^{*}

Edited by: Antonio D. Pereira, Universidade Federal Fluminense, Brazil

Reviewed by: Roberto Percacci, International School for Advanced Studies (SISSA), Italy; Nobuyoshi Ohta, Kindai University, Japan

This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

According to the asymptotic-safety conjecture, the gravitational renormalization group flow features an ultraviolet-attractive fixed point that makes the theory renormalizable and ultraviolet complete. The existence of this fixed point entails an antiscreening of the gravitational interaction at short distances. In this paper we review the state-of-the-art of phenomenology of Asymptotically Safe Gravity, focusing on the implications of the gravitational antiscreening in cosmology.

Similarly to the case of Quantum Chromodynamics (QCD), the gravitational interaction might exhibit an antiscreening behavior at high energies [

admit a Newton coupling _{min} ~ _{Pl}. In this case the resolution of the cosmological singularity is due to the violation of the energy conditions, thus invalidating one of the key assumptions leading to the Hawking-Penrose singularity theorems [

The “asymptotic freedom of gravity” discussed in Markov and Mukhanov [^{2} in

where η_{N} = _{k} log _{N} depends on _{N} depends on the RG scale _{N} = −2 for some values of the gravitational couplings. Assuming that a non-trivial fixed point indeed exists in the full (not truncated) theory space, the simple fact that η_{N} < 0 at the non-trivial fixed point implies that the dimensionfull Newton coupling decreases with the RG scale _{*}, where _{*} denotes the fixed-point value of the dimensionless Newton coupling.

Although in the context of asymptotically safe gravity (ASG) the gravitational antiscreening has a quantum origin and despite the different semantics [Markov's asymptotic freedom of

In this paper we review some of the main cosmological implications of ASG based on the running of the gravitational couplings. The rest of the present review is organized as follows. Section 2 summarizes the mechanism behind the renormalization group improvement and the scale-setting procedure. In sections 3 and 4 we review the main implications of the gravitational antiscreening in cosmology and inflation respectively. Finally, in section 5 we summarize the state-of-the-start of phenomenology of ASG, its main problems, and future perspectives.

One of the strengths of ASG is the possibility of constructing a quantum theory of gravity using the “language” of Quantum Field Theory—the standard framework to describe matter and all known fundamental interactions within the Standard Model of particle physics. On the one hand, this makes the connection between gravity and matter more straightforward than in other approaches to quantum gravity and allows to constrain the ultraviolet details of quantum gravity by verifying systematically its consistency with low-energy experiments and observations on the matter sector [see [

These are effectively classical field equations. Nonetheless, its solutions 〈_{μν}〉 actually incorporate all quantum gravitational effects.

The computation of the effective action comes along with several technical and conceptual issues. First, computing the effective action exactly would require either to solve the gravitational path integral over globally hyperbolic spacetimes or, equivalently, to solve the Functional Renormalization Group (FRG) equation for the effective average action Γ_{k} [

and take the limit

The ^{2} ≳ ^{2} are integrated out, resulting in the partially-quantized effective action Γ_{k}. In the limit _{0} coincides with the ordinary effective action. At the basis of the decoupling mechanism is the possibility that some infrared physical scales appearing in _{dec}—below which the running of Γ_{k} is essentially frozen. The “threshold effective action” Γ_{kdec} and the ordinary effective action are thus expected to be approximately the same. The identification of the infrared scale _{dec}, if any, could provide some of the (typically non-local) terms in Γ_{0}. An emblematic example of this mechanism is massless scalar electrodynamics, where _{0} is thus expected to involve an effective non-local interaction of the form ^{4}log(ϕ)-interaction appearing in the famous Coleman–Weinberg effective potential [_{dec} becomes a more involved task: _{dec} might be a complicated non-linear function of all these scales or, in the best case, it might be given by the one physical infrared scale which dominates over the others. Conversely, a decoupling scale might not exist at all: this is the case if there are no dominant infrared scales acting as an actual physical cutoff. Therefore, the existence of _{dec} and its specific form strongly depend on the physical system under consideration.

