Quantum corrections in Higgs inflation: the real scalar case

We revisit the problem of generality of Starobinsky and Higgs inflation. The known results obtained in the Einstein frame are generalized for the case of an arbitrary initial energy of the scalar field. These results are compared with the results obtained directly in the Jordan frame, which, to our knowledge, has not been thoroughly explored in the literature previously. We demonstrate that the qualitative picture of initial conditions zone in the $(\ensuremath{\phi},\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\phi}})$ plane, which leads to sufficient amount of inflation, is quite similar for both the frames in the case of Higgs inflation. For Starobinsky inflation, the conformal transformation between the frames relates the geometrical variables in the Jordan frame with the properties of an effective scalar field in the Einstein frame. We show that the transformation $(H,R)\ensuremath{\rightarrow}(\ensuremath{\phi},\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\phi}})$ is not regular everywhere, leading to some peculiarities in the zone of good initial conditions in the $(H,R)$ plane.

Higgs inflation and $R^2$-inflation (Starobinsky model) are two limits of the same quantum model, hereafter called Starobinsky-Higgs. We analyse the two-loop action of the Higgs-like scalar $\phi$ in the presence of: 1) non-minimal coupling ($\xi$) and 2) quadratic curvature terms. The latter are generated at the quantum level with $\phi$-dependent couplings ($\tilde\alpha$) even if their tree-level couplings ($\alpha$) are tuned to zero. Therefore, the potential always depends on both Higgs field $\phi$ and scalaron $\rho$, hence multi-field inflation is a quantum consequence. The effects of the quantum (one- and two-loop) corrections on the potential $\hat W(\phi,\rho)$ and on the spectral index are discussed, showing that the Starobinsky-Higgs model is in general stable in their presence. Two special cases are also considered: first, for a large $\xi$ in the quantum action one can integrate $\phi$ and generate a "refined" Starobinsky model which contains additional terms $\xi^2 R^2\ln^p (\xi \vert R\vert/\mu^2)$, $p=1,2$ ($\mu$ is the subtraction scale). These generate corrections linear in the scalaron to the "usual" Starobinsky potential and a "running" scalaron mass. Second, for a small fixed Higgs field $\phi^2 \ll M_p^2/\xi$ and a vanishing classical coefficient of the $R^2$-term, we show that the "usual" Starobinsky inflation is generated by the quantum corrections alone, for a suitable non-minimal coupling ($\xi$).

We consider the prescription dependence of the Higgs effective potential under the presence of general nonminimal couplings. We evaluate the fermion loop correction to the effective action in a simplified Higgs-Yukawa model whose path integral measure takes simple form either in the Jordan or Einstein frame. The resultant effective action becomes identical in both cases when we properly take into account the quartically divergent term coming from the change of measure. Working in the counterterm formalism, we clarify that the difference between the prescriptions I and II comes from the counter term to cancel the logarithmic divergence. This difference can be absorbed into the choice of tree-level potential from the infinitely many possibilities, including all the higher-dimensional terms. We also present another mechanism to obtain a flat potential by freezing the running of the effective quartic coupling for large field values, using the nonminimal coupling in the gauge kinetic function.

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 explicitly calculate the one-loop divergences in both parametrizations by using the generalized Schwinger-DeWitt algorithm and compare both results. We find that the quantum corrections do not coincide off shell and hence induce an off shell dependence on the parametrization. According to the equivalence theorem, the one-loop divergences should however coincide on shell. For a cosmological background, we show explicitly that the on shell equivalence is indeed realized by a nontrivial cancellation.

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.

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 Einstein frame. In particular, the Minkowski metric exists only in the Jordan frame. In this sense, the two frames are not equivalent.

