Einstein’s celebrated theory of gravitation can be presented in three forms: general relativity, teleparallel gravity, and the rarely considered before symmetric teleparallel gravity. Extending the latter, we introduce a new class of theories where a scalar field is coupled nonminimally to nonmetricity $Q$, which here encodes the gravitational effects like curvature $R$ in general relativity or torsion $T$ in teleparallel gravity. We point out the similarities and differences with analogous scalar-curvature and scalar-torsion theories by discussing the field equations, role of connection, conformal transformations, relation to $f(Q)$ theory, and cosmology. The equations for a spatially flat universe coincide with those of teleparallel dark energy, thus allowing us to explain accelerating expansion.

We consider a generalized teleparallel theory of gravitation, where the action contains an arbitrary function of the torsion scalar and a scalar field, $f(T,\varphi )$, thus encompassing the cases of $f(T)$ gravity and a nonminimally coupled scalar field as subclasses. The action is manifestly Lorentz invariant when besides the tetrad one allows for a 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.

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 antisymmetric 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 $f(T)$ gravity.

We consider the general scalar-tensor gravity without derivative couplings. By rescaling of the metric and reparametrization of the scalar field, the theory can be presented in different conformal frames and parametrizations. In this work we argue that while due to the freedom to transform the metric and the scalar field, the scalar field itself does not carry a physical meaning (in a generic parametrization), there are functions of the scalar field and its derivatives which remain invariant under the transformations. We put forward a scheme to construct these invariants, discuss how to formulate the theory in terms of the invariants, and show how the observables like parametrized post-Newtonian parameters and characteristics of the cosmological solutions can be neatly expressed in terms of the invariants. In particular, we describe the scalar field solutions in Friedmann-Lemaître-Robertson-Walker cosmology in Einstein and Jordan frames and explain their correspondence despite the approximate equations turning out to be linear and nonlinear in different frames.

We perform an analysis of models of chaotic inflation where the inflaton field $\varphi$ is coupled nonminimally to gravity via $\xi {\varphi }^n{g}^{\mu \nu }{R}_{\mu \nu }(\mathrm{\Gamma }),n>0$. We focus on the Palatini theory of gravity, i.e., the case where the assumptions of general relativity are relaxed (that of the connection being the Levi-Civita one) and the gravitational degrees of freedom are encoded in not only the metric but also the connection $\mathrm{\Gamma }$, which is treated as an independent variable. We show that in this case the famous attractor behavior of simple nonminimally coupled models of inflation is lost. Therefore the attractors are not universal, but their existence depends on the underlying theory of gravity in a subtle way. We discuss what this means for chaotic models and their observational consequences.

Seemingly unrelated models of inflation that originate from different physical setups yield, in some cases, identical predictions for the currently constrained inflationary observables. In order to classify the available models, we propose to express the slow-roll parameters and the relevant observables in terms of frame and reparametrization invariant quantities. The adopted invariant formalism makes manifest the redundancy that afflicts the current description of inflation dynamics and offers a straightforward way to identify classes of models which yield identical phenomenology. In this Letter, we offer a step-to-step recipe to recast every single field inflationary model in the proposed formalism, detailing also the procedure to compute inflationary observables in terms of frame and reparametrization invariant quantities. We hope that our results become the cornerstone of a new categorization of viable inflationary models and open the way to a deeper understanding of the inflation mechanism.