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A new gravitational action is presented that could be a fair minimal description of the effective gravitational action in the context of Asymptotic Safety. It generates consistent modified Einstein equations where $\Lambda$ and $G$ are functions of spacetime. In addition, these equations respect, without any ad hoc choices, the Bianchi identities. We analyse this action in a quantum cosmology scenario. Furthermore, assuming homogeneous and isotropic spacetime, the action gives a non singular cosmology with power law inflation. Tensor and scalar perturbations have been used to constraint the free parameters.

We propose an emergent cosmological model rooted in the Asymptotically Safe antiscreening behavior of the Newton constant at Planckian energies. Distinguishing itself from prior approaches, our model encapsulates the variable nature of $G$ through a multiplicative coupling within the matter Lagrangian, characterized by a conserved energy-momentum tensor. The universe emerges from a quasi-de Sitter phase, transitioning to standard cosmological evolution post-Planck Era. Our analysis demonstrates the feasibility of constraining the transition scale to nearly classical cosmology using Cosmic Microwave Background (CMB) data and the potential to empirically probe the antiscreening trait of Newton's constant, as predicted by Asymptotic Safety., Comment: 12 pages, 5 figures, comments welcome

We reply to the ``Comment'' on ``Regular evaporating black holes with stable cores'' by R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio, and M. Visser. As a key result, we show that the regime of mass-inflation identified in the comment connects smoothly to the late-time attractors discovered in our works [A. Bonanno et. al., Regular black holes with stable cores, Phys. Rev. D 103, 124027 (2021) and Regular evaporating black holes with stable cores, Phys. Rev. D 107, 024005 (2023)]. Hence, the late-time stability of regular black holes is not affected by this intermediate phase., Comment: 3 pages, 1 figure

Regular black hole spacetimes are obtained from an effective Lagrangian for Quantum Einstein Gravity. The interior matter is modeled as a dust fluid, which interacts with the geometry through a multiplicative coupling function denoted as $\chi$. The specific functional form of $\chi$ is deduced from Asymptotically Safe gravity, under the key assumption that the Reuter fixed point remains minimally affected by the presence of matter. As a consequence the gravitational coupling vanishes at high energies. The static exterior geometry of the black hole is entirely determined by the junction conditions at the boundary surface. Consequently, the resulting global spacetime geometry remains devoid of singularities at all times. This result offers a novel perspective on regular black holes in Asymptotically Safe gravity., Comment: 6 pages, 4 figures

Black holes encountered in general relativity are characterized by spacetime singularities hidden within an event horizon. These singularities provide a key motivation to go beyond general relativity and look for regular black holes where the spacetime curvature remains bounded everywhere. A prominent mechanism achieving this replaces the singularity by a regular patch of de Sitter space. The resulting regular geometries exhibit two horizons: the outer event horizon is supplemented by an inner Cauchy horizon. The latter could render the geometry unstable against perturbations through the so-called mass-inflation effect, i.e., an exponential growth of the mass function. This chapter reviews the mass-inflation effect for spherically symmetric black hole spacetimes contrasting the dynamics of the mass function for Reissner-Nordst\"om and regular black holes. We also cover recent developments related to the late-time attractors induced by Hawking radiation which exorcise the exponential growth of the spacetime curvature encountered in the standard mass-inflation scenario. In order to make the exposition self-contained, we also briefly discuss basic properties of regular black holes including their thermodynamics., Comment: 34 pages, Invited chapter for the edited book "Regular Black Holes: Towards a New Paradigm of the Gravitational Collapse" (Ed. C. Bambi, Springer Singapore, expected in 2023)

A feature shared by many regular black hole spacetimes is the occurrence of a Cauchy horizon. It is then commonly believed that this renders the geometry unstable against perturbations through the mass-inflation effect. In this work, we perform the first dynamical study of this effect taking into account the mass-loss of the black hole due to Hawking radiation. It is shown that the time-dependence of the background leads to two novel types of late-time behavior whose properties are entirely determined by the Hawking flux. The first class of attractor-behavior is operative for regular black holes of the Hayward and renormalization group improved type and characterized by the square of the Weyl curvature growing as $v^6$ at asymptotically late times. This singularity is inaccessible to a radially free-falling observer though. The second class is realized by Reissner-Nordstr{\"o}m black holes and regular black holes of the Bardeen type. In this case the curvature scalars remain finite as $v\rightarrow\infty$. Thus the Hawking flux has a profound effect on the mass-inflation instability, either weakening the effect significantly or even expelling it entirely., Comment: 17 pages, 9 figures