We will summarize recent results on the Hamiltonian equivalence between the Jordan and Einstein frames based on the analysis of Brans-Dicke theory for both cases ω ≠ − 3 2 and ω = − 3 2 We will introduce and perform ADM analysis for spherically symmetric solutions of gravity. We will discuss with particular care the problem of the boundary terms to be introduced in the general case of spherical symmetry. These two frames are connected through a Hamiltonian canonical transformation on the reduced phase space obtained by gauge fixing the lapse and the radial shift functions. We introduce and discuss two static solutions (Fisher, Janis, Newman and Winicour solution in the Einstein frame and Bocharova-Bronnikov-Melnikov-Bekenstein black hole solution in the Jordan frame)

String theory enjoys an elevated role among quantum gravity theories, since it seems to be the most consistent UV completion of general relativity and the Standard Model. However, it is hard to verify the existence of this underlying theory on terrestrial accelerators. One way to probe string theory is to study its imprints on the low-energy effective inflationary Lagrangian, which are quantified in terms of high energy correction terms. It is highly likely, thus, to find higher order curvature terms combined with string moduli, that is scalar fields, since both these types of interactions and matter fields appear in string theory. In this work we aim to stress the probability that the inflationary dynamics are controlled by the synergy of scalar fields and higher order curvature terms. Specifically, we shall consider a well motivated quantum corrected canonical scalar field theory, with the quantum corrections being of the R2 type. The reason for choosing minimally coupled scalar theory is basically because if scalar fields are evaluated in their vacuum configuration, they will either be minimally coupled or conformally coupled. Here we choose the former case, and the whole study shall be performed in the string frame (Jordan frame), in contrast to similar studies in the literature where the Einstein frame two scalar theory is considered. We derive the field equations of the quantum-corrected theory at leading order and we present the form the slow-roll indices obtain for the quantum corrected theory. We exemplify our theoretical framework by using the quadratic inflation model, and as we show, the R2 quantum corrected quadratic inflation model produces a viable inflationary phenomenology, in contrast with the simple quadratic inflation model.

We review the conformal equivalence in describing the background expansion of the universe by [Formula: see text] gravity both in the Jordan frame and the Einstein frame. In the Jordan frame, we present the general analytic expression for [Formula: see text] models that have the same expansion history as the [Formula: see text]CDM model. This analytic form can provide further insights on how cosmology can be used to test the [Formula: see text] gravity at the largest scales. Moreover we present a systematic and self-consistent way to construct the viable [Formula: see text] model in Jordan frame using the mass dilation rate function from the Einstein frame through the conformal transformation. In addition, we extend our study to the linear perturbation theories and we further exhibit the equivalence of the [Formula: see text] gravity presented in the Jordan frame and Einstein frame in the perturbed space–time. We argue that this equivalence has solid physics root.

In this Letter, we try to apply the unified first law to the “cosmic triad” vector field scenario both in the minimal coupling case and in the non-minimal coupling case. After transferring the non-minimally coupling action in the Jordan frame to the Einstein frame, the correct dynamical equation (Friedmann equation) is gotten in a thermal equilibrium process by using the already existing entropy while the entropy in the non-minimal coupled “cosmic triad” scenario has not been derived. And after transferring the variables back to the Jordan frame, the corresponding Friedmann equation is demonstrated to be correct. For complete arguments, we also calculate the related Misner–Sharp energy in the Jordan and Einstein frames.

We discuss wormhole solution in the Brans-Dicke theory with existence of relativistic radiation field. The solutions of wormhole are obtained in the Einstein frame. When the dilaton field σ = const, the solution of wormhole should be reduced to the well-known Tolman wormhole. Euclidean wormhole solution also has been obtained in the Einstein frame when σ ≠ const. For the comparison, we also derive the wormhole solution in the Jordan frame with existence of relativistic radiation field. However, the two frames, the Einstein frame and the Jordan frame, are not physically equivalent.

Nonlinear massive spin-2 field generated by higher derivative gravity

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

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 a generalized superconformal model that gives rise to a subcritical regime of $D$-term hybrid inflation. Exhibiting the model both in a Jordan frame and in the Einstein frame, the effective potential of the subcritical regime is derived in the Einstein frame. It turns out the inflaton-waterfall field dynamics leads to various types of inflaton potential. Consequently the tensor-to-scalar ratio is found to range from $10^{-4}$ ($10^{-3}$) to $0.1$ for getting 60 (50) $e$-folds before the end of inflation.