The procedure of identifying and replacing the RG scale _{dec}, is known as _{0}. Other forms of RG improvement are the RG improvement

We remark that the RG running of the gravitational couplings extrapolated from FRG computations relies on the use of Euclidean metrics. The implications of ASG obviously involve Lorentzian spacetimes. It is thereby assumed that the scaling of the couplings and the existence of an ultraviolet-attractive fixed point are not affected, at least qualitatively, by the metric signature. Hints that this might indeed be the case have been found in Manrique et al. [

ASG relies on the existence of an ultraviolet-attractive fixed point of the gravitational RG flow. In the Einstein-Hilbert truncation, the scaling of the dimensionless Newton coupling ^{2} and cosmological constant λ(^{−2} about the NGFP (_{*}, λ_{*}) reads

where _{i} are integration constants labeling all possible RG trajectories, _{i} are the eigenvectors of the stability matrix ∂_{gi}β_{j}|_{g*} constructed using the beta functions β_{j} of all dimensionless couplings _{j}, and (−θ_{i}) are its eigenvalues. The real part of the critical exponents θ_{i} determine the stability properties of the NGFP. In the case of pure gravity, the critical exponents θ_{1} and θ_{2} are typically a pair of complex conjugate numbers with positive real part: this implies that the NGFP is ultraviolet-attractive in the Einstein-Hilbert truncation. In extended truncations, involving higher-derivative operators, it has been shown that the NGFP comes with three relevant directions associated with the volume, _{i}) to be fixed by comparison with observations, e.g., by requiring that in the infrared _{1, 2} real [

Neglecting the running of the matter couplings, the scale-dependent Einstein-Hilbert action reads

At some intermediate scale

If there is no energy-momentum flow between the gravitational and matter components of the theory, i.e., if the energy-momentum tensor _{μν} is separately conserved, the cutoff function

The above equation provides a “consistency condition” [

CASE I:

Performing the RG improvement at the level of the field equations or solutions is equivalent to neglecting Δ_{μν}. Assuming that the matter energy-momentum tensor is covariantly conserved,

The solution to this equation depends on the form of the energy-momentum tensor. As it will be important in cosmology, let us focus on the case of a perfect fluid with energy density ρ and pressure

with

so that

In this setup the running gravitational couplings depend on the energy density ρ of the matter degrees of freedom: this is exactly the

The relation between _{dec} for the flow of the scale-dependent effective action ^{2} ~

In cosmology one can further limit the form of the effective metric to a Friedmann-Robertson-Walker (FRW) spacetime. Since _{μν} is assumed to be separately conserved, the energy density ρ obeys the standard (i.e., classical) conservation equations and thus

On the other hand, if _{μν} is not separately conserved, the Bianchi identities

do not add any additional constraint on the form of the cutoff function

CASE II:

If the RG improvement is performed at the level of the action, the field equations contain an additional contribution encoded in the gravitational energy-momentum tensor Δ_{μν}. Its variation reads [

The contracted Bianchi identities (8), together with the assumption that the energy-momentum tensor is separately conserved, thus yield the condition

Note that in this case, due to the presence of Δ_{μν}, the contribution from the energy-momentum tensor _{μν} cancels out. A scaling relation of the form ^{4} ~ ρ [_{μν} is negligible. We are thus left with the condition

Diffeomorphism invariance thus requires [

In the proximity to the NGFP, the couplings scale as in (11) and the constraint (18) gives

This condition should also hold in the more general case of _{k}(

It is worth noting that the replacement ^{2} ~

The RG improvement at the level of the field equations and at the level of the action lead to effective field equations which differ by a gravitational energy-momentum tensor Δ_{μν}. Based on the idea behind the decoupling mechanism, performing the RG improvement at the level of the action would seem to be more natural. However, the possibility of choosing between different forms of RG improvement is considered as a source of ambiguity. The identification of the decoupling scale _{dec} is a second possible source of ambiguity: in the case of gravity, there are multiple scales that could potentially act as a decoupling scale for the flow of the scale-dependent gravitational effective action. However, if the matter energy-momentum tensor is covariantly conserved, the form of the cutoff function

Phenomenological implications of ASG have been explored in the literature by means of RG-improved cosmological and astrophysical models. Although these models are not expected to provide precise and quantitative predictions, they are expected to capture qualitative features of the modifications of Einstein gravity induced by ASG.

Keeping strengths and limitations of the RG-improvement procedure in mind, in the next sections we will review some of the main phenomenological implications of ASG based on models of RG-improved cosmology.

