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Physics & Astronomy International Journal

Review Article Volume 2 Issue 4

Bulk viscous Bianchi–I cosmological model in f(R, T) gravity theory

Shri Ram,1 Surendra K Singh,2 Verma MK1

1Department of Mathematical Sciences, Banaras Hindu University, India
2Pt Ramadhar J Tiwari College of Polytechnic, India

Correspondence: Shri Ram, Department of Mathematical Sciences, Banaras Hindu University, Varanasi?2210 05, India, Tel 0542 2316 318

Received: December 09, 2017 | Published: July 25, 2018

Citation: Ram S, Singh SK, Verma MK. Bulk viscous Bianchi–I cosmological model in f(R, T) gravity theory. Phys Astron Int J. 2018;2(4):330-334. DOI: 10.15406/paij.2018.02.00106

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Abstract

This paper deals with a new spatially homogeneous Bianchi–I cosmological model of the universe filled with a bulk viscous fluid in f(R, T) gravity theory. We obtain exact solutions of the field equations by using a special law of variation of Hubble’s parameter for the average scale factor that yields a negative constant value of the deceleration parameter, which correspond to the model of the universe with a big–bang singularity at the initial time. The universe subsequently expands with power law expansion and gives essentially an empty universe for large time. The physical and dynamical properties of the model are discussed which are consistent with cosmological scenario of the present–day accelerated universe.

Keywords: bulk viscous, bianchi –I cosmology, f(R, T) gravity

Introduction

A wide range of recent cosmological observations1,2 has suggested that the universe in its present state is in the phase of accelerated expansion. The cause of this acceleration is supposed to be some kind of anti gravitational force. A large class of cosmological models has explained the acceleration of the universe in terms of a component with negative pressure, the so called dark energy (DE). The limitations of general relativity in providing satisfactory explanation of this phase of evolution have led cosmologists to adopt hypotheses and study their implications in this context. The hypotheses include those assigning (I) the time–dependence of the gravitational constant and cosmological term (II) some other geometries or physical fields associated with the universe and (III) modified or alternative theories of gravity. Modified gravity theories certainly provide a way of understanding the problem of DE and the possibility to reconstruct the gravitational field theories that would be capable to reproduce the late–time acceleration of the universe. Among several modified theories, f(R) gravity theory formulated by Nojiri et al.,3 is indeed a realistic alternative to general relativity consistent with DE. In f(R) gravity theory, the cosmic acceleration could be achieved by replacing Einstein–Hilbert action of general relativity with a general function of Ricci scalar R. Various attributes of cosmological models in f(R) gravity theory depicting time inflation and late–time cosmic acceleration have been investigated by a number of authors.4 A further extension of f(R) gravity theory is the f(R, T) gravity theory.5 In this theory, an arbitrary function of the Ricci scalar R and trace T of the energy momentum tensor represents the gravitation Lagrangian. Since the inception of this theory, its characteristics have been studied both with isotropic and anisotropic Bianchi models by several cosmologists which certainly promote understanding of many associated issues in respect of the accelerated expansion of the present–day universe. Houndjo6 has developed the cosmological reconstruction of f(R, T) gravity theory and discussed transaction of matter dominated phase to an acceleration phase. Further, Houndjo et al.,7 considered cosmological scenario based on f(R, T) reconstructed numerically from Holographic DE. The cosmic acceleration in this theory results not only from the geometrical contribution but also from the matter content. The role played by viscosity and the consequent dissipative mechanism in cosmology have been studied by several cosmologists.8–12 These dissipative processes may indeed responsible for smoothing out of initial anisotropy. Bulk viscosity is the only dissipative phenomena occurring in FRW models and is significant in causing the accelerated expansion of the universe known as inflationary phase as discussed by Setare et al.13 Several cosmologists have discussed the role of bulk viscosity in the early evolution of the universe in different physical contexts. The cosmological and astrophysical implications of f(R, T) gravity theory in the presence of perfects fluid and bulk viscous fluids have been studied by several cosmologists. Adhav et al.,14 derived an LRS Bianchi type–I model in f(R, T) gravity theory. Subsequently Reddy et al.,15,16 Shamir et al.,17 Chaubey et al.,18 Shri Ram et al.,19,20 Chandel et al.,21 etc. have discussed Bianchi models in the presence of perfect fluids in different physical contexts in the framework of f(R, T) gravity theory. Kiran et al.,22 has shown the non–existence of Bianchi type–III bulk viscous string cosmological model. Shri Ram et al.,23 investigated Bianchi type–I and V bulk viscous fluid cosmological models. Sahu et al.,24 discussed cosmic transits and anisotropic models of Bianchi type–III. Further, Sahoo et al,.25 studied cosmological models in f(R, T) theory with variable deceleration parameter. This motivates the theorists to construct various models in different Bianchi space–time in different contexts.26–29 Sahoo et al.,30 has investigated a locally rotationally symmetric Bianchi type–I cosmological model in the presence of one–dimensional cosmic strings within the framework of f(R, T) gravity theory. Recently, Sahoo et al.,31 have studied homogeneous and anisotropic locally rotationally symmetric (LRS) Bianchi type–I model with magnetized strange quark matter distribution and cosmological constant in f(R, T ) gravity. It deserves mention that Momeni et al.,32 have discussed cylindrically symmetric solutions of cosmic strings in gravity rainbow scenario and calculated the gravitational fields equations corresponding to energy dependent background.

In this paper, we investigate a new exact spatially homogeneous Bianchi type–I space–time in the presence of bulk viscous fluid in f(R, T) gravity theory. The paper is organized as follows: In Section 2, we present the metric and field equations of f(R, T) gravity theory. In Section 3, we obtain exact solution of the field equations by utilizing the special law of variation for the average scale factor that yields a negative constant value of the deceleration parameter. The resulting cosmological model is continuously expanding, shearing and accelerating universe which would essentially gives an empty space at late time. The other physical and geometrical features of the model are discussed in Section 4. Some concluding remarks are given in Section 5.

Metric and field equations

 We consider the Bianchi type–I space–time represented by the metric

d s 2 =d t 2 A 2 d x 2 B 2 d y 2 C 2 d z 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGKb Gaam4CaSWaaWbaaeqabaqcLbmacaaIYaaaaKqzGeGaaGypaiaadsga caWG0bWcdaahaaqabeaajugWaiaaikdaaaqcLbsacqGHsislcaWGbb WcdaahaaqabeaajugWaiaaikdaaaqcLbsacaWGKbGaamiEaSWaaWba aeqabaqcLbmacaaIYaaaaKqzGeGaeyOeI0IaamOqaSWaaWbaaeqaba qcLbmacaaIYaaaaKqzGeGaamizaiaadMhalmaaCaaabeqaaKqzadGa aGOmaaaajugibiabgkHiTiaadoealmaaCaaabeqaaKqzadGaaGOmaa aajugibiaadsgacaWG6bWcdaahaaqabeaajugWaiaaikdaaaaaaa@5A81@  (1)

Where, A, B and C are functions of cosmic time t only.

