This thesis investigates a toy model for inflation in a class of modified theories of gravity in the metric formalism. Instead of the standard procedure -- assuming a non-linear Lagrangian $f(R)$ in the Jordan frame -- we start from a simple ϕ^2 potential in the Einstein frame and investigate the corresponding $f(R)$ in the former picture. Such approach yields plenty of new pieces of information, namely a self-terminating inflationary solution with a linear Lagrangian, a robust criterion for stability of such theories, a dynamical effective potential for the Ricci scalar $R$, the addition of an ad-hoc Cosmological Constant in the Einstein frame leads to a Thermodynamical interpretation of this physical system, which allows further insight on its (meta)stability and evolution.
We study integrable models in the Bianchi I metric case with scalar fields minimally and non-minimally coupled with gravity and the correspondence between their general solutions. Using the model with a minimally coupled scalar field and a constant potential as an example, we demonstrate how to obtain the general solutions of the corresponding models in the Jordan frame.
Even though we know that physical observations are frame independent, the frame dependence of cosmological perturbations is relatively subtle and has led to confusion in the past. In this paper we show that while the (unobservable) matter power spectrum is frame dependent, the observable number counts are not. We shall also determine how the frame dependence of the power spectrum depends on scale.
We show, considering a specific f(R)-gravity model, that the Jordan frame and the Einstein frame are physically non-equivalent, although they are connected by a conformal transformation which yields a mathematical equivalence. Since all the calculations are performed analytically, this non-equivalence is shown in an unambiguous way.
Jordan and Einstein frame are studied under the light of Hamiltonian formalism. Dirac's constraint theory for Hamiltonian systems is applied to Brans-Dicke theory in the Jordan Frame. In both Jordan and Einstein frame, Brans-Dicke theory has four secondary first class constraints and their constraint algebra is closed. We show, contrary to what is generally believed, the Weyl (conformal) transformation, between the two frames, is not a canonical transformation, in the sense of Hamiltonian formalism. This addresses quantum mechanical inequivalence as well. A canonical transformation is shown.
We investigate how inflationary predictions are affected by the difference in the number of $e$-folds between the Jordan and Einstein frames. We study several test models in relation to a Jordan frame defined by the common Higgs-inflation-like non-minimal coupling to gravity and consider two different formulations of gravity: metric and Palatini. We find that the difference is quite contained in case of a metric Jordan frame while can be quite remarkable in case of a Palatini Jordan frame. We also discuss a way to overcome the discrepancy by introducing a frame invariant physical distance and …
In this work we shall study a new class of attractor models which we shall call generalized $R^p$-attractor models. This class of models is based on a generalization of the Einstein frame potential of $R^p$ $f(R)$ gravity models in the Jordan frame. We present the attractor properties of the corresponding non-minimally coupled Jordan frame theory, and we calculate the observational indices of inflation in the Einstein frame. As we show, there is a large class of non-minimally coupled scalar theories, with an arbitrary non-minimal coupling which satisfies certain conditions, that yield the same…
We discuss the conformal symmetry between Jordan and Einstein frames considering their relations with the metric and Palatini formalisms for modified gravity. Appropriate conformal transformations are taken into account leading to the evident connection between the gravitational actions in the two mentioned frames and the Hilbert-Einstein action with a cosmological constant. We show that the apparent differences between Palatini and metric formalisms strictly depend on the representation while the number of degrees of freedom is preserved. This means that the dynamical content of both formalis…
The present work shows that the mathematical equivalence of Jordan frame and its conformally transformed version, the Einstein frame, so far as Brans-Dicke theory is concerned, survives a quantization of cosmological models in the theory. We work with the Wheeler-deWitt quantization scheme and take up quite a few anisotropic cosmological models as examples. We effectively show that the transformation from Jordan to Einstein frame is a canonical one and hence two frames are equivalent description of same physical scenario.
This paper is devoted to the analysis of the covariant canonical formalism of $F(R)$ gravity in Einstein frame. We also find canonical transformation between covariant canonical formulation of F(R) gravity in Jordan frame and Einstein frames and we also determine corresponding generating function.