The inclusion of the Weyl squared term in the gravitational action is one of the most simple, yet non trivial modifications to General Relativity at high energies. Nevertheless the study of the spherically-symmetric vacuum solutions of this theory has received much attention only in recent times. A new type of asymptotically flat wormhole which does not match symmetrically at a finite radius with another sheet of the spacetime is presented. The outer spacetime is characterized by a newtonian potential with a Yukawa correction, and has gravitational properties that can be arbitrarily close to the ones of a Schwarzschild black hole. The internal spacetime instead possesses a singularity at $r=\infty$ with the topology of a 2-dimensional sphere. The expansion scalar of geodesics reaching this singularity diverges in a finite amount of proper time, with a striking resemblance with the future singularity of the Big Rip cosmological scenario. In terms of the external Yukawa hair and mass $M$, these new wormholes fill a large region of the two-dimensional parameter space of physical solutions with $M>0$. On the contrary black holes, both of Schwarzschild and non-Schwarzschild nature, are confined on a line. We argue that this type of wormholes are ideal black hole mimickers., Comment: 11 pages, 5 figures

A convincing explanation for the nature of the dark energy and dark matter is still missing. In recent works a RG-improved swiss-cheese cosmology with an evolving cosmological constant dependent on the \sch radius has been proven to be a promising model to explain the observed cosmic acceleration. In this work we extend this model to consider the combined scaling of the Newton constant $G$ and the cosmological constant $\Lambda$ according to the IR-fixed point hypothesis. We shall show that our model easily generates the observed recent passage from deceleration to acceleration without need of extra energy scales, exotic fields or fine tuning. In order to check the generality of the concept, two different scaling relations have been analysed and we proved that both are in very good agreement with $\Lambda$CDM cosmology. We also show that our model satisfies the observational local constraints on $\dot{G}/G$., Comment: 15 pages, 3 figures

In a recent work, it has been proposed that the recent cosmic passage to a cosmic acceleration era is the result of the existence of small anti-gravity sources in each galaxy and clusters of galaxies. In particular, a swiss-cheese cosmology model which relativistically integrates the contribution of all these anti-gravity sources on galactic scale has been constructed assuming the presence of an infrared fixed point for a scale dependent cosmological constant. The derived cosmological expansion provides explanation for both the fine tuning and the coincidence problem. The present work relaxes the previous assumption on the running of the cosmological constant and allows for a generic scaling around the infrared fixed point. Our analysis reveals in order to produce a cosmic evolution consistent with the best $\Lambda$CDM model, the IR-running of the cosmological constant is consistent with the presence of an IR-fixed point., Comment: 12 pages, 6 figures. Accepted in Universe, MDPI

The characterization of the gravitational field of isolated objects is still an open question in quadratic theories of gravity. We study static equilibrium solutions for a self-gravitating fluid in extensions of General Relativity including terms quadratic in the Weyl tensor $C_{\mu\nu\rho\sigma}$ and in the Ricci scalar $R$, as suggested by one-loop corrections to classical gravity. By the means of a shooting method procedure we link the total gravitational mass and the strength of the Yukawa corrections associated with the quadratic terms with the fluid properties at the center. It is shown that the inclusion of the $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$ coupling in the lagrangian has a much stronger impact than the $R^2$ correction in the determination of the radius and of the maximum mass of a compact object. We also suggest that the ambiguity in the definition of mass in quadratic gravity theories can conveniently be exploited to detect deviations from standard General Relativity., Comment: 18 pages, 7 figures, 1 table

We reconstruct the Lorentzian graviton propagator in asymptotically safe quantum gravity from Euclidean data. The reconstruction is applied to both the dynamical fluctuation graviton and the background graviton propagator. We prove that the spectral function of the latter necessarily has negative parts similar to, and for the same reasons, as the gluon spectral function. In turn, the spectral function of the dynamical graviton is positive. We argue that the latter enters cross sections and other observables in asymptotically safe quantum gravity. Hence, its positivity may hint at the unitarity of asymptotically safe quantum gravity., Comment: 22 pages, 8 figures, 2 tables