We assume that the cosmic matter is represented by the energy–momentum tensor of a bulk viscous fluid given by

T ij =(ρ+ p ¯ ) u i u j p ¯ g ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub WcdaWgaaqaaKqzadGaamyAaiaadQgaaSqabaqcLbsacaaI9aGaaGik aiabeg8aYjabgUcaRiqadchagaqeaiaaiMcacaWG1bWcdaWgaaqaaK qzadGaamyAaaWcbeaajugibiaadwhalmaaBaaabaqcLbmacaWGQbaa leqaaKqzGeGaeyOeI0IabmiCayaaraGaam4zaSWaaSbaaeaajugWai aadMgacaWGQbaaleqaaaaa@4EEB@  (2)

 Where the effective pressure p ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGWb Gbaebaaaa@3792@ is given by

p ¯ =pξ u ;i i . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGWb GbaebacaaI9aGaamiCaiabgkHiTiabe67a4jaadwhalmaaDaaabaqc LbmacaaI7aGaamyAaaWcbaqcLbmacaWGPbaaaKqzGeGaaGOlaaaa@4374@ (3)

Here p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGWb aaaa@377A@ is the isotropic pressure, p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGWb aaaa@377A@ the energy density of matter, ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEaaa@3848@ is the bulk viscous coefficient and u i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG1b qcfa4aaWbaaSqabeaajugWaiaadMgaaaaaaa@3A56@ the 4–velocity vector of the fluid satisfying u i u j =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG1b WcdaahaaqabeaajugWaiaadMgaaaqcLbsacaWG1bqcfa4aaSbaaSqa aKqzadGaamOAaaWcbeaajugibiaai2dacaaIXaaaaa@4044@ . A comma and semicolon denote ordinary and covariant differentiation respectively. On thermodynamically reasons the bulk viscosity coefficient is positive, assuring that the viscosity pushes the dissipative pressure p ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGWb Gbaebaaaa@3792@ towards negative values. However, the correction applied to the thermodynamic pressure p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGWb aaaa@377A@ due to bulk viscosity is very small. Therefore, the dynamics of cosmic evolution is not fundamentally influenced by the inclusion of viscous term in energy–momentum tensor.

The field equations in f(R, T) gravity theory with the special choice of the function

f(R,T)=R+2f(T) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb GaaGikaiaadkfacaaISaGaamivaiaaiMcacaaI9aGaamOuaiabgUca RiaaikdacaWGMbGaaGikaiaadsfacaaIPaaaaa@41A0@  (4)

 Are given by Reddy:33

R ij 1 2 R g ij =8π T ij +2 f ˙ (T) T ij +[2 p ¯ f ˙ (T)+f(T)] g ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsb WcdaWgaaqaaKqzadGaamyAaiaadQgaaSqabaqcLbsacqGHsisljuaG daWcaaGcbaqcLbsacaaIXaaakeaajugibiaaikdaaaGaamOuaiaadE galmaaBaaabaqcLbmacaWGPbGaamOAaaWcbeaajugibiaai2dacaaI 4aGaeqiWdaNaamivaSWaaSbaaeaajugWaiaadMgacaWGQbaaleqaaK qzGeGaey4kaSIaaGOmaiqadAgagaGaaiaaiIcacaWGubGaaGykaiaa dsfalmaaBaaabaqcLbmacaWGPbGaamOAaaWcbeaajugibiabgUcaRi aaiUfacaaIYaGabmiCayaaraGabmOzayaacaGaaGikaiaadsfacaaI PaGaey4kaSIaamOzaiaaiIcacaWGubGaaGykaiaai2facaWGNbqcfa 4aaSbaaSqaaKqzadGaamyAaiaadQgaaSqabaaaaa@66CF@ (5)

Where a dot denotes derivative with respect to T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub aaaa@375E@ . We further choose

f(T)=λT, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb GaaGikaiaadsfacaaIPaGaaGypaiabeU7aSjaadsfacaaISaaaaa@3DB8@  (6)

Where λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH7o aBaaa@3839@ is a constant. This gravity theory is equivalent to cosmological scenario with an effective cosmological term Λ H 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHBo atrqqr1ngBPrgifHhDYfgaiuaacqWF8iIocaWGibWcdaahaaqabeaa jugWaiaaikdaaaaaaa@4097@ , where H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGib aaaa@3752@ is the Hubble parameter.34 In commoving coordinates, the field equation (5) together with (2), (4) and (6) for the metric (1) yield the following system of equation:

B ¨ B + C ¨ C + B ˙ B C ˙ C =λρ(8π+3λ) p ¯ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGabmOqayaadaaakeaajugibiaadkeaaaGaey4kaSscfa4a aSaaaOqaaKqzGeGabm4qayaadaaakeaajugibiaadoeaaaGaey4kaS scfa4aaSaaaOqaaKqzGeGabmOqayaacaaakeaajugibiaadkeaaaqc fa4aaSaaaOqaaKqzGeGabm4qayaacaaakeaajugibiaadoeaaaGaaG ypaiabeU7aSjabeg8aYjabgkHiTiaaiIcacaaI4aGaeqiWdaNaey4k aSIaaG4maiabeU7aSjaaiMcaceWGWbGbaebacaaISaaaaa@537E@       (7)

A ¨ A + C ¨ C + A ˙ A C ˙ C =λρ(8π+3λ) p ¯ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGabmyqayaadaaakeaajugibiaadgeaaaGaey4kaSscfa4a aSaaaOqaaKqzGeGabm4qayaadaaakeaajugibiaadoeaaaGaey4kaS scfa4aaSaaaOqaaKqzGeGabmyqayaacaaakeaajugibiaadgeaaaqc fa4aaSaaaOqaaKqzGeGabm4qayaacaaakeaajugibiaadoeaaaGaaG ypaiabeU7aSjabeg8aYjabgkHiTiaaiIcacaaI4aGaeqiWdaNaey4k aSIaaG4maiabeU7aSjaaiMcaceWGWbGbaebacaaISaaaaa@537A@     (8)

A ¨ A + B ¨ B + A ˙ B ˙ AB =λρ(8π+3λ) p ¯ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGabmyqayaadaaakeaajugibiaadgeaaaGaey4kaSscfa4a aSaaaOqaaKqzGeGabmOqayaadaaakeaajugibiaadkeaaaGaey4kaS scfa4aaSaaaOqaaKqzGeGabmyqayaacaGabmOqayaacaaakeaajugi biaadgeacaWGcbaaaiaai2dacqaH7oaBcqaHbpGCcqGHsislcaaIOa GaaGioaiabec8aWjabgUcaRiaaiodacqaH7oaBcaaIPaGabmiCayaa raGaaGilaaaa@51A6@       (9)

