Reconstruction of modified f(R, T ) with Λ(T ) gravity in general class of Bianchicosmological models

Anirudh Pradhan1, Nasr Ahmed2,3, Bijan Saha4

1Department of Mathematics, Institute of Applied Sciences and Humanities, G. L. A. University, Mathura-281406, Uttar Pradesh, India

E-mail: pradhan.anirudh@gmail.com

2Mathematics Department, Deanery of Academic Services, Taibah University, Saudi Arabia

3Astronomy Department, National Research Institute of Astronomy and Geophysics, Helwan, Cairo, Egypt

E-mail: nasr.ahmed@nriag.sci.eg

4Laboratory of Information Technologies, Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region,Russia

E-mail: bijan@jinr.ru

Abstract

A new class of cosmological models in f(R, T ) modified theories of gravity (Harko et al. in Phys. Rev. D84:024020, 2011), where the gravitational Lagrangian is given by an arbitrary function of Ricci scalar R andthe trace of the stress-energy tensor T , have been investigated for a specific choice of f(R, T ) = f1(R)+f2(T )in a general class of Bianchi universe. Our intention is to reconstruct f(R, T ) models inspired by this speciallaw for the deceleration parameter (Akarsu and Dereli in Int. J. Theor. Phys. 51:612, 2012) which is linearin time with negative slope in connection with the theories of modified gravity. In the present study weconsider the cosmological constant Λ as a function of the trace of the stress energy-momentum-tensor, andsuch a model is nicknamed as “Λ(T ) gravity”. Such models may exhibit better consistency with the recentcosmological observations. We analyze the variation of pressure, energy density and cosmological term withcosmic time and discuss their physical consequences. We also discuss physical and geometric properties ofderived models.

Keywords: Bianchi universes, Modified gravity, Time dependent Λ-termPACS number: 04.50.Kd, 98.80.Es

1 Introduction

Modified theories of gravity have recently drawn much attention to explain the late time accelerated expansionof the universe and the existence of dark matter (DM) and dark energy (DE) [1]−[7]. Dark energy is relatedwith the modification of the Einstein gravity in such a way that it would give the gravitational alternative toDE. The modified theories of gravity justify the unification of DM and DE, deceleration to acceleration epochof the universe, description of hierarchy problem, dominance of effective DE, which help to solve the coincidenceproblem and many more. Currently, there are many candidates of dark energy, such as the cosmological con-stant, quintessence, phantom, quantum, nuclear binding energy and so on. Another alternative approach is tomodify general relativity itself. This is termed as ‘modified gravity’ approach which originates from the ideathat general relativity is incorrect in the cosmic scale and therefore need to be modified [8, 9, 10]. Noteworthyamongst them are the f(R) types theories of gravity formulated by Nojiri and Odintsov [11], El-Nabulsi [12],Capozziello et al. [13]; f(G) modified gravity discussed by Ghosh and Debnath [14], Sharif and Abbas [15] andthe f(R, T ) theory of gravity proposed by Harko et al. [8], where the gravitational Lagrangian is given by anarbitrary function of the Ricci scalar R and of the trace of the stress-energy tensor T . The cosmic accelerationin this modified f(R, T ) theory results not only from geometric contribution but also from the matter content.

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Recently, Chaubey and Sukla [16], Adhav [17], Samanta [18, 19], Samanta and Dhal [20], Chandel and Ram[21], Sharif et al. [22], Reddy et al. [23], Kiran and Reddy [24], Singh and Singh [25], Rao et al. [26], Sharmaan Singh [27], Singh and Sharma [28], Sahoo et al. [29] and references therein have studied cosmological modelsin f(R, T ) gravity in different Bianchi type space-times.

Bianchi universes play significant role in observational cosmology, since the WMAP data [30] seem to requirean addition to the standard cosmological model with positive cosmological constant that bears a likeness to theBianchi morphology [31]. After the discovery of CMB radiation, cosmology became a precise science. Experimen-tal studies of the isotropy of the Cosmic Microwave Background Radiation and speculation about the amountof helium formed at the early stages of the evolution of the universe have stimulated interest in anisotropiccosmological models. Since the observations of small anisotropies in CMBR [32] and large scale structure [33] itbecomes clear that a pure Friedman-Lemaitre-Robertson-Walker (FLRW) cosmology could not explain all theproperties of our Universe. It is therefore natural to consider anisotropic cosmological models that allow FLRWUniverse as special case. Recently, several authors [34−39] have studied anisotropic cosmological models anddiscussed the geometric and physical aspects of the universe in different context. Thus, the anisotropic Bianchimodels become of academical interest. Out of this, in the present study general class of Bianchi type space-timeis taken into consideration.

Inspired by the above discussions, in this paper, we propose to study cosmological model in f(R, T ) gravityin general class of Bianchi space-time by considering f(R, T ) = f1(R)+ f2(T ). In the present study we considerthe cosmological constant Λ as a function of the trace of the stress energy-momentum-tensor, and dub such amodel “Λ(T ) gravity” [40] where we have specified a certain form of Λ(T ).

