1 Introduction

# Locally Rotationally Symmetric Bianchi Type-I cosmological model in $f(T)$ gravity: from early to Dark Energy dominated universe

## Abstract

We study the locally rotational symmetry Bianchi type-I dark energy model in the framework of theory of gravity, where denotes the torsion scalar. A viable cosmological model is undertaken and the isotropization of this latter is checked, yielding a result that reflects the real evolution of our universe. Moreover, still in the anisotropic optic, a more complicated model is obtained from the cosmological reconstruction scheme and the analysis shows that the universe is more anisotropic at the beginning if the terms of higher order in are not considered. This means that the non-linear model should be favoured by observational data.

Locally Rotationally Symmetric Bianchi Type-I cosmological model in gravity: from early to Dark Energy dominated universe

M. E. Rodrigues 1, I. G. Salako 2, M. J. S. Houndjo 3 and J. Tossa 4,

Faculdade de Ciências Exatas e Tecnologia, Universidade Federal do Pará - Campus Universitário de Abaetetuba, CEP 68440-000, Abaetetuba, Pará, Brazil

Institut de Mathématiques et de Sciences Physiques (IMSP) - 01 BP 613 Porto-Novo, Bénin

Faculté des Sciences et Techniques de Natitingou - Université de Parakou - Bénin

Pacs numbers: 98.80.-k, 04.50.Kd, 04.20.Jb

## 1 Introduction

The discovery of the accelerated expansion of the late-time universe gives rise to a wide spectrum of various theories in order to explain this acceleration. In the general gravity (GR) scheme, the dark energy, known to have an exotic property such as negative pressure, is considered to be a first candidate to this acceleration of the universe [1]. Still in this spirit, and as alternatives to the GR, several theories based on the modification of the the curvature scalar have been developed with interesting results. The most important ones are the [2]-[5], [6]-[10], [11]-[18], [19]-[24], where , and are the curvature scalar, the Gauss-Bonnet invariant and the trace of the energy-momentum tensor, respectively.

Nevertheless, there exists another alternative to the dark energy, being the modification of the teleparallel theory equivalent to the GR (TEGR), namely theory of gravity, where denotes the torsion scalar. Instead of the Levi-Civita’s connection, as used in the GR and its different modified versions, the connection under consideration in is that of Weitzenbok. This theory has been introduced first by Ferraro [26] where they explained the modifications to GR and also the inflation. Afterwards, Ferraro and Bengochea considered the same models in the context of late cosmology to describe dark energy [27]. Other authors developed various cosmological and gravitational ideas, still in theory of gravity. In [28], Tamanini and Boehmer investigate the importance of choosing good tetrads for the study of the field equations of gravity. It is important to recall that the use of specific tetrads is extremely crucial of having a reasonable feature in this theory. Note that when the set of tetrads is not suitably chosen, the theory falls into the usual teleparallel one and there will be no reason of why having modified the teleparallel theory. This important scheme of choosing a suitable set of tetrads is also undertaken in this manuscript and presented in the section , where it will be observed that the algebraic function may be different from the linear form one, the teleparallel action term.

Universe is assumed to be homogeneous and isotropic at large scales. However, this is not the case if local consideration are made. In such a situation, the anisotropic effects that appears should not be described with the use of the common Friedmann-Robertson-Walker (FRW) metric. The simplest of anisotropic models usually used for describing the anisotropic effects are Bianchi type I (BI) spatially homogeneous models whose spatial sections are flat. The advantages in using anisotropic models are that they have significant role in the description of the early stage of the universe [50] and also are extremely useful in obtaining more general cosmological models that the isotropic FRW models. Anisotropy is completely supported in various angle scales by the Differential Microwave Radiometers on NASA’s Cosmic Background Explorer. Speculations indicate that the entire history of the cosmic evolution down to the recombination is hidden in anisotropies and they are conceived to be indicative the matter composing and the geometry of the universe. Important theoretical argument by Misner in 1968 and the modern experimental data defend the existence of anisotropic phase of the universe, which, later, turns into an isotropic one.

