Extended gravity theories and the Einstein--Hilbert action

We derive exact Friedmann--Robertson--Walker cosmological solutions in general scalar--tensor gravity theories, including Brans--Dicke gravity, for stiff matter or radiation. These correspond to the long or short wavelength modes respectively of massless scalar fields. If present, the long wavelength modes of such fields would be expected to dominate the energy density of the universe at early times and thus these models provide an insight into the classical behaviour of these scalar--tensor cosmologies near an initial singularity, or bounce. The particularly simple exact solutions also provide a useful example of the possible evolution of the Brans--Dicke (or dilaton) field, $\phi$, and the Brans--Dicke parameter, $\omega(\phi)$, at late times in spatially curved as well as flat universes. We also discuss the corresponding solutions in the conformally related Einstein metric.

Quantum fluctuations of scalar fields during inflation could determine the very large-scale structure of the universe. In the case of general scalar-tensor gravity theories these fluctuations lead to the diffusion of fundamental constants like the Planck mass and the effective Brans--Dicke parameter, $\omega$. In the particular case of Brans--Dicke gravity, where $\omega$ is constant, this leads to runaway solutions with infinitely large values of the Planck mass. However, in a theory with variable $\omega$ we find stationary probability distributions with a finite value of the Planck mass peaked at exponentially large values of $\omega$ after inflation. We conclude that general relativity is an attractor during the quantum diffusion of the fields.

Theories of physics can be considered viable if the initial value problem and the energy conditions are formulated self-consistently. The former allow a uniquely determined dynamical evolution of the system, and the latter guarantee that causality is preserved and that "plausible" physical sources have been considered. In this work, we consider the further degrees of freedom related to curvature invariants and scalar fields in Extended Theories of Gravity (ETG). These new degrees of freedom can be recast as effective perfect fluids that carry different meanings with respect to the standard matter fluids generally adopted as sources of the field equations. It is thus somewhat misleading to apply the standard general relativistic energy conditions to this effective energy-momentum, as the latter contains the matter content and a geometrical quantity, which arises from the ETG considered. Here, we explore this subtlety, extending on previous work, in particular, to cases with the contracted Bianchi identities with diffeomorphism invariance and to cases with generalized explicit curvature-matter couplings, which imply the non-conservation of the energy-momentum tensor. Furthermore, we apply the analysis to specific ETGs, such as scalar-tensor gravity, $f(R)$ gravity and modified Gauss-Bonnet gravity. Interesting results appear such as matter that may exhibit unusual thermodynamical features, for instance, and gravity that retains its attractive character in the presence of negative pressures; or alternatively, we verify that repulsive gravity may occur for standard matter.

The Kaluza-Klein compactification in the limit of large number of extra dimensions is studied. Starting point is the Einstein-Hilbert action plus cosmological constant in 4+D dimensions. It is shown that in the large D limit the effective four dimensional cosmological constant is of order 1/D whereas the size of the extra dimensions remains finite. A 't Hooft like large D expansion of the effective Lagrangian for the Kaluza-Klein scalar and gauge fields arising from the dimensional reduction is considered. It is shown that the propagator of the scalar field associated to the determinant of the metric of the extra dimensions is strongly suppressed. This is an interesting result as in standard Kaluza-Klein theory this scalar degree of freedom is responsible for the constraint on the gauge fields which makes it impossible to recover the usual Yang-Mills equations. Moreover in the large D limit it turns out that the ultraviolet divergences due to the interactions between gauge and scalar fields are softened.

Hybrid metric-Palatini gravity is a recently proposed theory, consisting of the superposition of the metric Einstein-Hilbert Lagrangian with an f(ℛ) term constructed à la Palatini. The theory predicts the existence of a long-range scalar field, which passes the Solar System observational constraints, even if the scalar field is very light, and modifies the cosmological and galactic dynamics.Thus, the theory opens new possibilities to approach, in the same theoretical framework, the problems of both dark energy and dark matter.In this work, we consider the generalized virial theorem in the scalar-tensor representation of the hybrid metric-Palatini gravity. More specifically, taking into account the relativistic collisionless Boltzmann equation, we show that the supplementary geometric terms in the gravitational field equations provide an effective contribution to the gravitational potential energy. We show that the total virial mass is proportional to the effective mass associated with the new terms generated by the effective scalar field, and the baryonic mass. In addition to this, we also consider astrophysical applications of the model and show that the model predicts that the mass associated to the scalar field and its effects extend beyond the virial radius of the clusters of galaxies. In the context of the galaxy cluster velocity dispersion profiles predicted by the hybrid metric-Palatini model, the generalized virial theorem can be an efficient tool in observationally testing the viability of this class of generalized gravity models.