A ˙ A B ˙ B + A ˙ A C ˙ C + B ˙ B C ˙ C =(8π+3λ)ρλ p ¯ . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGabmyqayaacaaakeaajugibiaadgeaaaqcfa4aaSaaaOqa aKqzGeGabmOqayaacaaakeaajugibiaadkeaaaGaey4kaSscfa4aaS aaaOqaaKqzGeGabmyqayaacaaakeaajugibiaadgeaaaqcfa4aaSaa aOqaaKqzGeGabm4qayaacaaakeaajugibiaadoeaaaGaey4kaSscfa 4aaSaaaOqaaKqzGeGabmOqayaacaaakeaajugibiaadkeaaaqcfa4a aSaaaOqaaKqzGeGabm4qayaacaaakeaajugibiaadoeaaaGaaGypai aaiIcacaaI4aGaeqiWdaNaey4kaSIaaG4maiabeU7aSjaaiMcacqaH bpGCcqGHsislcqaH7oaBceWGWbGbaebacaaIUaaaaa@5A48@ (10)

 An over dot denotes derivation with respect to t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b aaaa@377E@ .

Now we define certain physical and kinematical parameters associated with the metric (1). The spatial volume V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb aaaa@3760@ and the average scale factor a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGHb aaaa@376B@ is defined by

V= a 3 , a 3 =ABC. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb GaaGypaiaadggalmaaCaaabeqaaKqzadGaaG4maaaajugibiaaiYca caWGHbqcfa4aaWbaaSqabeaajugWaiaaiodaaaqcLbsacaaI9aGaam yqaiaadkeacaWGdbGaaGOlaaaa@4459@ (11)

 The expansion scalar θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ and the shear scalar σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHdp WCaaa@3848@ are given by

θ= u ;i i = A ˙ A + B ˙ B + C ˙ C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCcaaI9aGaamyDaSWaa0baaeaajugWaiaaiUdacaWGPbaaleaajugW aiaadMgaaaqcLbsacaaI9aqcfa4aaSaaaOqaaKqzGeGabmyqayaaca aakeaajugibiaadgeaaaGaey4kaSscfa4aaSaaaOqaaKqzGeGabmOq ayaacaaakeaajugibiaadkeaaaGaey4kaSscfa4aaSaaaOqaaKqzGe Gabm4qayaacaaakeaajugibiaadoeaaaaaaa@4C80@  (12)

σ 2 = 1 2 [ ( A ˙ A ) 2 + ( B ˙ B ) 2 + ( C ˙ C ) 2 ] 1 6 θ 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHdp WClmaaCaaabeqaaKqzadGaaGOmaaaajugibiaai2dajuaGdaWcaaGc baqcLbsacaaIXaaakeaajugibiaaikdaaaqcfa4aamWaaOqaaKqbao aabmaakeaajuaGdaWcaaGcbaqcLbsaceWGbbGbaiaaaOqaaKqzGeGa amyqaaaaaOGaayjkaiaawMcaaSWaaWbaaeqabaqcLbmacaaIYaaaaK qzGeGaey4kaSscfa4aaeWaaOqaaKqbaoaalaaakeaajugibiqadkea gaGaaaGcbaqcLbsacaWGcbaaaaGccaGLOaGaayzkaaWcdaahaaqabe aajugWaiaaikdaaaqcLbsacqGHRaWkjuaGdaqadaGcbaqcfa4aaSaa aOqaaKqzGeGabm4qayaacaaakeaajugibiaadoeaaaaakiaawIcaca GLPaaalmaaCaaabeqaaKqzadGaaGOmaaaaaOGaay5waiaaw2faaKqz GeGaeyOeI0scfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaaI2a aaaiabeI7aXTWaaWbaaeqabaqcLbmacaaIYaaaaKqzGeGaaGOlaaaa @6548@ (13)

 The generalized mean Hubble’s parameter H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGib aaaa@3752@ is defined by

H= a ˙ a = 1 3 ( H 1 + H 2 + H 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGib GaaGypaKqbaoaalaaakeaajugibiqadggagaGaaaGcbaqcLbsacaWG Hbaaaiaai2dajuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaio daaaqcfa4aaeWaaOqaaKqzGeGaamisaSWaaSbaaeaajugWaiaaigda aSqabaqcLbsacqGHRaWkcaWGibWcdaWgaaqaaKqzadGaaGOmaaWcbe aajugibiabgUcaRiaadIeajuaGdaWgaaWcbaqcLbmacaaIZaaaleqa aaGccaGLOaGaayzkaaaaaa@4EC1@ (14)

where H 1 = A ˙ A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGib WcdaWgaaqaaKqzadGaaGymaaWcbeaajugibiaai2dajuaGdaWcaaGc baqcLbsaceWGbbGbaiaaaOqaaKqzGeGaamyqaaaaaaa@3E2D@ , H 2 = B ˙ B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGib WcdaWgaaqaaKqzadGaaGOmaaWcbeaajugibiaai2dajuaGdaWcaaGc baqcLbsaceWGcbGbaiaaaOqaaKqzGeGaamOqaaaaaaa@3E30@ , H 3 = C ˙ C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGib WcdaWgaaqaaKqzadGaaG4maaWcbeaajugibiaai2dajuaGdaWcaaGc baqcLbsaceWGdbGbaiaaaOqaaKqzGeGaam4qaaaaaaa@3E33@ are directional Hubble’s parameter in the direction of x  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b aeaaaaaaaaa8qacaGGGcaaaa@38C6@ , y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG5b aaaa@3783@ and z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaa3=jugibi aadQhaaaa@38F6@ respectively. The anisotropy parameter Δ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHuo araaa@37EB@ is given by

Δ= 1 3 i=1 3 ( H i H H ) 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHuo arcaaI9aqcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaaIZaaa aKqbaoaaqahakeqaleaajugWaiaadMgacaaI9aGaaGymaaWcbaqcLb macaaIZaaajugibiabggHiLdqcfa4aaeWaaOqaaKqbaoaalaaakeaa jugibiaadIealmaaBaaabaqcLbmacaWGPbaaleqaaKqzGeGaeyOeI0 IaamisaaGcbaqcLbsacaWGibaaaaGccaGLOaGaayzkaaWcdaahaaqa beaajugWaiaaikdaaaqcLbsacaaIUaaaaa@527A@           (15)

 The deceleration parameter q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGXb aaaa@377B@ has the usual definition

q= a a ¨ a ˙ 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGXb GaaGypaiabgkHiTKqbaoaalaaakeaajugibiaadggaceWGHbGbamaa aOqaaKqzGeGabmyyayaacaWcdaahaaqabeaajugWaiaaikdaaaaaaK qzGeGaaGOlaaaa@4122@               (16)

The sign of q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGXb aaaa@377B@ indicates whether the model inflates or not. The positive sign of q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGXb aaaa@377B@ corresponds to a standard deceleration model whereas the negative sign indicates inflation.

