J. Comp. & Math. Sci. Vol.2 (4), 589-594 (2011)
Mathematical Study of Cosmological Models Static and Expanding Universe with Cosmological Constant R. K. DUBEY* and AJAI KUMAR TRIPATHI1 *Associate Professor Govt. P. G. Science College, Rewa, M. P. India Department of Mathematics Govt. P. G. College, Shahdol, M. P. India
1
ABSTRACT We have considered a cosmological model with a phenomenological model for the cosmological constant for the accelerating universe, the universe must be accelerating in the presence of positive cosmological constant .When evidence for an expanding universe began to accumulate, and the cosmological constant lost some importance. A 0 would however also be necessary if the Hubble time and astrophysically determined data led to different ages for the universe. PACS numbers: 98.80 k, 95.35. +d, 95.36 +X; Keywords: Cosmology, cosmological constant, dark energy.
Ń” =
1. INTRODUCTION Einstein ,s field equations are unable to describe a static universe without the term . Since in those days there was a strong belief in a static universe, Einstein introduced a free parameter in to his equations in 1917. The meaning of can be seen immediately if the field equation are written in the form Âľ Âľ
Âľ Âľ
Obviously has the same dimension as the energy –momentum tensor. According to the above the cosmological constant can be associated with the energy density є of the vacuum to give
Cosmological constant1 is currently seen as one of the most important and enigmatic objects in cosmology. The cosmological constant appears in the friedmann equation as an extra term, giving
(1)
Here has units [time]-2 The effect of can be seen more directly from the acceleration equation. Using the Friedmann equation as given above gives
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
(2)
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R. K. Dubey, et al., J. Comp. & Math. Sci. Vol.2 (4), 589-594 (2011)
A positive cosmological constant gives a positive contribution to , and so acts effectively as a repulsive force. In particular if the cosmological constant is sufficiently large, it can overcome the gravitational attraction represented by the first term and lead to an accelerating universe. Density parameter for the cosmological constant can be defined as â„Ś
(3)
From the friedmann equation we find â„Ś â„Ś 1
(4)
The condition to have a flat universe k = 0 gives. â„Ś â„Ś 1
Open universe : 0 â„Ś â„Ś 1 Flat universe : â„Ś â„Ś 1 Closed universe : â„Ś â„Ś 1
2. FLUID DESCRIPTION OF with
energy
density
Brings the friedmann equation in to the form
The cosmological constant has a negative effective pressure2, 3. This means that as the universe expands; work is done on the constant fluid. This permits its energy density to remain constant even though the volume of the universe is increasing. 3.
COSMOLOGICAL MODELS WITH
Under the assumption of an homogeneous isotropic universe. Einstein’s field equation reduces themselves to the Einstein –fiedmann –lemaintre equation
The relation between the density parameter and the geometry now becomes as
Fluid defined as π
Since is constant by definition we must have .
(5)
We can consider the fluid equation for . 3 0
2 3
(6) (7)
We consider first the static solutions ( = 0 ) .The equations are then written (for p=0) as 8 2
(8)
(9)
From the first equation it follows that 0 and therefore the second equation has a solution for k=1 .
(10)
From equation (10) represents the equilibrium condition for the universe. The attractive force due to has to exactly
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
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R. K. Dubey, et al., J. Comp. & Math. Sci. Vol.2 (4), 589-594 (2011)
compensate the repulsive effect of a positive cosmological constant in order to produce a static universe. We now consider non - static solutions. We can see a positive always leads to an acceleration of the expansion for positive and k = -1, 0 the solutions are always positive ,which therefore result in a continually expanding universe. for k = 1, there exists a critical value .
4 !
8 #$
11
For static ,expanding and contracting solutions all exists. Solutions without an initial singularity also exists. If 0 , as illustrated below in figure.
The behavior of scale factor with a non-vanishing cosmological constant
a(t )
0 , & ' 1,0 0 ,& 1
Solution of the Friedmann equation for 0 and 0 , & 1
a(t)
t
0 ; K= 1
Solution for K = 1 ;
0 ,
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
t
592 4.
