansion apart from Lambda decaying models. There are various models with f (R) and f (T ) (gravity models) explained by Jamil et al. [63], Jamil and Momeni [64], and Momeni and Azadi [65]. These models can explain the accelerated expansion of the universe, however, these models led to a singularity at late times. In this regard, the model with quark matter in f (R) gravity could be useful for the research interest since

N. Jain et al. · Kaluza–Klein Cosmological Model, Strange Quark Matter, and Time-Varying Lambda

quark matter behaves like the phantom type dark matter, and solutions for such models can reveal information about the inflation; and quark matter can be considered as a source of dark energy at the early universe as enlightened by Yilmaz et al. [66]. A modified gravity model at quantum chromodynamic (QCD) scale has also been discussed by Klinkhamer [67]. They claim that their model is advantageous over the Lambda cold dark matter (Λ CDM) model. 5. Conclusions In this paper, we derived exact solutions in generalized form for the Kaluza–Klein cosmological model in presence of quark matter for a flat, closed, and open universe. Our derived model is a non-singular expanding model, and it generalizes the work done

[1] T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. k1, 966 (1921). [2] O. Klein, Zeit. Phys. 37, 895 (1926). [3] J.-A. Gu, Proc. Inter. Symposium, Cos. & Part. Astrophysics (CosPA 02), Taipei, Academia Sinica, Taiwan 2003, pp. 124 – 133. [4] C. Ranjit, S. Chakraborty, and U. Debnath, Int. J. Theor. Phys. 52, 862 (2012). [5] S. Chakraborty, U. Debnath, M. Jamil, and R. Myrzakulov, Int. J. Theor. Phys. 51, 2246 (2012). [6] M. Sharif and A. Jawad, Astrophys. Space Sci. 337, 789 (2012). [7] M. Sharif and F. Khanum, Gen. Relat. Gravit. 43, 2885 (2011). [8] R. A. El-Nabulsi, Astrophys. Space Sci. 324, 71 (2009). [9] F. Grassi, CERN-TH.4682/86, 198704301 (1986). [10] M. Brilenkov, M. Eingorn, L. Jenkovszky, and A. Zhuk, J. Cosmol. Astropart. Phys. 8, 002 (2013). [11] D. N. Schramm, Proceedings of Dark Matter (Fermilab-conf.), 91 (20-A), Tennessee (1991). [12] K. S. Adhav and A. S. Nimkar, Int. J. Theor. Phys. 47, 2002 (2008). [13] C. Ozel, H. Kayhan, and G. S. Khadekar, Adv. Studies Theor. Phys. 4, 117 (2010). [14] R. Bali and A. Pradhan, Chin. Phys. Lett. 24, 585 (2007). [15] I. Yilmaz and A. Yavuz, Int. J. Mod. Phys. D 15, 477 (2006). [16] E. Witten, Phys. Rev. D 30, 272 (1984). [17] N. Itoh, Phys. Rev. 44, 291 (1970). [18] A. R. Bodmer, Phys. Rev. D 4, 1601 (1971).

by Ozel et al. [13]. The expressions for density and pressure are observed to be similar. It is also inferred that the model is accelerating at late times. The universe passes from a radiation-dominated phase to a matter-dominated phase, which is the present era, as density as well as pressure decreases exponentially. It is possible to have a phantom divide crossing in the present model, and it can explain eternal inflation. Due to the non-singularity behaviour of our model, it is advantageous over f (R) and f (T ) models. Acknowledgements Authors are thankful to IUCAA (India), IIT (Mumbai, India), and TIFR (India) for providing library facilities. Authors are also grateful to the referees for their valuable suggestions.

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[57] S. Perlmutter, S. Gabi, G. Goldhaber, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, R. Pain, C. R. Pennypacker, I. A. Small, R. S. Ellis, R. G. McMahon, B. J. Boyle, P. S. Bunclark, D. Carter, M. J. Irwin, K. Glazebrook, H. J. M. Newberg, A. V. Filippenko, T. Matheson, M. Dopita, and W. J. Couch, Astrophys. J. 483, 565 (1995). [58] A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, B. Leibundgut, M. M. Phillips, D. Reiss, B. P. Schmidt, R. A. Schommer, R. C. Smith, J. Spyromilio, C. Stubbs, N. B. Suntzeff, and J. Tonry, Astronom. J. 116, 1009 (1999). [59] R. R. Caldwell, Phys. Lett. B 545, 23 (2002) [Preprint arXiv:astro-phy /9908168]. [60] R. A. El-Nabulsi, Eur. Phys. J. P 128, 55 (2013). [61] R. A. El-Nabulsi, Can. J. Phys. 91, 623 (2013). [62] Y. Du, H. Zhang, and X. Li, Eur. Phys. J. C 71, 1660 (2011). [63] M. Jamil, D. Momeni, and R. Myrzakulov, Chin. Phys. Lett. 29, 109801 (2012). [64] M. Jamil, D. Momeni, M. Raza, and R. Myrzakulov, Eur. Phys. J. C 72, 1999 (2012). [65] D. Momeni and A. Azadi, Astrophys. Space Sci. 317, 231 (2008). [66] I. Yilmaz, H. Baysal, and C. Aktas, Gen. Relat. Gravit. 49, 1391 (2011). [67] F. R. Klinkhamer, Adv. Space Res. 49, 213 (2011).

