NATURE OF CONFORMALLY COUPLED SCALAR FIELD IN COSMOLOGICAL MODELS by IASET Journals

InternationalJournalofAppliedMathematics&

StatisticalSciences(IJAMSS)

ISSN(P):2319–3972;ISSN(E):2319–3980

NATUREOFCONFORMALLYCOUPLEDSCALARFIELDINCOSMOLOGICAL MODELS

MohammadShyfurRahmanChowdhury

DepartmentofBusinessAdministration,InternationalIslamicUniversityChittagong

ABSTRACT

Thecosmosappearstobeexpanding,accordingtorecentmeasurementsofsupernovaeandthecosmicmicrowave background.Gravitationalscalartensortheoriesproducecosmicmodelsthatlogicallyincorporatealate-timefast expansion.Inadifferenthypothesis,thestretchoftheuniverseisproducedusingascalarfield(quintessence),muchlike theearlyinflation.Inthisstudy,wefocusonaparticularcategoryofscalarfieldcosmologicalmodels.Ithasaquartic potentialandaconformallyconnectedscalarfield.Thecosmicdynamicsaredetailed,andthebrakingscaleisassessedin themodel.Forthevaluesofthescale,thataresupplied,theexpansionacceleratesinthelatestages.

KEYWORDS:CosmologicalModel,Deceleration,ScalarFieldParameter

ArticleHistory

Received:24Sep2022|Revised:07Oct2022|Accepted:12Oct2022

INTRODUCTION

Thelargelyacceptedtheoryofthenatureoftheuniverse'sstructureandevolutionhasbeenre-examinedinlightofrecent evidenceofthecosmos'acceleratedexpansion(Copelandetal.,2006;Perlmutteretal.,1999;Garnavichetal.,1998).The cosmologicalconstantisthemostlikelychoicetoexplaintheaccelerationofspace-time(Weinberg,1989).Considerthe negativepressurethatoccursfromanon-zerovacuumenergyasacounterparttothis(Linde,1982;Starobinski,1982; Guth,1981).Theideathattheuniverseiscomposedofauniquematerialtermedquintessence(SteinhardtandCaldwell, 1998)offersanalternateexplanationfortheobservedacceleration.Intriguingly,thepresenceofquintessencehasan impactonLogunovandcolleagues'relativistictheoryofgravitation(RTG):ratherthanexpandingmorequickly,the universeslowsdown,stops,andthencontractstoascalefactorminimumbeforebeginninganewcycleofexpansion (Chernin,2008;Gershteinetal.,2003).Itshouldbeemphasizedthatbefore,mostlyforphilosophicalreasons,the oscillatingnatureoftheuniverse'sevolutionwasinitiallyposited(MarkovandAman,1984).Thetransitthroughthe cosmicsingularityandtheriseinentropyfromcycletocyclecausetheoscillationmodeoftheFriedmannclosedmodelto bebrokenup(Tolmen,1949).

Inthisstudy,weexamineaclassofscalarfieldcosmologicalmodels.Thepaperisstructuredasfollows.The conformallylinkedscalarfieldcosmologymodelwiththebarotropicequationofstate,non-gravitationalmatter,andthe Higgspotentialisdiscussedinthenextsection.Theuniverse'sacceleratedexpansionmodelsarelisted.Thekeyfindingsof thepaperareoutlinedinSection3.

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Vol.11,Issue2,Jul–Dec2022;51–56 ©IASET

Mohammad Shyfur Rahman Chowdhury

CosmologicalMatterwithaScalar Field(Conformally Coupled)

The quartic potential and the conformally connected scalar field interact (Stanukovich and Melnikov, 1983; Melnikov, 2011;BronshteinandSemendjaev,1980)asshownbelow:

The followingfieldequationsareobtainedas:

Inthisinstance,theEq.(2.2)iscondensedbytaking(2.3)intoconsideration

The homogeneous and isotropic universe model is often thought to work well with geometry using the Friedmann-Robertson-Walker metric. The aforementioned equations consequently hold true for the flat model of matter havingthe equationofstateP= 

HerethescalefactorisR(t).One ofthe system'sequations,(2.6)-(2.10),isderivedfromthe others. Thefollowingexpressionforthedecelerationparameterisobtainedfrom(2.8)

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W= x gd L 12 ) 2( 1 R 12 2k 1 4 m 4 2 2                        (2.1)

, T 6 k 1 k R 2g 1 R mat 1 2 v v                 (2.2) 0, 3 6 R 3 2        (2.3) where v = 3 2 v 6 1 gv ()2 3  v + 3  gv2  + 12   4 gv, , Pg U P)U ( T v v mat v       (2.4)

R= kT= k(

(2.5)

3P)

, R R 2 /6) k 3(1 k R R 2 2 2 2               (2.6) , 3P R R 2 2 /6 k 3(1 k R R R 2R 2 2 2 2                         (2.7) 3P), 6( k R R R R 2 2    (2.8) 0 3 R R R R R R 3 3 2 2                     (2.9) ) 3(1 0 / ), ( 3          R P R R   (2.10)

Here the Hubble parameter isH = R/R , because it followsfrom the requirement q 0 thatthere isa chance that the cosmosisexpandinguniformlyorquicklyis

(Let us consider that the framework of General Relativity one has crit = 3H2/(8G)) The problem q is phrased as followsforthevariation of , afeature of cosmology:

1 2

, 6H

matter dominated( =0),

, 3H 2k 1 2  vacuummodel( = 1),

dominated( =1/3),

stifffluid( =1).

q=1,scalar fieldconquered( =0,P=0),

The Hubbleparameter can be statedusing(2.8)inthe followingway:

 isanintegrationconstant.

