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We propose a revised formulation of General Relativity for cosmological contexts, in which the Einstein constant varies with the energy density of the Universe. We demonstrate that this modification has no direct phenomenological impact on the Universe's dynamics or on particle motion within the expanding cosmos. Assuming a state close to vacuum, here defined by the vanishing product of the Einstein coupling constant and the Universe's energy density, we perform a Taylor expansion of the theory. In this framework, the vacuum energy problem is addressed, and an additional constant pressure term, which induces a Chaplygin-like contribution to the dark energy equation of state, arises in the late-time dynamics. The correction to the late-time Hubble parameter is investigated by comparing theoretical predictions with the late Universe observational data. Our findings indicate that the current value of the vacuum energy is consistent with zero. Alternatively, the expansion used in our formulation would no longer be valid if the current state significantly deviates from the assumed near-vacuum condition. Implications of the modified $\Lambda$CDM model with respect to the Hubble tension are also discussed., Comment: 17 pages, 3 figures

This paper focuses on the search for a coherent and consistent formulation to describe quantum gravity corrections to Quantum Field Theory. We implement two fundamental ingredients discussed in previous analyses: on one hand, the construction of the Wentzel-Kramer-Brillouin and Born-Oppenheimer picture for the Wheeler-DeWitt theory of gravity and matter by using a reference fluid as physical clock; on the other hand, we explicitly separate in the metric field its own purely classical contribution from the quantum graviton degrees of freedom. This allows to derive, by the expansion in a Planckian parameter, a unitary theory properly evaluating the Quantum Field Theory modifications in the considered regime. More specifically, we recover at zero order the standard theory on curved space-time after averaging over the graviton sector. The physical corrections are induced at first order by (quantum) gravitons, thought as the slow-varying component of the system, as opposed to the fast quantum matter and reference fluid. The genuine contribution due to gravitons, non-factorizable into an independent phase, provides a coherent quantum gravity modification on Quantum Field Theory. We show this by determining the predicted inflationary spectrum during an exact de Sitter phase of the primordial Universe, finding a breaking of the scale invariant morphology even when the inflation potential is modelled by a cosmological constant term. Remarkably, the behavior of such non-scale-invariant spectrum overlaps predictions previously obtained in literature by neglecting non-unitary terms emerging in those formulations. In this respect, the present model provides a satisfactory regularization of such approaches, by virtue of a more realistic construction of the physical clock for the total (gravity + matter) quantum dynamics., Comment: 18 pages, no figures

A precise interpretation of the Universe wave function is forbidden in the spirit of the Copenhagen School since a precise notion of measure operation cannot be satisfactorily defined. Here we propose a Bohmian interpretation of the isotropic Universe quantum dynamics, in which the Hamilton-Jacobi equation is restated by including quantum corrections, which lead to a classical trajectory containing effects of order $\hbar^2$. This solution is then used to determine the spectrum of gauge-invariant quantum fluctuations living on the obtained background model. The analysis is performed adopting the wave function approach to describe the fluctuation dynamics, which gives a time-dependent harmonic oscillator for each Fourier mode and whose frequency is affected by the $\hbar^2$ corrections. The properties of the emerging spectrum are discussed, outlining the modification induced with respect to the scale-invariant result and the hierarchy of the spectral index running is discussed., Comment: 19 pages, 4 figures; discussion expanded and references added; matching the published version

After a brief review of the different approaches to predict the possible quantum gravity corrections to quantum field theory, we discuss in some detail the formulation based on a Gaussian reference frame fixing. Then, we implement this scenario to the determination of the inflationary spectrum of primordial perturbations. We consider the quantization of an inhomogeneous free massless scalar field on a quasi-classical isotropic Universe, developing a WKB expansion of the dynamics at the next order in the Planckian parameter, with respect to the one at which standard QFT emerges. The quantum gravity corrections to the scale invariant spectrum are discussed in a specific primordial cosmological setting and then in a general minisuperspace formalism, showing that there is no mode-dependent effect and thus the scale invariant inflationary spectrum is preserved. Such result is discussed in connection to the absence of a matter backreaction on the gravitational background in the considered paradigm., Comment: 24 pages. Discussion expanded and references added; matches the published version