Int J Theor Phys (2013) 52:4416–4426 DOI 10.1007/s10773-013-1760-7

Implications of Time Varying Cosmological Constant on Kaluza-Klein Cosmological Model Namrata I. Jain · S.S. Bhoga · G.S. Khadekar

Received: 29 April 2013 / Accepted: 19 July 2013 / Published online: 15 August 2013 © Springer Science+Business Media New York 2013

Abstract In this paper, the cosmological model with variable Λ = α ṘR2 + β R12 in KaluzaKlein metric have been studied. Here α and β are dimensionless parameters. The solutions to Einstein field equations which assume that the Universe is filled with perfect fluid have been obtained by using the Gamma Law Equation p = (γ − 1)ρ; in which the parameter γ is constant and power law equation A(t) = R n (t)—where A(t) is scale factor for extra dimension and R(t) is scale factor for space dimensions. The fifth dimension for the radiation dominated phases is more prominent with this model. Other physical parameters i.e. density, pressure, deceleration parameter, Hubble parameter have been determined for this model. It is observed physical parameters depends upon constants α, β and n. Neo-classical tests have also been studied in this paper. Keywords Kaluza-Klein metric · Higher dimension · Cosmological model · Cosmological constant · FRW metric · Field equations

1 Introduction Since the past two decades the Cosmological constant Λ has received a lot of attention by researchers attempting to resolve some of the most important unsolved problems of the Universe.

N.I. Jain ( ) Dept. of Physics, M.D. College, Parel Mumbai, India e-mail: nam_jain@rediffmail.com S.S. Bhoga Dept. of Physics, RTM Nagpur University, Nagpur, India e-mail: msrl.physics1@gmail.com G.S. Khadekar Dept. of Mathematics, RTM Nagpur University, Nagpur, India e-mail: gkhadekar@yahoo.com

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The Cosmological constant which represents vacuum energy density actually originates from the concept of zero-point energy in quantum mechanics. The status and consequences of the cosmological constant were reviewed by Weinberg [1] which provoked scientists to look for its physical significance. The value of the cosmological constant has undergone a crisis since a discrepancy, of about ∼10120 , between observed value [2, 3] and predicted value by Particle Physics was detected. This was first pointed out by Barrow and Shaw and also by Caroll and Press [4, 5]. This is known as the Cosmological Constant Problem(CCP). It was concluded that the Cosmological constant is not really constant but varying. Many researchers suggested that CCP might be due to distribution of matter in the Universe. Current observations about distribution of matter in the Universe explain that 4 % consists of visible matter, 22 % of dark matter and 74 % of dark energy. The existence of dark energy, dark matter and its constitution is still a mystery which can be resolved by knowing the nature of Λ as it represents vacuum energy density. The Cosmological constant problem and its consequences on cosmology are investigated by many authors in various contexts [2–11]. That the Cosmological constant varies as 1/R(t)2 was first theorized by Chen and Wu [12] in 1990. Thereafter a number of 4D Cosmological Models with time variations of the Cosmological Constant were investigated during the last two decades [13–16]. These papers mainly emphasized on cosmological constant’s dependence on scale factor which explained the dynamic behavior of the Universe but were inadequate to explain features of 2 early Universe. Four dimensional cosmological models with Λ = α ṘR2 +β R12 were discussed by Carvalho et al. and other researchers [17–19] which depicted observational constraints of some physical parameters for certain ranges. Four dimensional Cosmological constants with time varying Λ were inadequate to explain behavior of Universe at its early stages. So the theory of higher dimension evolved which was first put forth by Kaluza [20] and Klein [21] independently-unifying gravitation with electromagnetism and gauge theories in particle physics. Kaluza-Klein Cosmology was main area of interest among many researchers for solving mysteries of Universe; especially for the study of early Universe. Some aspects of Kaluza-Klein Cosmology have also been highlighted by Srivastav [22], Overduin and Wesson [23], Servant and Tait [24]. With the help of extra dimensions, early phases of the universe have been studied by Chodos and Detweiler [25]. Recently 5D field equations and its solutions are turning into an area of interest in the study of particle interactions. Five dimensional cosmological models with modification of matter have been set up by Chatterjee et al. [26], and Fukui [27], Chatterjee et al. [28] to know the impact of Kaluza-Klein theory. Several researchers [29–39] attempted the study of five dimensional models with varying G (gravitational constant) and Λ. Rami [40] had analyzed cosmic implications of extra dimensions with time varying cosmological constant. 2 The solutions of Kaluza-Klein cosmological model with Λ ∝ ȧa 2 , Λ ∝ a12 , Λ ∝ äa , Λ ∝ ρ, where a is scale factor were studied [36–38] to relate age and density parameters with observational data. 2 It was found [38] that the models with Λ ∝ ȧa 2 , Λ ∝ äa , Λ ∝ ρ are dynamically equivalent for spatially flat Universe. 2 Cosmological model with Λ = α ṘR2 + β R12 has been explained by G.P. Singh et al. [41] in higher dimension space-time for matter-dominated phase as well as radiation- dominated phase of the Universe. Most of the models inferred age, acceleration of expanding Universe

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with certain constraints but the role of extra dimension with decay lambda in its generalized form which throws light on certain features of the Universe, can be discussed. 2 In this paper, we explain Kaluza-Klein cosmological model with Λ = α ṘR2 + β R12 to generalize its study which enables us to deal with physical parameters, irrespective of matter or radiation-dominated phases. This paper is organized as follows: In Sect. 2, we have obtained general field equations for the 5D FRW metric. In Sect. 3, the exact solutions of field equations, with varying cosmological constant are obtained. The solutions thus obtained will be used to find density, pressure, Hubble parameter, deceleration parameter etc. In Sect. 4 we discuss Neo classical tests; thereafter followed by concluding remarks in Sect. 5.