It is an old idea that the gravitational couplings could depend on the cosmic time and that this time-dependence could have implications in cosmology [

We first focus on the RG-improved cosmological dynamics in the NGFP regime, following the analysis in Bonanno and Reuter [

The matter degrees of freedom are encoded in a perfect fluid with energy-momentum tensor

In this subsection we focus on the case where the energy-momentum tensor is separately conserved, as originally assumed in Bonanno and Reuter [_{μν} is not covariantly conserved has been studied in [

where _{μν} = _{μν} + _{μ}_{ν} is the projection tensor onto the tangent 3-space orthogonal to the 4-velocity ^{μ} of an observer comoving with the cosmological fluid. Provided that ^{μ}, i.e.,

The consistency condition (22) can thus be used to constrain the form of the cutoff function _{t} is a positive constant [_{t} in terms of fixed-point quantities. As it will become clear soon, the scale-setting ^{−1} employed in Bonanno and Reuter [

In what follows we will only consider the case of a spatially flat universe, _{t} can then be eliminated by imposing the consistency condition (22), which leads to the relation

Using this expression for ξ_{t}, the family of cosmological solutions associated with the RG-improved system (24) reads [

where _{*}_{*}), which is known to be scheme independent [_{h}

This magic can only occur in the proximity of a critical fixed point, where physical quantities should vary as power laws of a unique scale. Away from the NGFP, the complete solution to the cosmological system (24) can only be obtained numerically. This has been done in Reuter and Saueressig [

The fixed-point scaling of gravitational couplings modifies the power-law scaling of the scale factor, Equation (26), so that also the causal structure of the spacetime is modified at early times. At it can be easily seen, provided that _{μν} discussed in section 2 is taken into account, i.e., when the RG improvement is performed at the level of the action [

The gravitational effective energy-momentum tensor Δ_{μν} is also crucial to make the cosmological evolution non-singular: in the setting introduced above, the RG-improved cosmological evolution (26) is still singular, as scale factor vanishes at _{μν}-term in the effective field Equations (7) might mimic the effect of higher-order operators in the gravitational effective action [_{μν}-term in the effective field Equations (7)] which allow for a non-singular cosmological evolution for any value of the spatial curvature

The results reviewed in the previous subsection are based on the assumption that the matter energy-momentum tensor

If the matter energy-momentum tensor

and can be written in the form

Identifying ^{3} as the energy encapsulated in the proper volume ^{3}, the latter equation assumes the form of the first law of thermodynamics

In classical cosmology

Specifically, the production of entropy during the expansion of the universe requires ^{4} and therefore ^{3}ρ^{3/4} + const. The precise behavior of _{h}

Focusing on the “NGFP era,” where _{h} ^{α}, with

and ^{−1} − 1. Imposing the consistency condition (22) fixes _{*} derived from FRG computations is of order _{tr}) = ξ_{h}_{tr}) ~ _{Pl} which, using the fact that ξ_{h} ~ 1, implies that the parameter α sets the ratio between the transition time _{tr} and the Planck time, _{tr} = α _{Pl}. We thus learn that if α > 1 the transition to the classical regime occurs before the Planck time _{tr} > _{Pl}.

In a RG-improved radiation-dominated epoch, the production of entropy is given by the power law

In particular, the condition

We highlight that the condition α > 1, necessary to generate entropy in the early-universe expansion, is the same condition needed in order to produce a period of power-law inflation. In a radiation-dominated epoch this is condition is satisfied for

The existence of an ultraviolet-attractive NGFP entails a weakening of the gravitational interaction at high energies. It is then natural to ask whether this weakening can lead to non-singular cosmologies. While a definite answer in the context of ASG is still out of reach, the mechanism underlying a possible singularity resolution might be captured by a simple model embedding the running of the gravitational couplings in the spacetime dynamics [

Following the discussion in section 2, the running of the gravitational couplings in the Einstein-Hilbert action generates an additional term Δ_{μν} in the modified field equations. As we have seen, neglecting this term and introducing the running couplings at the level of the field equations leads to a family of RG-improved cosmologies admitting a period of power-law inflation and explaining the entropy production in terms of the energy flow between the gravitational and matter sectors [_{μν} to the effective field Equations (7) might modify this conclusion.