2 Reconstruction of modified f(R, T ) with Λ(T ) gravity

The theory suggests a modified gravity action given by

S =1

16π

∫f(R, T )

√−gd4x+

∫Lm

√−gd4x, (1)

where f(R, T ) is an arbitrary function of the Ricci scalar, R, and the trace T of the stress-energy tensor ofthe matter, Tµν . Lm is the matter Lagrangian density. The stress-energy tensor of matter is defined as

Tµν = − 2√−g

δ√−gLm

δgµν, (2)

and its trace by T = gµνTµν . Assuming that the Lagrangian density Lm of matter depends only on themetric tensor components gµν and not on its derivatives leads to

Tµν = gµνLm − 2∂Lm

∂gµν(3)

By varying the action S with respect to the metric tensor components gµν , the gravitational field equationsof f(R, T ) gravity is obtained as

fR(R, T )Rµν − 1

2f(R, T )gµν + (gµν2−∇µ∇ν)fR(R, T ) (4)

= 8πTµν − fT (R, T )Tµν − fT (R, T )Θµν .

where 2 = ∇i∇i, fR(R, T ) = ∂f(R,T )∂R , fT (R, T ) = ∂f(R,T )

∂T and ∇i denotes the covariant derivative. Θµν is givenby

Θµν = −2Tµν + gµνLm − 2gαβ∂2Lm

∂gµν∂gαβ(5)

Using matter Lagrangian Lm, the stress-energy tensor of the matter Lagrangian is given by

Tµν = (ρ+ p)uµuν − pgµν (6)

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where uµ = (0, 0, 0, 1) is the four velocity in the moving coordinates which satisfies the conditions uµuµ = 1 anduµ∇νuµ = 0. ρ and p are the energy density and pressure of the fluid respectively. With the use of Eq. (5), weobtain for the variation of the stress-energy of a perfect fluid the expression

Θµν = −2Tµν − pgµν (7)

Since the field equations in f(R, T ) gravity also depend on the physical nature of the matter field (through thetensor Θµν), for each choice of f we can obtain several theoretical models. Three explicit specification of thefunctional form of f has been considered in [8]

f(R, T ) =

R+ 2f(T )f1(R) + f2(T )f1(R) + f2(R)f3(T )

The cosmological consequences for the first class was discussed in details by Chaubey and Shukla [16]. In thispaper we are considering the cosmological consequences of the second class for which f(R, T ) = f1(R) + f2(T ).The gravitational field equations (4) becomes

f′

1(R)Rµν − 1

2f1(R)gµν + (gµν2−∇µ∇ν)f

′

1(R) =

8πTµν + f′

2(T )Tµν +

(f

′

2(T )p+1

2f2(T )

)gµν , (8)

where the prime denotes differentiation with respect to the argument. The field equations of the standard f(R)gravity can be recovered for p = 0 (the dust case) and f2(T ) = 0. We consider a particular form of the functionsf1(R) = λ1R and f2(T ) = λ2T where λ1 and λ2 are arbitrary parameters. In this article we take λ1 = λ2 = λ(constant) so that f(R, T ) = λ(R+ T ). Equation (8) can now be written as

λRµν − 1

2λRgµν = 8πTµν + λTµν + λ(p+

1

2T )gµν . (9)

Taking into account that Gµν = Rµν − 12gµνR, we get

λGµν = 8πTµν + λTµν + λ

(p+

1

2T

)gµν . (10)

This could be rearranged as

Gµν −(p+

1

2T

)gµν =

8π + λ

λTµν . (11)

Recalling Einstein equations with cosmological constant

Gµν + Λgµν = 8πTµν . (12)

We choose a positive small value for the arbitrary λ so that we have the same sign of the RHS of Eqs. (11) and(12), we keep this choice of λ through the article. The term

(p+ 1

2T)can now be regarded as the negative of

the cosmological constant in this gravity model. i.e,

p+1

2T ≡ −Λ(T ). (13)

Considering the perfect fluid case, we have for our model

Λ =1

2(p− ρ). (14)

The above reconstruction of modified f(R, T ) gravity theory is also recently discussed by Ahmed and Pradhan[41]. They obtained a new class of Bianchi type V cosmological models in f(R, T ) modified gravity theory withΛ(T ).

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3 Metric and the basic equations

We use the following metric of general class of Bianchi cosmological model

ds2 = dt2 − a21dx2 − a22e

−2xdy2 − a23e−2mxdz2, (15)

where the functions a1(t), a2(t) and a3(t) are the three anisotropic directions of expansion in normal threedimensional space. Those three functions are equal in FRW models due to the radial symmetry and so we haveonly one function a(t) there. m = 0 corresponds to type III Bianchi models, m = 1 corresponds to type V ,m = −1 corresponds to type V I0, and all other m give V Ih, where m = h− 1. The average scale factor a, thespatial volume V and the average Hubble’s parameter H are defined as

a = (a1a2a3)1/3 , (16)

V = a3 = a1a2a3 , (17)

and

H =1

3

V

V=

1

3

(a1a1

+a2a2

+a3a3

)=

1

3(H1 +H2 +H3) , (18)

respectively with H1 = a1

a1, H2 = a2

a2and H3 = a3

a3where the dot denotes differentiation with respect to cosmic

time t.

Assuming G = c = 1 in proper unit, the cosmological field equations for energy momentum tensor given byEq. (6) and metric given by Eq. (15) read as

−m

a21+ H2 + H3 +H2

2 +H2H3 +H23 =

(8π + λ

λ

)p− Λ , (19)

−m2

a21+ H3 + H1 +H2

3 +H3H1 +H21 =

(8π + λ

λ

)p− Λ , (20)

− 1

a21+ H1 + H2 +H2

1 +H1H2 +H22 =

(8π + λ

λ

)p− Λ , (21)

−m2 +m+ 1

a21+H1H2 +H2H3 +H3H1 = −

(8π + λ

λ

)ρ− Λ , (22)

−(m+ 1)H1 +H2 +mH3 = 0 . (23)

The expression for the dynamical scalars such as the expansion scalar (θ), anisotropy parameter (Am) and theshear scalar (σ) are defined as usual:

θ = ui;i = (H1 +H2 +H3) = 3H, (24)

Am =1

3

3∑i=1

(Hi −H

H

)2

, (25)