Let'sexamineeachuniqueinstanceinturn:

(i) R = b, 2ct  H = c/R2results from the scalar curvature going to zero for the scalar field rules (, P = 0) or whenitcomestothe equationof stateP= /3

/(cR)with

i.e.for adequatelylargeRwe have  6/k

(ii) The systematic structure for the functions R,, H be unchanged, if the radiation is dominating (= 1/3) while the scalarfieldispresent,andequation(2.14)represents

so that

In both scenarios (i) and (ii), the universe's growth is continuously slowed down, andasit gets older,scalar fieldsoftenhaveconstant values.

(iii)

,whichmeansthatwegetthe Hubbleparameterforthe matter-dominantera,

and,correspondingly,

Nature of Conformally Coupled Scalar Field in Cosmological Models 53 q= 3P), ( 6H k 1 2  (2.11)

3P G 8 3H 2 2     (2.12)

q=1,

(2.13) q=

k 1



radiation

, 3H



H2 = , R 3R k 4 ) 3(1 0    

2 c1

  6/k, C 0 k/2 C 3 c 2 2 2 2 2     = /cR, c 6/k 1 (2.14)

 =c

  0 2 2 2 k k/2 c 3 c   , (2.15)

2 6/k 2 0

= /c

0,  = 0

4 3 0 4 2 R 3R k R 3 k H         (2.16)

//R

Mohammad Shyfur Rahman Chowdhury

Thefirstcomponent on the RHS of (2.16),whichdominates nearthe end of the cosmicexpansion (R),resultsinthe standardmatter-dominateddevelopmentandadeceleratedexpansion.Onehasq ½inthislimit(asinGR).

For >0,wehave the followingsolution

ItdirectlyintegratesEq.(2.16).Thesolutionis giveninthe formof

is used. At the beginning and ending of the cosmic growth, we have, respectively,R(t)

.The expansion alwaysslowsdowninthiscircumstance. For

Currently, the early-time dynamics are completely different, while the late-time evolution matches that of the earlier scenario. An asymptotic behaviour of R

and

min = 3|

| / (k

0) are the initial conditions for the expansion. Early on, the expansion is accelerated and the inequality 1  v  2replaces the prerequisite q  0.

Itiseasytofindthescalarfield'stemporalevolutioninthecasewhere  =0

withthe representation U =R.

For thescalarfield(2.16)fromhere,wefind

Where the scale factor  and v are related by the formulas (2.18 and (2.19) At the end of the expansion, one has  0 in both occurrences of the sign for one. For non-zero values of the parameter, the dynamics of the field at the beginning of the expansion are fundamentally different: for positive values, we obtain,   , 1  0, whereas the field tendstozerofornegativevalues,  0,1  0.

(iv)Wehave asetofequationsforthe equationforthe condition P = 

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R= 2, 1 v 2) (v v,t/t k 3 2 0      0 , (2.18) in

the

t2 = 6||3/2/(k0)2

 t1/2att  0,

R(t)  t2/3 att 

<0, R= 11 v 2) (v v,t/t k 3 2 0     ,0. (2.19)

which

notation

and

R

+ (k0)3 t2 / (42), t 

min





0, UR/R U   (2.20)

 =const                 0 for 1/r,1 0 for 1/r,0 v v v v (2.21)

3 2 R R R R 2 2    k (2.22) 0 3 R 3 3 0 4 0           2k k (2.23) Substituting v = R2 equation(2.23) becomes i2 4k 3 v 2 =  4 (2.24) Itbecomes R(t)= R0sinh1/2 v, v =2t 3/ k (2.26)

For >0

R(t)= R0cosh1/2 v(2.26)

For <0,where 2 0 4R = /( 3   k ,equation(2.23) isreducedtotheequation

where f = than for < 0 and f = cosh for > 0. The hyper geometric function is used to express the answer to equation (2.27). The Eq. (2.27) is reduced to when the scale factor is big, which corresponds to the later phases of the cosmic growth, t , one hasR  e/2,

andasaresult,for  one finds

HereC1,C2areconstants.The scalarfield vanishesatthislimit,one has q1.

CONCLUSION

Scalar fieldsare essential in hypothetical models that show the growth of the cosmos. Now, we have examined the cosmic dynamics of a specific class of scalar field models. We have demonstrated, by means of this model that the cosmos enters the phase of accelerated expansion when the dynamic term of the scalar field prevails during the phase of decelerating expansion. When the scalar field reaches a constant value in the latter stages, we see an exponential expansion. There is a description of the conditions that lead to an accelerated period of cosmic expansion. We have particularly shown that the expansion is always slowed down in the presence of radiation-type materials or a dominant scalar field. The collection of dust is real in this model. The cosmos starts to grow at in (2.16), starting from a determinate worth of the scale issue, initially expanding more swiftly before slowing down and ultimately arriving at a location where the scalar field tests to zero. The early universe mostly relies on the value of the incorporation constant ‘b’ on the RHS of the cosmological equations when the extra source is provided by the cosmological constant. The analogous scaling factor, which has a smallest non-zero value for negative ‘b’, is given by formulas (2.25) and (2.26). The scalar field diminishes exponentially attheendofthe cosmic expansion,andtheexpansionisnow drivenexponentiallybythecosmologicalconstant.

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Nature of Conformally Coupled Scalar Field in Cosmological Models 55

0 2 1 ) '( ) 2f( 3 ) "(          (2.27)

0 3 2 3 3         k k ,

, C2e C1e 1/2 t   

Mohammad Shyfur Rahman Chowdhury

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