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 how 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 non-linear in different frames.

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 )

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.

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 scalarcurvature and scalar-torsion theories by discussing the field equations, role of connection, conformal transformations, relation to f (Q) theory, and cosmology. The equations for spatially flat universe coincide with those of teleparallel dark energy, thus allowing to explain accelerating expansion. II. CONNECTIONS, GEOMETRIES, AND GRAVITATIONAL THEORIESA. Decomposition of affine connection

The teleparallel formulation of gravity theories reveals close structural analogies to electrodynamics, which are more hidden in their usual formulation in terms of the curvature of spacetime. We show how every locally Lorentz invariant teleparallel theory of gravity with second order field equations can be understood as built from a gravitational field strength and excitation tensor which are related to each other by a constitutive relation, analogous to the premetric construction of theories of electrodynamics. We demonstrate how the previously studied models of f (T) and f (Tax, Tten, Tvec) gravity as well as teleparallel dark energy can be formulated in this language. The advantage of this approach to gravity is that the field equations for different models all take the same compact form and general results can be obtained. An important new such result we find is a constraint which relates the field equations of the tetrad and the spin connection.