σ2 =1

2σijσ

ij =1

2σijσ

ij =1

2

[H2

1 +H22 +H2

3

]− θ2

6. (26)

With a known form of Λ(T ), we have a system of 5 differential equations with 5 unknowns a1, a2, a3, pand ρ. The solution is based on utilizing the linearly varying deceleration parameter q proposed by Akarsu andDereli [42] which gives

q = −aa

a2= −1 +

H

H2= −kt+ n− 1 (27)

where k and n are positive constants. The sign of q indicates whether the model inflates or not. Here q > 0(n > 1 + kt) corresponds to a decelerating model and −1 ≤ q < 0 (kt < n ≤ 1 + kt) describes an acceleratingmodel. Solving Eq. (27) for a(t) gives three different form of solutions for the scale factor:

a(t) = (nlt+ c1)1n , k = 0, n > 0 (28)

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a(t) = c2 exp[lt], k = 0, n = 0 (29)

a(t) = c3 exp

[2

ncoth

(kt

n− 1

)], k > 0, n > 1 (30)

where c1, c2 and c3 are constants of integration. Since Eq. (27) is an ordinary differential equation, to solve itwe have to assume some initial conditions, e.g., a(t = 0) = a0 and H(t = 0) = H0. Then inserting it into Eqs.(28), (29) and (30) we find c1 = an0 , c2 = a0 and c3 = a0 exp

[− 2

n coth (−1)], respectively.

From Eq. (28), it is clear that the condition for expanding universe is n(= q+kt+1) > 0. This law given byAkarsu and Dereli [42] also covers the special law of variation for Hubble’s parameter first proposed by Berman[43].

4 Cosmological solutions and discussions

In this section we discuss three different cases (i) k = 0, n > 0, (ii) k = 0, n = 0 and (iii) k > 0, n > 1 forphysically viable/plausible cosmological models.

Now from Eq. (23) one finds the relation between the metric functions:

a2am3

am+11

= const. (31)

Therefore, if one considers thata3 = V b = a3b, (32)

where b is any constant number, then in view of Eq. (16) or (17) all the metric functions a1, a2, a3 will becompletely determined in terms of V or a given by Eqs. (28), (29) or (30), respectively. These values of themetric functions could be inserted into the Einstein field equations to find the expressions for ρ and p. Takinginto account the expression for Λ given by Eq. (14) from the Einstein equations we find

Ap+Bρ = f1(a), (33)

−BP −Aρ = f2(a), (34)

where A = 8π/λ+ 1/2 and B = 1/2. Here f1(a) and f2(a) are to be determined from

f1(a) = − 1

a21+ H1 + H2 +H2

1 +H1H2 +H22 , (35)

and

f2(a) = −m2 +m+ 1

a21+H1H2 +H2H3 +H3H1. (36)

After a little manipulation from Eqs. (35) and (36) we find

ρ = −Bf1(a) +Af2(a)

A2 −B2, (37)

p =Af1(a) +Bf2(a)

A2 −B2, (38)

and

Λ =1

2

f1(a) + f2(a)

A−B. (39)

In what follows we consider the three cases given by Eqs. (28), (29) or (30) in details.

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Figure 1: Plot of p versus t in Model 1. Here λ = 0.1, c1 = 0.5, n = 0.5, b = 0.1 and l = 0.1.

Figure 2: Plot of ρ versus t in Model 1. Here λ = 0.1, c1 = 0.5, n = 0.5, b = 0.1 and l = 0.1.

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Figure 3: Plot of Λ versus t in Model 1. Here λ = 0.1, c1 = 0.5, n = 0.5, b = 0.1 and l = 0.1.

Figure 4: Plot of Energy condition versus t for m = 1 in Model 1. Here λ = 0.1, c1 = 0.5, n = 0.5, b = 0.1 andl = 0.1.

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Figure 5: Plot of Energy condition versus t for m = 0 in Model 1. Here λ = 0.1, c1 = 0.5, n = 0.5, b = 0.1 andl = 0.1.

Figure 6: Plot of Energy condition versus t for m = −1 in Model 1. Here λ = 0.1, c1 = 0.5, n = 0.5, b = 0.1and l = 0.1.

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4.1 Model 1

When k = 0, n > 0 and a3 = V b, where b be any constant, then from Eqs. (17), (23) and (28) we obtain

a1(t) = (nlt+ c1)3(1+mb−b)

n(m+2) , (40)

a2(t) = (nlt+ c1)3(1+m−b−2mb)

n(m+2) , (41)

a3(t) = (nlt+ c1)3bn . (42)

In this case the geometry of the universe given by Eq. (15) is reduced to

ds2 = dt2 − (nlt+ c1)6(1+mb−b)

n(m+2) dx2 − (nlt+ c1)6(1+m−b−2mb)

n(m+2) e−2mdy2 − (nlt+ c1)6bn e−2mxdz2. (43)

Using Eqs. (35)−(42), we obtain expressions for pressure, energy density and cosmological constant for modelgiven by (43) as

p =A

(A2 −B2)

− 1((nlt+ c1)

3+3mb−3bn(m+2)

)2 +(3 + 3mb− 3b)(3 + 3mb− 3b− 6mb)l2

(m+ 2)2(nlt+ c1)2

+3l2(1 +mb− b)(3 + 3mb− 3b− nm− 2n)

(m+ 2)2(nlt+ c1)2+

3l2(−1−m+ b+ 2mb)(−3− 3m+ 3b+ 6mb+ nm+ 2m)

(m+ 2)2(nlt+ c1)2

− 1

(160π + 1)

m2 +m+ 1((nlt+ c1)

3+3mb−3bn(m+2)

)2 − 1

(160π + 1)

