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.

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

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.Starobinsky model [10], non-minimal Higgs inflation [11], or generalized models with alphaattractors [12], which all deal with non-minimal couplings. By introducing a non-minimal coupling one works in the framework of the scalar-tensor theory (STG) [13][14][15][16] rather than in general relativity with a minimally coupled scalar field. Many modified gravity theories [17] can be recast into STG form, e.g. f (R) [18], scale-free [19], and non-local [20] theories.Minimally coupled (MC) and non-minimally coupled (NMC) theories are in principle different, because the latter permits transformations which consist of rescaling of the metric and reparametrization of the scalar field. This allows one to perform transformations between different conformal frames. For example, a NMC theory can be transformed to such a conformal frame, where the scalar field is minimally coupled either to curvature (the Einstein frame) or to matter (the Jordan frame). In the case of the standard MC theory these frames coincide and hence it contains only one metric, which corresponds to the measurable one [14,15,21]. In the case of NMC theories the choice of frame is arbitrary from the mathematical point of view, and one can work in either frame, but there is an ongoing debate on their physical equivalence [22][23][24].Since inflation is driven by the scalar field, one can neglect the matter terms yielding that for inflationary dynamics, the NMC theory in Einstein frame and the MC theory have identical equations. This provides a method for generalizing the standard slow-roll regime from the MC theory to the general STG. However, in the MC theory the Einstein and Jordan frames simply coincide, and generalization of the results from the MC theory to the Einstein frame of the NMC theory must be taken with care. For instance, in the NMC theory one finds that the number of inflationary e-folds in different frames is not equal [25,26] (but cf. [16]). Further, in the MC theory the measurement of the amplitude of tensor fluctuations would determine the scale of inflation, whereas in the case of the NMC theory it would not completely determine the scale of inflation [27].Regarding the Jordan frame as physical motivates one to consider the slow-roll inflation directly in the Jordan frame. This can be achieved by defining the generalized or extended slow-roll conditions [26,28,29]. In addition to the slow-roll parameters controlling the flatness of the effective potential, one has parameters for the n...

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 the Jordan frame Brans-Dicke type parametrization the diverging kinetic coupling function ω → ∞ can lead to the general relativity regime, however then the transformation functions to other parametrizations typically become singular, possibly spoiling the correspondence between different parametrizations. We give a detailed analysis of the transformation properties of the field equations with arbitrary metric 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 some other.

We extend the class of recently formulated scalar-nonmetricity theories by coupling a fiveparameter nonmetricity scalar to a scalar field and considering a mixed kinetic term between the metric and the scalar field. The symmetric teleparallel constraint is invoked by Lagrange multipliers or by inertial variation. The equivalents for the general relativity and ordinary (curvature-based) scalar-tensor theories are obtained as particular cases. We derive the field equations, discuss some technical details, e.g., debraiding, and formulate the Hamilton-like approach. CONTENTS