2 The Field Equations Let us consider 5D FRW line element which is given as follows ds = −dt + R (t) 2

2 dr 2 2 2 2 2 2 + r dθ + sin θ dϕ + A (t)dΨ (1 − kr 2 )

(2.1)

where k is curvature parameter which is equal to 0, 1, −1 for flat, closed and open universe respectively. R(t) and A(t) are scale factors. Ψ is fifth dimension. 2 We use varying cosmological constant as follows: Λ = α ṘR2 + β R12 , α, β, are free dimensionless parameters. The universe is filled with perfect viscous fluid satisfying the equation p = (γ − 1)ρ and energy momentum tensor of perfect fluid is represented by Tji = (ρ + p)ui uj − pgji

(2.2)

where ρ is the energy density of the cosmic matter and p is its pressure and ui is the five dimensional velocity vector such that ui uj = 1. Einstein field equation is given as follows 1 Gij = Rji − Rgji + Λgji = −8πGTji 2

(2.3)

while calculating Einstein field equations, we assume that , c, and 8πG = 1, where Rji is Ricci tensor, R is Ricci scalar, gji is metric element. We also have Tji = diag(ρ, −p, −p, −p, −p), where stress–energy tensor is Tji and ρ is the normal density, p is the normal pressure. With the help of Eq. (2.1) Einstein field equations obtained with the help of Riemannian geometry Eq. (2.3) are as follows G11 = 2

Ṙ Ȧ Ṙ 2 R̈ k Ä +2 + + + = −p + Λ R R A R2 R2 A

(2.4)

G44 = 3

Ṙ 2 R̈ k + 3 2 + 3 2 = −p + Λ R R R

(2.5)

G55 = 3

k Ṙ 2 Ṙ Ȧ +3 2 =ρ +Λ +3 R2 RA R

(2.6)

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Divergence of Einstein’s tensor implies 1 i i = −Tji + Λgji ;j = 0 Rj − Rgj 2 ;j

(2.7)

From above equation we get, 3Ṙ Ȧ ρ̇ + (p + ρ) + + Λ̇ = 0 R A

(2.8)

3Ṙ Ȧ + ∴ ρ̇ + Λ̇ = −(p + ρ) R A

(2.9)

In order to get exact solutions, we use Ansatz Power Law equation A = R n . Substituting it in Eq. (2.3) we get the following equation: 3(n + 1)

k Ṙ 2 +3 2 =ρ +Λ 2 R R

(2.10)

Differentiating above equation ∴ 6(n + 1)

R̈ Ṙ Ṙ 3 k Ṙ − 6(n + 1) − 6 2 = ρ̇ + Λ̇ R2 R3 R

(2.11)

∴ 6(n + 1)

R̈ Ṙ Ṙ 3 k Ṙ Ṙ − 6(n + 1) − 6 2 = −(p + ρ)(3 + n) R2 R3 R R

(2.12)

Using equation of state i.e. p = (γ − 1)ρ in above equation, it is rewritten as ∴ 6(n + 1)

R̈ Ṙ Ṙ 3 k Ṙ Ṙ − 6(n + 1) 3 − 6 2 = −γρ(3 + n) 2 R R R R

(2.13)

Using Eq. (2.6) for ρ in the above equation we get R̈ Ṙ Ṙ 3 k Ṙ Ṙ 2 k Ṙ ∴ 6(n + 1) 2 − 6(n + 1) 3 − 6 2 = −γ 3(n + 1) 2 + 3 2 − Λ (3 + n) (2.14) R R R R R R 2

In the next section we will consider cosmological constant as Λ = α ṘR2 + β R12 , α, β, are free parameters and solve above differential equation. 2

1 Ṙ 3 Time Varying Cosmological Constant: Λ = α R 2 + β R2

We use above form of lambda as it is the general cosmological constant. First term of above lambda which varies as H 2 , has been introduced to reconcile age parameter with observational data and the term proportional to R −2 is involved as it does not conflict with the high degree of isotropy of the cosmic background radiation. Substituting Λ in Eq. (2.14), simplifying it and hence rewriting it we get, [γ (n + 3) − 2]3(n + 1) − γ α(n + 3) Ṙ 2 γβ(n + 3) − 3k[γ (n + 3) − 2] R̈ + − =0 R 6(n + 1) R2 6(n + 1)R 2 (3.1)

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Let m=

[γ (n + 3) − 2]3(n + 1) − γ α(n + 3) 6(n + 1)

(3.2)

−3k[γ (n + 3) − 2] + γβ(n + 3) 6(n + 1)

(3.3)

k1 =

thus Eq. (2.13) now is rewritten as Ṙ 2 k1 R̈ +m 2 − 2 =0 R R R

(3.4)

With the help of simple mathematical manipulation we have first integral equation as Ṙ 2 = A1 R −2m +

k1 , m

here m > 0

(3.4a)

General solution of above equation is hyper-geometric function which is quite complicated and difficult to solve further so as to obtain final expression of scale factor. For simplicity, we assume m = 1 to find an expression for scale-factor. Thus we have Ṙ 2 = A1 R −2 + k1

(3.5)

To get constants we will put initial conditions as per present epoch. To find A1 let us define the terms R0 , H0 as scale factor and Hubble factor respectively for present Universe. This is necessary as we know that R(t) and H (t) are functions of time. Putting initial conditions we obtain A1 = H02 R04 − k1 R02

(3.5a)

To get R(t), we integrate the above equation. Thus expression for scale factor is given by R2 =

−A1 + k1 (t + c)2 k1

(3.6)

where c is constant of integration. To find c, we substitute constant A1 , R = R0 at t = t0 where t0 is the present epoch c=

1 k1

A1 + k1 R02 − t0

H0 R02 H02 R04 − t0 = − t0 2 k1 k1 −A1 H0 R02 2 R2 = + k1 (t − t0 ) + k1 k1 ∴c=

(3.7)

The Fifth dimension A can be written as

n −A1 H0 R02 2 2 A=R = + k1 (t − t0 ) + k1 k1 n

(3.8)

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The Universe passes through minimum at t = t0 − k01 0 when k1 > H02 R02 . The solutions are nonsingular only for certain range of parameter but they will be singular for k1 =