9l2(−1−m− 2b− 2mb− 2m2b+ 3m2 + 3m2b2 + 3b2

2(m+ 2)2(nlt+ c1)2

(44)

ρ =1

(A2 −B2)

− 1

2((nlt+ c1)

3+3mb−3bn(m+2)

)2 +(3 + 3mb− 3b)(3 + 3mb− 3b− 6mb)l2

2(m+ 2)2(nlt+ c1)2

+3l2(1 +mb− b)(3 + 3mb− 3b− nm− 2n)

2(m+ 2)2(nlt+ c1)2+

3l2(−1−m+ b+ 2mb)(−3− 3m+ 3b+ 6mb+ nm+ 2m)

2(m+ 2)2(nlt+ c1)2

+(80π +1

2)

− m2 +m+ 1((nlt+ c1)

3+3mb−3bn(m+2)

)2 − 9l2(−1−m− 2b− 2mb− 2m2b+ 3m2 + 3m2b2 + 3b2

(m+ 2)2(nlt+ c1)2

(45)

Λ =λ

4π

− m2 +m+ 1((nlt+ c1)

3+3mb−3bn(m+2)

)2 +3l2(b− 1)(n− 1)

(nlt+ c1)2

(46)

Figure 1 plots the variation of pressure p versus cosmic time t for λ = b = l = 0.1, c2 = n = 0.5 as arepresentative case in Model 1. From Fig. 1, it is observed that isotropic pressure p is increasing function oftime and remains negative for all three values m = 0, m = 1 and m = −1 corresponding to Bianchi type-III,V and V I0 respectively throughout the evolution as expected which in turn indicates accelerated expansion ofthe universe. From the discovery of the accelerated expansion of the universe, it is generally assumed that thiscosmic acceleration is due to some kind of energy-matter with negative pressure known as ‘dark energy’. Recentcosmological observations [1−7] strongly indicate that our universe is dominated by a component with negativepressure, dubbed as dark energy, which constitutes with ≃ 3/4 of the critical density. The cosmic accelerationis realized with negative pressure and positive energy density that violate the strong energy condition. Thisviolation gives a reverse gravitational effect. Due to this effect, the Universe gets a jerk and the transition fromthe earlier deceleration phase to the recent acceleration phase take place.

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In this paper the whole figures are drawn as a representative case with appropriate arbitrary choice of con-stants of integration. The values of parameters are given in figure’s caption. In what follows in figures anddiscussions here and later in entire manuscript, we have chosen axis level conversion as follows. Along x-axis, tdenotes cosmic evolution time which is generally measured in Gigayears (1 Gyr = 109 years). Along y-axis allphysical quantities like pressure p, energy density ρ, cosmological constant Λ etc. are measured in geometrizedunits, where speed of light c = Gravitational constant G = 1 in Einstein’s general relativity.

From Eq. (45), it is noted that the energy density ρ(t) is a decreasing function of time and it approaches asmall positive value at present epoch. Figure 2 depicts the variation of ρ versus t in Model 1. The behaviourenergy density is clearly shown in Fig. 2 for all three values m = 0, m = 1 and m = −1 as a representative casewith appropriate choice of constants of integration and other physical parameters using reasonably well knownsituations.

In recent time the Λ-term has attracted theoreticians and observers for various reasons. The nontrivial roleof the vacuum in the early universe generate a Λ-term that leads to inflationary phase. Observationally, thisterm provides an additional parameter to accommodate conflicting data on the values of the Hubble constant,the deceleration parameter, the density parameter and the age of the universe (for example, see the referencesGunn and Tinsley [44]; Wampler and Burke [45]. The behaviour of the universe in this model will be determinedby the cosmological term Λ; this term has the same effect as a uniform mass density ρeff = −Λ/4πG, whichis constant in time. A positive value of Λ corresponds to a negative effective mass density (repulsion). Hence,we expect that in the universe with a positive value of Λ, the expansion will tend to accelerate; whereas in theuniverse with negative value of Λ, the expansion will slow down, stop and reverse.

From Eq. (46), we see that the cosmological term Λ is a decreasing function of time. Figure 3 depicts thevariation of Λ versus cosmic time t for all three values m = 0, m = 1 and m = −1 corresponding to Bianchitype-III, V and V I0 respectively in Model 1. In Fig. 3 we have plotted the variation of Λ with t in Model 1.From this Fig. 3, we observe that for m = −1 i.e. in Bianchi-V I0 model, Λ(t) is positive decreasing function oftime and it approaches to zero at late time. But for m = 0 and m = 1, Λ is negative increasing function of timeand it approaches to zero at late time. Though Λ is negative at initial stage in Bianchi-III and V models butit is an increasing function of time. This results in the accelerating mode of expansion because in every nexttime we get a less positive pressure than the previous one. This is the physics behind it. Models with negativecosmological constant have been investigated by Yadav [46], Saha and Boyadjiev [47], Pedram et al. [48], Biswasand Mazumdar [49], and Jotania et al. [50]. Really at present the estimation of Λ is not only complicatedbut it is uncertain and indirect too. However, the Einstein-Maxwell theory indicates to a different approachwhich looks simpler and more significant, since a possibility is illustrated for Λ ≤ 0 i.e. for the case when thepresence of Λ decelerates the expansion of universe. Recent cosmological observations suggest the existence ofa positive cosmological constant Λ with magnitude Λ(Gℏ/c3) ≈ 10−123. These observations on magnitude andred-shift of type Ia supernova suggest that our universe may be an accelerating one with induced cosmologicaldensity through the cosmological Λ-term. Thus the nature of Λ in our derived models are supported by recentobservations.

