Dark energy models with variable equation of state parameter

1

arXiv:1006.5412v2 [gr-qc] 24 Sep 2010

DARK ENERGY MODELS WITH VARIABLE EQUATION OF STATE PARAMETER

Observations on large scale structure (LSS) [1] and cosmic microwave background radiation [2]−[6] indicate that the Universe is highly homogeneous and isotropic on large scales. Under standard assumptions on the matter content, in general relativistic cosmological models, isotropy is a special feature, requiring a high degree of fine tuning in order to reproduce the observed Universe. Recent years have witnessed the emergence of the idea of an accelerating Universe due to some observational results [7]−[10]. This signifies a remarkable shift in cosmological research from expanding Universe to accelerated expanding Universe. Now, the problem lies in detecting an exotic type of unknown repulsive force, termed as dark energy (DE) which is responsible for the accelerating phase of the Universe. The detection of DE would be a new clue to an old puzzle: the gravitational effect of the zero-point energies of particles and fields [11]. The total with other energies, that are close to homogeneous and nearly independent of time, acts as DE. The paramount characteristic of the DE is a constant or slightly changing energy density as the Universe expands, but we do not know the nature of DE very well (see [12]−[20] for reviews on DE). DE has been conventionally characterized by the equation of state (EoS) parameter ω = p/ρ which is not necessarily constant. The simplest DE candidate is the vacuum energy (ω = −1), which is argued to be equivalent to the cosmological constant (Λ) [21]. The other conventional alternatives, which can be described by minimally coupled scalar fields, are quintessence (ω > −1), phantom energy (ω < −1) and quintom (that can across from phantom region to quintessence region) as evolved and have time dependent EoS parameter. Some other limits obtained from observational results coming from SN Ia data [22] and SN Ia data collaborated with CMBR anisotropy and galaxy clustering statistics [23] are −1.67 < ω < −0.62 and −1.33 < ω < −0.79 respectively. However, it is not at all obligatory to use a constant value of ω. Due to lack of observational evidence in making a distinction between constant and variable ω, usually the equation of state parameter is considered as a constant [24, 25] with phase wise

Anil Kumar Yadav1 , Farook Rahaman2 and Saibal Ray3 1

2

Department of Physics, Anand Engineering College, Keetham, Agra -282 007, India E-mail: [email protected], [email protected]

Department of Mathematics, Jadavpur University, Kolkata - 700 032, India E-mail: farook [email protected] 3

Department of physics, Government College of Engineering and Ceramic Technology, Kolkata 700 010, India E-mail: [email protected]

Abstract The dark energy models with variable equation of state parameter ω are investigated by using law of variation of Hubble’s parameter that yields the constant value of deceleration parameter. Here the equation of state parameter ω is found to be time dependent and its existing range for this model is consistent with the recent observations of SN Ia data, SN Ia data (with CMBR anisotropy) and galaxy clustering statistics. The physical significance of the dark energy models have also been discussed.

Keywords : LRS Bianchi type V Universe, Dark Energy, EoS Parameter 1 corresponding

Introduction

author

1

energy momentum tensor of fluid is taken as   Tij = diag T00 , T11 , T22 , T33

value −1, 0, 1/3 and +1 for vacuum fluid, dust fluid, radiation and stiff fluid dominated universe, respectively. But in general, ω is a function of time or redshift [26]−[28]. For instance, quintessence models involving scalar fields give rise to time dependent EoS parameter ω [29]−[32]. Also some literature is available on models with varying fields, such as cosmological model with variable equation of state parameter in Kaluza-Klein metric and wormholes [33, 34]. In recent years various form of time dependent ω have been used for variable Λ models [35, 36]. Recently Ray et al [37], Akarsu and Kilinc [38] have studied variable EoS parameter for generalized dark energy model. Bianchi type-V universe is generalization of the open universe in FRW cosmology and hence its study is important in the study of DE models in a universe with non-zero curvature [39]. A number of authors such as Collins [40], Maartens and Nel [41], Wrainwright et al [42], Canci et al [43], Pradhan et al [44] and Yadav [45] have studied Bianchi type-V model in different physical contexts. Recently Yadav and Yadav [46] have studied anisotropic DE models with variable EoS parameter. In this paper, we have investigated the DE models with variable ω in Bianchi type-V Universe. This paper is organized as follows: The metric and field equation are presented in section 2. In section 3, we deal with the solution of field equations and discussion. Finally the result are discussed in section 4.

