Observational data from distant Type Ia supernovae (SNIa) revealed a hidden fact that the present Universe is expanding in an accelerated phase (Riess et al. 1998; Perlmutter et al. 1999; Kowalski et al. 2008). This important fact has been confirmed by other cosmological observations such as the cosmic microwave background (CMB) (Komatsu et al. 2009; Jarosik et al. 2011; Ade et al. 2016), large-scale structure (LSS), and baryonic acoustic oscillation (BAO) (Tegmark et al. 2004; Cole et al. 2005; Eisenstein et al. 2005; Percival et al. 2010; Blake et al. 2011; Reid et al. 2012), high-redshift galaxies (Alcaniz 2004), high-redshift galaxy clusters (Wang & Steinhardt 1998; Allen et al. 2004), and weak gravitational lensing (Benjamin et al. 2007; Amendola, Kunz & Sapone 2008; Fu et al. 2008). The accelerated phase of the expansion can be interpreted by modifying the standard theory of gravity on cosmological scales or by adding an exotic cosmic fluid with negative pressure, the so-called dark energy (DE) (Riess et al. 1998; Perlmutter et al. 1999; Kowalski et al. 2008). Historically, Einstein’s cosmological constant Λ with a constant equation of state (EoS) parameter equal to −1 is the first and simplest DE model to interpret the current accelerated phase of the expansion. The standard model of cosmology, the ΛCDM model, which accounts for about 70 per cent of the total energy density of the Universe from Λ and about 30 per cent from cold dark matter (CDM) is consistent with the most of the cosmological data. From a theoretical point of view, however, this model has two fundamental problems, namely the problem of fine-tuning and the problem of cosmic coincidence (Weinberg 1989; Sahni & Starobinsky 2000; Carroll 2001; Padmanabhan 2003; Copeland, Sami & Tsujikawa 2006). In addition, the standard model of cosmology suffers from the big tension between the local measurement values of Hubble constant H0 with that of the Planck CMB estimation. Moreover, there are big tensions between the Planck CMB data with weak lensing measurements and redshift surveys, concerning the value of non-relativistic matter density Ωm and the amplitude of the growth of perturbations σ8. The above observational tensions can make the standard model as an approximation of a general gravitational scenario yet to be found. For a recent review about the cosmological tensions of the standard model, we refer the reader to see (Abdalla et al. 2022). In this concern, a large family of dynamical DE models with time-varying EoS parameters has been proposed in the literature (for some earlier attemts, see Armendariz-Picon, Mukhanov & Steinhardt 2000; Caldwell 2002; Padmanabhan 2002; Elizalde, Nojiri & Odintsov 2004). Parallel to the solution of DE, the positive cosmic acceleration can be seen as the expression of a new theory of gravity on large cosmological scales. Indeed, modifying the standard Einstein–Hilbert action in the context of the Friedmann–Robertson–Walker (FRW) metric leads to the modified Friedmann equations, which can be used to justify the current accelerated expansion of the Universe without resorting to DE fluid. One of the most popular modified gravity theories is the f(R) scenario, in which the Lagrangian of the modified Einstein–Hilbert action is extended to the function of the Ricci scalar R (Capozziello, Stabile & Troisi 2007; Capozziello & Francaviglia 2008; Sotiriou & Faraoni 2010; Nojiri & Odintsov 2011). Besides the f(R) theories of gravity, the so-called f(T) theory of gravity is the other solution to solve the puzzle of cosmic acceleration. This theory is defined on the basis of the old definition of the teleparallel equivalent of general relativity (TEGR), first introduced by Einstein (Einstein 1928) and extended by (Hayashi & Shirafuji 1979; Maluf 1994). A comprehensive study on the theory of teleparallel gravity (TG) can be found in the recent review by (Bahamonde et al. 2021). In general, an extended theory of gravity involves curvature, torsion, and non-metricity components. If only torsion is non-vanishing, one can obtain the torsional teleparallel geometry which is the basic geometry for the most f(T) theories in literature. TG theory can be used to formulate a TEGR formalism, which is dynamically equivalent to GR but may have different behaviors for other scenarios, such as quantum gravity. The Horndeski gravity can also be formulated using the teleparallel geometry as a possible revival modes for regular Hordenski gravity models (see Bahamonde et al. 2021, for more detils). TG as a theory built on the tangent sapce must be invariant under general coordinate transformation and local lorentz transformation. In the first formulation of teleparallel theories, it was assumed that the spin connection was always zero and then torsion tensor depends on the tetrads. This torsion tensor is a particular case which is computed in the so-called Weitzenböck gauge (see section 2.2.3 of Bahamonde et al. 2021). By taking a local Lorentz transformation only in the tetrads, one can conclude that the torsion tensor is non-covariant quantity under the local Lorentz transformation. In this context, the action of TEGR has a total divergence term which can be removed as being a boundary term. Hence, the TEGR action is invariant under local Lorentz transformation up to a boundary term. In modified teleparallel theories of gravity like f(T) gravity, we have no boundary term anymore, meaning that f(T) gravity models break the local Lorentz invariance. Notice that the problem of breaking the local Lorentz invariance is related to the particular Weitzenböck gauge (spin connection zero) (Krššák & Saridakis 2016). If we take the simultaneous transformations in the tetrads and the spin connection, the teleparallel theories are then fully invariant (diffeomorphisms and local Lorentz). Thus, any action and consequently field equations constructed based on the torsion tensor will be fully invariant. In the context of cosmology, there is a large body of works that has examined the cosmological properties of various f(T) models. In this framework, the dynamics and various aspects of the Universe with homogeneous and isotropic background are studied in recent works (Bengochea & Ferraro 2009; Linder 2010; Wu & Yu 2011, 2010b; Bamba et al. 2011, 2012; Dent, Dutta & Saridakis 2011; Geng et al. 2011; Myrzakulov 2011; Zhang et al. 2011). In general, cosmological consequences for the various formulations of TG at both background and perturbation levels have been discussed in (Bahamonde et al. 2021). The f(T) models have been studied and constrained using the various cosmological data (Wu & Yu 2010a; Capozziello, Luongo & Saridakis 2015; Iorio, Radicella & Ruggiero 2015; Nunes, Pan & Saridakis 2016; Saez-Gomez et al. 2016). As a more recent study, we refer the work of Briffa et al. 2022, where the various versions of f(T) model have been studied observationally. Using the combinations of cosmological data including the cosmic chronometers, the SNIa observations from Pantheon catalogue, BAO observations, and different model-independent values for Hubble constant H0, Briffa et al. (2022) studied a detailed analysis of the impact of the H0 priors on the late time properties of f(T) cosmologies. They found a higher value of H0 compared to equivalent analysis without considering H0 priors. In addition, they showed that the f(T) model produces the higher value of H0 and slightly lower value of matter density Ωm0 as compared to standard ΛCDM model. In general, studies on the cosmological tensions and their possible alleviation in TG formalism have been addressed in section 10 of review article (Bahamonde et al. 2021). For a review of cosmological tensions H0 and σ8, we refer the reader to see the current studies in (Verde, Treu & Riess 2019; Di Valentino et al. 2021a, b, c; Perivolaropoulos & Skara 2021).
Citation: Padmanabhan 2002
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