Therefore, its quite reasonable to check the isotropization of any anisotropic model able to describe a stage of the evolution history of the universe. At a simple level, the isotropy of the present stage of the universe allows the BI model to be a prime candidate in the study of the possible effects of an anisotropy in the early universe on the current observational data.

Recently, Fadragas et al [40] performed a detailed dynamical analysis of anisotropic scalar-field cosmologies, and in particular of the most significant Kantowski-Sachs, Bianchi I and Bianchi III case, and as conclusion, they found a very rich behaviour, and amongst others the universe can result in isotropized solutions with observables in agreement with observations, such as de Sitter, quintessence-like or stiff-dark energy solutions. Fontanini et al considered the cosmic parallax effect in anisotropic cosmological models described by an axisymmetric homogeneous Bianchi I metric and discuss whether any observation of cosmic parallax would distinguish between different anisotropic evolutions [41]. Still in the framework of anisotropic models, Campanelli et al [42] analysed the magnitude-redshift data of type Ia supernovae included in the Union and Union2 compilations anisotropic Bianchi I cosmological model and in the presence of dark energy fluid anisotropic equation of state and found that the amount of deviation from isotropy of the equation of state of dark energy, the skewness, which should be restricted to the interval . Also, in order to well understand the notion of “real-time cosmology”, Quercellini et al undertook the real-time measurements of the overall redshift drift and angular separation shift in distant source, which allows the observer to trace the background cosmic expansion and large scale anisotropy, respectively [43, 44].

Following these evident features of the universe, we focus our attention in this work to the Locally Rotationally Symmetric (LRS) BI model which is a special case of BI one. Within this metric and the general field equations give rise to three independent equations. In the sequence, we present the generalisation of the anisotropic model within torsion consideration. As we mentioned above, the isotropization is an important aspect following the evolution history of the universe. This feature is extensively analysed in the framework of LRS-BI model in this work. The final and important point undertaken in this work is the reconstruction of model according to the LRS-BI metric. Across all the above studies, two interesting cases are considered. A linear form of the algebraic function and a more complicated case. Our results show that for the linear case, the real evolution of our universe holds whereas, for the more complicated one, the universe presents more anisotropy at the beginning only when the terms of higher order in fade away.

The paper is organized as follows. In Sec. 2 we present the preliminary definitions and equations of motion in gravity. The field equations are presented according to the LRS-BI metric in Sec. 3. The Sec. 4 is devoted to the generalisation of the anisotropic models with the torsion and also to the isotropization scheme. A linear case is developed in the Sec. 5 and our conclusions are presented in Sec. 6.

## 2 Preliminary definitions and equations of motion

As mentioned above, we define theory within Weitzenkock’s connection where the line element is described by

 dS2=gμνdxμdxν, (1)

being the components of the metric. One can describe the theory in the spacetime or in the tangent space. This allows to rewrite the line element (1) as follows

 dS2 = gμνdxμdxν=ηijθiθj, (2) dxμ = eμiθi,θi=eiμdxμ, (3)

where and or . We also express the square root of the metric determinant by and the matrix are called tetrads and play the crucial role in the dynamic fields of the theory.

By using theses fields, we define the Weitzenbock’s connection as

 Γαμν=eαi∂νeiμ=−eiμ∂νeαi. (4)

The geometrical objects of the spacetime are then constructed from this connection. The components of the tensor torsion are defined by the antisymmetric part of this connection

 Tαμν = (5)

The components of the contorsion tensor are defined as

 Kμνα = −12(Tμνα−Tνμα−Tμνα). (6)

In order to make more clear the definition of the scalar equivalent to the curvature scalar of RG, we first define a new tensor , constructed from the components of the tensors torsion and contorsion as

 Sμνα = 12(Kμνα+δμαTβνβ−δναTβμβ). (7)

We can now define the torsion scalar by the following contraction

 T=TαμνSμνα. (8)

The action of the theory is defined by generalizing the Teleparallel theory, as

 S=∫[T+f(T)+LMatter]ed4x. (9)

Here, denotes an algebraic function of the torsion scalar . Making the functional variation of the action (9) with respect to the tetrads, we get the following equations of motion [45, 46, 47]

 Sνρμ∂ρTfTT+[e−1eiμ∂ρ(eeαiSνρα)+TαλμSνλα](1+fT)+14δνμ(T+f)=4πTνμ, (10)

where is the energy momentum tensor, and . By setting , being a constant, the equations of motion (10) are exactly that of the Teleparallel theory with a cosmological constant, and this is dynamically equivalent to the GR. We see from this that the equations clearly depend on the choice made for the set of tetrads [48].