2

(2)

Then, one may parametrize it as follows,

Tij = diag [ρ, −px , −py , −pz ] = diag [1, −ωx, −ωy , −ωz ] ρ = diag [1, −ω, − (ω + δ) , − (ω + δ)] ρ

(3)

where ρ is the energy density of fluid, px , py and pz are the pressures and ωx , ωy and ωz are the directional EoS parameters along the x, y and z axes respectively. ω is the derivation-free EoS parameter of the fluid. We have parametrized the deviation from isotropy by setting ωx = ω and then introducing skewness parameter δ that are deviation from ω along y and z axis respectively. The Einstein field equations, in gravitational units (8πG = 1 and c = 1), are 1 Rij − Rgij = −Tij 2

(4)

where the symbols have their usual meaning. The Einstein’s field equation (4) for the Bianchi type V space-time (1), in case of (3), lead to the following system of equations 2

B44 3 B2 + 42 − 2 = ρ B B A

(5)

1 B2 B44 (6) + 42 − 2 = −ωρ B B A A44 B44 A4 B4 1 + + − 2 = − (ω + δ) ρ (7) A B AB A A4 B4 − =0 (8) A B Here, the sub indices 4 in A, B, and elsewhere denote differentiation with respect to t. Integrating equation (8), we obtain 2

The Metric and Field Equations

We consider LRS Bianchi type V metric in the form
(1) ds2 = −dt2 + A2 dx2 + B 2 e2x dy 2 + dz 2

A = kB

where A and B are the function of t only. The simplest generalisation of EoS parameter of perfect fluid may be to determine the EoS parameter separately on each spatial axis by preserving the diagonal form of the energy-momentum tensor in a consistent way with the considered metric. Thus, the

(9)

where k is the positive constant of integration. We substitute the value of equation (9) in equation (7) and subtract the result from equation (6), we obtain that the skewness parameter on z-axis is null i.e. δ=0 2

Thus system of equations from (5) - (8) may be re- From equations (14) and (15), we get duce to R4 = k1 R−n+1 (17) B44 3 B2 2 (10) + 42 − 2 = ρ B B A R44 = −k12 (n − 1)R−2n+1 (18) 2 B44 1 B4 2 (11) Using equations (14) and (18), equation (16) leads to + 2 − 2 = −ωρ B B A Now we have three linearly independent equations q = n − 1 (constant) (19) (9) - (11) and four unknown parameters (A, B, ω, ρ). Thus one extra condition is needed to solve the sys- The sign of q indicates whether the model inflates or tem completely. To do that, we have used the law not. The positive sign of q corresponds to standard of variation of Hubble’s parameter that yields a con- decelerating model where as the negative sign of q indicates inflation. It is remarkable to mention here stant value of deceleration parameter. The average scale factor of Bianchi type V metric is that though the current observations of SN Ia (Permutter et al 1999, Riess et al 1998) and CMBR favour given by
1 accelerating models i.e. q < 0. But both do not 2 2x 3 R = AB e (12) altogether rule out the decelerating ones which are We define, the generalised mean Hubble’s parameter also consistent with these observations (Vishwakarma 2003). H as 1 (13) From equation (17), we obtain the law of average H = (H1 + H2 + H3 ) 3 scale factor R as B4 A4  1 , H = H = are the directional where H1 = A 2 3 n B when n 6= 0 R = (Dtk1+t c1 ) (20) Hubble’s parameter in the direction of x, y and z c2 e when n = 0 respectively. Equation (13), may be reduces to where c1 and c2 are the constant of integration. From equation (20), for n 6= 0, it is clear that the   R4 1 A4 B4 (14) condition for accelerating expansion of Universe is H= = +2 R 3 A B 0 < n < 1. Here the line-element (1) is completely characterized by Hubble’s parameter H. Therefore, let us consider that mean Hubble parameter H is related to average scale factor R by following relation H = k1 R−n

3

(15)

3.1

Solutions of the Field Equations and Discussion Case (i): when n 6= 0

where k1 > 0 and n ≥ 0, are constant. Such type Equations (8), (14) and (20) lead to of relation have already been considered by Berman 1 [47] for solving FRW models. Later on many authors (21) B = l (Dt + c1 ) n (Singh et al [48, 49] and references therein) have studied flat FRW and Bianchi type models by using the From equations (9) and (21), we obtain special law of Hubble parameter that yields constant 1 value of deceleration parameter. A = L (Dt + c1 ) n (22) The deceleration parameter is defined as where l is the constant of integration and L = kl. Thus the Hubble’s parameter (H), scalar of expanRR44 q=− 2 (16) sion (θ), shear scalar (σ 2 ) and spatial volume (V ) are R 4