Starting from a re-analysis of previous work, we construct the proper low energy quantum field theory (QFT) limit of a full quantum gravity theory in the Born-Oppenheimer approach. We separate the gravitational sector into a classical background, given by a vacuum diagonal Bianchi I cosmology, and its quantum perturbations represented by the two graviton degrees of freedom; we further include quantum matter in the form of a test scalar field. We then implement a Born-Oppenheimer separation, where the gravitons and matter play the role of "slow" and "fast" quantum components respectively, and perform a Wentzel-Kramers-Brillouin (WKB) expansion in a Planckian parameter. The functional Schr\"odinger evolution for matter is recovered after averaging over quantum gravitational effects, provided that a condition is imposed on the gravitons' wave functional. Such a condition fixes the graviton dynamics and is equivalent to the purely gravitational Wheeler-DeWitt constraint imposed in previous approaches. The main accomplishment of the present work is to clarify that QFT in curved spacetime can be recovered in the low energy limit of quantum gravity only after averaging over the graviton degrees of freedom, in the spirit of effective field theory. Furthermore, it justifies a posteriori the implementation of the gravitational Wheeler-DeWitt equation on the "slow" gravitons' wave functional rather than assuming its validity a priori., Comment: 8 pages. Title updated and discussion expanded; matches published version

In this review, we analyse different aspects concerning the possibility to separate a gravity-matter system into a part which lives close to a quasi-classical state and a "small" quantum subset. The considered approaches are all relying on a WKB expansion of the dynamics by an order parameter and the natural arena consists of the Bianchi universe minisuperspace. We first discuss how, limiting the WKB expansion to the first order of approximation, it is possible to recover for the quantum subsystem a Schr\"odinger equation as written on the classical gravitational background. We give some applications for the quantum subsystem in the "corner" configuration of the Bianchi IX model, identifying the quantum variable in the small one of the two anisotropy degrees of freedom. The most surprising result is the possibility to get a non-singular Bianchi IX cosmology when the scenario is extrapolated backwards in time. We also provide some basic hints on the extension of this result to the generic cosmological solution. In the last part of the review, we consider the same scheme to the next order of approximation identifying the quantum subset as made of matter variables only. We deal with the very fundamental problem of non-unitary morphology of the quantum gravity corrections to quantum field theory discussing some proposed reformulations. Instead of constructing the time dependence via that one of the classical gravitational variables on the label time, we analyse a recent proposal to construct time by fixing a reference frame. This scheme can be reached both introducing the so-called "kinematical action", as well as by the well-known Kuchar-Torre formulation. In both cases, the Schr\"odinger equation, amended for quantum gravity corrections, has the same morphology and we provide a cosmological implementation of the model, to elucidate its possible predictions., Comment: References added and discussion expanded. Matches the published version

We analyze the effect induced on standard quantum field theory (in functional approach) by quantum gravity corrections to a pure classical background. In the framework of the Kucha\v{r} and Torre proposal for a gravity-matter theory constrained to a Gaussian reference frame, materialized as a fluid in the system evolution, we consider a Born-Oppenheimer separation of the system, regarding the gravity degrees of freedom as the slow varying component and the matter plus the Gaussian fluid as fast quantum coordinates. The slow gravity component obeys the Wheeler-DeWitt equation and we consider a Wentzel-Kramer-Brillouin expansion of its quantum dynamics via a Planckian parameter. The main issue of the proposed scenario is that, on one hand, we recover a modified quantum field theory in the presence of an Hermitian Hamiltonian (not affected by non-unitarity as in other approaches) and, on the other hand, we get the Gaussian fluid as a physical clock for such amended quantum theory (verifying the correct energy conditions). We also show its equivalence with the kinematical action method, used in a previous work, in the homogeneous setting. Then, we implement the proposed paradigm to describe the dynamics of a homogeneous free massless scalar field, living on an isotropic Universe, in the presence of a cosmological constant. We completely solve the dynamics up to the first order correction in the Planckian parameter to the standard quantum field theory. We determine the explicit form of the modified scalar field wave-function, due to quantum features of the cosmic scale factor evolution. Phenomenological considerations and discussions are provided., Comment: 13 pages in RevTeX format, 1 figure. Matches published version