It is assumed that the universe is homogeneous and isotropic and that the energy-momentum tensor is covariantly conserved. Introducing the running of the gravitational couplings at the level of the (Einstein-Hilbert) action yields the modified Friedmann equation [

where _{N} → 0 as the RG flow approaches the perturbative regime (_{Pl}), Δ vanishes in this limit. Going back in time, η_{G} varies from its classical value η_{N} = 0 to the fixed-point value η_{N} = −2 (reached when ^{2} ∝ ^{−2} was assumed. This approximation is valid if the scale factor undergoes a period of exponential growth at early times—an assumption that can be verified

in Equation (34), where (_{0}, Λ_{0}) are the low-energy values of the gravitational couplings, and assuming that the universe is initially dominated by radiation, it can be easily seen [

where

and _{b} is not real, the universe is singular and a period of exponential growth of the scale factor is not possible (unless other degrees of freedom are introduced). However this would invalidate the initial assumption that ^{2} ∝ ^{−2} and a separate analysis would be required. This model thus shows how the gravitational antiscreening, encoded in the RG running of the gravitational couplings and in the presence of additional terms in the effective Friedmann equation, could lead to non-singular cosmologies and a period of exponential growth of the universe at early times.

Primordial quantum fluctuations occurring in the pre-inflationary epoch have left indelible imprints, which we measure today in the form of tiny temperature anisotropies, δ^{−5}, in the CMB radiation: according to the standard cosmological model, the inhomogeneities in the CMB can be traced back to the primordial quantum fluctuations in the pre-inflationary era. These fluctuations were subsequently amplified and smoothed out by the exponential growth of the universe, thus resulting in small density fluctuations at the last scattering surface. The distribution of temperature anisotropies in the sky could thus give us indirect information on the physics of the very early universe.

In momentum space, the power spectra of scalar and tensorial perturbations are written as follows

where _{s} and the tensor-to-scalar ratio _{t}/_{s} can be obtained from observational data. In particular, the most recent observations to date [_{s} = 0.9649±0.0042 at 68% confidence level, and limit the tensor-to-scalar ratio to values _{s} = 1) is excluded.

The existence of a NGFP in the RG flow of gravity could provide a natural and intuitive explanation for the nearly-scale invariance of the power spectrum of temperature fluctuations in the CMB. Close to the NGFP, the effective background graviton propagator behaves as _{N} = −2 in the ultraviolet limit. In this case the background graviton propagator in coordinate space scales as

where

The spectral index _{s} defines the power-law scaling of the power spectrum, ^{−4} gives rise to a perfectly scale invariant power spectrum, with _{s} = 1 [

Among all proposed inflationary models [^{2}-term in the gravitational action. This is the minimal modification of Einstein gravity needed to produce inflation. From the point of view of ASG, focusing on an

should comprise all relevant couplings of the theory (with respect to the NGFP): according to the studies of the renormalization group flow of

are all fulfilled. We refer the reader to the original paper [^{23}^{30}^{2}, while its observed constant value is only reached at ^{−2}

In this subsection we review the results in Bonanno et al. [_{i} governing the scaling of the gravitational couplings in the vicinity of the NGFP. In turn, the specific values of the critical exponents depend on the number of scalar, Dirac and vectors fields in the theory [_{s} and tensor-to-scalar ratio

We restrict ourselves to the Einstein-Hilbert truncation, where the scaling of the gravitational couplings about the NGFP is that given in Equation (5). Following the discussion in section 2, close to the NGFP the consistency condition (16) imposes the scaling relation ^{2} = ξ

where _{RG}(

with the coefficients _{i} being defined by

Here _{Pl} is the reduced Planck mass, while the integration constants _{i}, the critical exponents θ_{i} and the eigenvectors _{i} are those introduced in section 2.2 (cf. Equation 5). The action

is the fixed-point action [_{k}(^{n}-operators are generated along the RG flow. The effective action in (43) is thus expected to capture key features of the gravitational RG flow: at the fixed point the action is _{i}. In what follows we will explore the consequences of this fact in inflationary cosmology [

Provided that

where the subscript “E” indicates that these quantities are computed using the metric ^{−2}, with

A period of exponential grow of the scale factor occurs if the dynamics of the scalar field ϕ is dominated by its potential energy

The violation of the slow-roll conditions, encoded in the equation ϵ(ϕ_{f}) = 1, defines the value of the field at the end of inflation, ϕ_{f} ≡ ϕ(_{f}). The initial condition ϕ_{i} ≡ ϕ(_{i}) is then obtained by fixing the number of e-folds

before the end of inflation. In the slow-roll approximation, the spectral index and tensor-to-scalar ratio characterizing the scalar power spectrum

Moreover, every inflationary model has to be “normalized” [_{s} of the scalar power spectrum (38) is

At the NGFP (_{RG}(

and therefore it would generate an exactly scale-invariant power spectrum, with _{s} = 1. This is compatible with the discussion made in section 4.1 and based on the scaling of the background graviton propagator at the NGFP [_{*}(ϕ) and reads