3

given by H=

k1 (Dt + c1 )

(23) 4

10

3k1 θ = 3H = (Dt + c1 ) σ2 =

(24)

3k12 − D2 (Dt + c1 )2

4

3

(25)

V2

3

V = Ll2 (Dt + c1 ) n e2x

(26) 1

Using equations (10), (20) and (21), the energy density of the fluid is obtained as

0 0.12

ρ=

3D2 n2 (Dt + c1 )2

−

9.2

3 2

L2 (Dt + c1 ) n

0.52

5.2

(27)

t

Using equations (11), (21), (22) and (27), the equation of state parameter ω is obtained as h 2 i 2(n−1) n n + 2n − 3 L2 D2 (Dt + c1 ) ω= (28) h i 2(n−1) 2 3 1 − Ln2 D2 (Dt + c1 ) n

1.2

0.92 x

Figure 1: The plot of spatial volume V for n = .8

Also we see that during hevolution ofi Universe, at an n (3−2n)L2 D2 2n−2 1 instant of time t = D − c1 , the n2

From equation (27), we note that ρ(t) is the decreasing function of time. This behaviour is clearly shown in Fig. 2, as a representative case with appropriate choice of constants of integration and other physical parameters using reasonably well known situations. From equation (28), it is observed that the equation of state parameter ω is time dependent, it can be function of redshift z or scale factor R as well. The redshift dependence of
ω can be linear like ω(z) = ω0 + ω 1 z with ω 1 = dω dz z = 0 ω1 z [52, 53]. [50, 51] or nonlinear as ω(z) = ω0 + 1+z The SN Ia data suggests that −1.67 < ω < −0.62 [22] while the limit imposed on ω by a combination of SN Ia data (with CMB anisotropy) and galaxy clustering statistics is −1.33 < ω < −0.79 [23]. So, if the present work is compared with experimental results mentioned above then, one can conclude that the limit of ω provided by equation (28) may accommodated with the acceptable range of EoS parameter. The value ω = −1 is the case of vacuum fluid dominated Universe. But here, we are dealing with the solution for n 6= 0, therefore in this case, vacuum fluid dominated universe is meaningless.

ω vanishes. Thus at this particular time, our model represents dusty Universe.

The critical density (ρc ) and density parameter (Ω) are given by 3k12 ρc = (29) (Dt + c1 )2 D2 L2 − n2 (Dt + c1 ) Ω= k12 L2 n2

2n−2 n

(30)

The variation of equation of state parameter (ω) with cosmic time (t) is depicted in figures 3 and 4 as a representative case with appropriate choice of constants. Fig. 3 and 4, clearly show that ω is evolving with negative value and the existing range of ω is in nice agreement with SN Ia data [22]. Thus our model is a realistic model. The plots of ω for n=0.5 and n=0.85 indicate that ω merge well with SN Ia and CMBR observations [Fig. 3 and Fig. 4]. The variation of density parameter (Ω) with time in ac4

t 1

2

3

4

5

K

1.0

0.6

0.8

K

0.8

r

0.6

K

1.0

w

0.4

K

1.2

0.2

K

1.4

1

2

3

4

5

6

t

n n

n = 0.9

= 0.85

n

n

= 0.95

K

= 0.97

1.6

= 0.99

Figure 4: The plot of EoS parameter (ω) vs. time (t) for n = 0.85

Figure 2: The plot of energy density (ρ) vs. time (t)

1.0

t 1

2

3

4

5

6

7

8

9

10 0.9

K

0.7

0.8

K

0.7

0.8 0.6

K

W

0.9

K

0.5

0.4

1.0

w

0.3

K

1.1

0.2

K

0.1

1.2

K

2

4

6

8

10

12

14

16

18

20

t

1.3

n = 0.85 n = 0.99

K

1.4

n

= 0.9

n

= 0.95

n = 0.97

Figure 5: The plot of density parameter (Ω) vs. time (t) for various value of n .

Figure 3: The plot of EoS parameter (ω) vs. time (t) for n = 0.5

5

V = L0 l02 e(3k1 t+2x)

t 0

K

1

2

3

4

5

6

7

8

9

Using equations (10), (31) and (32), the energy density of the fluid is obtained as
3 k12 L20 − e−2k1 t ρ= (37) L20

1.0

K

1.1

K

1.2

w

Using equations (11), (31), (32) and (37), the equation of state parameter ω is obtained as
e−2k1 t − 3k12 L20 ω= (38) 3 (k12 L20 − e−2k1 t )

K

1.3

K

1.4

K

The variation of the EoS parameter ω with cosmic time (t) is shown in Fig. 6. The value of ω is found to be negative which is supported by SN Ia data and galaxy clustering statistics [22, 23]. The critical density (ρc ) and density parameter (Ω) are given by (39) ρc = 3k12

1.5

K

1.6

Figure 6: The plot of EoS parameter (ω) vs. time (t) for n = 0 .

