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 relativity limit. Therefore the answer to the question whether general relativity is an attractor for the theory depends on the choice of the frame.
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…
Jordan-Brans-Dicke cosmology with a standard kinetic term for the scalar field and no mass term has the same radiation dominated solution as standard Einstein cosmology without the cosmological constant. Because of this, the primordial nucleosynthesis (Big - Bang nucleosynthesis) result obtained for standard cosmology remains the same for Jordan-Brans-Dicke cosmology. We show that Jordan-Brans-Dicke cosmology with a mass term for the scalar field as well as explaining dark energy for the present era, can also explain radiation dominated cosmology for the primordial nucleosynthesis era.
We consider the embedding of 3+1 dimensional cosmology in 4+1 dimensional Jordan-Brans-Dicke theory. We show that exponentially growing and power law scale factors are implied. Whereas the 4+1 dimensional scalar field is approximately constant for each, the effective 3+1 dimensional scalar field is constant for exponentially growing scale factor and time dependent for power law scale factor.
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…
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…
A perturbative analysis shows that black holes do not remember the value of the scalar field $ϕ$ at the time they formed if $ϕ$ changes in tensor-scalar cosmology. Moreover, even when the black hole mass in the Einstein frame is approximately unaffected by the changing of $ϕ$, in the Jordan-Fierz frame the mass increases. This mass increase requires a reanalysis of the evaporation of primordial black holes in tensor-scalar cosmology. It also implies that there could have been a significant magnification of the (Jordan-Fierz frame) mass of primordial black holes.
Scalar-Tensor theories of gravity can be formulated in different frames, most notably, the Einstein and the Jordan one. While some debate still persists in the literature on the physical status of the different frames, a frame transformation in Scalar-Tensor theories amounts to a local redefinition of the metric, and then should not affect physical results. We analyze the issue in a cosmological context. In particular, we define all the relevant observables (redshift, distances, cross-sections, ...) in terms of frame-independent quantities. Then, we give a frame-independent formulation of the …
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.
We consider the evolution of primordial black holes in a generalyzed Jordan-Brans-Dicke cosmological model where both the Brans-Dicke scalar field and its coupling to gravity are dynamical functions determined from the evolution equations. The evaporation rate for the black holes changes compared to that in standard cosmology. We show that accretion of radiation can proceed effectively in the radiation dominated era. The black hole lifetime shortens for low initial mass, but increases for high initial mass, and is thus considerably modified compared to the case of standard cosmology. We derive…
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 study the evolution of homogeneous and isotropic, flat cosmological models within the general scalar-tensor theory of gravity with arbitrary coupling function and potential. After introducing the limit of general relativity we describe the details of the phase space geometry. 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…
In the present article we study the cosmological evolution of a two-scalar field gravitational theory defined in the Jordan frame. Specifically, we assume one of the scalar fields to be minimally coupled to gravity, while the second field which is the Brans-Dicke scalar field is nonminimally coupled to gravity and also coupled to the other scalar field. In the Einstein frame this theory reduces to a two-scalar field theory where the two fields can interact only in the potential term, which means that the quintom theory is recovered. The cosmological evolution is studied by analyzing the equili…
We develop a gauge invariant canonical perturbation scheme for perturbations around symmetry reduced sectors in generally covariant theories, such as general relativity. The central objects of investigation are gauge invariant observables which encode the dynamics of the system. We apply this scheme to perturbations around a homogeneous and isotropic sector (cosmology) of general relativity. The background variables of this homogeneous and isotropic sector are treated fully dynamically which allows us to approximate the observables to arbitrary high order in a self--consistent and fully gauge …
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 a…
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.
In this paper we study a type of model for closed inflationary universe models using the Jordan-Brans-Dicke theory. Herein we determine and characterize the existence of $Ω>$1, together with the period of inflation. We have found that our model, which takes into account a Jordan-Brans-Dicke type of theory, is less restrictive than the one used in Einstein's theory of general relativity. Our results are compared to those found in Einstein's theory of Relativity.