A new set of field equations for a space-time dependent Newton's constant $G(x)$ and cosmological constant $\Lambda(x)$ in the presence of matter is presented. We prove that it represents the most general mathematically consistent, physically plausible, set of evolution equations assuming at most second derivatives in the dynamical variables. In the new Einstein's equations, only $\Lambda$-kinetic terms arise, while in the modified conservation equation, derivative terms of $G$ also appear. As an application, this formalism is applied in the context of the Asymptotic Safety scenario to the early universe, assuming a perfect fluid with a radiation equation of state. Cosmological solutions are obtained for all types of spatial curvature, displaying a variety of interesting cosmic evolutions. As an indication of such behaviours, bouncing solutions, recollapsing solutions or non-singular expanding solutions with a transient acceleration era are discussed in details., Comment: 21 pages, 12 figures, to appear in PRD

We consider Lifshitz-type scalar theories with explicit breaking of the Lorentz symmetry that, in addition, exhibit anisotropic scaling laws near the ultraviolet fixed point. Using the proper time regularization method on the spatial coordinates only, we derive the regularized form of the one-loop effective potential in such theories. We study the main features of the one-loop effective potential and, also, the RG flow of the scale-dependent potential both in the IR and UV regimes. The beta functions for the couplings are derived., Comment: 12 pages, 3 figures

Non-singular black hole geometries typically come with two spacetime horizons: an (outer) event horizon and an (inner) Cauchy horizon. This nurtures the speculation that they may be subject to a mass-inflation effect which renders the Cauchy horizon unstable. We analyze the dynamics associated with spherically symmetric, regular black holes taking the full backreaction between the infalling matter and geometry into account. On this basis, we identify the crucial features taming the growth of the mass function and diminishing the curvature singularity at the Cauchy horizon. It is demonstrated explicitly that the regular black hole solutions proposed by Hayward and obtained from Asymptotic Safety satisfy these properties., Comment: 8 pages, 2 figures, agrees with published version

Asymptotic safety is a theoretical proposal for the ultraviolet completion of quantum field theories, in particular for quantum gravity. Significant progress on this program has led to a first characterization of the Reuter fixed point. Further advancement in our understanding of the nature of quantum spacetime requires addressing a number of open questions and challenges. Here, we aim at providing a critical reflection on the state of the art in the asymptotic safety program, specifying and elaborating on open questions of both technical and conceptual nature. We also point out systematic pathways, in various stages of practical implementation, towards answering them. Finally, we also take the opportunity to clarify some common misunderstandings regarding the program., Comment: 23 pages plus references

A new class of relativistic stellar structure equations which include the effects of an energy-dependent Newton coupling is presented. Significant modifications in the mass-radius relation for neutron stars are possible only if the running of the Newton coupling due to quantum gravity occurs at low energies. A new Buchdahl limit is derived and its physical implications are discussed. In particular, sub-Planckian self-gravitating objects with arbitrarily small radii are possible., Comment: 19 pages, 3 figures. V2: minor changes, references added. Matches published version

The recent discovery of non-Schwarzschild black hole spacetimes has opened new directions of research in higher-derivative gravitational theories. However, despite intense analytical and numerical efforts, the link with the linearized theory is still poorly understood. In this work we address this point for the Einstein-Weyl Lagrangian, whose weak field limit is characterized by the standard massless graviton and a spin-2 ghost. We show that the strength of the Yukawa term at infinity determines the thermal properties of the black hole and the structure of the singularity near $r=0$. Moreover, inspired by recent results in the Asymptotic Safety scenario we investigate the consequences of an imaginary ghost mass. In this case we find a countable set of solutions all characterized by spatial oscillations of typical wavelength determined by the mass of the spin-2 field., Comment: 5 pages, 6 figures

We use the non-Gaussian fixed points (NGFPs) appearing in the renormalization group flow of gravity and gravity-matter systems to construct models of NGFP-driven inflation via a renormalization group improvement scheme. The cosmological predictions of these models depend sensitively on the characteristic properties of the NGFPs, including their position and stability coefficients, which in turn are determined by the field content of the underlying matter sector. We demonstrate that the NGFPs appearing in gravity-matter systems where the matter content is close to the one of the standard model of particle physics are the ones compatible with cosmological data. Somewhat counterintuitively, the negative fixed point value of the dimensionless cosmological constant is essential for these findings., Comment: 10 pages, 2 figures. V2: references added. Matches published version