In the context of a cubic Galileon model in which the Vainshtein mechanism suppresses the scalar field interactions with matter, we study low-density stars with slow rotation and static relativistic stars. We develop an expansion scheme to find approximated solutions inside the Vainshtein radius, and show that deviations from General Relativity (GR), while considering rotation, are also suppressed by the Vainshtein mechanism. In a quadratic coupling model, in which the scalarisation effect can significantly enhance deviations from GR in normal scalar tensor gravity, the Galileon term successfully suppresses the large deviations away from GR. Moreover, using a realistic equation of state, we construct solutions for a relativistic star, and show that deviations from GR are more suppressed for higher density objects. However, we found that the scalar field solution ceases to exist above a critical density, which roughly corresponds to the maximum mass of a neutron star. This indicates that, for a compact object described by a polytropic equation of state, the configuration that would collapse into a black hole cannot support a non-trivial scalar field.

The scalar and Einstein field equations are considered in the general symmetric comoving system. The self-gravitational interaction of the scalar field is expressed by identifying the energy momentum tensor with the one canonically associated to the scalar field. It is shown that the scalar field can depend only on the radial and time variables and that the energy momentum tensor is implicitly conserved. Two classes of solutions are then studied. A first class, that contains the static case previously considered and the complete time-dependent case, is determined by a priori assuming the unknown functions to depend on one only variable. This class is meaningful essentially for massless field. A second class of solutions refers to the a priori assumption that the scalar field alone depends on r or on t. In both cases some comments are developed for the massive scalar field and for a large class of massless fields we are able to determine the solutions.

We propose a cosmological framework in which neutrino masses evolve dynamically through coupling with a scalar field that simultaneously drives inflation. The neutrino mass is modeled as a power-law, exponential, or hybrid function of the scalar field, yielding an effective potential that includes neutrino backreaction. Starting from the Einstein–Hilbert action in a flat FLRW background, we derive the modified Friedmann and Klein–Gordon equations incorporating this coupling. Using the Fermi–Dirac integrals, we account for the continuous transition of neutrinos from relativistic to nonrelativistic regimes. The inflationary dynamics are investigated through the slow-roll parameters derived from the effective potential, together with the evaluation of the scalar spectral index , and the tensor-to-scalar ratio r for each model. The exponential and mixed MaVaN couplings emerge as the most flexible cases, allowing inflationary dynamics and neutrino mass variation to be accommodated within a single scalar field, only in a constrained region of the parameter space.

We present a stationary spherically symmetric solution of the Einstein equations, with a source generated by a scalar field of q-theory. In this theory Riemannian gravity, as described by the Einstein–Hilbert action, is coupled to a three—form field that describes the dynamical vacuum. Formally it behaves like a matter field with its own stress—energy tensor, equivalent to a scalar field minimally coupled to gravity. The asymptotically flat solutions obtained to the field equations represent black holes (BHs). For a sufficiently large horizon radius the energy density is localized within a thin spherical shell situated just outside of the horizon, analogous to a gravastar. The resulting solutions to the field equations, which admit this class of configurations, satisfy existence conditions that stem from the BH no—hair theorem, thanks to the presence of a region in space in which the energy density is negative.

The gravitational interaction of a scalar field, with allowance for the possible influence of the torsional and nonmetric nature of space-time, is investigated within the framework of the affine-metric theory of gravitation. The equations of the theory are derived from the variational principle. It is shown that in an affine-metric space, the combined Lagrangian of the gravitational and scalar fields with conformal coupling is reduced to the Lagrangian of the system of gravitational and axion fields in the general theory of relativity. All of the exact general solutions of the consistent system of equations of gravitational and scalar (massless) fields in the affine-metric space under consideration are obtained for all types of homogeneous Friedmann cosmological models, with the initial singularity being removed from some of them. Homogeneous, anisotropic cosmological models, for which all of the exact general solutions are also obtained, are investigated. Some of these models are nonsingular, and the effect of isotropization due to the torsional and nonmetric nature of space-time occurs for many of them.

