Equation of State LHS Left hand side RHS Right hand side STEGR Symmetric teleparallel equivalent of general relativity TEGB Teleparallel equivalent of the Gauss-Bonnet PPN Parameterized post-Newtonian formalism SN Supernova (we use abbreviations SNeIa and SNIa to refer to supernova type 1a) BAO Baryonic Acoustic Oscillations CC Cosmic Chronometers DE Dark energy HDE Holographic Dark Energy QCD Quantum Chromodynamics PlDE Pilgrim Dark Energy CMB Cosmic Microwave Background WMAP Wilkinson Microwave Anisotropy Probe SVT Scalar-Vector-Tensor VLBI Very Long Baseline Interferometry SH0ES Supernova H0 for the Equation of State TRGB Tip of the red-giant branch TDCOSMO Time-delay Cosmography vi

Teleparallel gravity theories employ a tetrad and a Lorentz spin connection as independent variables in their covariant formulation. In order to solve their field equations, it is helpful to search for solutions which exhibit certain amounts of symmetry, such as spherical or cosmological symmetry. In this article we present how to apply the notion of spacetime symmetries known from Cartan geometry to teleparallel geometries. We explicitly derive the most general tetrads and spin connections which are compatible with axial, spherical, cosmological and maximal symmetry. For homogeneous and isotropic spacetime symmetry we find that the tetrads and spin connection found by the symmetry constraints are universal solutions to the anti-symmetric part of the field equations of any teleparallel theory of gravity. In other words, for cosmological symmetry we find what has become known as "good tetrads" in the context of f (T )

General Relativity and the ΛCDM framework are currently the standard lore and constitute the concordance paradigm. Nevertheless, long-standing open theoretical issues, as well as possible new observational ones arising from the explosive development of cosmology the last two decades, offer the motivation and lead a large amount of research to be devoted in constructing various extensions and modifications.All extended theories and scenarios are first examined under the light of theoretical consistency, and then are applied to various geometrical backgrounds, such as the cosmological and the spherical symmetric ones. Their predictions at both the background and perturbation levels, and concerning cosmology at early, intermediate and late times, are then confronted with the huge amount of observational data that astrophysics and cosmology are able to offer recently. Theories, scenarios and models that successfully and efficiently pass the above steps are classified as viable and are candidates for the description of Nature.We list the recent developments in the fields of gravity and cosmology, presenting the state of the art, high-lighting the open problems, and outlining the directions of future research.

We consider a generalized teleparallel theory of gravitation, where the action contains an arbitrary function of the torsion scalar and a scalar field, f (T, φ), thus encompassing the cases of f (T ) gravity and nonminimally coupled scalar field as subclasses. The action is manifestly Lorentz invariant when besides the tetrad one allows for flat but nontrivial spin connection. We derive the field equations and demonstrate how the antisymmetric part of the tetrad equations is automatically satisfied when the spin connection equation holds. The spin connection equation is a vital part of the covariant formulation, since it determines the spin connection associated with a given tetrad. We discuss how the spin connection equation can be solved in general, and provide the cosmological and spherically symmetric examples. Finally we generalize the theory to an arbitrary number of scalar fields.

In the attempts to apply Finsler geometry to construct an extension of general relativity, the question about a suitable generalization of the Einstein equations is still under debate. Since Finsler geometry is based on a scalar function on the tangent bundle, the field equation which determines this function should also be a scalar equation. In the literature two such equations have been suggested: the one by Rutz and the one by one of the authors. Here we employ the method of canonical variational completion to show that Rutz equation can not be obtained from a variation of an action and that its variational completion yields the latter field equations. Moreover, to improve the mathematical rigor in the derivation of the Finsler gravity field equation, we formulate the Finsler gravity action on the positive projective tangent bundle. This has the advantage of allowing us to apply the classical variational principle, by choosing the domains of integration to be compact and independent of the dynamical variable. In particular in the pseudo-Riemannian case, the vacuum field equation becomes equivalent to the vanishing of the Ricci tensor. * manuel.hohmann@ut.ee † christian.pfeifer@ut.ee ‡ nico.voicu@unitbv.ro arXiv:1812.11161v3 [gr-qc] 1 Oct 2019 3 The ones considered in [44,45] do not fit in our definition. We do not consider these Finsler spacetimes since for them the curvature tensor, which defines the dynamics of Finsler spacetimes, is not necessarily defined for all physical observer directions, which in our definition is given by the conic subbundle T . The definition could be relaxed so as to include the possibility of having an observer direction where curvature is not defined, but in this case, a thorough analysis of whether the evolution of spacetime is causal, as seen by the respective observer, is needed. This is the subject for future work.