The universal character of the gravitational interaction provided by the equivalence principle motivates a geometrical description of gravity. The standard formulation of general relativity a la Einstein attributes gravity to the spacetime curvature, to which we have grown accustomed. However, this perception has masked the fact that two alternative, though equivalent, formulations of general relativity in flat spacetimes exist, where gravity can be fully ascribed either to torsion or to nonmetricity. The latter allows a simpler geometrical formulation of general relativity that is oblivious to the affine spacetime structure. Generalizations along this line permit us to generate teleparallel and symmetric teleparallel theories of gravity with exceptional properties. In this work we explore modified gravity theories based on nonlinear extensions of the nonmetricity scalar. After presenting some general properties and briefly studying some interesting background cosmologies (including accelerating solutions with relevance for inflation and dark energy), we analyze the behavior of the cosmological perturbations. Tensor perturbations feature a rescaling of the corresponding Newton's constant, while vector perturbations do not contribute in the absence of vector sources. In the scalar sector we find two additional propagating modes, hinting that f(Q) theories introduce, at least, 2 additional degrees of freedom. These scalar modes disappear around maximally symmetric backgrounds because of the appearance of an accidental residual gauge symmetry corresponding to a restricted diffeomorphism. We finally discuss the potential strong coupling problems of these maximally symmetric backgrounds caused by the discontinuity in the number of propagating modes.

We consider a general theory of all possible quadratic, first-order derivative terms of the non-metricity tensor in the framework of Symmetric Teleparallel Geometry. We apply the Noether Symmetry Approach to classify those models that are invariant under point transformations in a cosmological background and we use the symmetries of these models to reduce the dynamics of the system in order to find analytical solutions.

This article presents an exhaustive classification of metric-affine theories according to their scale symmetries. First it is clarified that there are three relevant definitions of a scale transformation. These correspond to a projective transformation of the connection, a rescaling of the orthonormal frame, and a combination of the two. The most general second order quadratic metric-affine action, including the parity-violating terms, is constructed in each of the three cases. The results can be straightforwardly generalised by including higher derivatives, and implemented in the general metric-affine, teleparallel, and symmetric teleparallel geometries.

In the recently introduced gauge theory of translations, dubbed Coincident General Relativity (CGR), gravity is described with neither torsion nor curvature in the spacetime affine geometry. The action of the theory enjoys an enhanced symmetry and avoids the second derivatives that appear in the conventional Einstein-Hilbert action. While it implies the equivalent classical dynamics, the improved action principle can make a difference in considerations of energetics, thermodynamics and quantum theory. This paper reports on possible progress in those three aspects of gravity theory. In the so-called purified gravity, (1) energy-momentum is described locally by a conserved, symmetric tensor, (2) the Euclidean path integral is convergent without the addition of boundary or regulating terms and (3) it is possible to identify a canonical frame for quantization.

A Cartan geometry of the General Linear symmetry is formulated by dividing out the displacements from the group. The resulting action is quadratic in curvature, polynomial in all the (minimal) variables, and describes an observer space that-in the symmetry-broken phase-reproduces the predictions of General Relativity in the presence of dark matter.

The geometrical nature of gravity emerges from the universality dictated by the equivalence principle. In the usual formulation of General Relativity, the geometrisation of the gravitational interaction is performed in terms of the spacetime curvature, which is now the standard interpretation of gravity. However, this is not the only possibility. In these notes, we discuss two alternative, though equivalent, formulations of General Relativity in flat spacetimes, in which gravity is fully ascribed either to torsion or to non-metricity, thus putting forward the existence of three seemingly unrelated representations of the same underlying theory. Based on these three alternative formulations of General Relativity, we then discuss some extensions.

The observer's frame is the more elementary description of the gravitational field than the metric. The most general covariant, even-parity quadratic form for the frame field in arbitrary dimension generalises the New General Relativity by nine functions of the d'Alembertian operator. The degrees of freedom are clarified by a covariant derivation of the propagator. The consistent and viable models can incorporate an ultra-violet completion of the gravity theory, an additional polarisation of the gravitational wave, and the dynamics of a magnetic scalar potential.

In a talk at the conference Geometrical Foundations of Gravity at Tartu, 2017, it was suggested that the affine spacetime connection could be associated with purely fictitious forces. This leads to gravitation in a flat and smooth geometry. Fermions are found to nevertheless couple with the metrical connection and a phase gauge field. The theory is reviewed in this proceeding in a Palatini, and in a metric-affine gauge formulation.

The metric-affine variational principle is applied to generate teleparallel and symmetric teleparallel theories of gravity. From the latter we discover an exceptional class which is consistent with a vanishing affine connection. Based on this remarkable property, this work proposes a simpler geometrical formulation of general relativity that is oblivious to the affine spacetime structure, thus fundamentally depriving gravity of any inertial character. The resulting theory is described by the Hilbert action purged from the boundary term and is more robustly underpinned by the spin-2 field theory, where an extra symmetry is now manifest, possibly related to the double-copy structure of the gravity amplitudes. This construction also provides a novel starting point for modified gravity theories, and the paper presents new and simple generalizations where analytical self-accelerating cosmological solutions arise naturally in the early-and late-time Universe.

Cosmological perturbations are considered in f (T) and in scalar-torsion f (phi) T teleparallel models of gravity. Full sets of linear perturbation equations are accurately derived and analysed at the relevant limits. Interesting features of generalisations to other teleparallel models, spatially curved backgrounds, and rotated tetrads are pointed out.