Ω=

4

celerating mode of Universe is clearly shown in Fig. 5.

3.2

(36)

Equations (8), (15) and (21) lead to (31)

From equations (9) and (31), we obtain A = L0 ek1 t

(32)

where l0 is constant of integration and L0 = kl0 . Thus the Hubble’s parameter (H), scalar of expansion (θ), shear scalar (σ 2 ), and spatial volume (V ) are given by H = k1 (33) θ = 3k1

(34)

σ 2 = 2k12

(35)

(40)

Concluding Remarks

In this paper, we have studied dark energy model with variable EoS parameter ω. The model is derived by using law of variation of Hubble’s parameter that yields a constant value of deceleration parameter. We have studied two cases, (3.1) and (3.2) for n 6= 0 and n = 0 respectively. In both the cases, ω is found to be time varying and negative which is consistent with recent observations [22, 23]. In case (i), we have shown that, ω is evolving from ω < −1 and finally end up with ω > −1, representing two phases of universe i.e. ω < −1 (Phantom fluid dominated universe) and ω > −1 (quintessence),[Fig. 3 and Fig. 4]. Where as in case (ii), for n = 0, from equation (38), we have obtained that, at cosmic time t = 0, ω < −1 and when t → ∞, ω → −1. Therefore, one can conclude that at early stage, universe was dominated by phantom fluid and at late time it will be vacuum fluid dominated universe [Fig. 6]. Also it is seen that that in both cases, σθ = constant, therefore the proposed models do not approach to

Case(ii): when n = 0 B = l0 ek1 t

k12 L20 − e−2k1 t L20 k12

6

[2] C. L. Bennett et al., Astrophys. J. Suppl. Ser. 148 1 (2003).

isotropy at any time. Here the age of universe is given by   1 T0 = H0−1 k1

[3] D. N. Spergel et al., Astrophys. J. Suppl. Ser. 148 175 (2003). [4] D. N. Spergel et al., Astrophys. J. Suppl. 170 377 (2007).

which is also differ from the present estimate i.e. T0 = H0−1 ≈ 14Gyr. But if we take, k1 = 1 then our model (n 6= 0) is in good agreement with present age of universe.

[5] C. B. Collins and S. W. Hawking, Astrophys. J. 180 317 (1973). [6] D. Eardley, E. Liang and R. Sachs, J. Math. Phys. 13 99 (1992).

The main fetures of the models are as follows

[7] J. Dunlop et al., Nature 381 581 (1996).

• Though there are many candidates such as cosmological constant, vacuum energy, space-time curvature, cosmological nuclear energy, etc as reported in the vast literature for DE, the proposed models in this paper at least present a new candidate (EoS parameter) as possible suspect of the dark energy.

[8] H. Spinard et al., Astrophys. J. 484 581 (1997). [9] A. G. Riess et al., Astron. J. 116 1009 (1998). [10] S. J. Permitter et al.,Astrophys. J. 517 565 (1999). [11] Y. B. Zel’dovich, JETP letters 6 316 (1967); Sov. Phys. Uspekhi 11 381 (1968).

• The dark energy models are based on exact solution of Einstien’s field equations for the Bianchi type V space-time filled with perfect fluid. To our knowledge, the literature hardly witnessed this sort of exact solution for Bianchi type V space-time. So the derived DE models add one more feather to the literature. The DE models presents the dynamics of EoS parameter ω provided by equations (28) and (38) may accommodated with the acceptable range −1.67 < ω < −0.62 of SN Ia data (Knop et al 2003).

[12] V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. D 9 373 (2000). [13] P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75 559 (2003). [14] U. Alam et al., J. Cosmol. Astropart. Phys. 0406 008 (2004). [15] V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. D 15 2105 (2006).

Acknowledgements

[16] E. J. Copeland et al., Int. J. Mod. Phys. D 15 1753 (2006).

One of Authors (AKY) would like to thank The Institute of Mathematical Science (IMSc), Chennai, India for providing facility and support where part of this work was carried out.

[17] T. Padmanabhan, Gen. Rel. Grav. 40 529 (2008). [18] M. S. Turner and D. Huterer, J. Phys. Soc. Jap. 76 111015 (2007).

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