Accelerating universe in hybrid and logarithmic teleparallel gravity

https://doi.org/10.1016/j.dark.2020.100551Get rights and content

Abstract

Teleparallel gravity is a modified theory of gravity for which the Ricci scalar R of the underlying geometry in the action is replaced by an arbitrary functional form of torsion scalar T. In doing so, cosmology in f(T) gravity becomes greatly simplified owing to the fact that T contains only the first derivatives of the vierbeins. The article exploits this appealing nature of f(T) gravity and present cosmological scenarios from hybrid and logarithmic teleparallel gravity models of the form f=emTTn and f=Dlog(bT) respectively, where m, n, D and b are free parameters constrained to suffice the late time acceleration. We employ a well motivated parametrization of the deceleration parameter having just one degree of freedom constrained with a χ2 test from 57 data points of Hubble data set in the redshift range 0.07<z<2.36, to obtain the expressions of pressure, density and EoS parameter for both the teleparallel gravity models and study their temporal evolution. We find the deceleration parameter to experience a signature flipping for the χ2 value of the free parameter at ztr0.6 which is consistent with latest Planck measurements. Next, we present few geometric diagnostics of this parametrization to understand the nature of dark energy and its deviation from the ΛCDM cosmology. Finally, we study the energy conditions to check the consistency of the parameter spaces for both the teleparallel gravity models. We find the SEC to violate for both the models which is an essential recipe to obtain an accelerating universe.

Introduction

Several Observations reveal that the universe is accelerating for the very second time in its 13.7 billion year long lifetime [1]. It has now been agreed that a cosmological entity with almost three-quarters of the energy budget of the universe coupled with a EoS parameter ω1 is required to suffice the observations. In this spirit, several interesting proposals have been reported to expound this conundrum [2].

One of the most interesting proposal refuting the existence of dark energy are the ‘modified theories of gravity’. In modified gravity theories, dark energy is purely geometric in nature and is connected to novel dynamical terms following modification of the Einstein–Hilbert action [3]. Many such theories such as f(R) gravity, f(G) gravity, f(R,T) gravity, etc. have widespread use in modern cosmology (For a recent review on modified gravity see [4]. Also see [5] for some interesting cosmological applications of modified gravity).

Teleparallel gravity is a well established and well motivated modified theory of gravity inspired from f(R) gravity [6] (See [7] for a review on teleparallel gravity). In teleparallel gravity, the Ricci scalar R of the underlying geometry in the action is replaced by an arbitrary functional form of torsion scalar T. Thus, in teleparallel gravity, instead of using the torsionless Levi-Civita connection (which is usually assumed in GR), the curvatureless Weitzenböck connection is employed in which the corresponding dynamical fields are the four linearly independent verbeins, and T is related to the antisymmetric connection following from the non-holonomic basis [3], [8].

Linear f(T) gravity models are the teleparallel equivalent of GR (TEGR) [9]. Nonetheless, f(T) gravity differ significantly from f(R) gravity in the fact that the field equations in f(T) gravity are always at second-order compared to the usual fourth-order in f(R) gravity. This owes to the fact that the torsion scalar contains only the first derivatives of the vierbeins and thus makes cosmology in f(T) gravity much simpler. However, Despite being a second-order theory, very few exact solutions of the field equations have been reported in literature. Power law solutions in FLRW spacetime have been reported in [10], while for anisotropic spacetimes in [11]. Solutions for Bianchi I spacetime and static spherically spacetimes can be found in [3] and [12] respectively.

Since cosmology in f(T) gravity is much simpler compared to other modified gravity theories, it has been employed to model inflation [13], late time acceleration [14] and big bounce [15]. The instability epochs of self-gravitating objects coupled with anisotropic radiative matter content and the instability of cylindrical compact object in f(T) gravity have been discussed in Ref. [16], [17].

The manuscript is organized as follows: In Section 2 we present an overview of f(T) gravity. In Section 3 we describe the kinematic variables obtained from a parametrization of deceleration parameter used to obtain the exact solutions of the field equations. In Section 4 we present the hybrid and logarithmic teleparallel gravity models and obtain the expressions of pressure, density and EoS parameter. In Section 5 we present some geometric diagnostics of the parametrization of deceleration parameter. In Section 6 we study the energy conditions for both the teleparallel gravity models. In Section 7 we obtain some observational bounds on the free parameters of the parametrization by performing a chi-square test using Hubble datasets with 57 datapoints, Supernovae datasets consisting of 580 data points from Union2.1 compilation datasets and Baryonic Acoustic Oscillation (BAO) datasets. Finally, in Section 8 we present our results and conclusions.