R02 (2t0 H0 −1) t02

. From Eq. (3.8) we see that if n < 1 then fifth dimension becomes small

as time evolves. k = 0 i.e. for flat universe, we have −k1 = Equation (3.8) takes the form as R2 =

γβ . (n+2)

γβ −A1 (n + 2) H0 R02 (n + 2) 2 + (t − t0 ) + γβ (n + 2) γβ

(3.9)

We can find other Physical quantities as follows. To find density and pressure if k1 = 1 and m = 1, α and β are obtained as follows α=

[γ (n + 3) − 4]3(n + 1) γ (n + 3)

(3.10)

β=

3k[γ (n + 3) − 2] + 6(n + 1) γ (n + 3)

(3.11)

In this case we simplify Eq. (3.6) and expression for R(t) is given by

2 1 R = −A1 + (t − t0 ) + H0 R02 2

2 n A = R n = −A1 + (t − t0 ) + H0 R02 2 ∴ Λ(t) =

(α + β)[(t − t0 ) + H0 R02 ]2 − βA1 [{(t − t0 ) + H0 R02 }2 − A1 ]2

(3.12) (3.13) (3.14)

As per the field equation, we get expression for density as ρ(t) =

[(3(n + 1) − α) + (3k − β)][(t − t0 ) + H0 R02 ]2 − (3k − β)A1 [{(t − t0 ) + H0 R02 }2 − A1 ]2

p = (γ − 1)ρ

(3.15) (3.16)

Volume of the Universe can be calculated as V = R 3 A

2 3+n 2 V = −A1 + (t − t0 ) + H0 R02

(3.17)

Hubble parameter and deceleration parameter are obtained as follows H (t) =

(t − t0 ) + H0 R02 Ṙ = R [{(t − t0 ) + H0 R02 }2 − A1 ]2

(3.18)

From here, we also see that H (t) ∝ t −1 R R̈ A1 q(t) = − 2 = 2 2 2 k1 [(t − t0 ) + H0 R0 ] Ṙ If k1 is equal to one then q(t) → t −2 . This shows that acceleration is decreasing.

(3.19) (3.20)

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Fig. 1 Graph of density vs. cosmic time

Fig. 2 Graph of q(t) vs. cosmic time

The graph of density and deceleration parameter are also shown below in Fig. 1 and in Fig. 2 for arbitrary values of α, β and n i.e. α = 0.5, β = −4, n = 0.5, A1 = 0.5, t0 = H0 R02 , k1 = 1. Earlier the acceleration was high and it reduces. This shows the Universe’s transition from inflationary phase to steady state phase. From above graph it is concluded that density is quite high during the early stages of the Universe. It is also seen that for all types of the Universe density decreases as t → ∞. Age of the Universe can be calculated by assuming q = q0 and H = H0 , so that from Eqs. (3.4) and (3.4a) we have dR (3.21) t = (H0 R0 )−1

1 − qm0 + qm0 ( RR0 )2m

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Above equation reduces to a simpler form, much similar to that discussed by Carvalho et al. [17]. 1 m+1 , assuming constant of integration c = 0. If q0 = m then, H0 t = m+1 R (m+1)R0

This shows that the age of the Universe is affected by the values of n and α, although it seems to be independent of β values; but under certain conditions age parameter is free from β. In the following section some of physical parameters under Neo-classical tests have been determined which help us to deal with the horizon problem.

4 Neo-classical Tests 4.1 Proper Distance D(z) It is necessary to establish causality connection between observer & source at any time. Thus let an observer at r at time t receiving signal from source which is at distance r = r1 at t = t1 R then the proper distance between the source and observer is given by d(z) = R0 R 0 RdRṘ . First integral solution for k1 = 0 and if m = 1 is given by Ṙ 2 = A1 R −2m d(z) =

d(z) =

1 √ R0 [R0m − R m ] m A1

1 √ R0m+1 1 − (1 + z)−m m A1

where R0 1+z= = R For m = 1, d(z) =

√1 R 2 [1 − (1 + z)−1 ]. A1 0

m+1 t0 t

(4.1)

(4.2)

This is independent of γ .

4.2 Luminosity Distance In theoretical Cosmology Luminosity Distance provides distribution of light to an observer who is at a distance from the source; obeying inverse square law for its intensity. In other words, it is defined as amount of light received from a distant object. If dL is the luminosity distance to the object, then dL =

L 4πl

where L is the total energy emitted by the source per unit time, ‘l’ is the apparent luminosity of the object. Therefore we arrive at dL = (1 + z)d(z)

(4.3)

using equation (4.1) we can calculate Luminosity distance. We see that d(z) depends upon red shift. This is also an increasing function of ‘z’.

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4.3 Angular Diameter Distance It is a measure how large an object appears to be. The angular diameter dA of a light source at proper distance d is given by dA = d(z)(1 + z)−1 = dL (1 + z)−2

(4.4)

From Eq. (4.1) we get the expression for angular diameter as dA = d(z)(1 + z)−1 =

1 √

m A1

R0m+1 (1 + z)−1 − (1 + z)−m−1

(4.5)

We can also see that dA is decreasing function of z. It is also seen that Luminosity Distance and angular diameter have different dependence on red shift. While Luminosity distance is directly proportional to z; angular diameter is a decreasing function of red shift, which shows that objects appear smaller as distance of object increases. This is consistent with the experimental result. 4.4 Look Back Time The time in the past when the light we now receive from a distant object was emitted is called the Look Back Time. How long ago the light was emitted (the look back time) depends on the dynamics of the universe. The radiation travel time (or look back time) (t − t0 ) for photon emitted by a source at instant t and received at t0 is given by t − t0 =

m+1

1 dR R m+1 =√ R0 − R m+1 = √ 0 1 − z−(m+1) A1 (m + 1) A1 (m + 1) Ṙ

(4.6)

Using binomial expansion, we find red shift can be a measure of look back time. For z < 1, Luminosity distance and angular diameter are proportional to z. These results are similar to that obtained by Carmeli [42] in his research work.

