Accelerating universe in hybrid and logarithmic teleparallel gravity
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 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 gravity, gravity, 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 gravity [6] (See [7] for a review on teleparallel gravity). In teleparallel gravity, the Ricci scalar of the underlying geometry in the action is replaced by an arbitrary functional form of torsion scalar . 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 is related to the antisymmetric connection following from the non-holonomic basis [3], [8].
Linear gravity models are the teleparallel equivalent of GR (TEGR) [9]. Nonetheless, gravity differ significantly from gravity in the fact that the field equations in gravity are always at second-order compared to the usual fourth-order in gravity. This owes to the fact that the torsion scalar contains only the first derivatives of the vierbeins and thus makes cosmology in 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 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 gravity have been discussed in Ref. [16], [17].
The manuscript is organized as follows: In Section 2 we present an overview of 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 datapoints, Supernovae datasets consisting of 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 gravity
The action in teleparallel gravity is represented as where and is Newtonian gravitational constant. The gravitational field in this framework arises due to torsion defined as The contracted form of torsion tensor reads varying the action , where represent the matter Lagrangian yields the field equations as where
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 where and constants. Interestingly, this model takes power-law and exponential forms depending on the values of and . Particularly:
- •
For Eq. (16) reduces to (power law).
- •
For and , Eq. (16) reduces to (exponential).
Using Eq. (16) in Eqs. (7), (8), the expressions of energy density , pressure and EoS parameter reads respectively as
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 CDM, SCDM, HDE, CG and Quintessence. With that reasoning, [22] proposed the diagnostics where and are defined as Different combinations of and represent different dark energy models. Particularly:
- •
For CDM .
- •
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 datapoints, Supernovae datasets consisting of 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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