Scalar gravitational waves and Einstein frame

The conformal transformations presented in the previous chapter establish a mathematical equivalence between scalar-tensor theories and Einstein’s general relativity, with the modifications dictated by the anomalous coupling of the Einstein frame scalar to ordinary matter and its consequences. A priori, the mathematical equivalence need not be a physical one. The literature using conformal transformations in cosmology and in alternative theories of gravity is plagued by confusion and ambiguity on the issue of the physical equivalence of the two conformal frames, or which frame should be regarded as the physical one. As a consequence, the conformal transformation technique is often misused and its results are given erroneous interpretations. Incompatible points of view occurring in the literature support the following ideas [613, 363] : (first viewpoint) that the Jordan and the Einstein frame are physically equivalent, (second viewpoint) that the two frames are inequivalent the Jordan frame is the physical one, and the Einstein frame is unphysical, (third viewpoint) that the two frames are inequivalent — the Jordan frame is unphysical, and the Einstein frame is physical.

We study charged boson stars in scalar-tensor (ST) gravitational theories. We analyze the weak field limit of the solutions and analytically show that there is a maximum charge to mass ratio for the bosons above which the weak field solutions are not stable. This charge limit can be greater than the GR limit for a wide class of ST theories. We numerically investigate strong field solutions in both the Brans-Dicke and power law ST theories. We find that the charge limit decreases with increasing central boson density. We discuss the gravitational evolution of charged and uncharged boson stars in a cosmological setting and show how, at any point in its evolution, the physical properties of the star may be calculated by a rescaling of a solution whose asymptotic value of the scalar field is equal to its initial asymptotic value. We focus on evolution in which the particle number of the star is conserved and we find that the energy and central density of the star decrease as the cosmological time increases. We also analyze the appearance of the scalarization phenomenon recently discovered for neutron stars configurations and, finally, we give a short discussion on how making the correct choice of mass influences the argument over which conformal frame, the Einstein frame or the Jordan frame, is physical.

We consider a general scalar-tensor theory of gravity and review briefly different forms it can be presented (different conformal frames and scalar field parametrizations). We investigate the conditions under which its field equations and the parametrized post-Newtonian parameters coincide with those of general relativity. We demonstrate that these so-called limits of general relativity are independent of the parametrization of the scalar field, although the transformation between scalar fields may be singular at the corresponding value of the scalar field. In particular, the limit of general relativity can equivalently be determined and investigated in the commonly used Jordan and Einstein frames.

Dynamics in conformally connected frames are often referred to as 'equivalent'. In the context of cosmology, while the perturbations remain invariant for a single scalar field model, the background equations differ and therefore the dynamics. However, since the background dynamics are not the same, it is not clear whether the attractor nature of the solutions remains the same in all conformally connected frames, i.e., a stable solution in one frame implies an equivalent stable solution in another frame. To answer the question, in this work, we first consider power law cosmology in the Brans-Dicke theory as well as in the conformal Einstein frame. We show that, in this case, the attractor behavior is indeed equivalent under conformal transformation, i.e., an attractor solution in one frame implies an attractor solution in another frame. However, the decay rates of the deviations from the fixed points are different in the two frames. We are able to relate the behavior and find that the difference is due to the difference in e-fold 'clocks' in different frames, i.e., $\Delta N$ in different frames differ from one another. We show that the behavior is indeed true for any model in conformally connected frames and obtain the general 'equivalence' relation. In the context of inflation, we consider two models: Starobinsky and chaotic inflation, and explicitly point out the differences in these two frames. We show that the duration of inflation in any model in the Jordon frame is always higher than the Einstein frame.

In the bibliography, a certain confusion arises in what regards to the classification of the gravitational theories into scalar–tensor theories (STT) and general relativity with a scalar field either minimally or nonminimally coupled to matter. Higher-derivatives Horndeski and beyond Horndeski theories that at first sight do not look like STT only add to the confusion. To further complicate things, the discussion on the physical equivalence of the different conformal frames in which a given scalar–tensor theory may be formulated, makes even harder to achieve a correct classification. In this paper, we propose a specific criterion for an unambiguous identification of STT and discuss its impact on the conformal transformations issue. The present discussion carries not only pedagogical but also scientific interest since an incorrect classification of a given theory as a scalar–tensor theory of gravity may lead to conceptual issues and to the consequent misunderstanding of its physical implications.

An explicit proof of the vanishing of the covariant divergence of the energy–momentum tensor in modified theories of gravity is presented. The gravitational action is written in arbitrary dimensions and allowed to depend nonlinearly on the curvature scalar and its couplings with a scalar field. Also the case of a function of the curvature scalar multiplying a matter Lagrangian is considered. The proof is given both in the metric and in the first-order formalism, i.e. under the Palatini variational principle. It is found that the covariant conservation of energy–momentum is built in to the field equations. This crucial result, called the generalized Bianchi identity, can also be deduced directly from the covariance of the extended gravitational action. Furthermore, in all of these cases, the freely falling world lines are determined by the field equations alone and turn out to be the geodesics associated with the metric compatible connection. The independent connection in the Palatini formulation of these generalized theories does not have a similar direct physical interpretation. However, in the conformal Einstein frame a certain bi-metricity emerges into the structure of these theories.