5 Discussion It is observed that the model is quite different from 4D models in principle. We obtain significant difference in the solutions of the present model from the models with variation of Λ as H 2 , R̈R etc. Since m and k1 depends upon γ , hence physical parameters—density and pressure will be different for matter as well as radiation-dominated phase. Present Cosmological model is nonsingular model. From Eq. (3.15), it is observed that a five-dimensional model reduces to a four dimensional model as t → ∞ and from equation (3.14), it is found that Λ ∝ t −2 . The model with Λ ∝ t −2 has been accepted by many authors. It is also found that for matter-dominated phase γ = 1, α = 3(n − 1)(n + 1)/(n + 3), β = 3(k + 2)(n + 1)/(n + 3) and to have positive α, β, n = 1, −1 or −3. For radiation-dominated phase we have γ = 4/3, so α = 3n(n + 1)/(n + 3) and β = 3[k(2n + 3) − (n + 1)]/2(n + 3). In both cases we find that n = −3. It has been also observed that the model becomes steady state model as time increases. One can also conclude that all physical parameters depend on n which is the index factor of R(t).

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For n = 0.5 behavior of density for flat or non-flat Universe is similar. In Eq. (3.13) q reaches constant value. This result may be able to explain the slowing down of cosmic acceleration as explained by Varun Sahani [43]. From Eqs. (3.9) and (3.11) and from graphs we observe that normal density decreases more rapidly than effective density for flat as well as non-flat models. Since effective density is equal to normal matter density and vacuum density therefore if free parameters α and β > 3(n + 1) then we get early Universe situation. We also found Volume and age parameter which depend upon H0 and R0 . Under Neo-classical tests physical parameters like proper distance, Luminosity distance, angular diameter distance and look-back time have also been calculated. Acknowledgements I highly appreciate the help rendered by Dr. Farookh Rahman by providing valuable comments and guiding me from time to time. The facilities provided by IUCAA library were wonderful. I also thank IIT and TIFR for permitting me use of their library facilities. I also acknowledge the funding provided by UGC under Minor Research Project.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

Weinberg, S.: Gravitation and Cosmology. Wiley, New York (1972) Perlmutter, et al.: Astrophys. J. 517, 565 (1999) Reiss, et al.: Astron. J. 116, 1009 (1998) Barrow, J., Shaw, D.J.: gr-qc/1105.3106 (2011) Caroll, Press: Annu. Rev. Astron. Astrophys. 30, 499 (1992) Ratra, B., Peebles, P.J.E.: Phys. Rev. D 37, 3406 (1988) Dolgov, A.D.: Phys. Rev. D 55, 5881 (1997) Padmanabhan, T.: Phys. Rep. 380, 235 (1990). hep-th/0212290 Ponce de Leon: Class. Quantum Gravity 27, 095002 (2010) Romano, Chen: Int. J. Mod. Phys. B 20, 2823 (2011) Nakamura, K., et al.: JPG 37, 075021 (2010) Chen, Wu: Phys. Rev. D 41, 695 (1990) Sahni, V.: Pramana 55(4), 559 (2000) Sahni, V., Shafiero, Starobinsky, A.A.: Phys. Rev. D 78, 103502 (2008) Pavon, D.: Phys. Rev. D 43, 375 (1991) Overduin, Cooperstock: astro-ph/9805260v1 (1998) Carvalho, S.C., Lima, J.A.S., Waga, I.: Phys. Rev. D 46, 2404 (1992) Lima, J.A.S., Trodden, M.: Phys. Rev. D 53, 4280 (1996) Carneiro, S., Lima, J.A.: Int. J. Mod. Phys. A 20, 2465 (2005) Kaluza, T.: Sitz. Preuss. Akad. Wiss. Phys. Math. K 1, 966 (1921) Klein, O.: Z. Phys. 37, 875 (1996) Overduin, J., Wesson, P.S.: gr-qc/9805018v1 (1998) Srivastav, S.K.: Pramana 49, 323 (1997) Servant, G., Tait, T.: hep-ph/0206071v2 (2003) Chodos, A., Detweiler, S.: Phys. Rev. D 21, 2167 (1980) Chatterjee, S.: Astron. Astrophys. 179, 1 (1987) Fukui, T.: Gen. Relativ. Gravit. 19, 43 (1987) Chatterjee, S., Banerjee, A., Bhui, B.: Phys. Lett. A 149, 91 (1990) Pande, H.C., et al.: Indian J. Pure Appl. Math. 31, 161 (2000) Bhowmik, et al.: Indian J. Pure Appl. Math. 34, 1189 (2003) Pradhan, A., Chauhan, Q.S.: Astrophys. Space Sci. (2010). doi:10.1007/s10509-10-0478-8 Carneiro, S.: gr-qc/0307114v1 (2003) Ma, Y.-Z.: astro-ph/0708.3606v6 (2008) Singh, C.P.: Astrophys. Space Sci. 331, 337 (2011) Arbab, A.I.: astro-ph/0212565v1 (2002) Khadekar, G.S., Kondwar, G.L., Kambdi, V., Ozel, C.: Int. J. Mod. Phys. B 47, 2444 (2008) Khadekar, G.S., Patki, V.: Int. J. Theor. Phys. 47, 1751 (2008)

4426 38. 39. 40. 41. 42. 43.