Exact solution

 We obtain solution of the field equations (7)–(10) which are four equations in six unknowns A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb aaaa@374B@ , B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb aaaa@374C@ , C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGdb aaaa@374D@ , ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHbp GCaaa@3845@ , P ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGqb Gbaebaaaa@3772@ and ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEaaa@3848@ . Therefore to find a deterministic solution we shall need two extra conditions on physical ground of the problem or for simply for mathematical convenience.

Subtracting (7) from (8), (8) from (9) and (9) from (7) and integrating the results, we obtain

A ˙ A B ˙ B = k 1 a 3 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGabmyqayaacaaakeaajugibiaadgeaaaGaeyOeI0scfa4a aSaaaOqaaKqzGeGabmOqayaacaaakeaajugibiaadkeaaaGaaGypaK qbaoaalaaakeaajugibiaadUgalmaaBaaabaqcLbmacaaIXaaaleqa aaGcbaqcLbsacaWGHbWcdaahaaGcbeqaaKqzadGaaG4maaaaaaqcLb sacaaISaaaaa@47A3@        (17)

B ˙ B C ˙ C = k 2 a 3 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGabmOqayaacaaakeaajugibiaadkeaaaGaeyOeI0scfa4a aSaaaOqaaKqzGeGabm4qayaacaaakeaajugibiaadoeaaaGaaGypaK qbaoaalaaakeaajugibiaadUgajuaGdaWgaaWcbaqcLbmacaaIYaaa leqaaaGcbaqcLbsacaWGHbWcdaahaaqabeaajugWaiaaiodaaaaaaK qzGeGaaGilaaaa@482C@    (18)

A ˙ A C ˙ C = k 3 a 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGabmyqayaacaaakeaajugibiaadgeaaaGaeyOeI0scfa4a aSaaaOqaaKqzGeGabm4qayaacaaakeaajugibiaadoeaaaGaaGypaK qbaoaalaaakeaajugibiaadUgajuaGdaWgaaWcbaqcLbmacaaIZaaa leqaaaGcbaqcLbsacaWGHbWcdaahaaqabeaajugWaiaaiodaaaaaaa aa@46E6@           (19)

where k 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb WcdaWgaaqaaKqzadGaaGymaaWcbeaaaaa@3995@ , k 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb WcdaWgaaadbaGaaGOmaaqabaaaaa@3869@ and k 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb WcdaWgaaqaaKqzadGaaG4maaWcbeaaaaa@3997@  are constants of integration . Using (17), (18), (19) in (13) and simplifying, we find that

σ= k a 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHdp WCcaaI9aqcfa4aaSaaaOqaaKqzGeGaam4AaaGcbaqcLbsacaWGHbWc daahaaqabeaajugWaiaaiodaaaaaaaaa@3ECD@   (20)

 Where the constant k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb aaaa@3775@ is given by k 2 = k 1 2 + k 2 2 + k 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb WcdaahaaqabeaajugWaiaaikdaaaqcLbsacaaI9aGaam4AaSWaa0ba aeaajugWaiaaigdaaSqaaKqzadGaaGOmaaaajugibiabgUcaRiaadU galmaaDaaabaqcLbmacaaIYaaaleaajugWaiaaikdaaaqcLbsacqGH RaWkcaWGRbWcdaqhaaqaaKqzadGaaG4maaWcbaqcLbmacaaIYaaaaa aa@4CB8@ . Equations (17), (18) and (19) suggest that

A n =BC, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb WcdaahaaqabeaajugWaiaad6gaaaqcLbsacaaI9aGaamOqaiaadoea caaISaaaaa@3D34@  (21)

n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb aaaa@3778@ is a positive constant. In view of (21), we can write

B= A n/2 D,C= A n/2 D 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb GaaGypaiaadgealmaaCaaabeqaaKqzadGaamOBaiaai+cacaaIYaaa aKqzGeGaamiraiaaiYcacaWGdbGaaGypaiaadgeajuaGdaahaaWcbe qaaKqzadGaamOBaiaai+cacaaIYaaaaKqzGeGaamiraSWaaWbaaeqa baqcLbmacqGHsislcaaIXaaaaaaa@49AB@  (22)

Where D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGeb aaaa@374E@ is a function of time. Substituting (22) in to (18), we obtain

D ˙ D = K a 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGabmirayaacaaakeaajugibiaadseaaaGaaGypaKqbaoaa laaakeaajugibiaadUeaaOqaaKqzGeGaamyyaSWaaWbaaeqabaqcLb macaaIZaaaaaaaaaa@3FC6@                 (23)

Where K= 1 2 k 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb GaaGypaKqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaaGOmaaaa caWGRbWcdaWgaaqaaKqzadGaaGOmaaWcbeaaaaa@3E74@ . From (23) we can determine D(t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGeb GaaGikaiaadshacaaIPaaaaa@39AC@ if the average scale a(t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGHb GaaGikaiaadshacaaIPaaaaa@39C9@ is explicitly known. We assume that the Hubble parameter H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGib aaaa@3752@ to be related to the average scale factor a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGHb aaaa@376B@ by the relation

H=m a 1/m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGib GaaGypaiaad2gacaWGHbWcdaahaaqabeaajugWaiabgkHiTiaaigda caaIVaGaamyBaaaaaaa@3E9F@            (24)

 Where m>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb GaaGOpaiaaicdaaaa@38F9@ is a constant.35 Combining (14) and (24) and solving the resulting equation, we get

a(t)=(t+c ) m ,c=constant. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGHb GaaGikaiaadshacaaIPaGaaGypaiaaiIcacaWG0bGaey4kaSIaam4y aiaaiMcajuaGdaahaaWcbeqaaKqzadGaamyBaaaajugibiaaiYcaca qGJbGaaeypaiaabogacaqGVbGaaeOBaiaabohacaqG0bGaaeyyaiaa b6gacaqG0bGaaeOlaaaa@4CB1@ (25)

 This gives a constant value of the deceleration parameter as

q=1+ 1 m . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGXb GaaGypaiabgkHiTiaaigdacqGHRaWkjuaGdaWcaaGcbaqcLbsacaaI Xaaakeaajugibiaad2gaaaGaaGOlaaaa@3F01@  (26)

Since recent observational data indicates that the universe is accelerating and the value of deceleration parameter lies somewhere in the range 1<q<0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaIXaGaaGipaiaadghacaaI8aGaaGimaaaa@3B69@ , so we have m>1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb GaaGOpaiaaigdaaaa@38FA@ for the accelerating universe.