The expressions for physical parameters such as spatial volume (V ), mean Hubble’s parameter (H), expansionscalar (θ), shear scalar (σ) and anisotropy parameter (Am) for model given by Eq. (43) are given by

V = (nlt+ c1)3n , (47)

θ = 3H =3l

(nlt+ c1), (48)

Am =2(3b− 1)2(m2 +m+ 1)

(m+ 2)2, (49)

σ2 =3l2(3b− 1)2(m2 +m+ 1)

(m+ 2)2(nlt+ c1)2. (50)

We observe that the spatial volume (V ) is zero at t = t0, where t0 = − c1nl and expansion scalar (θ) is infinite,

which shows that the universe given by Eq. (43) starts evolving with zero volume at t = t0 with an infinite rateof expansion. The scale factors also vanishes at t = t0 and hence the model has a point type singularity [51]

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at the initial epoch. The pressure, energy density, cosmological term, Hubble’s parameter, shear scalar divergeat the initial singularity. The universe exhibits power-law expansion after big bang impulse. As t increases thescale factors and spatial volume increase but the expansion scalar decreases. Thus the rate of expansion slowsdown with increase in time. We also observe that p, ρ, Λ, H and σ2 decrease as t increases. Thus the modelgiven by Eq. (43) represents shearing, non-rotating and expanding model of the universe with a big bang start.The anisotropy parameter is uniform in the models. For b = 1

3 , the anisotropy dies out and the models becomeisotropic.

Energy conditions:The weak energy conditions (WEC) and dominant energy conditions (DEC) are given by

(i) ρ ≥ 0, (ii) ρ− p ≥ 0 and (iii) ρ+ p ≥ 0.

The strong energy conditions (SEC) are given by ρ+ 3p ≥ 0.

The left hand side of energy conditions have been graphed in Figs. 4, 5, 6 for m = 1, m = 0 and m = −1respectively in Model 1. From these figures, we observe that

• The WEC and DEC for the derived model are satisfied in all three Bianchi-III, V and V I0 models ofuniverse.

• The SEC is also satisfied in Bianchi-V model only (Fig. 4). But SEC is violated by Bianchi-III (Fig. 5)and V I0 (Fig. 6) models as expected.

It has been shown by Wald [52] that under very general conditions all Bianchi cosmologies (except BianchiIX) with a cosmological constant and an energy momentum tensor satisfying the strong and dominant energyconditions, will unavoidably enter a phase of exponential expansion. With the help of this result Jensen andStein-Schabes [53] showed that if the number e-folds the Universe expands during its exponential phase is givenby N then it will take a time of the order t ≃ e2N

√Λ for anisotropy to have any effect on the observable universe.

One remarkable result is the independence of this result from the type or magnitude of the initial anisotropy.

4.2 Model 2

When k = 0, n = 0, and a3 = V b, where b be any constant, then from Eqs. (17), (23) and (29) we get

a1(t) = c3(1+mb−b)

m+2

2 exp

(3l(1 +mb− b)

m+ 2t

), (51)

a2(t) = c3(1+m−b−2mb)

m+2

2 exp

(3l(1 +m− 2mb− b)

m+ 2t

), (52)

a3(t) = c3b2 exp (3blt). (53)

In this case the geometry of the universe given by Eq. (15) is modified into the form

ds2 = dt2 − c6(1+mb−b)

m+2

2 exp

(6l(1 +mb− b)

m+ 2t

)dx2

− c6(1+m−b−2mb)

m+2

2 exp

(6l(1 +m− 2mb− b)

m+ 2t

)dy2 − c6b2 exp (6blt)dz2. (54)

From Eqs. (35)−(39) and (51)−(53), one can obtain expressions for pressure, energy density and cosmologicalconstant for model given by Eq. (54) as follows

p =A

(A2 −B2)

− 1(c2

3+3mb−3 bm+2

)2 (e

(3+3mb−3 b)ltm+2

)2

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.

+9l2(3 + 3m− 6b− 6mb− 3m2b+ 3mb2 + 3m2b2 + 3b2 +m2)

(m+ 2)2+

1

160π + 1×

m2 +m+ 1(c2

3+3mb−3 bm+2

)2 (e

(3+3mb−3 b)ltm+2

)2 − 9l2(−1−m− 2b− 2mb− 2m2b) + 3mb2 + 3m2b2 + 3b2

2(m+ 2)2

(55)

ρ =1

(A2 −B2)

− 1

2(c2

3+3mb−3 bm+2

)2 (e

(3+3mb−3 b)ltm+2

)2+9l2(3 + 3m− 6b− 6mb− 3m2b+ 3mb2 + 3m2b2 + 3b2 +m2)

2(m+ 2)2+ (80π +

1

2)×

m2 +m+ 1(c2

3+3mb−3 bm+2

)2 (e

(3+3mb−3 b)ltm+2

)2 − 9l2(−1−m− 2b− 2mb− 2m2b) + 3mb2 + 3m2b2 + 3b2

(m+ 2)2

(56)

Λ =λ

4π

− m2 +m+ 2(c2

3+3mb−3 bm+2

)2 (e

(3+3mb−3 b)ltm+2

)2 + 9l2(b− 1)

(57)

The expressions for physical parameters such as spatial volume (V ), mean Hubble’s parameter (H), expansionscalar (θ), shear scalar (σ) and anisotropy parameter (Am) for model given by Eq. (54) are given by

V = c32e(3lt), (58)

θ = 3H = l, (59)

Am =2

(m+ 2)2[(1 +m)(1− 6b) + 3mb(3b− 2m) + 9b2(m2 + 1) +m2

], (60)

σ2 =9

(m+ 2)4[3(1 +m)− 9b{2(1 +m)− 3mb}]2 + 9

(m+ 2)4[m2(1− 6b) + 9b2(m2 + 1)

]2, (61)

The variations of pressure and energy density versus t have been observed for all three values m = 0, m = 1and m = −1 corresponding to Bianchi type-III, V and V I0 respectively in Model 2. The nature of both pand ρ are found to be same as in Model 1 in previous case. We have also observed that in all three cases thevariation of Λ is negative increasing function of time and reaches to a small positive value at late time (i.e., atpresent epoch). The recent cosmological observations admit a small positive value of Λ at present epoch. Theanisotropic parameter is constant through out the evolution of the universe. The figures are not given here forthis case.