R. K. Dubey, et al., J. Comp. & Math. Sci. Vol.2 (4), 589-594 (2011)
Acceleration of the Universe
In 1998 observation of Type Ia supernova [4,5] suggested that the expansion of universe is speeding up this suggests 0
12
Where is the acceleration rate of the universe . Now from equation
3
And the relationship between p and P = )
(13)
(14)
We can study the nature of the accelerating universe in some detail. Since when the universe expands ,the density of dark matter declines more quickly than the density of dark energy6, the dark energy dominates in the later times .for example if the volume of the universe doubles, then the density of dark matter will be halved but that of the dark energy will not change very much. Actually we can even consider it as a constant. A particular case is the cosmological constant equation (12) shows, if the dark energy dominates, we have *
4 3 + 0 3 3
Since a 0 , we find
3 Which means 3p π
0
15 16
17
Now we can discuss it in three different cases 0, 0 01 0
Ď€
0,
3p 2
, π
, 0
, 03 0
(18)
Defining the new energy density and pressure as (19) ′ = π π ′ ′
) p=p′
(20)
(21)
We find that the acceleration equation can be re-written as 4 ′ 3 ′ 5 (22)
Or
41 3)′ 5 ′
(23)
Because 0 , we find that )′
π π
(24)
Which is the basic condition for the universe to be accelerating. The de Sitter universe (1917) is a cosmological model in which the universe is empty (p=0, 0 and flat (k=0). From equation (19)and (20) we get p′ = - ′. which on substitution in
/ /
gives 6
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
R. K. Dubey, et al., J. Comp. & Math. Sci. Vol.2 (4), 589-594 (2011)
log a =
/
a = A exp !
/ 6$
6 log :
(25)
In the de Sitter vacuum universe test particle move away from each other because of the repulsive gravitational effect of the positive cosmological constant .The de Sitter model was only of marginal historical interest until the last 15 years or so. In recent years however it has been a major component of inflationary universe models in which for a certain interval of time the expansion assumes an exponential character of the type (6).
models when the galaxies are very far apart after a period of expansion, the attractive force of the matter becomes weak and eventually the repulsive force due to a positive cosmological constant take over, and one gets asymptotically the de sitter behavior. REFERENCES 1.
2. 3.
4. a (t)
5.
Behaviour of a(t) in the de Sitter model 5. CONCLUSION In this paper we have formulated a set of concept on the cosmological constant we have also discussed cosmological model with non – static (accelerating expansion) . It is clear from the existence of the Einstein static model, a positive cosmological constant as introduced here represents a repulsive force. So that the attractive force of the matter is balanced by this repulsive force in the Einstein model, In the dynamic
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6.
7.
8.
Moss, I. and Sahni, V. Anisotropy in the Chaotic inflationary Universe Phys. Lett. B 178 159 – 62 (1986). S. Perlmutter, et al., Nature (London) 391, 51 (1998). Neta A. Bahcall, J. P. Ostriker, S. Perlmutter and P. J. Steinhardt, Science 284 ,1481-1488 (1999). Goldhaber, G. and Perlmutter, S. “A study of 42 type Ia supernovae and a resulting measurement of Omega (M) and Omega (Lambda) “ Physics Reports –Review section of Physics letters.307(1-4), 325-331 Dec (1998). Garnavich, P.M., Kirshner, R. P., Challis, P, et al.” Constraints on cosmological models from Hubble space Telescope observations of High – Z Supernovae” , Astophys. J. 493 L 53, arXiv;astro-ph/9710123 (1998). D. Garfinkle, “Inhomogeneous space times as a dark energy model”, arXiv:gr qc/ 0605088. Rendall, A. D. Mathematical properties of cosmological models with accelerated expansion. Electronic archive www.arxiv.org/gr-qc/0408072 (2004). Starobinsky, A. A. Isotropization of Arbitrary Cosmological Expansion Given an Effective Cosmological constant. JETP Lett., 37, 66-67 (1983).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
594 9.
R. K. Dubey, et al., J. Comp. & Math. Sci. Vol.2 (4), 589-594 (2011)
Wald, R. M. Asymptotic behaviour of homogeneous cosmological models with cosmological constant. Phys. Rev., D28, 2118-2120 (1983). 10. G. F. Hinshaw, “Tests of the Big Bang: Expansion”, 2008. 11. R. Horvat, Phys. Rev. D 70, 087301 (2004). 12. C.J. Feng, Phys. Lett. B 663, 367 (2008).
13. L.X. Xu, W.B. Li, J.B. Lu, Eur. Phys. J. C 60, 135 (2009). 14. Y. Gong, Phys. Rev. D 61, 043505 (2000). 15. Y. Gong, Phys. Rev. D 70, 064029 (2004). 16. M. R. Setare, Phys. Lett. B 644, 99 (2007). 17. N. Banerjee, D. Pavon, Phys. Lett. B 647, 447 (2007).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)