_{1}

^{*}

In this paper, the dynamical behavior of an anisotropic universe in an extended gravity e.g. the f ( R, T ) theory of gravity is studied. We use f ( R, T ) = R + 2 μT , where R is the Ricci scalar, T is the trace of energy-momentum tensor and μ is a constant. Two cosmological models are constructed using the power law expansion and hybrid law cosmology in Bianchi type I universe, where the matter field is considered to be a perfect fluid. It is found that in both the cases the anisotropic behavior is in agreement with the observational results. The state finder diagnostic pair and energy conditions are also calculated and analyzed.

Einstein’s General Theory of Relativity (GR) is one of the revolutionary concepts in modern physics. This equation results out of the outstanding interplay

between matter and space-time. Based on Einstein equation G μ ν − 1 2 g μ ν R = κ T i j ,

the presence of energy-momentum distribution has shown how gravity influenced space-time and how the curvature of space-time acts on gravity simultaneously. While (GR) is successful in several physical aspects, some astrophysical and cosmological issues still have not found their appropriate explanation by GR. Late time cosmic acceleration, is the most significant result that is indescribable in theory of GR. Of the proposed alternatives of the general theory of gravity, modified gravity theory is found to be one of the more appealing solutions. The recent cosmological observations of Type Ia supernovae (SN Ia) [

An unprecedented view of our dark universe has been shown by the recent discovery of the gravitational wave by LIGO (Laser Interferometer Gravitational- Wave Observatory). With the help of precise measurement of advanced LIGO, we will be able to test the differences between Einstein’s General Theory of Relativity and it’s extended theories [

Primary ideas of generalization of the GR have gone beyond the Einstein-Hilbert action. The generalization of interest here is including the term of f ( R ) theory of gravity in the Einstein-Hilbert action. The presence of a late time cosmic acceleration of the universe can indeed be explained by f ( R ) gravity [

Bamba [

Most studies on f ( R , T ) gravity are with isotropic universes. However, Mishra et al. [

Naser [

Cosmology in f ( R , T ) gravity. Several cosmological models were analysed on the perspective aspect of the anisotropic universe by considering recent observation of supernovae type I as a perfect standard candle and studied their spectrum [

Simultaneously [

The arbitrariness in the choice of different functional forms of f ( R , T ) to the need for constraints. Some constraints on f ( R ) gravity from the energy conditions have been considered by Santos et al. [

energy is possible through its equation of state parameter ω = p ρ . If the value of

ω = − 1 this is equivalent to the cosmological constant Λ . For minimally

coupled scalar field such as quintessence ( − 1 ≤ ω ≤ − 1 3 ) and phantom ( ω ≤ − 1 ).

From results of supernovae data CMBR anisotopy, the values are − 1.67 ≤ ω ≤ − 0.62 and − 1.33 ≤ ω ≤ − 0.79 respectively.The notion of energy conditions have been used to explore viable models of gravity in [

It is to be noted that perfect fluid is one of the matter of the universe which go through variation and contraction of the action. At the back drop of our mind, we would be interested to see that f ( R , T ) would gradually be fitted into GR equation. The present paper is arranged as follows. In Section II, we present the field equations obtained from the substitution of the Bianchi type I space-time metric in the f ( R , T ) gravity. Some cosmological parameters of this approach are also defined. In Section III, we have constructed the cosmological models with the power law cosmology and hybrid cosmology. The physical behaviors of the models along with the energy conditions are described in Section IV. Concluding remarks are given in Section V.

In this section, we will derive the field equations of f ( R , T ) gravity in the framework of Bianchi type I space-time and energy momentum tensor in the form of perfect fluid with the choice of the functional f ( R , T ) = R + 2 μ T , where μ is an arbitrary constant.The action in the f ( R , T ) theory of gravity can be written as .

S = ∫ ( 1 16 π f ( R , T ) + L m ) − g d 4 x (1)

with f ( R , T ) be a function of R and T, g is the determinant of the metric and L m is the matter Lagrangian density. The energy-momentum tensor can be

defined as T i j = − 2 − g δ ( − g L m ) δ g i j . It is to be mentioned that here the matter

Lagrangian is assumed to be dependent only on the metric tensor g i j and not on its derivatives. So, the stress-energy tensor can be expressed as :

T i j = g i j L m − δ L m δ g i j . (2)

Now, varying action (1) with respect to the metric component g i j , the field equations of f ( R , T ) gravity can be obtained as

f ( R , T ) R i j − 1 2 f ( R , T ) g i j + ( g i j □ − ∇ i ∇ j ) f R ( R , T ) = 8 π T i j − f T ( R , T ) T i j − f T ( R , T ) Θ i j , (3)

with R i j being the Ricci tensor, where Θ i j = − 2 T i j + g i j L m − 2 g α β ∂ 2 L m ∂ g i j ∂ g α β . Here, f R = ∂ ∂ R , f T = ∂ ∂ T and □ ≡ ∇ i ∇ j , with ∇ i being the covariant derivative.