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Khadekar, G.S., Kambdi, V.: Int. J. Theor. Phys. 48, 3147 (2009) Singh, G.P., et al.: Rom. J. Phys. 53, 607 (2008) Rami, A.: Braz. J. Theor. Phys. 39, 574 (2009) Singh, G.P., et al.: Fizika B 15, 23 (2006) Carmeli, M.: astro-ph/0111071v2 (2001) Sahni, V., Shafiero, Starobinsky, A.A.: Phys. Rev. D 78, 103502 (2008)

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Kaluza Klein Cosmological Model with String, SQM and Time Varying  1

Namrata Jain,2S.S. Bhoga,3G.S. Khadekar

Department of Physics, M. D. College , Parel , Mumbai-400 012 2 Department of Physics, RTM Nagpur University, Nagpur-4400033 Department of Mathematics, RTM Nagpur University, Nagpur-4400033

ABSTRACT In this paper, exact solutions of Einstein field equations of five dimensional Kaluza-Klein cosmological model with strange quark matter and string cloud have been obtained by using  = αH2 +βR-2, where R(t) is scale factor and H is Hubble parameter. To get solutions, Quark matter is assumed to behave like perfect fluid and its equation of state is given by p=1/3(-4BC). In order to obtain exact solutions, extra dimension is assumed to be equal to nth power of R(t) where n < 1. This assumption is followed because the Universe is anisotropic at early stages, although as per cosmic principle, it is believed to be homogeneous and isotropic. The exact solutions thus obtained led to anisotropic and non singular model. Keywords:String cosmology. cosmological model . cosmological constant . strange quark matter. Kaluza Klein metric.

1. INTRODUCTION Recently, lot of attention has been gained by string cosmology, which is the key factor for getting information about early Universe. Since Big-Bang, early universe phenomenology is major curiosity among many cosmologists. The idea of string theory came up to describe events of early stage of evolution of the Universe. At the very early stages of evolution of the Universe, it is assumed that when the Universe passed through phase transition at critical temperature, Symmetry of the Universe is broken spontaneously. This gave rise to topological defects, such as Magnetic monopoles, Strings and Domain walls. The strings are actually the consequences of Quantum field theory, which were used to explain hadron scattering through Feynman diagrams. The concept of strings in Feynman diagrams was first explained by Gabriele Veneziano [1] in 1968. Cosmic strings are topologically stable defects which might be found during a phase transition of the early Universe (Kibble [2]). It is also believed that strings are density perturbations which lead to the formation of galaxies (Zel’dovich [3]). These strings are also useful in the study of gravitational effects as they have stress energy and can be coupled to gravitational effects (Letelier [4]). The gravitational effects of the strings can be studied, if strings are supposed to be consisted of string cloud with particles. Stachel [5] had also explained string dust model where null strings were taken to be massive. Vilenkin [6] had studied the gravitational effects of domain walls with strings. Strings are also considered important for discussing quantum gravity and useful in obtaining information about graviton. Gasperini [7] reviewed the string cosmology for pre Big-Bang scenario and concluded

that the strings can alone be useful to explain early universe phenomena.

In this paper, we investigate higher dimensional string cosmological model in presence of quark matter and time varying cosmological constant in Kaluza- Klein metric. After the Big-Bang, Universe had undergone another phase transition i.e. Quark phase to hadron phase when TC ~200MeV. This state is also called as QuarkGluon -Plasma (QGP) state. This phase transition which plays an important role at early Universe had been discussed by Witten [8], Fahri & Jaffe [9]. They also pointed out the role of Quark matter at early Universe. Gerlach [10], Ivaneckaet. al [11], Bodmer [12] , Itoh [13] had discussed the importance of Quarks during phase transition. The concept of Quark matter had emerged from the Quantum Chromo-dynamics (QCD). It is also thought that there is possibility of so called Quark star or compact star which is smaller than neutron star and which is supported by degenerated quark pressure. It is plausible to attach Quark matter to strings as strings are free to vibrate in different modes and different modes represent different particles, and also different modes of vibrations are observed as different masses or spins. Charged strange quark matter, attached to string cloud in cylindrical space time admitting conformal motion, had been studied by Mak & Harko [14] . Sanjay Oli [15] , Pradhan A. et.al [16], Bali & Pradhan [17] , Mahanta et .al [18], Singh & N. K. Sharma [19], Banerjee et .al [20], Yavuz I. and Tarhan I.[21] had discussed the string cosmological model attached to quark matter in Bianchi space time. The papers by S. K. Tripathi [22], D.R. K. Reddy [23], M.Glovanini [24], Kanti Jotania [25] , Bijan

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Saha et. al [26] , G. P. Singh & T. Singh [27] had studied Bianchi type string cosmological model in presence of either electro-magnetic or magnetic field in different context. The papers by D.D. Pawar [28], Bali Raj [29] had obtained the solutions of Einstein field equationsfor the string cosmological model with viscous fluid and bulkviscous fluid respectively. While Xing-Xang Wang [30] had discussed BI string with Bulk Viscosity & magnetic field.

problem with observational data, lambda had been further generalized by Carvalho & Lima [46] as

Since concept of higher dimension has originated from string cosmology so it plays an important role in the study of early Universe. Higher dimensions also gained attention for unifying gravitation and particle interaction, electromagnetism, gauge theories etc. which were first put forth by Kaluza [31] and Klein [32] independently.

Motivated by above discussion, in this paper, exact solutions of Einstein field equations of Kaluza Klein Cosmological model with string in presence of quark matter and time varying Cosmological Constant have been obtained.