The existence of spacetime singularities is one of the biggest problems of nowadays physics. According to Penrose, each physical singularity should be covered by a "cosmic censor" which prevents any external observer from perceiving their existence. However, classical models describing the gravitational collapse usually results in strong curvature singularities, which can also remain "naked" for a finite amount of advanced time. This proceedings studies the modifications induced by Asymptotically Safe Gravity on the gravitational collapse of generic Vaidya spacetimes. It will be shown that, for any possible choice of the mass function, Quantum Gravity makes the internal singularity gravitationally weak, thus allowing a continuous extension of the spacetime beyond the singularity., Comment: 13 pages, 5 figures

The Asymptotically Safe Gravity provides a framework for the description of gravity from the trans-Planckian regime to cosmological scales. According to this scenario, the cosmological constant and Newton's coupling are functions of the energy scale whose evolution is dictated by the renormalization group equations. The formulation of the renormalization group equations on foliated spacetimes, based on the Arnowitt-Deser-Misner (ADM) formalism, furnishes a natural way to construct the RG energy scale from the spectrum of the laplacian operator on the spatial slices. Combining this idea with a Renormalization Group improvement procedure, in this work we study quantum gravitational corrections to the Einstein-Hilbert action on Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) backgrounds. The resulting quantum-corrected Friedmann equations can give rise to both bouncing cosmologies and emergent universe solutions. Our bouncing models do not require the presence of exotic matter and emergent universe solutions can be constructed for any allowed topology of the spatial slices., Comment: 8 pages, 1 figure. V2: references added

Asymptotic Safety, based on a non-Gaussian fixed point of the gravitational renormalization group flow, provides an elegant mechanism for completing the gravitational force at sub-Planckian scales. At high energies the fixed point controls the scaling of couplings such that unphysical divergences are absent while the emergence of classical low-energy physics is linked to a crossover between two renormalization group fixed points. These features make Asymptotic Safety an attractive framework for cosmological model building. The resulting scenarios may naturally give rise to a quantum gravity driven inflationary phase in the very early universe and an almost scale-free fluctuation spectrum. Moreover, effective descriptions arising from an renormalization group improvement permit a direct comparison to cosmological observations as, e.g. Planck data., Comment: Invited review for the special issue "Testing quantum gravity with cosmology" to appear in Compte Rendus Physique A

We study the quantum gravity modification of the Kuroda-Papapetrou model induced by the running of the Newton's constant at high energy in Quantum Einstein Gravity. We argue that although the antiscreening character of the gravitational interaction favours the formation of a naked singularity, quantum gravity effects turn the classical singularity into a "whimper" singularity which remains naked for a finite amount of advanced time., Comment: 10 pages, 6 figures; v3: version accepted for publication in CQG

Asymptotically Safe theories of gravity have recently received much attention. In this work we discuss a class of inflationary models derived from quantum-gravity modification of quadratic gravity according to the induced scaling around the non-Gaussian fixed point at very high energies. It is argued that the presence of a three dimensional ultraviolet critical surface generates operators of non-integer power of the type $R^{2-\theta/2}$ in the effective Lagrangian, where $\theta>0$ is a critical exponent. The requirement of a successful inflationary model in agreement with the recent Planck 2015 data puts important constraints on the strenght of this new type of couplings., Comment: 6 pages, 4 figures. Physics Letters B, to appear

We discuss the stabilization of the conformal factor by higher derivative terms in a conformally reduced $R+R^2$ Euclidean gravity theory. The flat spacetime is unstable towards the condensation of modes with nonzero momentum, and they "condense" in a modulated phase above a critical value of the coupling $\beta$ of the $R^2$ term. By employing a combination of variational, numerical and lattice methods we show that in the semiclassical limit the corresponding functional integral is dominated by a single nonlinear plane wave of frequency $\approx 1/\sqrt{\beta} \lp$. We argue that the ground state of the theory is characterized by a spontaneous breaking of translational invariance at Planckian scales., Comment: 24 pages, 8 figures

The scaling properties of quantum gravity are discussed by employing a class of proper-time regulators in the functional flow equation for the conformal factor within the formalism of the background field method. Renormalization group trajectories obtained by projecting the flow on a flat topology are more stable than those obtained from a projection on a spherical topology. In the latter case the ultraviolet flow can be characterized by a Hopf bifurcation with an ultraviolet attractive limiting cycle. Although the possibility of determining the infrared flow for an extended theory space can be severely hampered due to the conformal factor instability, we present a robust numerical approach to study the flow structure around the non-gaussian fixed point as an inverse-problem strategy. In particular it is shown the possibility of having a spontaneous breaking of the diffeomorphism invariance can be realized with non-local functionals of the volume operator., Comment: 16 pages, 5 figures; replaced with journal version