The energy conditions for m = 1, m = 0 and m = −1 have also been tested in Model 2. We have observedthat

• The WEC and DEC for the derived model are satisfied in all three Bianchi-III, V and V I0 models ofuniverse.

• The SEC is also satisfied in Bianchi-V model only. But SEC is violated by Bianchi-III and V I0 modelsas expected.

All cosmological parameters are constant at t = 0. Thus, the universe starts with a non-singular state att = 0. As t increases, the scale factors and spatial volume increase exponentially. The universe exhibits uniformexponential expansion, expansion scalar being a constant. As t → ∞, the scale factors and volume of theuniverse become infinitely large. The pressure, energy density and Hubble’s factors become constant such thatp = −ρ. This shows that at late time the universe is dominated by vacuum energy, which drives the expansionof the universe. The model represents a shearing, non-rotating and expanding universe.

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4.3 Model 3

When k > 0, n > 1 and a3 = V b, where b be any constant, then from Eqs. (17), (23) and (30) we obtain

a1(t) = c3(1+mb−b)

m+2

3 exp

[6(1 +mb− b)

n(m+ 2)coth

(kt

n− 1

)], (62)

a2(t) = c3(1+mb−b−2mb)

m+2

3 exp

[6(1 +m− b− 2mb)

n(m+ 2)coth

(kt

n− 1

)], (63)

a3(t) = c3b3 exp

[6b

ncoth

(kt

n− 1

)]. (64)

In this case the geometry of the universe given by Eq. (15) reduces to

ds2 = dt2 − c6(1+mb−b)

m+2

3 exp

[12(1 +mb− b)

n(m+ 2)coth

(kt

n− 1

)]dx2

− c6(1+mb−b−2mb)

m+2

3 exp

[12(1 +m− b− 2mb)

n(m+ 2)coth

(kt

n− 1

)]dy2 − c6b3 exp

[12b

ncoth

(kt

n− 1

)]. (65)

From Eqs. (35)−(39) and (62)−(64), the expressions for pressure, energy density and cosmological constantfor model given by Eq. (65) are obtained as

p =A

(A2 −B2)

−1(

c33+3mb−3 b

m+2

)2(e

(6+6mb−6 b)arctanh( ktn

−1)n(m+2)

)2 − 12(−3 + 3b− kt+ n)(1 +mb− b)

(m+ 2)(kt− 2n)2t2

+12(−1−m+ b+ 2mb)(−ktm− 2kt+ nm+ 2n− 3− 3m+ 3b+ 6mb)

(m+ 2)2(kt− 2n)2t2+

1

80π + 12

×

−1

2

1(c3

3+3mb−3 bm+2

)2(e

(6+6mb−6 b)arctanh( ktn

−1)n(m+2)

)2 − 18(−1−m− 2b− 2mb− 2m2b+ 3mb2 + 3m2b2 + 3b2)

(m+ 2)2t2(kt− 2n)2

(66)

ρ =1

(A2 −B2)

−1

2

1(c3

3+3mb−3 bm+2

)2(e

(6+6mb−6 b)arctanh( ktn

−1)n(m+2)

)2 − 6(−3 + 3b− kt+ n)(1 +mb− b)

(m+ 2)(kt− 2n)2t2

+3(−1−m+ b+ 2mb)(−ktm− 2kt+ nm+ 2n− 3− 3m+ 3b+ 6mb)

(m+ 2)2(kt− 2n)2t2+ (80π +

1

2)×

− 1(c3

3+3mb−3 bm+2

)2(e

(6+6mb−6 b)arctanh( ktn

−1)n(m+2)

)2 − 36(−1−m− 2b− 2mb− 2m2b+ 3mb2 + 3m2b2 + 3b2)

(m+ 2)2t2(kt− 2n)2

(67)

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Λ =λ

4π

−m2 +m+ 2(

c33+3mb−3 b

m+2

)2(e

(6+6mb−6 b)arctanh( ktn

−1)n(m+2)

)2 +12(−1 + nb− kbt)

t2(kt− 2n)2

(68)

The expressions for physical parameters such as spatial volume (V ), mean Hubble’s parameter (H), expansionscalar (θ), shear scalar (σ) and anisotropy parameter (Am) for model given by Eq. (65) are given by

V = c33 exp

[6

ncoth3

(kt

n− 1

)], (69)

θ = 3H =6

(2n− kt)t, (70)

Am =12[m2(3b− 1)2 +m(b− 1)2 + (3b− 1)2

](m+ 2)2

, (71)

σ2 =12[m2(3b− 1)2 +m(b− 1)2 + (3b− 1)2

]t2(2n− kt)2(m+ 2)2

. (72)

The nature of variations of pressure and energy density with t for all three values m = 0, m = 1 and m = −1corresponding to Bianchi type-III, V and V I0 respectively in Model 3 are found to be same as in Model 1 inprevious case. We also observe that for m = 0 i.e. in Bianchi-III model, Λ(t) is positive decreasing functionof time and it approaches to zero at late time. But for m = 1 (Bianchi V ) and m = −1 (Bianchi V I0), Λ isnegative increasing function of time and it approaches to zero at late time. The nature of Λ, either negativeor positive are is already discussed in detail in previous subsection 4.1. The anisotropic parameter is constantthrough out the evolution of the universe. Figures are not given here. We observe that

• The WEC and DEC for the derived model are satisfied in all three Bianchi-III, V and V I0 models ofuniverse.