We find a black hole solution with non-Abelian field in Brans-Dicke theory.\nIt is an extension of non-Abelian black hole in general relativity. We discuss\ntwo non-Abelian fields: "SU(2)" Yang-Mills field with a mass (Proca field) and\nthe SU(2)$\\times$SU(2) Skyrme field. In both cases, as in general relativity,\nthere are two branches of solutions, i.e., two black hole solutions with the\nsame horizon radius. Masses of both black holes are always smaller than those\nin general relativity. A cusp structure in the mass-horizon radius\n($M_{g}$-$r_{h}$) diagram, which is a typical symptom of stability change in\ncatastrophe theory, does not appear in the Brans-Dicke frame but is found in\nthe Einstein conformal frame. This suggests that catastrophe theory may be\nsimply applied for a stability analysis as it is if we use the variables in the\nEinstein frame. We also discuss the effects of the Brans-Dicke scalar field on\nblack hole structure.\n

We revisit singularities of two distinct kinds in the Cauchy problem of general scalar-tensor theories of gravity (previously discussed in the literature), and of metric and Palatini f(R) gravity, in both their Jordan and Einstein frame representations. Examples and toy models are used to shed light onto the problem and it is shown that, contrary to common lore, the two conformal frames are equivalent with respect to the initial value problem.

In this paper, we make a detailed side-by-side comparison between Jordan and Einstein frames in the context of cosmic magnetogenesis. We have computed the evolution of the vector potential in each frame along with some observables such as the spectral index and the magnetic field amplitude. We found that contrary to the Einstein frame, the electric and magnetic energy densities in the Jordan Frame do not depend on any parameter associated with the scalar field. Furthermore, in the Einstein frame, and assuming scale invariance for the magnetic field, most of the total energy density contribution comes from the electric and magnetic densities. Finally, we show the ratio between magnetic field signals in both frames printed in the CMB.

We investigate the initial data for localized gravitational waves in space-time with a cosmological constant $\ensuremath{\Lambda}$. Using a conformal transformation, we find that the Hamiltonian and momentum constraints in the conformal frame turn out to be the same as those for a vacuum space-time without $\ensuremath{\Lambda}$. As initial data, we consider Brill's waves in the conformal frame and discuss the trapped and antitrapped surfaces in the physical frame. Just as Brill's wave in asymptotically flat space-time, the gravitational of our case is positive; however, the waves with a large gravitational mass do not always provide trapped surfaces in contrast with the case of $\ensuremath{\Lambda}=0$. The large amount of gravitational waves, hence, does not seem to be an obstacle to the cosmic no-hair conjecture.

Parametric resonance in the Einstein frame: The Jordan-frame Doppelgänger

The geodesic deviation equation (GDE) describes the tendency of objects to accelerate towards or away from each other due to spacetime curvature. The GDE assumes that nearby geodesics have a small rate of separation, which is formally treated as the same order in smallness as the separation itself. This assumption is discussed in various papers but is not articulated in any standard textbooks on general relativity. Relaxing this assumption leads to the generalized geodesic deviation equation (GGDE). We elucidate the distinction between the GDE and the GGDE by explicitly computing the relative acceleration between timelike geodesics in two-dimensional de Sitter spacetime. We do this by considering a fiducial geodesic and a secondary geodesic (both timelike) that cross with nonzero speed. These geodesics are spanned by a spacelike geodesic, whose tangent evaluated at the fiducial geodesic defines the separation. The second derivative of the separation describes the relative acceleration between the fiducial and secondary geodesics. Near the crossing point, where the separation between the timelike geodesics is small but their rates of separation can be large, we show that the GGDE holds but the GDE fails to apply.

The quantization of gravity is still an unsolved problem. However, thus far in all quantized models there is a perturbative regime when the fields behave as free fields. Such a quantization scheme is based on the observation that there are particles associated to the quantized fields. The detection of gravitational waves gives a support for the conjecture that there is a regime of quantum gravity when the gravitational waves can be treated as a stream of particles. After a short introduction to classical gravity we quantize the linearized gravity in the transverse-traceless gauge. We discuss the geodesic equation in a falling frame of another particle (the geodesic deviation equation) as an analog of the particle equation of motion in quantum electrodynamics. We derive a stochastic equation for a particle moving in an environment of thermal gravitons.

The duality between a higher-curvature f(R) gravity model and a scalar-tensor theory helps to bring out the role of the additional degree of freedom originating from the higher-derivative terms in the gravity action. Such a degree of freedom which appears as a scalar field has been shown to have multiple implications in the cosmological/astrophysical scenario. The present work proposes a novel generalization to this correspondence between f(R) gravity and a dual scalar-tensor theory when the affine connection is considered to have an antisymmetric part. It turns out that the f(R) action in the presence of spacetime torsion can be recast to a coupled scalar-tensor theory with a 2-rank massless antisymmetric tensor field in the Einstein frame, where the scalar field gets coupled with the antisymmetric field through derivative coupling(s).

Using the reconstruction method, we investigate which $F(R)$ theories, with or without the presence of matter fluids, can produce the matter bounce scenario of holonomy corrected Loop Quantum Cosmology. We focus our study in two limits of the cosmic time, the large cosmic time limit and the small cosmic time limit. For the former, we found that, in the presence of non-interacting and non-relativistic matter, the $F(R)$ gravity that reproduces the late time limit of the matter bounce solution is actually the Einstein-Hilbert gravity plus a power law term. In the early time limit, since it corresponds to large spacetime curvatures, assuming that the Jordan frame is described by a general metric that when it is conformally transformed to the Einstein frame, produces an accelerating Friedmann-Robertson-Walker metric, we found explicitly the scalar field dependence on time. After demonstrating that the solution in the Einstein frame is indeed accelerating, we calculate the spectral index derived from the Einstein frame scalar-tensor counterpart theory of the $F(R)$ theory and compare it with the Planck experiment data. In order to implement the resulting picture, we embed the $F(R)$ gravity explicitly in a Loop Quantum Cosmology framework by introducing holonomy corrections to the $F(R)$ gravity. In this way, the resulting inflation picture corresponding to the $F(R)$ gravity can be corrected in order it coincides to some extent with the current experimental data.