This mass depends on fixed-point quantities only via the universal product (λ_{*}_{*}) [

In the model (43) the departure from the exact scale invariance is due to the departure of the RG flow from the NGFP. Lowering the RG scale down toward the infrared, the gravitational Lagrangian is modified by the operators in

Its form is determined by the critical exponents θ_{i}, which are real numbers in the case of the most commonly studied gravity-matter systems [_{i}) > 0. As we are interested in the case of gravity-matter systems, we will only focus on the case where the critical exponents are real. It is assumed, in a first approximation, that the energy-density of the inflation field ϕ dominates. Under this assumption, the other matter fields do not contribute to the inflationary dynamics [

It is crucial to note that if all critical exponents are θ_{i} > 4, the effective Lagrangian

where ^{−p} is the dominant correction in _{RG}(^{−p} are suppressed when _{i} = 4, the model (43) gives rise to an inflationary scenario compatible with the Planck data only under specific conditions [

Spectral index and tensor-to-scalar ratio induced by the family of theories in Equation (57) as a function of the power index _{s},

The agreement with the Planck data thus requires that at least one of the critical exponents is θ_{i} < 4. This condition is realized, e.g., when gravity is minimally-coupled to the fields of the Standard Model, at least in the approximation where these fields are free [_{i} > 4 would not be compatible with observational data. In this sense, the Planck data on the CMB anisotropies could be used to constrain the primordial matter content of the universe [

The scalar potential _{1} = θ_{2}. All functions _{*} for ϕ ≫ _{Pl}. In fact, as soon as θ_{i} ≠ 0, the coupling of the ^{2}-term in Equation (43) is not modified by the presence of the additional operators in ^{2} operator is a running quantity and therefore also the value of the scalar potential at ϕ ≫ _{Pl} should vary along the flow. In other words, in a more elaborate model accounting for the running of the coupling _{plateau} ≠ _{*}: this decoupling would allow to set the initial conditions for inflation at Planckian scales and, at the same time, to reproduce the correct amplitude of scalar perturbations at the horizon exit [

Inflationary potentials _{1} = θ_{2} [_{s} = 1. When the RG flow departs from the NGFP, additional operators are generated by the flow. These operators break the perfect scale invariance realized at the NGFP and destabilize the fixed-point potential, _{*}→_{*} + δ_{Pl}. The scalar field ϕ thus acquires a RG-induced kinetic energy, _{i}. In particular, the case θ_{1} = θ_{2} = 2 reproduces the well-known Starobinsky model.

As the RG flow moves way from the NGFP, the scalar potential _{1} = θ_{2} = 2 gives the scalar potential

i.e., a Starobinsky-like potential in the presence of an effective cosmological constant

in good agreement with the current observational data. In the case θ_{1} ≈ 2 and θ_{2} ≈ 4, realized when gravity is minimally coupled with the (free) matter fields of the Standard Model [^{−p}, with _{1} = θ_{2} = 2). In the next subsection we will see how this scenario is modified when the RG-improved effective action is obtained by starting from the quadratic gravity action (41).

In Bonanno and Platania [_{k}, λ_{k}, β_{k}), with β_{k} =

In order to derive an analytical form for the inflationary potential in the Einstein frame, the running of the gravitational couplings is approximated by [

where μ is a reference scale and the three parameters (_{0}, _{1}, _{2}), corresponding to the three relevant directions of the theory, identify the RG trajectories terminating at the NGFP in the ultraviolet limit. These are free parameters of the theory and must be fixed by comparing the results with observations. Using the cutoff ^{2} = ξ

with

The inflationary scenario generated in this model can be studied in the Einstein frame, where the

with the dimensionless couplings Λ and α given by ^{3/2}-term in the effective action, and the standard Starobinsky model is recovered by setting α = Λ = 0. Both functions _{±}(ϕ) define a two-parameters family of potentials, parameterized by the couple (α, Λ). The common feature of these potentials is the existence of a plateau for large positive values of the field ϕ, with _{k} now allows for an effective potential with _{plateau} ≠ _{*}, as mentioned in the previous section.