Asymptotically safe theories of gravitation have received great attention in recent times. In this framework an effective action embodying the basic features of the renormalized flow around the non-gaussian fixed point is derived and its implications for the early universe are discussed. In particular, a "landscape" of a countably infinite number of cosmological inflationary solutions characterized by an unstable de Sitter phase lasting for a large enough number of e-folds is found., Comment: 5 pages, to appear as a Rapid Communication in Physical Review D

We discuss a mechanism that induces a time-dependent vacuum energy on cosmological scales. It is based on the instability induced renormalization triggered by the low energy quantum fluctuations in a Universe with a positive cosmological constant. We employ the dynamical systems approach to study the qualitative behavior of Friedmann-Robertson-Walker cosmologies where the cosmological constant is dynamically evolving according with this nonperturbative scaling at low energies. It will be shown that it is possible to realize a "two regimes" dark energy phases, where an unstable early phase of power-law evolution of the scale factor is followed by an accelerated expansion era at late times., Comment: 26 pages, 4 figures. To appear in New Journal of Physics

We discuss the existence of inflationary solutions in a class of renormalization group improved polynomial f(R) theories, which have been studied recently in the context of the asymptotic safety scenario for quantum gravity. These theories seem to possess a nontrivial ultraviolet fixed point, where the dimensionful couplings scale according to their canonical dimensionality. Assuming that the cutoff is proportional to the Hubble parameter, we obtain modified Friedmann equations which admit both power law and exponential solutions. We establish that for sufficiently high order polynomial the solutions are reliable, in the sense that considering still higher order polynomials is very unlikely to change the solution., Comment: 25 pages, 6 figures; accepted for publication in Clas.Quant.Grav

In recent years it has emerged that the high energy behavior of gravity could be governed by an ultraviolet non-Gaussian fixed point of the (dimensionless) Newton's constant, whose behavior at high energy is thus {\it antiscreened}. This phenomenon has several astrophysical implications. In particular in this article recent works on renormalization group improved cosmologies based upon a renormalization group trajectory of Quantum Einstein Gravity with realistic parameter values will be reviewed. It will be argued that quantum effects can account for the entire entropy of the present Universe in the massless sector and give rise to a phase of inflationary expansion. Moreover the prediction for the final state of the black hole evaporation is a Planck size remnant which is formed in an infinite time., Comment: 28 pages, 6 figures. Invited talk at Workshop on Continuum and Lattice Approaches to Quantum Gravity. Sept. 2008, Brighton UK. To appear in the Proceedings

This paper studies the cosmological equations for a scalar field Phi in the framework of a quantum gravity modified Einstein--Hilbert Lagrangian where G and Lambda are dynamical variables. It is possible to show that there exists a Noether symmetry for the point Lagrangian describing this scheme in a FRW universe. Our main result is that the Noether Symmetry Approach fixes both Lambda = Lambda(G) and the potential V = V(Phi) of the scalar field. The method does not lead, however, to easily solvable equations, by virtue of the higher dimensionality of the reduced configuration space involved, the additional variable being the running Newton coupling., Comment: 10 pages, Revtex4

Renormalization group (RG) improved cosmologies based upon a RG trajectory of Quantum Einstein Gravity (QEG) with realistic parameter values are investigated using a system of cosmological evolution equations which allows for an unrestricted energy exchange between the vacuum and the matter sector. It is demonstrated that the scale dependence of the gravitational parameters, the cosmological constant in particular, leads to an entropy production in the matter system. The picture emerges that the Universe started out from a state of vanishing entropy, and that the radiation entropy observed today is essentially due to the coarse graining (RG flow) in the quantum gravity sector which is related to the expansion of the Universe. Furthermore, the RG improved field equations are shown to possess solutions with an epoch of power law inflation immediately after the initial singularity. The inflation is driven by the cosmological constant and ends automatically once the RG running has reduced the vacuum energy to the level of the matter energy density., Comment: 57 pages, 7 figures