From (23) and (25), we obtain

D=Mexp{ K (13m) (t+c) 1 (13m) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGeb GaaGypaiaad2eaciGGLbGaaiiEaiaacchajuaGdaGadaGcbaqcfa4a aSaaaOqaaKqzGeGaam4saaGcbaqcLbsacaaIOaGaaGymaiabgkHiTi aaiodacaWGTbGaaGykaaaacaaIOaGaamiDaiabgUcaRiaadogacaaI Paqcfa4aaWbaaSqabeaadaWcaaqaaKqzadGaaGymaaWcbaqcLbmaca aIOaGaaGymaiabgkHiTiaaiodacaWGTbGaaGykaaaaaaaakiaawUha caGL9baaaaa@52C2@ (27)

 Where M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb aaaa@3757@  is the constant of integration. Also, equation (11) and (25) lead to

A= ( t+c ) 3m n+1 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb GaaGypaKqbaoaabmaakeaajugibiaadshacqGHRaWkcaWGJbaakiaa wIcacaGLPaaalmaaCaaabeqaamaalaaabaqcLbmacaaIZaGaamyBaa WcbaqcLbmacaWGUbGaey4kaSIaaGymaaaaaaqcLbsacaaIUaaaaa@45B9@      (28)

 Substituting (27) in (22), we find that the solutions for the scale function B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb aaaa@374C@ and C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGdb aaaa@374D@ as

B=M ( t+c ) 3mn 2(n+1) exp[ K (13m) ( t+c ) 13m ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb GaaGypaiaad2eajuaGdaqadaGcbaqcLbsacaWG0bGaey4kaSIaam4y aaGccaGLOaGaayzkaaWcdaahaaqabeaadaWcaaqaaKqzadGaaG4mai aad2gacaWGUbaaleaajugWaiaaikdacaaIOaGaamOBaiabgUcaRiaa igdacaaIPaaaaaaajugibiGacwgacaGG4bGaaiiCaKqbaoaadmaake aajuaGdaWcaaGcbaqcLbsacaWGlbaakeaajugibiaaiIcacaaIXaGa eyOeI0IaaG4maiaad2gacaaIPaaaaKqbaoaabmaakeaajugibiaads hacqGHRaWkcaWGJbaakiaawIcacaGLPaaalmaaCaaabeqaaKqzadGa aGymaiabgkHiTiaaiodacaWGTbaaaaGccaGLBbGaayzxaaqcLbsaca aISaaaaa@6127@                 (29)

C=M ( t+c ) 3mn 2(n+1) exp[ K (13m) ( t+c ) 13m ]. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGdb GaaGypaiaad2eajuaGdaqadaGcbaqcLbsacaWG0bGaey4kaSIaam4y aaGccaGLOaGaayzkaaqcfa4aaWbaaSqabeaadaWcaaqaaKqzadGaaG 4maiaad2gacaWGUbaaleaajugWaiaaikdacaaIOaGaamOBaiabgUca RiaaigdacaaIPaaaaaaajugibiGacwgacaGG4bGaaiiCaKqbaoaadm aakeaajugibiabgkHiTKqbaoaalaaakeaajugibiaadUeaaOqaaKqz GeGaaGikaiaaigdacqGHsislcaaIZaGaamyBaiaaiMcaaaqcfa4aae WaaOqaaKqzGeGaamiDaiabgUcaRiaadogaaOGaayjkaiaawMcaaKqb aoaaCaaaleqabaqcLbmacaaIXaGaeyOeI0IaaG4maiaad2gaaaaaki aawUfacaGLDbaajugibiaai6caaaa@63C2@              (30)

provided m 1 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb GaeyiyIKBcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaaIZaaa aaaa@3C86@ . Without lose of any generality we can take m=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb GaaGypaiaaigdaaaa@38F9@ . Hence, the metric of our solution can be written in the form

d s 2 =d T 2 T 6m n+1 d x 2 T 3mn n+1 exp{ 2K (13m) T (13m) }d y 2 T 3mn n+1 exp{ 2K (13m) T (13m) }d z 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGKb Gaam4CaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaaGypaiaa dsgacaWGubWcdaahaaqabeaajugWaiaaikdaaaqcLbsacqGHsislca WGubWcdaahaaqabeaadaWcaaqaaKqzadGaaGOnaiaad2gaaSqaaKqz adGaamOBaiabgUcaRiaaigdaaaaaaKqzGeGaamizaiaadIhalmaaCa aabeqaaKqzadGaaGOmaaaajugibiabgkHiTiaadsfalmaaCaaabeqa amaalaaabaqcLbmacaaIZaGaamyBaiaad6gaaSqaaKqzadGaamOBai abgUcaRiaaigdaaaaaaKqzGeGaciyzaiaacIhacaGGWbqcfa4aaiWa aOqaaKqbaoaalaaakeaajugibiaaikdacaWGlbaakeaajugibiaaiI cacaaIXaGaeyOeI0IaaG4maiaad2gacaaIPaaaaiaadsfalmaaCaaa beqaaKqzadGaaGikaiaaigdacqGHsislcaaIZaGaamyBaiaaiMcaaa aakiaawUhacaGL9baajugibiaadsgacaWG5bWcdaahaaqabeaajugW aiaaikdaaaqcLbsacqGHsislcaWGubWcdaahaaqabeaadaWcaaqaaK qzadGaaG4maiaad2gacaWGUbaaleaajugWaiaad6gacqGHRaWkcaaI XaaaaaaajugibiGacwgacaGG4bGaaiiCaKqbaoaacmaakeaajuaGda WcaaGcbaqcLbsacqGHsislcaaIYaGaam4saaGcbaqcLbsacaaIOaGa aGymaiabgkHiTiaaiodacaWGTbGaaGykaaaacaWGubWcdaahaaqabe aajugWaiaaiIcacaaIXaGaeyOeI0IaaG4maiaad2gacaaIPaaaaaGc caGL7bGaayzFaaqcLbsacaWGKbGaamOEaSWaaWbaaeqabaqcLbmaca aIYaaaaaaa@96E9@        (31)

 Where T=t+c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub GaaGypaiaadshacqGHRaWkcaWGJbaaaa@3AE8@ .