The energy momentum tensor incorporates the interaction of the gravitational field with the matter ones and is defined as

 Tνμ=diag(1,−ωx,−ωy,−ωz)ρ. (11)

Here, () are the parameters of equations of state related to the pressures , and .

## 3 Field equations for LRS Bianchi type-I model

Let us first establish the equations of motion of a set of diagonal tetrads using the Cartesian coordinate metric, for describing LRS Bianchi type-I model. The LRS Bianchi type-I metric reads

 dS2=dt2−A2(t)dx2−B2(t)(dy2+dz2), (12)

Let us choose the following set of diagonal tetrads related to the metric (57)

 [eaμ]=diag[1,A,B,B]. (13)

The determinant of the matrix (13) is . The components of the torsion tensor (5) for the tetrads (13) are given by

 T101=˙AA,T202=˙BB,,T303=˙BB, (14)

and the non null components of the corresponding contorsion tensor are

 K011=˙AA,K022=˙BB,,K033=˙BB. (15)

The components of the tensor , in (7), are given by

 S101=˙BB,S202=12(˙AA+˙BB),S303=12(˙AA+˙BB). (16)

By using the components (14) and (16), the torsion scalar (8) is given by

 T=−2(2˙A˙BAB+˙B2B2). (17)

The equations of motion corresponding to LRS Bianchi type-I model are obtained by

 16πρ = (f+T)+4(1+fT)⎡⎣(˙BB)2+2˙A˙BAB⎤⎦, (18) 16πpx = (f+T)+4(1+fT)⎡⎣¨BB+(˙BB)2+˙A˙BAB⎤⎦+4˙BB˙TfTT, (19)
 16πpy = (f+T)+2(1+fT)⎡⎣¨AA+¨BB+(˙BB)2+3˙A˙BAB⎤⎦+2(˙AA+˙BB)˙TfTT, (20) py = pz.

We parametrize it as follows

 Tνμ = diag(1,−ωx,−ωy,−ωz)ρ, (21) = diag(1,−ω,−(ω+δ),−(ω+δ))ρ, (22)

where is the energy density of the fluid; and are the pressures and and are the directional equation of state (EoS) parameters of the fluid. Now, parametrizing the deviation from isotropy by setting   and then introducing skewness parameter that is the deviations from respectively on both the and axes. Here and are not necessarily constants and can be functions of the cosmic time .

The equations of motion corresponding to LRS Bianchi type-I model for an anisotropic fluid are obtained by

 16πρ = (f+T)+4(1+fT)⎡⎣(˙BB)2+2˙A˙BAB⎤⎦, (23) −16πωρ = (f+T)+4(1+fT)⎡⎣¨BB+(˙BB)2+˙A˙BAB⎤⎦+4˙BB˙TfTT, (24) −16π(ω+δ)ρ = (f+T)+2(1+fT)⎡⎣¨AA+¨BB+(˙BB)2+3˙A˙BAB⎤⎦+2(˙AA+˙BB)˙TfTT. (25)

The conservation of the energy-momentum tensor of the fluid parametrized in (22), i.e , leads to the following equation:

 ˙ρ+[˙AA(1+ω)+2˙BB(1+(ω+δ))]ρ=0. (26)

## 4 Generalized anisotropic models with Torsion and Isotropization

Before determining the parameters and whose involved in the isotropization, we will determine the Hubble parameters in the direction of

 Hx=˙AA,Hy=˙BB,Hz=˙BB. (27)

The main Hubble parameter is given by

 H=13˙VV = 13(˙AA+˙BB+˙BB) (28) = 13(˙AA+2˙BB), (29)

with

 V=AB2, (30)

being the spatial volume of the universe. The rate of expansion is evaluated by anisotropy parameter given as

 Δ=133∑i=1(Hi−HH)2, (31)

where .