The theory described by the sum of the Einstein - Hilbert action and the action of a conformal scalar field possesses the duality symmetry, which includes some special conformal transformation of the metric, and also inversion of the scalar field and of the gravitational constant. In the present paper the conformal duality is generalized for an arbitrary spacetime dimension and for the general sigma-model type conformal scalar theory. We also consider applying the conformal duality to the investigation of quantum gravity in the strong curvature regime. The trace of the first coefficient of the Schwinger - DeWitt expansion is derived and its dependence on the gauge fixing condition is considered. Finally, we discuss the way to extract the gauge-fixing independent result and also its possible physical applications.

The work is devoted to a description of the dynamics of proper time as a function of the evolutionary parameter,which can be separated from the Dirac-ADM metric in the Hamiltonian version of the general theory of relativity (GTR) and in conformally invariant, Lichnerowicz variables. In these variables, the GTR is equivalent to the conformally invariant, Penrose-Chernikov-Tagirov, scalar-field theory, with a quantity proportional to the scale factor playing the role of the scalar field. Identifying such a scalar field with the modulus of the Higgs field in the standard model, which unifies the electrvweak and strong interactions, makes it possible to formulate a unified,conformally invariant theory in which the vacuum average of the scalar field is determined by integrals of motion of cosmological evolution.

The gravitational interaction, as described by the Einstein–Cartan theory, is shown to emerge as the by-product of the spontaneous symmetry breaking of a gauge symmetry in a pre-geometric four-dimensional spacetime. Starting from a formulation a ` la Yang–Mills on an SO ( 1 , 4 ) or SO ( 3 , 2 ) principal bundle and not accounting for a spacetime metric, the Einstein–Hilbert action is recovered after the identification of the effective spacetime metric and spin connection for the residual SO ( 1 , 3 ) gauge symmetry of the spontaneously broken phase—i.e. the stabiliser of the SO ( 1 , 4 ) or SO ( 3 , 2 ) gauge group. Thus, the two fundamental tenets of general relativity, i.e. diffeomorphism invariance and the equivalence principle, can arise from a more fundamental gauge principle. The two mass parameters that characterise Einstein gravity, namely the Planck mass and the cosmological constant, are likewise shown to be emergent. The phase transition from the unbroken to the spontaneously broken phase is expected to happen close to the Planck temperature. This is conjectured to be dynamically driven by a scalar field that implements a Higgs mechanism, hence providing mass to new particles, with consequences for cosmology and high-energy physics. The couplings of gravity to matter are discussed after drawing up a dictionary that interconnects pre-geometric and effective geometric quantities. In the unbroken phase where the fundamental gauge symmetry is restored, the theory is potentially power-counting renormalisable without matter, offering a novel path towards a UV completion of Einstein gravity.

In this work, we analyze the Einstein-scalar-Gauss–Bonnet (EsGB) theory of gravity in a cosmological context using the formalism of dynamical systems. We obtain the equations of motion of the theory and introduce an appropriate set of dynamical variables to allow for a direct comparison with the results from General Relativity (GR). We observe that the cosmological phase space features the same set of fixed points as in standard GR, i.e., radiation-dominated, matter-dominated, curvature-dominated, and exponentially-accelerated solutions independently of the values of the coupling function and the scalar field. Furthermore, the radiation-dominated fixed points are repellers and the exponentially accelerated fixed points are attractors in the phase space, thus allowing for cosmological solutions behaving qualitatively similar to the $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> CDM model, i.e., transitioning from a radiation-dominated phase into a matter-dominated phase, and later into a late-time cosmic acceleration phase supported by the scalar field potential. Following a reconstruction method through which we produce the cosmological solutions in the GR limit of the theory and introduce them into the general EsGB dynamical system, a numerical integration of the dynamical system shows that the EsGB theory provides cosmological solutions indistinguishable from those of the standard $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> CDM model, compatible with the current observations from the Planck satellite and weak-field solar system dynamics, while maintaining the scalar field and the coupling function finite and regular throughout the entire time evolution.

Dynamical aspects of an anisotropic cosmological model are investigated in an extended gravity theory. A hybrid scale factor is employed to investigate the cosmic dynamics. The hybrid scale factor simulates a cosmic transit behaviour from a decelerated phase of expansion to an accelerated phase. The parameters of the hybrid scale factors have been constrained from the cosmic transit behaviour and the signature flipping behaviour of the deceleration parameter. The evolutionary aspect of the model has been discussed through the dynamical evolution of the equation of state parameter and the energy conditions. From the present model, scalar fields have been reconstructed using certain reconstruction techniques. Different diagnostic methods have been applied to analyse the viability of the constructed model. The present model almost looks like a cosmological constant for a substantial cosmic time zone and does not show any slowing down feature in near future.