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…
We discuss cosmological implications of nonlinear supersymmetric(NLSUSY) general relativity(GR) of the form of Einstein-Hilbert(EH) action for empty spacetime, where NLSUSY GR is obtained by the geomtrical arguments on new spacetime just inspired by NLSUSY. The new action of NLSUSY GR is unstable and breaks down spontaneously to EH action with Nambu-Goldstone(NG) fermion matter. We show that NLSUSY GR elucidates the physical meanings of the cosmologically important quantities, e.g., the spontaneous SUSY breaking scale, the cosmological constant, the dark energy and the neutrino mass and descri…
We show that if the Lagrangean for a scalar field coupled to General Relativity only contains derivatives, then it is possible to completely deparametrise the theory. This means that 1.Physical observables, i.e. functions which Poisson commute with the spatial diffeomorphism and Hamiltonian constraints of General Relativity, can be easily constructed. 2. The physical time evolution of those observables is generated by a natural physical Hamiltonian which is (constrained to be) positive. The mechanism by which this works is due to Brown and Kuchař. In order that the physical Hamiltonian is clos…
We investigate a particular type of classical nonsingular bouncing cosmology, which results from general relativity if we allow for degenerate metrics. The simplest model has a matter content with a constant equation-of-state parameter and we get the modified Hubble diagrams for both the luminosity distance and the angular diameter distance. Based on these results, we present a Gedankenexperiment to determine the length scale of the spacetime defect which has replaced the big bang singularity. A possibly more realistic model has an equation-of-state parameter which is different before and afte…
We derive the equations of motion for scalar metric perturbations in a particular nonsingular bouncing cosmology, where the big bang singularity is replaced by a spacetime defect with a degenerate metric. The adiabatic perturbation solution is obtained for nonrelativistic hydrodynamic matter. We get the same result by working with conformal coordinates. This last method is also valid for vector and tensor metric perturbations, and selected results are presented. We, finally, discuss several new effects from the linear perturbations of this nonsingular bouncing cosmology, such as across-bounce …
We investigate a field theoretical approach to the Jordan-Brans- Dicke (JBD) theory extended with a particular potential term on a cosmological background by starting with the motivation that the Higgs field and the scale factor of the universe are related. Based on this relation, it is possible to come up with mathematically equivalent but two different interpretations. From one point of view while the universe is static, the masses of the elementary particles change with time. The other one, which we stick with throughout the manuscript, is that while the universe is expanding, particle mass…
Advances in our understanding of the origin, evolution and structure of the universe have long been driven by cosmological perturbation theory, model building and effective field theory. In this review, we introduce numerical relativity as a powerful new complementary tool for fundamental cosmology. To illustrate its power, we discuss applications of numerical relativity to studying the robustness of slow contraction and inflation in homogenizing, isotropizing and flattening the universe beginning from generic unsmooth initial conditions. In particular, we describe how recent numerical relativ…
We investigate which Jordan frame $F(R)$ gravity can describe a Type IV singular bouncing cosmological evolution, with special emphasis given near the point at which the Type IV singularity occurs. The cosmological bounce is chosen in such a way so that the bouncing point coincides exactly with Type IV singularity point. The stability of the resulting $F(R)$ gravity is examined and in addition, we study the Einstein frame scalar-tensor theory counterpart of the resulting Jordan frame $F(R)$ gravity. Also, by assuming that the Jordan frame metric is chosen in such a way so that, when conformall…
There are, at present, several gravitational and cosmological anomalies; the dark energy problem, the lambda problem, accelerating cosmological expansion, the anomalous Pioneer spacecraft acceleration, a spin-up of the Earth and an apparent variation of G observed from analysis of the evolution of planetary longitudes. These conundrums may be resolved in the theory of Self Creation Cosmology, in which the Principle of Mutual Interaction subsumes both Mach's Principle and the Local Conservation of Energy. The theory is conformally equivalent to General Relativity in vacuo with the consequence t…
The FRW model with non-minimally coupled massive scalar field has been investigated in LQC framework. Considered form of the potential and coupling allows applications to Higgs driven inflation. Out of two frames used in the literature to describe such systems: Jordan and Einstein frame, the latter one is applied. Specifically, we explore the idea of the Einstein frame being the natural 'environment' for quantization and the Jordan picture having an emergent nature. The resulting dynamics qualitatively modifies the standard bounce paradigm in LQC in two ways: (i) the bounce point is no longer …
By using a bottom-up reconstruction technique for non-minimally coupled scalar-tensor theories, we realize the Einstein frame attractor cosmologies in the $Ω(ϕ)$-Jordan frame. For our approach, what is needed for the reconstruction method to work is the functional form of the non-minimal coupling $Ω(ϕ)$ and of the scalar-to-tensor ratio, and also the assumption of the slow-roll inflation in the $Ω(ϕ)$-Jordan frame. By appropriately choosing the scalar-to-tensor ratio, we demonstrate that the observational indices of the attractor cosmologies can be realized directly in the $Ω(ϕ)$-Jordan frame.…
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…
We revisit the question of frame equivalence in Quantum Field Theory in the presence of gravity, a situation of relevance for theories aiming to describe the early Universe dynamics and Inflation in particular. We show that in those cases, the path integral measure must be carefully defined and that the requirement of diffeomorphism invariance forces it to depend non-trivially on the fields. As a consequence, the measure will transform also non-trivially between different frames and it will induce a new finite contribution to the Quantum Effective Action that we name frame discriminant. This n…
The Averaging problem in general relativity and cosmology is discussed. The approach of macroscopic gravity to resolve the problem is presented. An exact cosmological solution to the equations of macroscopic gravity is given and its properties are discussed. Contents: 1. Introduction to General Relativity 2. General Relativity -> Relativistic Cosmology 3. Introduction to Relativistic Cosmology 4. Relativistic Cosmology -> Mathematical Cosmology 5. Averaging Problem in Relativistic Cosmology 6. History of the Averaging Problem 7. Averaging and FLRW Cosmologies 8. Averaging Problem in General Re…
In the homogeneous and isotropic Friedmann-Robertson-Walker minisuperspace model, it is known that there are no Euclidean wormhole solutions in the pure gravity system. Here it is demonstrated explicitly that in Taub cosmology, which is one of the simplest anisotropic cosmology models, wormhole solutions do exist in pure general relativity in both classical and quantum contexts.