By introducing (29) and (29) into 31, the anisotropy parameter can be written as following

 Δ=29H2(Hx−Hz)2. (32)

The scalar expansion and the shear can be defined as

 θ(t)=˙AA+2˙BB, (33) σ(t)=1√3(˙AA−˙BB). (34)

We now we have three linearly independent equations (23)-(26) and five unknown functions and . Beside to this, we make use of an additional condition which allows us to choose the volumetric law expansion for solving the system completely. The following two different volumetric expansion laws have been used laws:

 V=c1e3H0t, (35)

where and are two positive constants. Because of the expansion of the universe, one can calculate the deceleration parameter as

 q=(ddtH−1(t))−1. (36)

Using equations (23), (29), (32) and the identity

 H2y+2HxHy=3H2[1−Δ(t)2], (37)

the energy density can be re-written as

 3H2=8πρeff=8π(ρ+ρT+ρanis), (38) ρT=−116π[f(T)+12H2fT(T)], (39) ρanis=316πH2Δ(t)[1+2fT(T)], (40)

where the subscript expresses the anisotropic component. By considering the particular case , one regains the anisotropic LRS-BI model in GR. When one considers the particular case (implying in (32)), , and is re-obtained in the framework of flat FRW metric. If all the densities are different from zero, one gets a theory that takes into account the anisotropic contribution and the contribution of the terms non-linear in torsion scalar.

In the next section, we will deal with the particular case , in order to re-obtain the anisotropic dark energy results, as an analogy of the GR.

Now, let us define the necessary condition for an anisotropic model of the universe presenting isotropization. First, let us explain the term “isotropization”. The isotropization is the process in which a possible anisotropic universe can adequate with the current observational data, being homogeneous and isotropic; such an universe is said becoming isotropic by the isotropization process. This process is really realised in the inflationary phase and modifies any geometry into an isotropic one and any fluid into an homogeneous one.

The criterion for having the possibility of such a model are known for several years, through the work of Collins and Hawking [49], where they established the following conditions: when taking the limit , one has the function , in (22), , in (32) and 5 tends to zero. This is clear when one makes in the definition of energy momentum tensor (22), obtaining a perfect fluid, and when in (32), leading to , which corresponds to the flat FRW metric in (57).

## 5 The Linear Case, f(T)=0

For this work, we have used the model with the exponential expansion given by the relation (35). By using equations (30) and (35), we can obtain:

 A(t)=c1e3H0tB2. (41)

By making use of (24),   (25), (41) after solving (23)-(26), we obtain:

 B(t)=eH0t(κH0+3λe−3H0t)1/3, (42)

 A(t)=c1eH0t(κH0+3λe−3H0t)−2/3. (43)

By introducing (42) and (43) into (29) and (32), we obtain the mean Hubble parameter

 H(t)=H0, (44)

and the anisotropy parameter of the expansion as:

 Δ(t)=18H20λ2(κe3H0t+3H0λ)2. (45)

By using equation (23), we can obtain the energy density of the fluid as:

 ρ1(t)=3κH20e3H0t(κe3H0t+6H0λ)8π(κe3H0t+3H0λ)2. (46)

The deviated part of the anisotropic EoS parameter is obtained by using equations (27), (46) as following:

 ω1(t)=−3κH20e3H0t(κe3H0t+6H0λ)8π(κe3H0t+3H0λ)2ρ1(t). (47)

In particular, one has a specific value in our linear case. We can determine the deviation-free part of anisotropic EoS parameter by making use (29), (46) and(47) as following:

 δ1(t)=−27H20λ2e−3H0tκ(κe3H0t+6H0λ). (48)

One can calculate the functions and , with (33) and (34), yielding

 θ(t)=3H0,σ(t)=3√3H20λ3H0λ+κe3H0t. (49)

This result can be interpreted as an universe in an accelerated expansion , with a constant mean Hubble parameter and a volume involving with the cosmic time , as shown in the Figure . Due to the fact that , on has an universe of type de Sitter, where the effective density in (38) is constant, mimicking the cosmological constant.