In order to fulfill the slow-roll conditions, the dynamical evolution of the inflation field must start from a quasi-deSitter state at _{i}) ~ _{plateau}, and then proceed toward ϕ ≪ _{Pl}. The inflationary dynamics depends on the values of (α, Λ). For any (α, Λ), the potential _{±}(ϕ) can either develop a minimum (_{min} ≤ 0. The case _{min} > 0 leads instead to eternal inflation, as shown in the right panel of

Scalar potential _{−}(ϕ) (blue thick line) and slow-roll function ϵ(ϕ) (red thin line) for α = −10 and Λ = −2 (potential unbounded from below, left panel) and Λ = 2 (potential with a minimum and _{min} > 0, right panel) [_{f} such that ϵ(ϕ(_{f})) = 1 (thin dashed line) and ϵ(ϕ) > 1 for any _{f}. The dynamics induced by the potential _{−}(ϕ) in the right panel keeps the dynamical field ϕ(

We now focus on the class of inflationary potentials providing a well defined exit from inflation by violation of the slow-roll conditions, followed by a phase of parametric oscillations of the inflation field [_{+}(ϕ), with α ∈ [1, 3] and Λ ∈ [0, 1.5] [

This figure depicts the potential _{+}(ϕ) for Λ = 1.4 and various values of α in the physically-interesting range, α ∈ [1, 3] [

As already mentioned, the constants (α, Λ) parameterize the deviations from the Starobinsky model due to the RG running of the gravitational couplings. It is thereby interesting to understand whether these modifications can affect the form of the power spectrum of temperature fluctuations in the CMB, and if the values of the spectral index _{s} and tensor-to-scalar ratio _{s} ∈ [0.965, 0.972], in agreement with the value extracted from the Planck data, _{s} = 0.968±0.006. The tensor-to-scalar ratio is always compatible with their upper limit, but it is slightly higher than the one predicted within the Starobinsky model.

Values of the spectral index _{s} and tensor-to-scalar ratio

_{s} |
_{s} |
_{s} |
|||||
---|---|---|---|---|---|---|---|

1.0 | 0.965 | 0.0069 | 0.968 | 0.0058 | 0.971 | 0.0050 | |

0 | 1.8 | 0.966 | 0.0074 | 0.969 | 0.0063 | 0.972 | 0.0055 |

2.6 | 0.967 | 0.0076 | 0.969 | 0.0065 | 0.972 | 0.0056 | |

1.0 | 0.965 | 0.0070 | 0.968 | 0.0059 | 0.971 | 0.0051 | |

1 | 1.8 | 0.966 | 0.0074 | 0.969 | 0.0063 | 0.972 | 0.0055 |

2.6 | 0.967 | 0.0076 | 0.969 | 0.0065 | 0.972 | 0.0056 |

_{s} ∈ [0.965, 0.972], is in agreement with the one obtained by the Planck Collaboration, n_{s} = 0.968 ± 0.006, and the tensor-to-scalar ratio is always compatible with their upper limit, r < 0.11

The phenomenological consequences of Asymptotically Safe Gravity (ASG) are typically investigated within models that take the running of the gravitational couplings into account. Based on the decoupling mechanism [

ASG is based on the existence of an interacting fixed point which is attained by the gravitational RG flow in the ultraviolet limit. The scale invariance of gravity at high energies and the consequent gravitational antiscreening can be regarded as the hallmarks of ASG. In particular the antiscreening character of gravity, rendering the gravitational interaction weaker at high energies, could lead to non-singular cosmological solutions: the classical singularity could be replaced by a bounce or an emergent universe [^{2}-coupling could also provide a mechanism to set the initial condition for inflation at trans-planckian scales, while being able to reproduce the amplitude of the scalar power spectrum at the horizon exit [

Most of the results listed above have been obtained by including the running of the gravitational couplings within the Einstein-Hilbert truncation. As argued in Lehners et al. [

One of the main problems of models of “RG-improved cosmologies” is the identification of the physical cutoff acting as a decoupling scale [

While it is expected that the RG-improved models capture the qualitative features of the quantum modifications of General Relativity according to ASG, quantitative results require the knowledge of the fully-quantum gravitational effective action. This is expected to be non-local, due to the resummation of quantum fluctuations on all scales. Progress in this direction has been made in Codello and Jain [

The author confirms being the sole contributor of this work and has approved it for publication.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We acknowledge financial support by the Baden-Württemberg Ministry of Science, Research, and the Arts and by Ruprecht-Karls-Universität Heidelberg. The author also thanks A. Bonanno and A. Eichhorn for their comments on this manuscript. The research of AP is supported by the Alexander von Humboldt Foundation.

^{2}GRAVITY

^{2}phase diagram of quantum Einstein gravity and its spectral dimension