We consider the correspondence between the Jordan frame and the Einstein frame descriptions of scalar-tensor theory of gravitation. We argue that since the redefinition of the scalar field is not differentiable at the limit of general relativity the correspondence between the two frames is lost at this limit. To clarify the situation we analyse the dynamics of the scalar field in different frames for two distinct scalar-tensor cosmologies with specific coupling functions and demonstrate that the corresponding scalar field phase portraits are not equivalent for regions containing the general re…
It is shown that the Jordan frame and its conformally transformed version, the Einstein frame of nonminimally coupled theories of gravity, are actually equivalent at the quantum level. The example of the theory taken up is the Brans-Dicke theory, and the wave packet calculations are done for a homogeneous and isotropic cosmological model in the purest form of the theory, i.e., in the absence of any additional matter sector. The calculations are clean and exact, and the result obtained are unambiguous.
With an explicit example, we show that Jordan frame and the conformally transformed Einstein frames clearly lead to different physics for a non-minimally coupled theory of gravity, namely Brans-Dicke theory, at least at the quantum level. The example taken up is the spatially flat Friedmann cosmology in Brans-Dicke theory.
We investigate the dynamics of $f(R)$ gravity in Jordan and Einstein frames. First, we perform a phase-space singularities analysis in both frames. We show that, typically, anisotropic singularities are absent in the Einstein frame, whereas they may appear in the Jordan frame. We conciliate this apparent inconsistency by showing that the necessary conditions for the existence of the Einstein frame are namely the same ones assuring the absence of the anisotropic singularities in the Jordan frame. In other words, we show that, at least in the context of Bianchi I cosmologies, the Einstein frame …
It is well known that, in contrast to general relativity, there are two conformally related frames, the Jordan frame and the Einstein frame, in which the Brans-Dicke theory, a prototype of generic scalar-tensor theory, can be formulated. There is a long standing debate on the physical equivalence of the formulations in these two different frames. It is shown here that gravitational deflection of light to second order accuracy may observationally distinguish the two versions of the Brans-Dicke theory.
We study the evolution of homogeneous and isotropic, flat cosmological models within the general scalar-tensor theory of gravity with arbitrary coupling function and potential and scrutinize its limit to general relativity. Using the methods of dynamical systems for the decoupled equation of the Jordan frame scalar field we find the fixed points of flows in two cases: potential domination and matter domination. We present the conditions on the mathematical form of the coupling function and potential which determine the nature of the fixed points (attractor or other). There are two types of fix…
We study the relation between the Jordan-Einstein frame transition and the possible description of the crossing of singularities in flat Friedmann universes, using the fact that the regular evolution in one frame can correspond to crossing singularities in the other frame. We show that some interesting effects arise in simple models such as one with a massless scalar field or another wherein the potential is constant in the Einstein frame. The dynamics in these models and in their conformally coupled counterparts are described in detail, and a method for the continuation of such cosmological e…
We study capability of $f(R)$ gravity models to allow crossing the phantom boundary in both Jordan and Einstein conformal frames. In Einstein frame, these models are equivalent to Einstein gravity together with a scalar field minimally coupled to gravity. This scalar degree of freedom appears as a quintessence field with a coupling with the matter sector. We investigate evolution of the equation of sate parameter for some cosmologically viable $f(R)$ gravity models in both conformal frames. This investigation (beyond mere theoretical arguments) acts as an operational tool to distinguish physic…
Distinguishing conformally coupled frames from the tree-level perturbative observables (scalar spectral index $n_{\rm s}$ and tensor-to-scalar ratio $r$) is challenging in cosmology as they are nearly identical. However, since the background evolution in these two frames differs significantly, we can look for potential signatures in the reheating constraints to discriminate these frames. In this work, we study the reheating phase in these frames and find that the difference in the inflationary energy scales in these frames contributes to a significant difference in the reheating e-folding numb…
We examine the possibility of static wormhole solutions in the vacuum Brans-Dicke theory both in the original (Jordan) frame and in the conformally rescaled (Einstein) frame. It turns out that, in the former frame, wormholes exist only in a very narrow interval of the coupling parameter, viz., -3/2<omega<-4/3. It is shown that these wormholes are not traversable in practice. In the latter frame, wormhole solutions do not exist at all unless energy conditions are violated by hand.