The scale factors in the directions and (also ) are increasing, as represented in the Figure . At the beginning, the universe increases more rapidly in the direction , and later being equal to the others (isotropization).

This is an isotropization model because the functions , in (48), and , in (45), tend to zero as the cosmic time goes toward infinity ( and ). They are the anisotropic contributions for this model. The functions and are described in the Figure . One can see in these expressions that the isotropization is obtained at the limit . We clearly see that it is the parameter which governs the isotropization of the universe in this case. If goes to infinity in (45), one gets the maximal anisotropy, so that . Hence, in our case.

One can define a function which gives the anisotropic measure by

 M(t)=σ(t)θ(t)=√3H0λ3H0λ+κe3H0t. (50)

The curve that shows the behaviour of the function is described by the Figure , so that, the anisotropy fades away when the time goes to infinity.

In the next section we will perform the reconstruction of the function for a specific case of an anisotropic geometry. The analysis of the matter content of this present section also will be done in the next section for a whole comparison.

## 6 Reconstruction for LRS Bianchi type-I model in f(T) Gravity

Here, we will consider that the anisotropic geometry is given by the same functions and in (42) and (43). Hence, we will get modifications in the matter content of the universe (the anisotropy of the fluid and the new contribution of the non-linear terms of ). One has the same functions for the volume (35), the mean Hubble parameter (44), the scalar expansion and the shear in (49). Therefore, the model is clearly an isotropization model for the universe.

For finding a good model for this theory, we will use the so-called reconstruction method, because the algebraic function is reconstructed according to the geometry and the matter content of the universe.

By introducing (42) and (43) into (23), we obtain the energy density of the fluid for the non-linear case

 ρ2(t)=116π[6H20+f+12H20fT−54H40λ2(1+2fT)(e3H0tκ+3H0λ)2]. (51)

After solving (27) and (29), we obtain the deviation-free part of anisotropic EoS parameter for the non-linear case:

 ω2(t)=116πρ2(t){−6H20+54H40λ2(e3H0tκ+3H0λ)2−f−12H20[1−9H20λ2(e3H0tκ+3H0λ)2]fT +1296H60κ2λ2fTTe6H0t(e3H0tκ+3H0λ)4}, (52)

and the skewness parameter for the non-linear case as following

 δ2(t) = 116πρ2(t)[−6H20−f−12H20fT−5832H80λ4fTT(e3H0tκ+3H0λ)4−1944H70λ3fTT(e3H0tκ+3H0λ)3 (54) −54H40λ2(2+fT−24H20fTT)(e3H0tκ+3H0λ)2−16πρ2(t)ω2(t)].

By replacing (42) and (43) into (17), we obtain the following relation:

 t(T)=1H0log⎡⎢ ⎢ ⎢⎣(3λH0κ)1/3⎛⎜ ⎜⎝√6H0√6H20+T−1⎞⎟ ⎟⎠1/3⎤⎥ ⎥ ⎥⎦, (55)

which leads to the equation of conservation for energy-momentum tensor

 −52488H100λ5[(6H20+T)−√6H0√(6H20+T)][f′(T)−f′′(T)](6H20+T)2=0, (56)

where the solution is given by:

 f(T)=T0+T1eT, (57)

where and are integration constants. The equations (51) and (52) become

 ρ2(t) = 116π[T0+T1exp[−6H20κe6H0t(e3H0tκ+6H0λ)(e3H0tκ+3H0λ)2] (58) +H20(6+12T1exp[−6H20κe6H0t(e3H0tκ+6H0λ)(e3H0tκ+3H0λ)2] +54H40λ2(−1−2T1exp[−6H20κe6H0t(e3H0tκ+6H0λ)(e3H0tκ+3H0λ)2])(e3H0tκ+3H0λ)2)],
 ω2(t)=−Xα, (59)

