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 consider first generation scalar–tensor theories of gravitation in a completely generic form, keeping the transformation functions of the local rescaling of the metric and the scalar field redefinition explicitly distinct from the coupling functions in the action. It is well known that in Jordan frame Brans–Dicke type parametrization, the diverging kinetic coupling function can lead to the general relativity regime, however the transformation functions to other parametrizations then typically become singular, possibly spoiling the correspondence between different parametrizations. We give a detailed analysis of the transformation properties of field equations with arbitrary metrics and also in the Friedmann cosmology, and provide sufficient conditions under which the correspondence between different parametrizations is retained, even if the transformation is singular. It is interesting to witness the invariance of the notion of the general relativity regime and the correspondence of the perturbed cosmological equations as well as their solutions in different parametrizations, despite the fact that in some cases the perturbed equation turns out to be linear in one parametrization and nonlinear in another.

We calculate the parametrized post-Newtonian parameters $\gamma$ and $\beta$ for scalar-tensor gravity with a generic coupling function $\omega$ and scalar potential $V$ in the Jordan conformal frame in the case of a static spherically symmetric source. Since the potential generally introduces a radial dependence to the effective gravitational constant as well as to $\gamma$ and $\beta$, we discuss the issue of defining these parametrized post-Newtonian parameters and compare our expressions with previous calculations in simpler cases. We confront our results with current observational constraints on the values of $\gamma$ and $\beta$ and thus draw restrictions on the form of the functions $\omega$ and $V$ around their asymptotic background values.

A general scalar–tensor theory can be formulated in different parametrizations that are related by a conformal rescaling of the metric and a scalar field redefinition. We compare formulations of slow-roll regimes in the Einstein and Jordan frames using quantities that are invariant under the conformal rescaling of the metric and transform as scalar functions under the reparametrization of the scalar field. By comparing spectral indices, calculated up to second order, we find that the frames are equivalent up to this order, due to the underlying assumptions.

We compute the parametrized post-Newtonian parameter $\gamma$ in the case of a static point source for multiscalar-tensor gravity with completely general nonderivative couplings and potential in the Jordan frame. Similarly to the single massive field case $\gamma$ depends exponentially on the distance from the source and is determined by the length of a vector of nonminimal coupling in the space of scalar fields and its orientation relative to the mass eigenvectors. Using data from the Cassini tracking experiment, we estimate bounds on a general theory with two scalar fields. Our formalism can be utilized for a wide range of models, which we illustrate by applying it to nonminimally coupled Higgs SU(2) doublet, general hybrid metric-Palatini gravity, linear (${\square }^{-1}$) and quadratic (${\square }^{-2}$) nonlocal gravity.

We consider homogeneous and isotropic cosmological models in the framework of three geometrical theories of gravitation. In Einstein’s general relativity, they are given in terms of the curvature of the Levi-Civita connection in torsion-free metric spacetimes; in the teleparallel equivalent of general relativity, they are given in terms of the torsion of flat metric spacetimes; and in the symmetric teleparallel equivalent of general relativity, they are given in terms of the nonmetricity of flat torsion-free spacetimes. We argue that although these three formulations seem to be different, the corresponding cosmological models are in fact equivalent and their choice is conventional.