We consider the homogeneous and anisotropic Bianchi universe as

d s 2 = d t 2 − b 1 2 d x 2 − b 2 2 ( d y 2 + d z 2 ) (4)

where the metric potentials b 1 and b 2 are function of the cosmic time. The energy-momentum tensor is

T i j = ( ρ + p ) u i u j − p g i j . (5)

Here, ρ and p, respectively, denote the matter-energy density and proper pressure of the matter field and u i = ( 0,0,0,1 ) is the four-velocity vector of the fluid in a co-moving coordinate system that satisfies u i u i = 1 . Now, by assuming the matter Lagrangian as L m = − p , the field equations Equation (3) for the choice f ( R , T ) = R + 2 μ T can be reduced to

R i j − 1 2 R g i j = ( 8 π + 2 μ ) T i j + Λ g i j . (6)

where, Λ = ( ρ − p ) μ . In the co-moving coordinate system, the field equations are

2 b ¨ 2 b 2 + b ˙ 2 2 b 2 2 = − ( 8 π + 3 μ ) p + ρ μ , (7)

b ˙ 1 b ˙ 2 b 1 b 2 + b ¨ 2 b 2 + b ¨ 1 b 1 = − ( 8 π + 3 μ ) p + ρ μ , (8)

2 b ˙ 1 b ˙ 2 b 1 b 2 + b ˙ 2 2 b 1 2 = ( 8 π + 3 μ ) ρ − p μ , (9)

An over dot on the field variables represent the ordinary derivative with respect to the cosmic time. In order to understand the model physically, we will

consider the directional Hubble rates in the form H x = b ˙ 1 b 1 and H y = H z = b ˙ 2 b 2 ,

which subsequently yield the mean Hubble parameter as H = a ˙ a = 1 3 ( H x + 2 H y ) , with (a) denotes the mean scale factor of the universe. So, the set of field Equations (7)-(9) can be expressed in term of the scale factor (a) as:

3 k + 2 [ 2 a ¨ a + ( 5 − 2 k k + 2 ) ( a ˙ a ) 2 ] = μ ρ − β p (10)

3 k + 2 [ ( k + 1 ) a ¨ a + ( 2 k 2 + 1 k + 2 ) ( a ˙ a ) 2 ] μ ρ − β p (11)

9 ( 2 k + 1 ) ( k + 2 ) 2 ( a ˙ a ) 2 = β ρ − μ p , (12)

where we have defined β ≡ 8 π + 3 μ . By performing algebraic manipulation on Equations (10)-(12), the pressure (p) and energy density ( ρ ) of the model concerning the scale factor can be expressed respectively as

p = ( 6 β k + 2 ) a ¨ a + 9 ( k + 2 ) 2 [ 3 β − μ − 2 3 β ( k + 2 ) − 2 k μ ] ( a ˙ a ) 2 (13)

and

ρ = ( 6 β k + 2 ) a ¨ a + 9 ( k + 2 ) 2 [ 3 μ − β − 2 3 μ ( k + 2 ) − 2 k β ] ( a ˙ a ) 2 (14)

Subsequently, with respect to the scale factor, the equation of state (EoS) parameter ω is obtained as

ω = p ρ = ( 6 β k + 2 ) a ¨ a + 9 ( k + 2 ) 2 [ 3 β − μ − 2 3 β ( k + 2 ) − 2 k μ ] ( a ˙ a ) 2 ( 6 β k + 2 ) a ¨ a + 9 ( k + 2 ) 2 [ 3 μ − β − 2 3 μ ( k + 2 ) − 2 k β ] ( a ˙ a ) 2 (15)

We also define an effective cosmological constant (ECC) for the model as

Λ = 2 μ μ + β [ ( 3 k + 2 ) a ¨ a + ( a ˙ a ) 2 6 k + 2 ] (16)

In this section, we will discuss the cosmological features of the model with the scale factor being in the form of the power law and hybrid law cosmologies.