Physical features of the model

We discuss the physical and kinematical features of the cosmological model represented by the line–element (31). The expression for energy density and effective pressure as calculated from (7)–(10) are

ρ= 1 (8π+3λ) 2 λ 2 [(8π+3λ){ 9 m 2 n(n+2) 4(n+ 1) 2 T 2 + 3mk (n+1) T (3m+1) K 2 T 6m } λ{ 9 m 2 (n+2) 2 4(n+ 1) 2 T 2 + 3m(3n+2) 2(n+ 1) 2 T 2 + 2 K 2 T 6m 3m T 2 }], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajugibi abeg8aYjaai2dajuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaa iIcacaaI4aGaeqiWdaNaey4kaSIaaG4maiabeU7aSjaaiMcalmaaCa aabeqaaKqzadGaaGOmaaaajugibiabgkHiTiabeU7aSLqbaoaaCaaa leqabaqcLbmacaaIYaaaaaaajugibiaaiUfacaaIOaGaaGioaiabec 8aWjabgUcaRiaaiodacqaH7oaBcaaIPaqcfa4aaiWaaOqaaKqbaoaa laaakeaajugibiaaiMdacaWGTbWcdaahaaqabeaajugWaiaaikdaaa qcLbsacaWGUbGaaGikaiaad6gacqGHRaWkcaaIYaGaaGykaaGcbaqc LbsacaaI0aGaaGikaiaad6gacqGHRaWkcaaIXaGaaGykaKqbaoaaCa aaleqabaqcLbmacaaIYaaaaKqzGeGaamivaSWaaWbaaeqabaqcLbma caaIYaaaaaaajugibiabgUcaRKqbaoaalaaakeaajugibiaaiodaca WGTbGaam4AaaGcbaqcLbsacaaIOaGaamOBaiabgUcaRiaaigdacaaI PaGaamivaSWaaWbaaeqabaqcLbmacaaIOaGaaG4maiaad2gacqGHRa WkcaaIXaGaaGykaaaaaaqcLbsacqGHsisljuaGdaWcaaGcbaqcLbsa caWGlbWcdaahaaqabeaajugWaiaaikdaaaaakeaajugibiaadsfalm aaCaaabeqaaKqzadGaaGOnaiaad2gaaaaaaaGccaGL7bGaayzFaaaa baqcLbsacqGHsislcqaH7oaBjuaGdaGadaGcbaqcfa4aaSaaaOqaaK qzGeGaaGyoaiaad2galmaaCaaabeqaaKqzadGaaGOmaaaajugibiaa iIcacaWGUbGaey4kaSIaaGOmaiaaiMcajuaGdaahaaWcbeqaaKqzad GaaGOmaaaaaOqaaKqzGeGaaGinaiaaiIcacaWGUbGaey4kaSIaaGym aiaaiMcalmaaCaaabeqaaKqzadGaaGOmaaaajugibiaadsfalmaaCa aabeqaaKqzadGaaGOmaaaaaaqcLbsacqGHRaWkjuaGdaWcaaGcbaqc LbsacaaIZaGaamyBaiaaiIcacaaIZaGaamOBaiabgUcaRiaaikdaca aIPaaakeaajugibiaaikdacaaIOaGaamOBaiabgUcaRiaaigdacaaI Paqcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacaWGubqcfa4aaW baaSqabeaajugWaiaaikdaaaaaaKqzGeGaey4kaSscfa4aaSaaaOqa aKqzGeGaaGOmaiaadUealmaaCaaabeqaaKqzadGaaGOmaaaaaOqaaK qzGeGaamivaSWaaWbaaeqabaqcLbmacaaI2aGaamyBaaaaaaqcLbsa cqGHsisljuaGdaWcaaGcbaqcLbsacaaIZaGaamyBaaGcbaqcLbsaca WGubWcdaahaaqabeaajugWaiaaikdaaaaaaaGccaGL7bGaayzFaaqc LbsacaaIDbGaaGilaaaaaa@CDF6@ (32)

p= 1 (8π+3λ) 2 λ 2 [λ{ 9 m 2 n (n+1) 2 T 2 + 9 m 2 n 2 4(n+ 1) 2 T 2 K 2 T 6m } (8π+3λ){ 9 m 2 (n+1) 2 T 2 + 9 m 2 n 2 4(n+ 1) 2 T 2 + 2 K 2 T 6m 3m T 2 }]. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajugibi aadchacaaI9aqcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaaI OaGaaGioaiabec8aWjabgUcaRiaaiodacqaH7oaBcaaIPaqcfa4aaW baaSqabeaajugWaiaaikdaaaqcLbsacqGHsislcqaH7oaBlmaaCaaa beqaaKqzadGaaGOmaaaaaaqcLbsacaaIBbGaeq4UdWwcfa4aaiWaaO qaaKqbaoaalaaakeaajugibiaaiMdacaWGTbqcfa4aaWbaaSqabeaa jugWaiaaikdaaaqcLbsacaWGUbaakeaajugibiaaiIcacaWGUbGaey 4kaSIaaGymaiaaiMcajuaGdaahaaWcbeqaaKqzadGaaGOmaaaajugi biaadsfalmaaCaaabeqaaKqzadGaaGOmaaaaaaqcLbsacqGHRaWkju aGdaWcaaGcbaqcLbsacaaI5aGaamyBaSWaaWbaaeqabaqcLbmacaaI YaaaaKqzGeGaamOBaSWaaWbaaeqabaqcLbmacaaIYaaaaaGcbaqcLb sacaaI0aGaaGikaiaad6gacqGHRaWkcaaIXaGaaGykaKqbaoaaCaaa leqabaqcLbmacaaIYaaaaKqzGeGaamivaSWaaWbaaeqabaqcLbmaca aIYaaaaaaajugibiabgkHiTKqbaoaalaaakeaajugibiaadUeajuaG daahaaWcbeqaaKqzadGaaGOmaaaaaOqaaKqzGeGaamivaKqbaoaaCa aaleqabaqcLbmacaaI2aGaamyBaaaaaaaakiaawUhacaGL9baaaeaa jugibiabgkHiTiaaiIcacaaI4aGaeqiWdaNaey4kaSIaaG4maiabeU 7aSjaaiMcajuaGdaGadaGcbaqcfa4aaSaaaOqaaKqzGeGaaGyoaiaa d2gajuaGdaahaaWcbeqaaKqzadGaaGOmaaaaaOqaaKqzGeGaaGikai aad6gacqGHRaWkcaaIXaGaaGykaKqbaoaaCaaaleqabaqcLbmacaaI YaaaaKqzGeGaamivaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaaaaju gibiabgUcaRKqbaoaalaaakeaajugibiaaiMdacaWGTbWcdaahaaqa beaajugWaiaaikdaaaqcLbsacaWGUbqcfa4aaWbaaSqabeaajugWai aaikdaaaaakeaajugibiaaisdacaaIOaGaamOBaiabgUcaRiaaigda caaIPaWcdaahaaqabeaajugWaiaaikdaaaqcLbsacaWGubWcdaahaa qabeaajugWaiaaikdaaaaaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqz GeGaaGOmaiaadUealmaaCaaabeqaaKqzadGaaGOmaaaaaOqaaKqzGe GaamivaKqbaoaaCaaaleqabaqcLbmacaaI2aGaamyBaaaaaaqcLbsa cqGHsisljuaGdaWcaaGcbaqcLbsacaaIZaGaamyBaaGcbaqcLbsaca WGubWcdaahaaqabeaajugWaiaaikdaaaaaaaGccaGL7bGaayzFaaqc LbsacaaIDbGaaGOlaaaaaa@C9AD@ (33)