The $f(R)$ gravity and scalar-tensor theory are known to be equivalent at the classical level. We study if this equivalence is valid at the quantum level. There are two descriptions of the scalar-tensor theory in the Jordan and Einstein frames. It is shown that these three formulations of the theories give the same determinant or effective action on shell, and thus they are equivalent at the quantum one-loop level on shell in arbitrary dimensions. We also compute the one-loop divergence in $f(R)$ gravity.
We construct a dark energy model where a scalar field non-minimally coupled to gravity plays the role of the dark component. We compare cosmological consequences of this non-minimal coupling of the scalar field and gravity in the spirit of the dark energy paradigm in Jordan and Einstein frames. Some important issues such as phantom divide line crossing, existence of the bouncing solutions and the stability of the solutions are compared in these two frames. We show that while a non-minimally coupled scalar field in the Jordan frame is a suitable dark energy component with capability to realize …
In this note we consider the issue of the classical equivalence of scale-invariant gravity in the Einstein and in the Jordan frames. We first consider the simplest example $f(R)=R^{2}$ and show explicitly that the equivalence breaks down when dealing with Ricci-flat solutions. We discuss the link with the fact that flat solutions in quadratic gravity have zero energy. We also consider the case of scale-invariant tensor-scalar gravity and general $f(R)$ theories. We argue that all scale-invariant gravity models have Ricci flat solutions in the Jordan frame that cannot be mapped into the Einstei…
We investigate the classical stability of Schwarzschild black hole in Jordan and Einstein frames which are related by the conformal transformations. For this purpose, we introduce two models of the Brans-Dicke theory and Brans-Dicke-Weyl gravity in Jordan frame and two corresponding models in the Einstein frame. The former model is suitable for studying the massless spin-2 graviton propagating around the Schwarzschild black hole, while the later is designed for the massive spin-2 graviton propagating around the black hole. It turns out that the black hole (in)stability is independent of the fr…
The conformal equivalence between Jordan frame and Einstein frame can be used in order to search for exact solutions in general theories of gravity in which scalar fields are minimally or nonminimally coupled with geometry. In the cosmological arena a relevant role is played by the time parameter in which dynamics is described. In this paper we discuss such issues considering also if cosmological Noether symmetries in the ``point--like'' Lagrangian are conformally preserved. Through this analysis and through also a careful analysis of the cosmological parameters Ωand Λ, it is possible to contr…
We study the finite time singularity correspondence between the Jordan and Einstein frames for various $F(R)$ gravity theories. Particularly we investigate the ordinary pure $F(R)$ gravity case and the unimodular $F(R)$ gravity cases, in the absence of any matter fluids. In the ordinary $F(R)$ gravity cases, by using specific illustrative examples, we show that it is possible to have various correspondences of finite time singularities, and in some cases it is possible a singular cosmology in one frame might be non-singular in the other frame. In the unimodular $F(R)$ gravity case, the unimodu…
The Hamiltonian of any quantum cosmological model must obey some conditions in order to be self-adjoint, or to admit self-adjoint extensions. We will consider in this paper the self-adjoint character of the Hamiltonian resulting from the Brans-Dicke theory, both in Einstein and Jordan frames. In both cases we find the wave-function solutions in order to obtain concrete predictions for the evolution of the Universe. We show that the specific cosmological scenarios obtained depend very weakly on the conditions of having a self-adjoint operator. The problem of equivalence between Einstein and Jor…
Vacuum Brans-Dicke theory can be self-consistently described in two frames, the Jordan frame (JF) and the conformally rescaled Einstein frame (EF), the transformations providing an easy passage from one frame to the other at the level of actions and solutions. Despite this, the conformal frames are inequivalent describing different geometries. It is shown that the predictions of the weak field lensing (WFL) observables in the EF are different from those recently obtained in the JF for the vacuum Brans-Dicke class 1 solution. The value of the Brans-Dicke coupling parameter $ω$ from the Cassini …