The static spheroidal solutions of Jordan-Brands-Dicke theory (JBD) are studied. We consider the effect of the anisotropic stresses of scalar field on the shape of JBD self-graviting objects. It is shown that scalar fields can have significant effect on the structure and properties of self-graviting objects. In contrast with general relativity in JBD theory there are nonflat static spheroidal solutions.
In these lectures my aim is to review enough of conformal differential geometry in four dimensions to give an account of Penrose's conformal cyclic geometry.
We present a new approach to early universe cosmology. Inflation is replaced by a phase transition in which both matter and geometry are created simultaneously. We calculate the spectrum of metric perturbations and show that it is flat. We then argue that as a consequence of the dynamic nature of the phase transition the spectrum is likely not completely flat but tilted. We argue that the tilt is related to $η$, one of the critical exponents characterizing the phase transition. This exponent generically lies between 0.03 and 0.06. It thus coincides with the observed tilt of the perturbation sp…
The widely accepted description of the present cosmic acceleration and the neutrino oscillations in terms of a cosmological constant and neutrino masses, respectively, seems conceptually not very satisfactory. From a phenomenological perspective, the possibility that those phenomena are instead the first windows to new physics at very low energies is explored.
We discuss dynamical systems approaches and methods applied to flat Robertson-Walker models in $f(R)$-gravity. We argue that a complete description of the solution space of a model requires a global state space analysis that motivates globally covering state space adapted variables. This is shown explicitly by an illustrative example, $f(R) = R + αR^2$, $α> 0$, for which we introduce new regular dynamical systems on global compactly extended state spaces for the Jordan and Einstein frames. This example also allows us to illustrate several local and global dynamical systems techniques involving…
Cosmology is a field of physics in which the use of General Relativity theory is indispensable. However, a cosmology based on Newtonian gravity theory for gravity is possible in certain circumstances. The applicability of Newtonian theory can be substantially extended if it is modified in such way that pressure has a more active role as source of the gravitational field. This was done in the neo-Newtonian cosmology. The limitation on the construction of a Newtonian cosmology, and the need for a relativistic theory in cosmology are reviewed. The neo-Newtonian proposal is presented, and its cons…
In this article we propose some Maple procedures, for teaching purposes, to study the basics of General Relativity (GR) and Cosmology. After presenting some features of GRTensorII, a package specially built to deal with GR, we give two examples of how one can use these procedures. In the first example we build the Schwarzschild solution of Einstein equations, while in the second one we study some simple cosmological models.