• The SEC is also satisfied in Bianchi-V model only. But SEC is violated by Bianchi-III and V I0 modelsas expected.

5 Conclusion

In this paper, a general class of Bianchi type cosmological models is studied in presence of perfect fluid andvariable cosmological constant in f(R, T ) theory of gravity recently proposed by Harko et al. [8], where thegravitational Lagrangian is given by an arbitrary function of the Ricci scalar R and of the trace of the stress-energy tensor T . For each choice of the function f(R, T ), we get different theoretical models. Three exam-ples have been given by Harko et al. [8] (i) f(R, T ) = R + 2f(T ), (ii) f(R, T ) = f1(R) + f2(T ) and (iii)f(R, T ) = f1(R) + f2(R)f3(T ). The cosmological consequences of the case (i) have been discussed before byseveral authors. The case (ii) has been considered first time in studying F (R, T ) modified theory for general classof Bianchi type space-time. In this paper, the gravitational field equation has been established by taking case(ii) into consideration. To find the deterministic solution, we have considered a special law for the decelerationparameter proposed by Akarsu and Dereli [42] which is linear in time with a negative slope. This law redressthe law of Berman [43] where deceleration parameter was constant. From general class of Bianchi universe, werecover additional classes of Bianchi models for different values of m as follows: Bianchi type-III correspondsto m = 0, Bianchi type-V corresponds to m = 1, Bianchi type-V I0 corresponds to m = −1 and all other val-ues of m bring forth Bianchi type-V Ih. The models represent an expanding, shearing, non-rotating and universe.

We have chosen the general class of Bianchi type space-time, so the anisotropic parameter Am should not beequal to zero. In our analysis it is very small and constant which impose the restriction on the numerical valuesof constants that is why we have choosen the small values of constants for plotting the figures.

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In Model 1, in Bianchi type-V I0 and in Model 3, in Bianchi type-III models, the cosmological constant is adecreasing function of time and it approaches a small positive value at late time (i.e. present epoch). It has beenfound that in other cases Λ is a negative increasing function of time and it converges to a small positive valueat late time. The nature of decaying vacuum energy density Λ(t) in our derived models is supported by recentcosmological observations. These observations on magnitude and red-shift of type Ia supernova suggest thatour universe may be an accelerating one with induced cosmological density through the cosmological Λ-term atpresent epoch.

Observations of 16 type Ia Supernovae made by Hubble Space Telescope (HST) [5] modified earlier astro-nomical results and provided conclusive evidence for deceleration prior to cosmic acceleration caused by darkenergy in the recent past. The acceleration is realized with negative pressure and positive energy density thatviolates strong energy condition (SEC) [54−56]. This violation of SEC gives reverse gravitational effect. Dueto this effect, universe gets a jerk and transition from deceleration to acceleration takes place. It is worth tomention here that in our derived Models 1, 2 and 3, strong energy condition is not satisfied.

Thus, the solutions demonstrated in this paper may be useful for better understanding of the characteristicof a general class of Bianchi type cosmological models in the evolution of the universe within the framework off(R, T ) gravity theory.

Acknowledgments

One of the Authors (A. Pradhan) would like to thank the Laboratory of Information Technologies, Joint Institutefor Nuclear Research, Dubna, Moscow Region, Russia for providing facility and supports where part of this workwas done. This work is also supported in part by the University Grants Commission, New Delhi, India underthe grant Project F.No. 41-899/2012 (SR). The authors thank the anonymous referee for his fruitful comments.

References

[1] S. Perlmutter et al. (Supernova Cosmology Project Collaboration), Nature 391, 51 (1998).

[2] S. Perlmutter et al. (Supernova Cosmology Project Collaboration), Astrophys. J. 517, 565 (1999).

[3] S. Perlmutter et al., Astrophys. J. 598, 102 (2003).

[4] A. G. Riess et al. (Supernova Search Team Collaboration), Astron. J. 116, 1009 (1998).

[5] A. G. Riess et al. (Supernova Search Team Collaboration), Astrophys. J. 607, 665 (2004).

[6] J. L. Tonry et al. (Supernova Search Team Collaboration), Astrophys. J. 594, 1 (2003).

[7] A. Clocchiatti et al. (High Z SN Search Collaboration), Astrophys. J. 642, 1 (2006).

[8] T. Harko, F. S. N. Lobo, S. Nojiri, S. D. Odintsov, Phys. Rev. D 84, 024020 (2011).

[9] S. Nojiri, S. D. Odintsov, Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007).

[10] S. Nojiri, S. D. Odintsov, Phys. Rev. D 565, 59 (2011).

[11] S. Nojiri, S. D. Odintsov, Phys. Rev. D 68, 123512 (2003).

[12] R. A. El-Nabulsi, Eur. Phys. J. Plus 126, 114 (2011).

[13] S. Capozziello, N. Carlevaro, M. De Laurentis, M. Lattanzi, G. Montani, Eur. Phys. J. Plus 128, 155(2013).

[14] R. Ghosh, U. Debnath, Eur. Phys. J. Plus 129, 81 (2014).

[15] M. Sharif, G. Abbas, Eur. Phys. J. Plus 128, 102 (2013).

[16] R. Chaubey, A. K. Shukla, Astrophys Space Sci. 343, 415 (2013).

15

Page 15 of 17C

an. J

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s. D

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oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y Sa

n D

iego

(U

CSD

) on

11/

15/1

4Fo

r pe

rson

al u

se o

nly.