Three new classes (II-IV) of solutions of the vacuum low energy effective string theory in four dimensions are derived. Wormhole solutions are investigated in those solutions including the class I case both in the Einstein and in the Jordan (string) frame. It turns out that, of the eight classes of solutions investigated (four in the Einstein frame and four in the corresponding string frame), massive Lorentzian traversable wormholes exist in five classes. Nontrivial massless limit exists only in class I Einstein frame solution while none at all exists in the string frame. An investigation of t…
In this work, we present a complete analysis of the quantisation of the classical Brans-Dicke Theory using the method of affine quantisation in the Hamiltonian description of the theory. The affine quantisation method is based on the symmetry of the phase-space of the system, in this case the (positive) half-plane, that is identified with the affine group. We consider a FLRW-type spacetime, and since the scale factor is always positive, the affine method seems to be more suited than the canonincal quantisation for our Quantum Cosmology. We find the wave function of the Brans-Dicke universe, an…
The potential realization of a gravitational wave (GW) astronomy in next years is a great challenge for the scientific community. By giving a significant amount of new information, GWs will be a cornerstone for a better understanding of the universe and of the gravitational physics. In this paper the author shows that the GW astronomy will permit to solve a captivating issue of gravitation as it will be the definitive test for the famous "Einstein frame versus Jordan frame" controversy. In fact, we show that the motion of the test masses, i.e. the beam splitter and the mirror in the case of an…
It is well known that the Jordan and Einstein frames are equivalent to each other in classical Brans-Dicke theory, provided that one and the same metric is employed for the physical space-time. Nevertheless, it is shown in this paper by cosmological models that the loop quantisation in the two different frames will lead to inequivalent effective theories. Analytical solutions have been found in both frames for the effective loop quantum Brans-Dicke cosmology without potential in: (i) vacuum case, (ii) additional massless scalar field case. In Einstein frame the analytical solution for the Bran…
In the framework of a general scalar-tensor theory, we investigate the equivalence between two different parametrizations of fields that are commonly used in cosmology - the so-called Jordan frame and Einstein frame. While it is clear that both parametrizations are mathematically equivalent at the level of the classical action, the question about their mathematical equivalence at the quantum level as well as their physical equivalence is still a matter of debate in cosmology. We analyze whether the mathematical equivalence still holds when the first quantum corrections are taken into account. …
We give a critical analysis of projective relativity theory. Examining Kaluza's own intention and the following development by Klein, Jordan, Pauli, Thiry, Ludwig and others, we conclude that projective relativity was abused in its own terms. Much more in the case of newer higher dimensional Kaluza--Klein theories with non-Abelian gauge groups. Reviewing the projective formulation of the Jordan isomorphy theorem yields some hints how one can proceed in a different direction. We can interpret the condition R5_{μν}=0 not as a field equation in a 5-dimensional Riemannian space, e.g. as vacuum Ein…
A general framework of the novel matter coupling in the Einstein gravity is introduced. We firstly prove that a class of theories whose Hamiltonian constraint is given by an arbitrary function $f(H_g)$, where $H_g$ is the Hamiltonian constraint of general relativity (GR), is equivalent to GR in the vaccum. A novel Jordan frame is defined when GR is rewritte in terms of one of its equivalents in this class. The transformation between the novel Jordan frame and the Eintein frame is a redifinition of lapse. We discuss two types of consistency condition for matter to couple to Einstein gravity in …
The scalar-tensor theory can be formulated in both Jordan and Einstein frames, which are conformally related together with a redefinition of the scalar field. As the solution to the equation of the scalar field in the Jordan frame does not have the one-to-one correspondence with that in the Einstein frame, we give a criterion along with some specific models to check if the scalar field in the Einstein frame is viable or not by confirming whether this field is reversible back to the Jordan frame. We further show that the criterion in the first parameterized post-Newtonian approximation can be d…
Scalar tensor theories of gravity can be formulated in the Einstein or in the Jordan frame, which are related by the conformal transformations. Although the two frames are describe the same physics, and are equivalent, the stability of the field equations in two frames are not the same. Here we implement dynamical system and phase space approach as a robustness tool to investigate this issue. We concentrate on the Brans Dicke theory, but the results can easily be generalized. Our analysis show that while there is one-to-one correspondence between critical points in two frames and each critical…
