L.R.S. Bianchi type II Stiff Fluid cosmological model with Decaying Vacuum Energy Density $\Lambda$ in general relativity

Locally rotationally symmetric (L.R.S.) Bianchi type II stiff fluid cosmological model is investigated. To get the deterministic model of the universe, we have assumed a condition $A=B^{m}$ between metric potentials $A,~B$ where $n$ is the constant. It is shown that the vacuum energy density $\Lambda$ is positive and proportional to $\frac{1}{t^{2}}$. The values of deceleration parameter $q$, matter-energy density $\Omega_{m}$ and dark-energy density $\Omega_{\Lambda}$ are found to be in good agreement with the values obtain from 5-years WMAP observations. the predicted value of the jerk parameter agrees with the SNLS SNIa and X-ray galaxy cluster distance data but does not with the SNIa gold sample data. In general, the model represent accelerating, shearing and non-rotating universe.The physical and geometrical behavior of these models are also discussed.


Introduction
One of the outstanding problems in cosmology is the so called "cosmological constant" problem. Recent observations of type Ia supernovae (the Supernova Cosmology Project and the High-Z Supernova Team) [1]- [4] presented evidence that the expansion of the universe is accelerating. These teams have measured the distances to cosmological supernovae by using the fact that the intrinsic luminosity of Type Ia supernovae is closely correlated to their decline rate from maximum brightness, which can be independently measured. These measurements, combined with red-shift data for the supernovae, led to the prediction of an accelerating universe. Both team obtained Ω m ≈ 0. 3 and Ω Λ ≈ 0.7 and strongly ruled out the traditional (Ω m , Ω Λ )= (1, 0) universe. This value of the density parameter Ω Λ corresponds to a cosmological constant that is small, nevertheless, nonzero and positive. Cosmological (or vacuum energy) constant [5]- [8] is one of the most theoretical candidate for dark energy. Unfortunately there is a huge difference of order 10 120 between observational (Λ ∼ 10 −55 cm −2 ) and the particle physics prediction value for Λ. This discrepancy is known as cosmological constant problem.
Carmeli and Kuzmenko [9] have Recently shown that the cosmological relativistic theory [10] predicts Λ = 1.934 × 10 −35 s −2 which is in agreement with the measurements recently obtained by the High-z Supernova Team and Supernova Cosmological Project [1]- [4]. There have been Several ansätz suggested in which the Λ term decays with time [11]- [25]. Chen and Wu [21] have suggested The special ansätz Λ ∝ R −2 (where R is the scale factor of the Robertson-Walkermetric) which has been modified by several authors [26]- [31]. However, an accelerating universe can not be predicted by all vacuum decaying cosmological models. Several authors have argued in favor of the dependence Λ ∝ 1 t 2 in different context [13]- [15]. The relation Λ ∝ 1 t 2 seems to play a major role in cosmology [15]. Recently Pradhan et al [32] have obtained some LRS Bianchi type II bulk viscous fluid universe with decaying vacuum energy density. A convenient method to describe models close to ΛCDM is based on the cosmic jerk parameter j, a dimensionless third derivative of the scale factor with respect to the cosmic time [33], [34]. A deceleration-to-acceleration transition occurs for models with a positive value of j 0 and negative q 0 . Flat ΛCDM models have a constant jerk j = 1.
Stiff fluid cosmological models create more interest in the study because for these models, the speed of light is equal to speed of sound and its governing equations have the same characteristics as those of gravitational field (Zeldovich [35]). Barrow [36] has discussed the relevance of stiff equation of state ρ = p to the matter content of the universe in the early state of evolution of universe. Wesson [37] has investigated an exact solution of Einsteins field equation with stiff equation of state. Mohanty et al. [38] have investigated cylindrically symmetric Zeldovich fluid distribution in General Relativity. Götz [39] obtained a plane symmetric solution of Einsteins field equation for stiff perfect fluid distribution. Bali and Tyagi [40] have investigated Bianchi type I magnetized stiff fluid cosmological model in General Relativity.
In this latter, a new anisotropic L.R.S. (Locally Rotationally Symmetric) Bianchi type II stiff fluid cosmological model with variable Λ has been investigated by assuming a supplementary condition A = B m between metric potentials A, B where n is the constant. The out line of the paper is as follows: In Section 2, the metric and the field equations are described. Section 3 deals with the solutions of the field equations. In Subsections (3.1) some physical and geometric properties of the model are described. Section 4 the jerk parameter of this model is driven. Finally, conclusions are summarized in the last Section 5.