The characteristic formalism in numerical relativity, which has been developed to study gravitational waves, and the observer metric approach in observational cosmology both make use of coordinate systems based on null cones. In this paper, these coordinate systems are compared and it is then demonstrated how characteristic numerical relativity can be used to investigate problems in observational cosmology. In a numerical experiment using the characteristic formalism, it is shown how the historical evolution of a LTB universe compares to that of the $Λ$CDM model given identical observational d…
In an attempt to generalize general relativity, we propose a new Hermitian theory of gravity. Space-time is generalized to space-time-momentum-energy and both the principles of general covariance and equivalence are extended. The theory is endowed with a Hermitian metric on a complex manifold. The Hermitian metric contains, apart from the symmetric metric, an anti-symmetric part, which describes dynamical torsion. The causality structure is changed in a way such that there is a minimal time for events to be in causal contact and a maximal radius for a non-local instantaneous causally related v…
General Very Special Relativity (GVSR) is the curved space-time of Very Special Relativity (VSR) proposed by Cohen and Glashow. The geometry of GVSR possesses a line element of Finsler Geometry proposed by Bogoslovsky. We calculate the Einstein field equations and derive a modified FRW cosmology, for an osculating Riemannian space. The Friedman equation of motion leads to an explanation of the cosmological acceleration in terms of an alternative non-Lorentz invariant theory. A first order approach for a primordial spurionic vector field introduced into the metric, gives back an estimation of t…
A five dimensional brane cosmology with non-minimally coupled scalar field to gravity has been considered in a Jordan-Brans-Dicke frame. We derive an effective four dimensional field equations on a 3+1 dimensional brane where the fifth dimension has been assumed to have an orbifold symmetry. We have noticed that the evolution equation for the matter component stuck to the brane is non-trivially coupled to the scalar field living on the brane and the bulk. Finally we discuss some cosmological consequences of this set-up.
In this paper, we describe physical effects occurring in the regularized Robertson-Walker spacetime which can reveal the presence of the defect. Our analysis is based on two main physical quantities: the compressive forces acting on (human) observers and the energy possessed by massive particles and photons during their dynamical evolution. In Sec. 3, we claim that with a characteristic length scale of the order of the Planck length compressive forces become so intense near the defect that no (human) observer is able to cross it. In Sec. 4, we show that the energy exhibits an unusual character…
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. …
This year marks the hundredth anniversary of Einstein's 1915 landmark paper "Die Feldgleichungen der Gravitation" in which the field equations of general relativity were correctly formulated for the first time, thus rendering general relativity a complete theory. Over the subsequent hundred years physicists and astronomers have struggled with uncovering the consequences and applications of these equations. This contribution, which was written as an introduction to six chapters dealing with the connection between general relativity and cosmology that will appear in the two-volume book "One Hund…
Formalising the logical dependence of physical quantities on material referents of scale, I show that both Hubble's law and the cosmological constant are in fact exactly replicable by a spatial contraction of referents locally on earth, and that the Pioneer anomaly is irrefutable indication that this is the case. The formalism literally embodies Feynman's ``hot-plate'' model, predictably yielding a logical derivation of the relativity postulates, and importantly, suffices to demonstrate the inherent logical consistency of general relativity and quantum mechanics, whose foundations I have shown…
The main consequences of de Sitter Special Relativity to the Standard Cosmological Model of Physical Cosmology are examined. The cosmological constant Lambda appears, in this theory, as a manifestation of the proper conformal current, which must be added to the usual energy-momentum density. As that conformal current itself vanishes in absence of sources, Lambda is ultimately dependent on the matter content, and can in principle be calculated. A present-day value very close to that given by the crossed supernova/BBR data is obtained through simple and reasonable approximations. Also a primeval…
The averaging problem in general relativity is briefly discussed. A new setting of the problem as that of macroscopic description of gravitation is proposed. A covariant space-time averaging procedure is described. The structure of the geometry of macroscopic space-time, which follows from averaging Cartan's structure equations, is described and the correlation tensors present in the theory are discussed. The macroscopic field equations (averaged Einstein's equations) derived in the framework of the approach are presented and their structure is analysed. The correspondence principle for macros…
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 find that general relativity can be naturally free of cosmological singularities. Several nonsingular models are currently available that either assume ad hoc matter contents, or are nonsingular only over a sector of solution space of zero measure, or depart drastically from general relativity at high energies. After much uncertainty over whether cosmological inflation could help solve the initial-singularity problem, the prevailing belief today is that general relativistic cosmology, with inflation or without, is endemically singular. This belief was reinforced by recent singularity theore…
New Planck scale physics may solve the singularity problems of classical general relativity and may lead to interesting consequences for very early Universe cosmology. Two approaches to these questions are reviewed in this article. The first is an effective action approach to including the effects of Planck scale physics in the basic framework of general relativity. It is shown that effective actions with improved singularity properties can be constructed. The second approach is based on superstring theory. A scenario which eliminates the big bang singularity and possibly explains the dimensio…
We perform three-dimensional numerical relativity simulations of homogeneous and inhomogeneous expanding spacetimes, with a view towards quantifying non-linear effects from cosmological inhomogeneities. We demonstrate fourth-order convergence with errors less than one part in 10^6 in evolving a flat, dust Friedmann-Lemaitre-Roberston-Walker (FLRW) spacetime using the Einstein Toolkit within the Cactus framework. We also demonstrate agreement to within one part in 10^3 between the numerical relativity solution and the linear solution for density, velocity and metric perturbations in the Hubble …