In recent years, gravitational models motivated by quantum corrections to gravity which introduce higher order terms like $R^{2}$ or terms in which the Riemann tensor is not symmetric have been studied by several authors in the form of a general Brans-Dicke type model containing the Ricci scalar, the Holst term and the Nieh-Yan invariant. In this paper we focus on the less explored Jordan frame of such theories and in the comparison between both this frame and the Einstein one. Furthermore, we discuss the role of the transformation of the torsion under conformal transformations and show that t…
The Jordan frame action for general disformal theories is presented and studied for the first time, motivated by several unresolved mysteries that arise when working in the Einstein frame. We present the Friedmann equations and, specialising to exponential functions, study the late-time cosmology using dynamical systems methods and by finding approximate solutions. Our analysis reveals that either the disformal effects are irrelevant or the universe evolves towards a phantom phase where the equation of state of dark energy is $-3$. There is a marginal case where the asymptotic state of the uni…
Non-minimal scalar-tensor (ST) theories may admit an Einstein frame representation, where gravity is described by the Einstein-Hilbert action plus the scalar sector. Some STs become just {\em minimal} Einstein-scalar (MES) theories, notable examples are Brans-Dicke and $Rϕ^2$. ST theories with {\em derivative} coupling can also be reduced to an Einstein frame by disformal transformations, but, as a rule, their scalar sector will contain higher derivative terms. Here we draw attention to a new Palatini kinetically coupled theory which can be reduced to pure MES by an invertible disformal transf…
We study the thermodynamical aspects of $f(R)$ gravity in the Jordan and the Einstein frame, and we investigate the corresponding equivalence of the thermodynamical quantities in the two frames. We examine static spherically symmetric black hole solutions with constant Ricci scalar curvature $R$, and as we demonstrate, the thermodynamical quantities in the two frames are equivalent. However, for the case of black holes with non-constant scalar curvature $R$, the thermodynamical equivalence of the two frames is no longer valid. In addition, we examine cosmological solutions with non-trivial cur…
We study gravitational interaction of Higgs boson through the unique dimension-4 operator $ξH^†H R$, with $H$ the Higgs doublet and $R$ the Ricci scalar curvature. We analyze the effect of this dimensionless nonminimal coupling $ξ$ on weak gauge boson scattering in both Jordan and Einstein frames. We explicitly establish the longitudinal-Goldstone boson equivalence theorem with nonzero $ξ$ coupling in both frames, and analyze the unitarity constraints. We study the $ξ$-induced weak boson scattering cross sections at O(1-30)TeV scales, and propose to probe the Higgs-gravity coupling via weak bo…
The Jordan-Brans-Dicke theory of gravitation, which promotes the gravitational constant to a dynamical scalar field, predicts a value for the Eddington-Robertson post-Newtonian parameter gamma that is significantly different from the general relativistic value of unity. This contradicts precision solar system measurements that tightly constrain gamma around 1. We consider a modification of the theory, in which the scalar field is sourced explicitly by matter. We find that this leads to a modified expression for the gamma-parameter. In particular, a specific choice of the scalar current yields …
We consider the quantum gravity and cosmology of a Jordan-Brans-Dicke theory, predicted by string effective actions. We study its canonical formalism and find that the constraint algebra is that of general relativity, as a consequence of the general covariance of scalar-tensor theories. We also analyze the problem of boundary conditions and propose that they must be imposed in the Jordan frame, in which particles satisfy the strong equivalence principle. Specifically, we discuss both Hartle-Hawking and wormhole boundary conditions in the context of quantum cosmology. We find quantum wormhole s…
The accelerated expansion of the universe demands presence of an exotic matter, namely the dark energy. Though the cosmological constant fits this role very well, a scalar field minimally coupled to gravity, or quintessence, can also be considered as a viable alternative for the cosmological constant. We study $f(R)$ gravity models which can lead to an effective description of dark energy implemented by quintessence fields in Einstein gravity, using the Einstein frame-Jordan frame duality. For a family of viable quintessence models, the reconstruction of the $f(R)$ function in the Jordan frame…