There are many reporting which studied higher dimension for exploring the idea of Brane Cosmology, scalar-tensor field theory, parameterization of mass etc. Paul Wesson [33] has enlightened Kaluza-Klein theory so as to develop the new idea of Space-Time - Matter theory and hence applied to many phenomenon i.e. particle interactions, Gravitation electromagnetism etc. Study of Higher dimension cosmological model with the string cloud and strange quark matter will be fruitful for bringing information at early universe stage. Cosmological constant ‘’ has played the key role in the study of Universe as it represents vacuum energy. Observational data of SN Ia by Perlmutter [34] and Reiss [35] in 1998 when combined with CMB measurements implied that Universe is accelerating, which indicate towards finite and small positive value of Cosmological Constant. However, this value differs from the value which was predicted by Standard model of Particle Physics. This problem is now known as Cosmological Constant Problem (CCP). Chen and Wu [36] suggested that time varying cosmological constant can solve this problem by assuming ansatz R-2 where R(t) is scale factor. There are number of papers which studied cosmological models with time varying cosmological constant in different context. A fine review had been done by Overduin and Cooperstock [37], Varun Sahani [38] and T. Padmanabhan [39], Arbab I. Arbab[40]. Recently it has been suggested in many papers that the value of Cosmological Constant  may be large at the time of evolution of Universe, when strings are also assumed to be dominating. There are papers by Pradhan et. al [41] , R.K. Tiwari [42], Abbasi & Rajmi [43], Anil Yadav [44] which had discussed string cosmological model with time varying cosmological constant . Ray, Rahman & Mukhopadhay [45] had explained scenarios of Cosmic strings with variable Cosmological constant by assuming =3r-2 ,=8πρ, ρs =3r-2 here r has its usual meaning. Later on, to reconcile age parameter and to deal with low density

 = αH2 +βR-2 . Yilmitz [47], Katore [48] had explained string cosmological model in presence of quark matter in kaluzaklein metric without cosmological constant.

The paper is organized as follows. In section II, Einstein field equations of string cosmological model in Kaluza Klein metric are obtained. In Section III we derive solutions for string cosmological model in presence of quark matter and time varying cosmological constant. Expressions for densities, pressure and other physical parameters are also obtained with the help of solutions. With the help of expressions of physical parameters, it is possible to draw some conclusions on the cosmological model and its behavior which seems to describe physical situations at early stage.

2. EINSTEIN FIELD EQUATIONS The line element for five dimensional KaluzaKlein model is given by

   dr 2 2 2 2 ds =-dt +R  t    +r 2 dθ2 +sin 2θdφ2 2   1-kr 









  2 2  +A  t  dΨ   (1)

Where k is curvature parameter, which is equal to 0,1,-1 for flat, closed and open universe respectively. R(t) and A(t) are scale factors. Ψ is fifth dimension. As per Cosmic principle, ћ= c= 8G= 1 is assumed. i i i i Tj =  ρ + p  U U j -p g j + λ x x j

(2)

Where p is isotropic pressure ;  is the proper energy density for a cloud of strings with particles attached to them ;  is the string tension density; Ui = (0,0,0,0,1) is five velocity and time like vector ; xi is a unit space like vector such that xixi = 1 in the direction of

xi   4i

which

represents the directions of cloud i.e. directions of anisotropy. We also consider UiUi = -1 and UiUj = -1 , Uixj = 0.

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Proper energy density for strings attached to particles can also be written as =p +. As per Bag model, quark density p = q + BC and quark pressure pq = ρq/3, pp = pq –Bc, where Bc is the Bag constant. As per Bag model, it is the difference between the energy density of the perturbative and non- perturbative QCD vacuum. Thus, proper density is given by =q + BC + .

per Bag model, we consider Equation Of State (EOS) for quark matter, which is given by,

To obtain Einstein field equations, consider the following equation

To obtain the scale factors R (t) and A (t), we solve field equations as follows. From equations (4) and (5), we have following relations

i i 1 i i G j= R j- R g j+ Λ g j 2

1 3

(3) 2

 ρ-4Bc 

Where

i j

R is Ricci tensor, R is Ricci scalar, g

i Tji = diag (ρ,-p,-p,-p, -p+), T j is the stress –energy tensor. From equation (1) Einstein field equations are obtained as follows

(4)

(5)

(9)

λ=

2 2R 2k 2RA A + 2 2 RA A R R

(10)

We will also derive an expression for physical variables, expansion factor and shear scalar in five dimensional metric , which are defined as θ=3

(11)

2 3 R A σ =  -  8R A

2 (12)

(6) Using equation of state for quark matter , we get ,

Conservation of energy momentum tensor is given by



2 A R A R k -2 -2 =-( p + ρ ) 2 2 A R A R R

i j

i i i is metric element .We know that G j = Tj + Λg j , where

2 RA R k A 1 R G1 =2 +2 + + + =-p+Λ 2 2 R RA R A R 2 R k 4 R G 4 =3 +3 +3 =-p+λ+Λ 2 2 R R R 2 R R A k 5 G5 =3 +3 +3 = ρ+Λ 2 2 R A R R

(8)

  

i i i T j  Λg j ; j  T j ; 0 eff j

2 A R A R k 4 ρ 4 Bc -2 -2 =+ 2 2 A R A R 3 3 R

(13)

Substituting equation (6) in above equation and simplifying it, we obtain following equation as,

From here, we obtain,  A  3R A   ρ  (p  ρ)  Λλ  0 A  R A

(7)

Above equation can be obtained from field equation also. In above equation, we have three equations and R, A, p,  , ,  are unknowns. To solve above equations explicitly, we need three more equations. We assume generalized  as a function of time, i.e. 1 2 2 R R Since Quarks are considered to be mass less particle and quark fluid is supposed to be perfect fluid as Λ=α

+β

k 4 Λ 4 Bc (14) = 2 2 3 R A R A 3 R R In order to solve above equation, we assume ansatz A=Rn. This power law equation is considered due to the fact that there is still anisotropy for the flat and homogeneous Universe and ij (shear tensor), so we use polynomial relation between metric coefficients. Thus Equation (14) is simplified as follows, 2

( n+2)

R A

2 R k 4 Λ 4 Bc 2 +( n +2n+2) +2 = 2 2 3 R 3 R R

(15)

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(n+2)

(n +2n+2) R

4Bc k 4Λ = 2 (n+2) R 3(n+2) 3(n+2) (16) 2

In next section we will take time variable cosmological constant which was generalized, first put forth by Carvalho and Lima [46]. We will derive the solution of above field equation and determine physical parameters i.e. density, pressure, string tension density, quark density, quark pressure etc.