We find exact cosmological solutions when the Newton parameter and the cosmological term are dynamically evolving in a renormalization-group improved Hamiltonian approach. In our derivation we use the Noether symmetry approach, leading to an interesting variable transformation which yields exact and general integration of the cosmological equations. The functional dependence of Lambda on G is determined by the method itself, therefore generalizing previous results on symmetry principles in cosmology. We find new functional relations between Lambda and G, jointly with power-law inflation for pure gravity., Comment: 18 pages, 2 figures. In the final version, physical interpretation and presentation have been improved in all sections

Models of gravity with variable G and Lambda have acquired greater relevance after the recent evidence in favour of the Einstein theory being nonperturbatively renormalizable in the Weinberg sense. The present paper applies the Arnowitt-Deser-Misner (ADM) formalism to such a class of gravitational models. Are modified action functional is then built which reduces to the Einstein-Hilbert action when G is constant, and leads to a power-law growth of the scale factor for pure gravity and for a massless Phi**4 theory in a Universe with Robertson-Walker symmetry, in agreement with the recently developed fixed-point cosmology. Interestingly, the renormalization-group flow at the fixed point is found to be compatible with a Lagrangian description of the running quantities G and Lambda., Comment: 17 pages, Revtex-4, 2 figures. In the final version, the Hamiltonian analysis has been amended and the presentation has been improved

We have investigated the turbulent mean-field dynamo action in protoneutron stars that are subject to convective and neutron finger instabilities. While the first one develops mostly in the inner regions of the star, the second one is favoured in the outer regions, where the Rossby number is much smaller and a mean-field dynamo action is more efficient. By solving the mean-field induction equation we have computed the critical spin period below which no dynamo action is possible and found it to be ~1 s for a wide range of stellar models. Because this critical period is substantially longer than the characteristic spin period of very young pulsars, we expect that a mean-field dynamo will be effective for most protoneutron stars., Comment: A&A Letters, in press

The structure of the Cauchy Horizon singularity of a black hole formed in a generic collapse is studied by means of a renormalization group equation for quantum gravity. It is shown that during the early evolution of the Cauchy Horizon the increase of the mass function is damped when quantum fluctuations of the metric are taken into account., Comment: 15 Pages, one figure. Minor changes in the presentation, to appear on Phys.Rev.D

A class of exact solutions of the field equations with higher derivative terms is presented when the matter field is a pressureless null fluid plus a Maxwellian static electric component. It is found that the stable solutions are black holes in anti de Sitter background. The issue of the stability of the Cauchy horizon is discussed., Comment: REVTex, 13 pages, to appear on the proceedings of the workshop on " The Internal Structure of Black Holes and Spacetime Singularities", held at Technion -Israel Institute of Technology-, Haifa,Israel,June 29-July 3, 1997

A renormalization group study of a scalar theory coupled to gravity through a general functional dependence on the Ricci scalar in the action is discussed. A set of non-perturbative flow equations governing the evolution of the new interaction terms generated in both local potential and wavefunction renomalization is derived. It is shown for a specific model that these new terms play an important role in determining the scaling behavior of the system above the mass of the inflaton field., Comment: 26 pages, RevTeX, five figures

The structure of the Cauchy horizon of a charged rotating black hole is analyzed under the combined effect of an ingoing and outgoing flux of gravitational waves. In particular, by means of an axisymmetric realization of the Ori model, the growth of the mass parameter near the Cauchy horizon is studied in the slow rotation approximation. It is shown that the mass-parameter inflates, while the angular momentum per unit mass deflates, but initial deviations from spherical symmetry survive., Comment: 6 pages, REVTex, one uuencoded figure

The Kadanoff-Wilson renormalization group approach for a scalar self-interacting field theor generally coupled with gravity is presented. An average potential that monitors the fluctuations of the blocked field in different scaling regimes is constructed in a nonflat background and explicitly computed within the loop-expansion approximation for an Einstein universe. The curvature turns out to be dominant in setting the crossover scale from a double-peak and a symmetric distribution of the block variables. The evolution of all the coupling constants generated by the blocking procedure is examined: the renormalized trajectories agree with the standard perturbative results for the relevant vertices near the ultraviolet fixed point, but new effective interactions between gravity and matter are present. The flow of the conformal coupling constant is therefore analyzed in the improved scheme and the infrared fixed point is reached for arbitrary values of the renormalized parameters., Comment: 18 pages, REVTex, two uuencoded figures. (to appear in Phys. Rev. D15, July) Transmission errors have been corrected