The Metric and Field Equations
The metric for LRS Bianchi type II in an orthogonal frame is given by where the Cartan bases θ i are given by Here, A and B are the time-dependent metric functions. Assuming (x, y, z) as local coordinates, the differential one forms ω i are given by The Einstein's cosmological field equations are given by (with 8πG = 1 and c = 1) We consider the energy-momentum tensor in the form Hence, for energy-momentum tensor and LRS Bianci type II The Einstein's field equations (4) leads to the following system of equations: where an overdot stands for the first and double overdot for second derivative with respect to t. The spatial volume for LRS B-II is given by We define a = (A 2 B) 1 3 as the average scale factor of LRS B-II model (1) so that the Hubble's parameter is given by We define the generalized mean Hubble's parameter H as where H x =Ḃ B , H y = H z =Ȧ A are the directional Hubble's parameters in the directions of x, y and z respectively.
The deceleration parameter q is conventionally defined by The scalar expansion θ, shear scalar σ 2 and the average anisotropy parameter Am are defined by where

Solution of the Field Equations
The field equations (6)-(8) are a system of three equations with five unknown parameters A, B, p, ρ, Λ. Two additional constraint relating these parameters are required to obtain explicit solutions of the system. Following Bali and Jain [41] and Pradhan et al. [42], I assume that the expansion (θ) in the model is proportional to the eigen value σ 1 1 of the shear tensor σ i j . This condition leads to where m is a constant. In order to overcome the under-determinacy we have here because of the five un-known involved in three independent field equations, I assume that the fluid obeys the stiff fluid equation of state i.e.
To get deterministic solution in terms of cosmic time t , we suppose k = 0. In this case (20) takes the form To get deterministic solution, we assume m = 3 2 . In this case eq. (21), reduces to dB Integrating eq. (22) we obtain and Eqs. (23) and (24) show that Λ > 0. In this case the LRS Bianchi type II space-time can be written as

The Geometric and Physical Significance of Model
The energy density (ρ), the pressure p and the vacuum energy density (Λ) for the model (25) are given by From (26), we see that energy conditions, ρ ≥ 0 is satisfied under condition  The expressions for Hubble parameter H, the scalar of expansion θ, magnitude of shear σ 2 , the average anisotropy parameter A m and the proper volume V for the model (25) are given by From (30) and (31) we get The deceleration parameter is given by If we put the value of Λ from eq. (27) we observe that From eq. (36) we observe that our model is in accelerating phase and it's behavior is almost same as the de-sitter universe.
Using equations (26)- (29) we can obtain the matter-energy density Ω m and dark-energy density Ω Λ as Ω m = 9 8 and From eqs. (37) and (38) we observe that the values of matter-energy density Ω m and dark-energy density Ω Λ are in good agreement with the values obtain from 5-years WMAP observations for ΛCMD model [43]. The compression of these parameters is shown in table. 1.

The jerk Parameter of the Model
The jerk parameter in cosmology is defined as the dimensionless third derivative of the scale factor with respect to cosmic time and in terms of the scale factor to cosmic time where the 'dots' and 'primes' denote derivatives with respect to cosmic time and scale factor, respectively. The jerk parameter appears in the fourth term of a Taylor expansion of the scale factor around a 0 where the subscript 0 shows the present value. One can rewrite eq. (39) as Using eqs. (29) and (35) in (44) we find Now, putting the valu of Λ from eq. (27) in eq. (43) we obtain This value does not overlap with the value j = 2.16 +0.81 −0.75 obtained from the combination of three kinematical data sets: the gold sample of type Ia supernovae [44], the SNIa data from the SNLS project [45], and the X-ray galaxy cluster distance measurements [46]. However, it is in consistent with two of the three data sets separately: the SNLS SNIa set gives j = 1.32 +1.37 −1.21 and the cluster set gives j = 0.51 +2.55 −2.00 , and it is the gold sample data that yields larger j = 2.75 +1.22 −1.10 [46].

Concluding Remarks
A new cosmological model based on LRS Bianchi type II cosmological models with decaying vacuum energy is obtained. The model (25) starts with a big bang at t = 0. The expansion in the model decreases as time increases. The proper volume of the model increases as time increases. Since σ θ is constant the model does not approach isotropy. There is a point type singularity in the model at t = 0 [47]. It is shown that Λ ∝ 1 t 2 . Therefor, as t → 0, Λ → ∞ and when t → ∞ then Λ → 0. In Brans-Dicke theories the relation like equation (27) can be finds when one supposes variable gravitational and cosmological "constant" [13], [15] and [17]. Berman [48] also has derived this relation in general relativity. A positive cosmological constant or equivalently the negative deceleration parameter is required to solve the age parameter and density parameter. The values of deceleration parameter q, matter-energy density Ω m , dark-energy density Ω Λ and the jerk parameter for this model are found to be in good agreement with the present values of these parameters obtained from observations. It is reasonable to say that a cosmological model is required to explain acceleration in the present universe. Therefor, the theoretical model found in this paper is in agreement with the recent observations. acknowledgements Author would like to thank the Islamic Azad university, Mahshahr branch for providing facility and support where this work was carried out.