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 has significantly increased in popularity in recent decades, bringing attention to Einstein’s other theory of gravity. In this Review, we give a comprehensive introduction to how teleparallel geometry is developed as a gauge theory of translations together with all the other properties of gauge field theory. We also related this form of geometry to the broader metric-affine approach to forming gravitational theories where we describe a systematic way of constructing consistent teleparallel theories that respect certain physical conditions such as local Lorentz invariance. We first use teleparallel gravity to formulate a teleparallel equivalent of general relativity which is dynamically equivalent to general relativity but which may have different behaviors for other scenarios, such as quantum gravity. After setting this foundation, we describe the plethora of modified teleparallel theories of gravity that have been proposed in the literature. We attempt to connect them together into general classes of covariant gravitational theories. Of particular interest, we highlight the recent proposal of a teleparallel analogue of Horndeski gravity which offers the possibility of reviving all of the regular Horndeski contributions. In the second part of the Review, we first survey works in teleparallel astrophysics literature where we focus on the open questions in this regime of physics. We then discuss the cosmological consequences for the various formulations of teleparallel gravity. We do this at background level by exploring works using various approaches ranging from dynamical systems to Noether symmetries, and more. Naturally, we then discuss perturbation theory, firstly by giving a concise approach in which this can be applied in teleparallel gravity theories and then apply it to a number of important theories in the literature. Finally, we examine works in observational and precision cosmology across the plethora of proposal theories. This is done using some of the latest observations and is used to tackle cosmological tensions which may be alleviated in teleparallel cosmology. We also introduce a number of recent works in the application of machine learning to gravity, we do this through deep learning and Gaussian processes, together with discussions about other approaches in the literature.

We apply Gaussian processes (GP) in order to impose constraints on teleparallel gravity and its f(T) extensions. We use available H(z) observations from (i) cosmic chronometers data (CC); (ii) Supernova type Ia (SN) data from the compressed pantheon release together with the CANDELS and CLASH multi-cycle treasury programs; and (iii) baryonic acoustic oscillation (BAO) datasets from the sloan digital sky survey. For the involved covariance functions, we consider four widely used choices, namely the square exponential, Cauchy, Matérn and rational quadratic kernels, which are consistent with one another within 1σ confidence levels. Specifically, we use the GP approach to reconstruct a model-independent determination of the Hubble constant H 0, for each of these kernels and dataset combinations. These analyses are complemented with three recently announced literature values of H 0, namely (i) Riess H 0 R = 74.22 ± 1.82 k m s − 1 M p c − 1 ; (ii) H0LiCOW collaboration H 0 HW = 73 . 3 − 1.8 + 1.7 k m s − 1 M p c − 1 ; and (iii) Carnegie–Chicago Hubble programme H 0 TRGB = 69.8 ± 1.9 k m s − 1 M p c − 1 . Additionally, we investigate the transition redshift between the decelerating and accelerating cosmological phases through the GP reconstructed deceleration parameter. Furthermore, we reconstruct the model-independent evolution of the dark energy equation of state, and finally reconstruct the allowed f(T) functions. As a result, the ΛCDM model lies inside the allowed region at 1σ in all the examined kernels and datasets, however a negative slope for f(T) versus T is slightly favoured.

We consider a generic cosmological model which allows for non-gravitational direct couplings between dark matter and dark energy. The distinguishing cosmological features of these couplings can be probed by current cosmological observations, thus enabling us to place constraints on this generic interaction which is composed of the conformal and disformal coupling functions. We perform a global analysis in order to independently constrain the conformal, disformal, and mixed interactions between dark matter and dark energy by combining current data from: Planck observations of the cosmic microwave background radiation anisotropies, a combination of measurements of baryon acoustic oscillations, a supernovae Type Ia sample, a compilation of Hubble parameter measurements estimated from the cosmic chronometers approach, direct measurements of the expansion rate of the Universe today, and a compilation of growth of structure measurements. We find that in these coupled dark energy models, the influence of the local value of the Hubble constant does not significantly alter the inferred constraints when we consider joint analyses that include all cosmological probes. Moreover, the parameter constraints are remarkably improved with the inclusion of the growth of structure data set measurements. We find no compelling evidence for an interaction within the dark sector of the Universe.

Abstract. In the search for an explanation for the current acceleration of the Universe, scalar fields are the most simple and useful tools to build models of dark energy. This field, however, must in principle couple with the rest of the world and not necessarily in the same way to different particles or fluids. We provide the most complete dynamical system analysis to date, consisting of a canonical scalar field conformally and disformally coupled to both dust and radiation. We perform a detailed study of the existence and stability conditions of the systems and comment on constraints imposed on the disformal coupling from Big-Bang Nucleosynthesis and given current limits on the variation of the fine-structure constant.

We consider a cosmology in which dark matter and a quintessence scalar field responsible for the acceleration of the Universe are allowed to interact. Allowing for both conformal and disformal couplings, we perform a global analysis of the constraints on our model using Hubble parameter measurements, baryon acoustic oscillation distance measurements, and a Supernovae Type Ia data set. We find that the additional disformal coupling relaxes the conformal coupling constraints. Moreover, we show that, at the background level, a disformal interaction within the dark sector is preferred to both ΛCDM and uncoupled quintessence, hence favoring interacting dark energy.

We study a theory in which the electromagnetic field is disformally coupled to a scalar field, in addition to a usual non-minimal electromagnetic coupling. We show that disformal couplings modify the expression for the fine-structure constant, α. As a result, the theory we consider can explain the non-zero reported variation in the evolution of α by purely considering disformal couplings. We also find that if matter and photons are coupled

A violation of the distance-duality relation is directly linked with a temporal variation of the electromagnetic fine-structure constant. We consider a number of well-studied f(T) gravity models and we revise the theoretical prediction of their corresponding induced violation of the distance-duality relationship. We further extract constraints on the involved model parameters through fine-structure constant variation data, alongside with supernovae data, and Hubble parameter measurements. Moreover, we constrain the evolution of the effective f(T) gravitational constant. Finally, we compare with revised constraints on the phenomenological parametrisations of the violation of the equivalence principle in the electromagnetic sector.