Section snippets

Overview of f(T) gravity

The action in teleparallel gravity is represented as S=116πG[T+f(T)]ed4x,where e=det(eμi)=g and G is Newtonian gravitational constant. The gravitational field in this framework arises due to torsion defined as Tμνγeiγ(μeνiνeμi).The contracted form of torsion tensor reads T14TγμνTγμν+12TγμνTνμγTγμγTννμ.varying the action S+Lm, where Lm represent the matter Lagrangian yields the field equations as e1μ(eeiγSγμν)(1+fT)(1+fT)eiλTμλγSγνμ+eiγSγμνμ(T)fTT+14eiν[T+f(T)]=k22eiγTγ(M)ν,where fT=d

Kinematic variables

The system of field equations described above has only two independent equations with four unknowns. To solve the system completely and in order to study the temporal evolution of energy density, pressure and EoS parameter, we need two more constraint equations (extra conditions). In literature, there are several arguments to choose these equations (see [18] for details). The method is well known as the model independent way approach to study cosmological models that generally considers a

Hybrid teleparallel gravity

For the first case, we presume the functional form of teleparallel gravity to be f(T)=emTTn,where m0 and n constants. Interestingly, this model takes power-law and exponential forms depending on the values of n and m. Particularly:

  • For m=0 Eq. (16) reduces to f(T)=Tn (power law).

  • For and n=0, Eq. (16) reduces to f(T)=emT (exponential).

Using Eq. (16) in Eqs. (7), (8), the expressions of energy density ρ, pressure p and EoS parameter ω reads respectively as ρ=3K+6n(K)ne6mK12n+6nK, p=2αKetαβα

Statefinder diagnostics

Due to the fact that the number of dark energy models are quite large and increasing on a daily basis, it becomes absolutely necessary to find a method to distinguish a particular model from the well established DE models like the ΛCDM, SCDM, HDE, CG and Quintessence. With that reasoning, [22] proposed the {r,s} diagnostics where r and s are defined as r=ä̇aH3, s=r13q12,q12.Different combinations of r and s represent different dark energy models. Particularly:

  • For ΛCDM (r=1,s=0).

  • For SCDM

Energy conditions

Based upon the Raychaudhuri equation, the energy conditions are essential to describe the behavior of the compatibility of timelike, lightlike or spacelike curves [25] and often used to understand the dreadful singularities [26]. Energy conditions in teleparallel gravity have been studied in [27]. Energy conditions also provide the corners in parameter spaces since they violate, for instance, in presence of singularities. They are defined as:

  • Strong energy conditions (SEC): Gravity is always

Observational constraints

In order to find the best fit value of the model parameters of our obtained models, we need to constrain the parameters with some available datasets. Here, we use three datasets, namely, Hubble datasets with 57 datapoints, Supernovae datasets consisting of 580 data points from Union2.1 compilation datasets and Baryonic Acoustic Oscillation (BAO) datasets. We use the Bayesian statistics for our analysis.

Results and discussions

The manuscript communicates the phenomena of late time acceleration in the framework of hybrid and logarithmic teleparallel gravity. To obtain the exact solutions of the field equations, we employ a parametrization of deceleration parameter first proposed in [20]. In this section, we shall discuss the energy conditions and the cosmological viability of the underlying teleparallel gravity models.

In Section 6, we show the temporal evolution of SEC, NEC and WEC for both the teleparallel gravity

CRediT authorship contribution statement

Sanjay Mandal: Writing - original draft. Snehasish Bhattacharjee: Conceptualization, Methodology, Writing - original draft. S.K.J. Pacif: Data curation, Formal analysis. P.K. Sahoo: Writing - review & editing, Validation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

S.M. acknowledges Department of Science & Technology (DST), Govt. of India, New Delhi, for awarding Junior Research Fellowship (File No. DST/INSPIRE Fellowship/2018/IF180676). SB thanks Biswajit Pandey for helpful discussions. PKS acknowledges CSIR, New Delhi, India [http://dx.doi.org/10.13039/501100012522] for financial support to carry out the Research project [No. 03(1454)/19/EMR-II Dt.02/08/2019]. We are very much grateful to the honorable referee and the editor for the illuminating

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