 For the specification of ξ(t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEcaaIOaGaamiDaiaaiMcaaaa@3AA6@ , we assume that the fluid obeys the equation state

p=γρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGWb GaaGypaiabeo7aNjabeg8aYbaa@3BA8@   (34)

 Where 0γ1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIWa GaeyizImQaeq4SdCMaeyizImQaaGymaaaa@3D0B@  is a constant. Then from (3), we obtain

ξ(t)= 1 3m[(8π+3λ ) 2 λ 2 ] [γ(8π+3λ){ 9 m 2 n (n+1) 2 T + 9 m 2 n 2 4(n+1)T K 2 T 6m1 } γλ{ 9 m 2 (n+1) 2 T + 9 m 2 n 2 4 (n+1) 2 T + 2 K 2 T 6m1 3m T }λ{ 9 m 2 (n+1) 2 T + 9 m 2 n 2 4 (n+1) 2 T 2 K 2 T 6m1 } λ(8π+3λ){ 9 m 2 (n+1) 2 T + 9 m 2 n 2 4(n+ 1) 2 T + 2 K 2 T 6m1 3m T }. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajugibi abe67a4jaaiIcacaWG0bGaaGykaiaai2dajuaGdaWcaaGcbaqcLbsa caaIXaaakeaajugibiaaiodacaWGTbGaaG4waiaaiIcacaaI4aGaeq iWdaNaey4kaSIaaG4maiabeU7aSjaaiMcalmaaCaaabeqaaKqzadGa aGOmaaaajugibiabgkHiTiabeU7aSLqbaoaaCaaaleqabaqcLbmaca aIYaaaaKqzGeGaaGyxaaaacaaIBbGaeq4SdCMaaGikaiaaiIdacqaH apaCcqGHRaWkcaaIZaGaeq4UdWMaaGykaKqbaoaacmaakeaajuaGda WcaaGcbaqcLbsacaaI5aGaamyBaSWaaWbaaeqabaqcLbmacaaIYaaa aKqzGeGaamOBaaGcbaqcLbsacaaIOaGaamOBaiabgUcaRiaaigdaca aIPaWcdaahaaqabeaajugWaiaaikdaaaqcLbsacaWGubaaaiabgUca RKqbaoaalaaakeaajugibiaaiMdacaWGTbWcdaahaaqabeaajugWai aaikdaaaqcLbsacaWGUbWcdaahaaqabeaajugWaiaaikdaaaaakeaa jugibiaaisdacaaIOaGaamOBaiabgUcaRiaaigdacaaIPaGaamivaa aacqGHsisljuaGdaWcaaGcbaqcLbsacaWGlbqcfa4aaWbaaSqabeaa jugWaiaaikdaaaaakeaajugibiaadsfalmaaCaaabeqaaKqzadGaaG Onaiaad2gacqGHsislcaaIXaaaaaaaaOGaay5Eaiaaw2haaaqaaKqz GeGaeyOeI0Iaeq4SdCMaeq4UdWwcfa4aaiWaaeaadaWcaaqaaKqzGe GaaGyoaiaad2gajuaGdaahaaqabKqbGeaajugWaiaaikdaaaaajuaG baqcLbsacaGGOaGaaiOBaiabgUcaRiaaigdacaGGPaqcfa4aaWbaae qajuaibaqcLbmacaaIYaaaaKqzGeGaamivaaaacqGHRaWkjuaGdaWc aaqaaKqzGeGaaGyoaiaad2gajuaGdaahaaqabKqbGeaajugWaiaaik daaaqcLbsacaWGUbqcfa4aaWbaaeqajuaibaqcLbmacaaIYaaaaaqc fayaaKqzGeGaaGinaiaacIcacaGGUbGaey4kaSIaaGymaiaacMcaju aGdaahaaqabKqbGeaajugWaiaaikdaaaqcLbsacaWGubaaaiabgUca RKqbaoaalaaabaqcLbsacaaIYaGaam4saKqbaoaaCaaabeqaaKqzad GaaGOmaaaaaKqbagaajugibiaadsfajuaGdaahaaqabKqbGeaajugW aiaaiAdacaWGTbGaeyOeI0IaaGymaaaaaaqcLbsacqGHsisljuaGda WcaaqaaKqzGeGaaG4maiaad2gaaKqbagaajugibiaadsfaaaaajuaG caGL7bGaayzFaaqcLbsacqGHsislcqaH7oaBjuaGdaGadaqaamaala aabaqcLbsacaaI5aGaamyBaSWaaWbaaKqbGeqabaqcLbmacaaIYaaa aaqcfayaaKqzGeGaaiikaiaac6gacqGHRaWkcaaIXaGaaiykaKqbao aaCaaabeqcfasaaKqzadGaaGOmaaaajugibiaadsfaaaGaey4kaSsc fa4aaSaaaeaajugibiaaiMdacaWGTbqcfa4aaWbaaeqajuaibaqcLb macaaIYaaaaKqzGeGaamOBaKqbaoaaCaaabeqcfasaaKqzadGaaGOm aaaaaKqbagaajugibiaaisdacaGGOaGaaiOBaiabgUcaRiaaigdaca GGPaqcfa4aaWbaaeqajuaibaqcLbmacaaIYaaaaKqzGeGaamivaaaa cqGHsisljuaGdaWcaaqaaKqzGeGaaGOmaiaadUeajuaGdaahaaqabK qbGeaajugWaiaaikdaaaaajuaGbaqcLbsacaWGubqcfa4aaWbaaeqa juaibaqcLbmacaaI2aGaamyBaiabgkHiTiaaigdaaaaaaaqcfaOaay 5Eaiaaw2haaaGcbaqcLbsacqGHsislcqaH7oaBcaaIOaGaaGioaiab ec8aWjabgUcaRiaaiodacqaH7oaBcaaIPaqcfa4aaiWaaOqaaKqbao aalaaakeaajugibiaaiMdacaWGTbWcdaahaaqabeaajugWaiaaikda aaaakeaajugibiaaiIcacaWGUbGaey4kaSIaaGymaiaaiMcalmaaCa aabeqaaKqzadGaaGOmaaaajugibiaadsfaaaGaey4kaSscfa4aaSaa aOqaaKqzGeGaaGyoaiaad2galmaaCaaabeqaaKqzadGaaGOmaaaaju gibiaad6galmaaCaaabeqaaKqzadGaaGOmaaaaaOqaaKqzGeGaaGin aiaaiIcacaWGUbGaey4kaSIaaGymaiaaiMcajuaGdaahaaWcbeqaaK qzadGaaGOmaaaajugibiaadsfaaaGaey4kaSscfa4aaSaaaOqaaKqz GeGaaGOmaiaadUeajuaGdaahaaWcbeqaaKqzadGaaGOmaaaaaOqaaK qzGeGaamivaSWaaWbaaeqabaqcLbmacaaI2aGaamyBaiabgkHiTiaa igdaaaaaaKqzGeGaeyOeI0scfa4aaSaaaOqaaKqzGeGaaG4maiaad2 gaaOqaaKqzGeGaamivaaaaaOGaay5Eaiaaw2haaKqzGeGaaGOlaaaa aa@3E6A@ (35)