where

 X=[6H20+T0+T1exp[−6H20κe6H0t(e3H0tκ+6H0λ)(e3H0tκ+3H0λ)2]− (60) 1296T1H60κ2λ2exp[−6H0(e6H0t(H0−t)κ2+6H0κλe3H0t(H0−t)−9H20tλ2)(e3H0tκ+3H0λ)2](e3H0tκ+3H0λ)4−54H40λ2(e3H0tκ+3H0λ)2+ (61) 12H20κT1(e3H0tκ+6H0λ)exp[−6H0(e6H0t(H0−t)κ2+6H0κλe3H0t(H0−t)−9H20tλ2)(e3H0tκ+3H0λ)2]], (62)
 α=T0+T1exp[−6H20κe6H0t(e3H0tκ+6H0λ)(e3H0tκ+3H0λ)2] (63) +H20(6+12T1exp[−6H20κe6H0t(e3H0tκ+6H0λ)(e3H0tκ+3H0λ)2] (64) +54H40λ2(e3H0tκ+3H0λ)2(−1−2T1exp[−6H20κe6H0t(e3H0tκ+6H0λ)(e3H0tκ+3H0λ)2]). (65)

The expression for the skewness parameter (54) is given by

 δ2(t)=−ZQ, (66)

where

 Z=162H40λ2(κ2exp[6H0(κ2(H0+t)e6H0t+6H0(H0+t)e3H0tλκ+9tH20λ2)(e3H0tκ+3H0λ)2] (67) +T1κ2e6H0t+6H0κλexp[3H0(κ2(2H0+t)e6H0t+6H0(2H0+t)e3H0tλκ+9tH20λ2)(e3H0tκ+3H0λ)2] (68) −6H0κλT1e3H0t(6H20−1)+9H20λ2exp[−6H20κe6H0t(e3H0tκ+6H0λ)(e3H0tκ+3H0λ)2]+9H20T1λ2), (69)

and

 Q=(e3H0tκ+3H0λ)2(κ2(6H20+T0)exp[6H0(κ2(H0+t)e6H0t+6H0(H0+t)e3H0tλκ+9tH20λ2)(e3H0tκ+3H0λ)2] (70) +T1κ2e6H0t(1+12H20)+6H0κλ(6H20+T0)exp[3H0(κ2(2H0+t)e6H0t+6H0(2H0+t)e3H0tλκ+9tH20λ2)(e3H0tκ+3H0λ)2] (71) +6H0κλT1e3H0t(1+12H20)+9H20λ2T0exp[6H20κe3H0t(e3H0tκ+6H0λ)(e3H0tκ+3H0λ)2]+9H20T1λ2). (72)

With the expressions of the energy densities (46) and (58), one represents this function in the Figure . Now, one can interpret the densities contributions. The density contributes lower than the energy density , for the anisotropic expansion of the universe. Indeed, one calculates the diverse contributions of the effective energy densities in (38).

With (44), (45), (57), (39) and (40), we have the density due to the torsion scalar terms and the density due to the anisotropy.

 ρT(2)(t)=−116πexp[−6H20κe3H0t(κe3H0t+6H0λ)(κe3H0t+3H0λ)2], (73) ρA(2)(t)=27H40λ28π(1+2T1exp[−6H20κe3H0t(κe3H0t+6H0λ)(κe3H0t+3H0λ)2])(κe3H0t+3H0λ)2. (74)

The densities of the linear case are given by

 ρT(1)(t)=0, (75) ρA(1)(t)=27H40λ28π(κe3H0t+3H0λ)2. (76)

We represent the curves of the energy densities and at the left hand side in the Figures . We recall that at the left hand side there are two overlapping curves despite of seeming have one. In order to clearly show that there are effectively two different curves, within the same parameters and plotting in the range , the curve at the right hand side, it appears clearly that the two curves are distinct. One sees directly that the most important contribution is that of the model , i.e, the non-linear in