In the previous section, we have expressed the physical parameters of the model with respect to the Hubble parameter. In order to study the dynamics of the universe, we need to know the parameters with respect to the cosmic time, hence

we consider here the power law function in the form a = t m 3 . This type of

assumption is quite helpful to understand the background cosmology, which can also be checked in the solutions for the standard cosmology Friedman equations. In order to employ some amount of anisotropic, we have assumed here H x = k H y , where (k) is a constant. It can be seen that when k = 1 , the space time reduces to standard FRW model. With these assumptions, we obtain the

directional and mean Hubble parameters, respectively, as H x = m k ( k + 2 ) t , H y = H z = m ( k + 2 ) t and H = 1 3 ( H x + 2 H y ) = m 3 t . So, the pressure and energy density can be obtained as

p = m ( 4 ( 4 + 2 k − 3 m ) π + ( 6 − 4 m + k ( 3 + m ) ) μ ) 4 ( 2 + k ) 2 t 2 ( 2 π + μ ) ( 4 π + μ ) (17)

ρ = − 2 m μ ( k + 2 ) + m 2 ( − 8 π − 2 k ( 8 π + 3 μ ) ) ( 2 + k ) 2 t 2 ( μ 2 − ( 8 π + 3 μ ) 2 ) (18)

The graphical behavior of the pressure and energy density have been represented in

shown for k = 0.12 and various range of m: 1.42, 1.5, 1.6, 1.8, 2 and μ: 0.01, 1.15, 1.8, 0.07, 0.001 as anisotropy parameter and scaling constant where have shown with Red, Black, Orange, Green and Blue respectively. Pressure minimized on

m > 8 π + 3 μ 6 π + 2 μ and 0 < k < − 16 π + 12 m π − 6 μ + 4 m μ 8 π + 3 μ + m μ .

Subsequently the EoS parameter and ECC can be obtained as

ω = 4 π ( 4 + 2 k − 3 m ) + μ ( 6 − 4 m + k ( 3 + m ) ) 4 m π ( 1 + 2 k + ( 2 + k + 3 k m ) μ (19)

Λ = μ m ( m − 1 ) 2 t 2 ( 2 + k ) ( 2 π + μ ) (20)

In

In

The power-law and exponential law cosmologies are having constant deceleration parameter and only can be used to describe the epoch based evolution of the universe. These cosmologies do not exhibit the transition of the universe from deceleration to acceleration. So, in this section, in order to show the early deceleration and late time acceleration, we have considered a scale

factor in the form a = e γ 3 t t α 3 known as hybrid scale factor [

This scale factor is having two components: one component behavior is like exponential universe and the other is the power law expansion. Moreover, in this scale factor the exponential component dominates at late phase while in the early phase of cosmic evolution the power law component dominates. It can be noted that when γ = 0 , the scale factor reduces to a power law expansion whereas when for α = 0 , only the exponential law can be recovered. For this

scale factor, the Hubble rate is H ( t ) = α 3 t + γ 3 , H x = k k + 2 ( γ + α t ) , H y = H z = 1 k + 2 ( γ + α t ) . The pressure and energy density for the hybrid scale factor can be respectively obtained as:

p = − 2 ( 2 + k ) α β + ( α + t γ ) 2 ( 3 β − μ ) − 2 k ( α + t γ ) 2 μ ( 2 + k ) 2 t 2 ( − β 2 + μ 2 ) (21)

ρ = − ( 2 k β + β − 3 μ ) ( α + t γ ) 2 + 2 ( 2 + k ) α μ ( 2 + k ) 2 t 2 ( − β 2 + μ 2 ) (22)

In

− 16 π + 12 π β + 6 μ + 4 β μ 8 π + 3 μ + β μ < k < 12 π + 4 μ μ

the pressure raised up to its lowest negative value.

From

w = − − 2 ( 2 + k ) α β + ( α + t γ ) 2 ( 3 β − μ ) − 2 k ( α + t γ ) 2 μ ( 2 k β + β − 3 μ ) ( α + t γ ) 2 + 2 ( 2 + k ) α μ (23)

Λ = 2 ( ( − 1 + α ) α + 2 t γ α + t 2 γ 2 ) μ ( 2 + k ) t 2 ( α + μ ) (24)

The scalar expansion and deceleration parameter of the model can be obtained

as θ = 3 H ( t ) = m t and q = − 1 + 3 m . We can infer that the scalar expansion

vanishes at late time and deceleration parameter remains constant at −1. It can be noted that when m > 3 , the model decelerates and accelerates for m < 3 .