 The physical and kinematical parameter has the values given by the following expression:

V= T 3m , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb GaaGypaiaadsfalmaaCaaabeqaaKqzadGaaG4maiaad2gaaaqcLbsa caaISaaaaa@3D4F@  (36)

θ= 3m T , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCcaaI9aqcfa4aaSaaaOqaaKqzGeGaaG4maiaad2gaaOqaaKqzGeGa amivaaaacaaISaaaaa@3E10@  (37)

σ= k T 3m . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHdp WCcaaI9aqcfa4aaSaaaOqaaKqzGeGaam4AaaGcbaqcLbsacaWGubqc fa4aaWbaaSqabeaajugWaiaaiodacaWGTbaaaaaajugibiaai6caaa a@4187@ (38)

 The directional Hubble’s parameters and the average Hubble parameter are given by

H 2 = 3mn 2(n+1)T + K T 3m , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGib WcdaWgaaqaaKqzadGaaGOmaaWcbeaajugibiaai2dajuaGdaWcaaGc baqcLbsacaaIZaGaamyBaiaad6gaaOqaaKqzGeGaaGOmaiaaiIcaca WGUbGaey4kaSIaaGymaiaaiMcacaWGubaaaiabgUcaRKqbaoaalaaa keaajugibiaadUeaaOqaaKqzGeGaamivaKqbaoaaCaaaleqabaqcLb macaaIZaGaamyBaaaaaaqcLbsacaaISaaaaa@4DFD@  (40)

H 2 = 3mn 2(n+1)T K T 3m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGib WcdaWgaaqaaKqzadGaaGOmaaWcbeaajugibiaai2dajuaGdaWcaaGc baqcLbsacaaIZaGaamyBaiaad6gaaOqaaKqzGeGaaGOmaiaaiIcaca WGUbGaey4kaSIaaGymaiaaiMcacaWGubaaaiabgkHiTKqbaoaalaaa keaajugibiaadUeaaOqaaKqzGeGaamivaKqbaoaaCaaaleqabaqcLb macaaIZaGaamyBaaaaaaaaaa@4CC3@ (41)
And
H= m T . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGib GaaGypaKqbaoaalaaakeaajugibiaad2gaaOqaaKqzGeGaamivaaaa caaIUaaaaa@3C6C@  (42)

The anisotropic parameter Λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHBo ataaa@37FA@ has the value

Δ= 1 2 ( n2 n+1 ) 2 + 2 K 2 3 m 2 T 6m2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqucLb sacaaI9aqcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaaIYaaa aKqbaoaabmaakeaajuaGdaWcaaGcbaqcLbsacaWGUbGaeyOeI0IaaG OmaaGcbaqcLbsacaWGUbGaey4kaSIaaGymaaaaaOGaayjkaiaawMca aSWaaWbaaeqabaqcLbmacaaIYaaaaKqzGeGaey4kaSscfa4aaSaaaO qaaKqzGeGaaGOmaiaadUealmaaCaaabeqaaKqzadGaaGOmaaaaaOqa aKqzGeGaaG4maiaad2galmaaCaaabeqaaKqzadGaaGOmaaaajugibi aadsfalmaaCaaabeqaaKqzadGaaGOnaiaad2gacqGHsislcaaIYaaa aaaajugibiaai6caaaa@5946@ (43)

We observe that the spatial volume and the three scale factors are zero at T=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub GaaGypaiaaicdaaaa@38DF@ . At this epoch the energy density and thermodynamic pressure assume infinite values. The expansion scalar and shear scalar are infinite at T=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub GaaGypaiaaicdaaaa@38DF@

Therefore; this model describes a continuously expanding shearing and accelerating universe with a big–bang start at T=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub GaaGypaiaaicdaaaa@38DF@ . The energy density and pressure are infinite at T=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub GaaGypaiaaicdaaaa@38DF@  and are decreasing functions with the passage of time, and ultimately tend to zero for large time. The bulk viscosity coefficient decreases as time increases and tend to zero. Thus, the model would essentially give an empty universe for large time. The fluid is highly anisotropic near the initial singularity. The anisotropy in the model is maintained throughout the passage of time. The ratio σ θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeq4WdmhakeaajugibiabeI7aXbaaaaa@3B3F@ tends to zero as T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub GaeyOKH4QaeyOhIukaaa@3ABC@ , which indicates that the shear scalar tends to zero faster than the expansion scalar. The directional Hubble’s parameters are different in three spatial directions for finite time and tend to zero for large time.

Conclusion

Evolution of spatially homogeneous Bianchi type–I cosmological model is studied in the presence of a bulk viscous fluid within the framework of f(R, T) gravity theory. We have obtained exact solutions of the field equations by using the special law of variation for Hubble’s parameter that yields a negative constant value of the deceleration parameter. The resulting cosmological model corresponds to continuously expanding, shearing and accelerating universe which would essentially gives an empty space for large time. The nature of the physical and kinematical parameters is discussed at the initial singularity and at infinity. The effect of the bulk viscosity is to produce a change in perfect fluid and hence exhibits the essential influence on the character of the solution. This effect is clearly visible in the expression of the isotropic pressure. Initially the fluid is highly viscous and viscosity decreases monotonically as time increases and becomes negligible for large time.

Acknowledgements

The authors would like to place on record their sincere thanks for honorable referees for their valuable suggestions which have helped in improving the quality of the paper.

Conflict of interest

The author declares there is no conflict of interest.

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