2C0  k1 R = - k 2  k 2 2

 k  Sinh 2k 2 (t+c)- 1 k2 

2C0  k1 a= - k 2  k 2

(21) 2

 3(n+2)C0  BC  - Let ,  =  2β  β   k1 φ= 2k 2 (t+c) and k 3  , Now equation (21) is k2 rewritten as follows

3. SOLUTIONS OF EINSTEIN FIELD R2 1 EQUATIONS FOR Λ = α 2 + β 2 R R To find the exact solution of field equation for flat 2 R 1 Universe, let k=0. Substituting Λ=α & k=0 in +β 2 2 R R the equation (16), We get,

(3n +6n+6-4 α ) R

3(n+2)

2 2

(-4β )

4BC 1 = 3(n+2) R 2 3(n+2) (17)

2 R = a S i n h φ - k3

(22)

1 R(t)= aSinhφ-k3  2

(23)

Knowing R(t) we calculate other physical parameters as follows

n A(t)= aSinh -k3 2





(24)

3 ( n + 1 ) -α k 2a 2C o s h 2φ + ( -2 β ) ( a S i n h φ -k 3 )

ρ(t)=

2 ( a S i n h φ -k 3 )

In above equation we assume that,

2 (25)

m

2 3n  6n  6  4α 3(n  2)

p(t)= (ρ-4Bc ) 3

4β , , k1 = 3(n+2)

4BC , 3(n  2) Equation (17) now will be simplified as

and k 2 

R R

2 2

-k1

1 -k = 0 2 2 R

Λ(t)=

(19)

For simplicity let us assume m=1, the solution of above equation has been obtained as k 2 2 C R =k1 + 2 R + 0 (20) 2 2 R Where C0 is constant of integration. Hence scale factor R(t) for Kaluza – Klein metric is obtained as follows.



aCosh aSinh -k3



-1

(27)

α  k 2a 2C o s h 2φ + 2 β ( a S i n h φ - k 3 )

(18)

First order integral solution of above equation is given by k2 2 k 2 C R = 1+ R + 0 2m m (m+1) R

H(t)=

(26)

q(t)=-

2 ( a S i n h φ - k3 )

a + k 3S i n h φ RR =2 2 R aCosh φ

(28)

(29)

To find string tension, density consider equation (9), since k=0 equation (8) is rewritten as 2 2R 2RA A 2 R R RA A on substitution of A(t) = Rn(t) , we get R R2   λ = (1 -n )  + (2 + n ) 2 R R   λ=

(30)

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 k ( n + 1 ) a 2C o s h 2 φ +   2   2 ak S i n h φ ( aS i n h φ -k )  2 3  λ = ( 1 -n )    2 ( aS i n h φ -k3 )2      

(31)

From above equation n 1.To find quark density we consider proper density as =q + BC + . - =q + BC.  k (n 2 +3n+2-α )a 2 C osh 2φ  2   [2 β +(1-n)2ak S i nhφ](aS i nhφ-k )  2 3  -B ρq =  c   2(aS i nh φ-k 3 )2      

(32)

 k ( n 2 + 3 n + 2 - α ) a 2 C o s h 2φ   2  ρq  [ 2 β + ( 1 -n ) 2 a k S i n h φ ] ( a S i n h φ -k )  B 2 3 c pq = = - 3 2 3  6 ( a S i n h φ -k 3 )       (33) Expression for expansion factor is found out with the help of equation (11) as follows, R R

= ( 3 + n ) a 2 k 2 C o s h φ ( a S i n h φ -k 3 )

-1 (34)

Expression for shear scalar can be found using equation (12) , 2 3 2R σ = ( 1 -n )   8 R

 σ =

3 16

2 2 2 -2 ( 1 -n ) a k 2C o s h φ ( a S i n h φ -k 3 )

It is also observed from Eq. (31) that string density ‘λ’ decreases with time and it is always positive. From equations (34) & (35), we find that σ l i m  c o n s t a n t and, hence it leads to anisotropic t  θ model.

5. CONCLUSION

Quark pressure can be obtained as follows.

θ=(3+n)

From Eq. (29), at t→ 0, q(t)= -1. This implies that the model is de-Sitter in the early stage of the evolution of the Universe. We also observe that when t→ ∞, q(t)= 0. This implies that the model reaches to steady state at late time, so Universe expansion rate falls down if time increases.

In this paper, we derived exact solutions for the higher dimensional Cosmological model with strange quark matter and a variable cosmological term . The model is also non singular and represents inflationary phase, if k1 is assumed to be zero. It is the generalized model in which dimensionless parameter α & β defines the behavior of the model. For real solution we must have,

2 2 3C0 (n+2)  β  β > and n-2 .  i.e. C0 > Bc 3(n+2)BC  Bc  With =0, our work generalizes the recent work done by Ozel et.el [49] in absence of string cloud . The model is expanding and it disappears at late time. Our derived model also generalizes the work of Yilmitz [47] in absence of any pressure and density.

ACKNOWLEDGMENT We highly appreciate the facilities given by IUCAA library. Also, thanks to TIFR library and Haffkin’s Institute for the facilities they provide. My sincere gratitude to Dr. Anirudh Pradhan for providing me his valuable suggestions during this work.

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4. DISCUSSION From equation (25), we observe that the total energy density decreases with time and it is always positive. From equation (26), we observe that in absence of BC, our model reaches to the radiation dominated phase. From Eq. (28), we observe that the cosmological constant decreases with time and approaches to a small positive value at late time, which is in good agreement with the recent astrophysical observations ( Perlmutter [34], Reiss [35]) .

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