The problem of time emerging in the canonical quantization procedure of gravity signals a necessity to properly define a relational time parameter. Previous approaches, which are here briefly discussed, make use of the dependence of the quantum system on semiclassical gravitational variables in order to define time. We show that such paths, despite the following studies, lead to a non-unitary evolution. We propose a different model for the quantization of the gravity-matter system, where the time parameter is defined via an additional term, i.e. the kinematical action, which acts as a clock for quantum matter. The procedure here used implements a Born-Oppenheimer-like separation of the system, which maintains covariance under the foliation of the gravitational background and keeps the correct classical limit of standard quantum field theory on a fixed background. It is shown with a WKB expansion that quantum gravity corrections to the matter dynamics arise at the next order of expansion, and such contributions are unitary, signalling a striking difference from previous proposals. Applications to a cosmological model are presented and the analogies of the kinematical term with an incoherent dust are briefly discussed., Comment: 10 pages. Proceeding of the Quantum Gravity Phenomenology Parallel Session, 16th Marcel Grossmann Meeting, July 2021

The problem of time is one of the most relevant open issues in canonical quantum gravity. Although there is a huge literature about this topic, a commonly accepted solution has not been found yet. Here, we focus on the semiclassical approach to the problem of time, that has the main goal of reproducing quantum field theory on a fixed WKB background accounting also for quantum gravity corrections. We analyze the different choices of the expansion parameter and discuss the problems arising in previous proposals, where a non-unitary evolution emerges as an effect of quantum gravity corrections. In this work we develop a new approach to solve this problem by performing the WKB expansion with the introduction of the so-called kinematical action as a clock for quantum matter, that allows to recover a unitary dynamics., Comment: 24 pages, no figures. Little adjustments, improved bibliography

We introduce a generalized trace functional TR in the spirit of Kontsevich and Vishik's canonical trace for classical SG-pseudodifferential operators on R^n and suitable manifolds, using a finite-part integral regularization technique. This allows us to define a zeta-regularized determinant det A for classical parameter-elliptic SG-operators A of order (\mu,m), with \mu>0, m\ge0. For m=0, the asymptotics of TR exp(-tA) as t\to 0 and of TR (\lambda-A)^{-k}$ as |\lambda|\to\infty are derived. For suitable pairs (A,A_0) we show that det A/det A_0 coincides with the so-called relative determinant det(A,A_0)., Comment: Slightly revised; accepted for publication in Math. Nachrichten

We deal with the asymptotic behaviour for $\lambda\to+\infty$ of the counting function $N_P(\lambda)$ of certain positive selfadjoint operators $P$ with double order $(m,\mu)$, $m,\mu>0$, $m\not=\mu$, defined on a manifold with ends $M$. The structure of this class of noncompact manifolds allows to make use of calculi of pseudodifferential operators and Fourier Integral Operators associated with weighted symbols globally defined on $\mathbb{R}^n$. By means of these tools, we improve known results concerning the remainder terms of the Weyl Formulae for $N_P(\lambda)$ and show how their behaviour depends on the ratio $\frac{m}{\mu}$ and the dimension of $M$., Comment: Final version, 30 pages

Let M be a closed manifold. We show that the Kontsevich-Vishik trace, which is defined on the set of all classical pseudodifferential operators on M, whose (complex) order is not an integer greater than or equal to -dim M, is the unique functional which (i) is linear on its domain, (ii) has the trace property and (iii) coincides with the L^2-operator trace on trace class operators. Also the extension to even-even pseudodifferential operators of arbitrary integer order on odd-dimensional manifolds and to even-odd pseudodifferential operators of arbitrary integer order on even-dimensional manifolds is unique.