The volume scale factor V and the ratio of anisotropy A = 1 3 Σ ( Δ H i H ) 2 can be respectively calculated as t m and 2 ( k − 1 ) 2 ( k + 2 ) 2 . As usual for k = 1 , the model

reduces to an isotropic universe. The state finder diagnostic pair which give some insight to the geometrical part of the model. The pair for the power law

cosmology can be obtained as r = ( m − 3 ) ( m − 6 ) m 2 , s = r − 1 3 ( q − 1 / 2 ) = 2 m .

For hybrid cosmology θ = β t + γ and q = − 1 + 3 β ( β + γ t ∗ γ ) 2 .

shown evolution of deceleration parameter in terms of cosmic time for transition from deceleration to acceleration phase. In this model we have

positive values of q in the range of 3 β β + γ t > 1 for decelerated universe and

facing to negative values of q for accelerated universe at the late time for

3 β β + γ t ≤ 1 . Similarly for a hybrid cosmology, the pair can be obtained as

r = 1 − 9 α ( − 2 + α + t γ ) ( α + γ t ) 3 , and s = r − 1 3 ( q − ( 1 2 ) ) = − 3 α ( − 2 + α − t γ ) ( α + t γ ) 3 ( − 3 2 + 3 α ( α + t γ ) 2 )

Energy conditions as a sets of linear equations put some additional constraints on the model. However Equation (25) has demonstrated that the field equation has been physically approved. Energy conditions derived as:

{ NullEnergyConditions ( NEC ) : ρ + p ≥ 0 WeakEnergyCondition ( WEC ) : ρ + p ≥ 0, ρ ≥ 0 StrongEnergyCondition ( SEC ) : ρ + 3 p ≥ 0, ρ + p ≥ 0 DominantEnergyCondition ( DEC ) : ρ ± p ≥ 0, ρ ≥ 0 (25)

Now, the energy conditions for power law cosmology can be obtained as:

ρ + p = 2 m ( 2 + k + ( k − 1 ) m ) ( 2 + k ) 2 t 2 ( β − μ ) (26)

ρ + 3 p = 2 m ( ( 6 − 4 m + k ( m + 3 ) ) β + ( 2 + k + 3 m k ) μ ) ( 2 + k ) 2 t 2 ( β 2 − μ 2 ) (27)

ρ − p = 2 m ( m − 1 ) ( 2 + k ) t 2 ( β + μ ) (28)

In

Similarly, for a hybrid cosmology, the energy conditions for the model can be calculated

ρ + p = 2 α ( 2 + k + ( − 1 + k ) α ) + 4 ( − 1 + k ) t α γ + 2 ( − 1 + k ) t 2 γ 2 ( 2 + k ) 2 t 2 ( β − μ ) (29)

3 p + ρ = 2 ( α β ( 3 ( 2 + k ) + 2 ( − 4 + k ) t γ ) + α ( 2 + k + 6 k t γ ) μ + α 2 ( ( − 4 + k ) β + 3 k μ ) + t 2 γ 2 ( ( − 4 + k ) β + 3 k μ ) ) ( 2 + k ) 2 t 2 ( β 2 − μ 2 ) (30)

ρ − p = 2 ( ( − 1 + α ) α + 2 t α γ + t 2 γ 2 ) ( 2 + k ) t 2 ( β + μ ) (31)

Another notable cases is evaluating the effectiveness of higher curvature on the dynamical variables of spherically compact stars which filled by an imperfect fluid as matter content has been investigated on [

The work done in this paper is to investigate the cosmological model obtained in f ( R , T ) gravity in Bianchi type I space-time with the functional f ( R , T ) = R + 2 μ T . The dynamical parameters are calculated and analyzed with a new mathematical formalism in power law cosmology and hybrid scale factor cosmology. We have shown that there exist various values for inputs parameters which the four energy conditions may be satisfied simultaneously. The state finder diagnostic pair and energy conditions are also studied. The results obtained in both the models are in accordance with the observational findings of accelerated expansion of the universe. Also the result of this work could be extended to a new approach of extended theories of gravity such as f ( R , □ R , T ) that could lead to unveil the secret of dark energy in modern cosmology. Hence we addressed to some successful studies on this formalism as [

I would like to thank Professor.Bindu. A. Bambah for her expert advice and encouragement throughout this research paper. The author is thankful to the anonymous referees for constructive comments concerning our manuscript suggestions for the improvement of the paper.

The authors declare no conflicts of interest regarding the publication of this paper.

Esmaeili, F.MD. (2018) Dynamics of Bianchi I Universe in Extended Gravity with Scale Factors. Journal of High Energy Physics, Gravitation and Cosmology, 4, 716-730. https://doi.org/10.4236/jhepgc.2018.44040