Thi

s Ju

st-I

N m

anus

crip

t is

the

acce

pted

man

uscr

ipt p

rior

to c

opy

editi

ng a

nd p

age

com

posi

tion.

It m

ay d

iffe

r fr

om th

e fi

nal o

ffic

ial v

ersi

on o

f re

cord

.

[17] K. S. Adhav, Astrophys. Space Sci. 339, 365 (2012).

[18] G. C. Samanta, Int. J. Theor. Phys. 52, 2303 (2013).

[19] G. C. Samanta, Int. J. Theor. Phys. 52, 2647 (2013).

[20] G. C. Samanta, S. N. Dhal, Int. J. Theor. Phys. 52, 1334 (2013).

[21] S. Chandel, S. Ram, Indian J. Phys. 87, 1283 (2013).

[22] M. Sharif, S. Rani, R. Myrzakulov, Eur. Phys. J. Plus 128, 123 (2013).

[23] D. R. K. Reddy, R. S. Kumar, T. V. P. Kumar, Int. J. Theor. Phys. 52, 239 (2013).

[24] M. Kiran, D. R. K. Reddy, Astrophys. Space Sci. 346, 521 (2013).

[25] C. P. Singh, V. Singh, Gen. Relativ. Gravit. 46, 1696 (2014).

[26] V. U. M. Rao, K. V. S. Sireesha, D. C. P. Rao, Eur. Phys. J. Plus 129, 17 (2014).

[27] N. K. Sharma, J. K. Singh, Int. J. Theor. Phys. 53, 2912 (2014).

[28] J. K. Singh, N. K. Sharma, Int. J. Theor. Phys. 53, 1424 (2014).

[29] P. K. Sahoo, B. Mishra, G. C. Reddy, Eur. Phys. J. Plus 129, 49 (2014).

[30] G. Hinshaw et al., Astrophys. J. Suppl. 180, 225 (2009).

[31] J. Hoftuft et al., Astrophys. J. 699, 985 (2009).

[32] J. Dunkley et al., Astrophys. J. 701, 1804 (2009).

[33] M. Tegmark et al. (SDSS collaboration), Phys. Rev. D 69, 103501 (2004).

[34] G. P. Singh, A. Y. Kale, Eur. Phys. J. Plus 120, 83 (2011).

[35] A. Pradhan, H. Amirhashchi, Mod. Phys. Lett. A 26, 2261 (2011).

[36] S. Kumar, O. Akarsu, Eur. Phys. J. Plus 127, 64 (2012).

[37] C. Chawla, R. K. Mishra, A. Pradhan, Eur. Phys. J. Plus 127, 137 (2012).

[38] A. Pradhan, R. Zia and R. P. Singh, Indian J. Phys. 87, 1157 (2013).

[39] M. Sharif, I. Shafique, Eur. Phys. J. Plus 129, 26 (2014).

[40] N. J. Poplawski, arXiv:gr-qc/0608031 (2006).

[41] N. Ahmed, A. Pradhan, Int. J. Theor. Phys. 53, 289 (2014).

[42] O. Akarsu, T. Dereli, Int. J. Theor. Phys. 51, 612 (2011).

[43] M. S. Berman, Nuovo Cimento B 74, 182 (1983).

[44] J. Gunn and B. M. Tinsley, Nature 257, 454 (1975).

[45] E. J. Wampler, W. L. Burke, in New Ideas in Astronomy eds. F. Bertola, J. W. Sulentic, B. F. Madore(Cambridge University Press, p. 317, 1988).

[46] A. K. Yadav, Elect. J. Theor. Phys. 10, 169 (2013).

[47] B. Saha, T. Boyadjiev, Phys. Rev. D 69, 124010 (2004).

[48] P. Pedram, M. Mirzaei, S. Jalalzadeh, S.S. Gousheh, Gen. Rel. Grav. 40, 1663 (2008).

[49] T. Biswas, A. Mazumdar, Phys. Rev. D 80, 023519 (2009).

[50] K. Jotania, P. Yadav, S. A. Faruki, Int. J. Theor. Phys. 50, 1424 (2011).

16

Page 16 of 17C

an. J

. Phy

s. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y Sa

n D

iego

(U

CSD

) on

11/

15/1

4Fo

r pe

rson

al u

se o

nly.

Thi

s Ju

st-I

N m

anus

crip

t is

the

acce

pted

man

uscr

ipt p

rior

to c

opy

editi

ng a

nd p

age

com

posi

tion.

It m

ay d

iffe

r fr

om th

e fi

nal o

ffic

ial v

ersi

on o

f re

cord

.

[51] M. A. H. MacCallum, Commun. Math. Phys. 20, 57 (1971).

[52] R. M. Wald, Phys. Rev. D 28, 2118 (1983).

[53] L. G. Jensen and J. A. Stein-Schabes, Fermi Nat. Accel. Lab. Pub-86/48-A March (1986).

[54] B. Ratra, P. J. E. Peebles, Rev. Mod. Phys. 75, 559 (2003).

[55] V. Sahni, M. Sami, T. Souradeep, Phys. Rev. D 65, 023518 (2002).

[56] M. Sami, N. Dadhich, T. Shiromizu, Phys. Lett. B 568, 118 (2003).

17

Page 17 of 17C

an. J

. Phy

s. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y Sa

n D

iego

(U

CSD

) on

11/

15/1

4Fo

r pe

rson

al u

se o

nly.

Thi

s Ju

st-I

N m

anus

crip

t is

the

acce

pted

man

uscr

ipt p

rior

to c

opy

editi

ng a

nd p

age

com

posi

tion.

It m

ay d

iffe

r fr

om th

e fi

nal o

ffic

ial v

ersi

on o

f re

cord

.