- freely available
- re-usable

*Galaxies*
**2014**,
*2*(2),
275-291;
doi:10.3390/galaxies2020275

^{1}

^{2}

^{*}

## Abstract

**:**We explore the possible cosmological consequences of a running Newton’s constant, $G(\square )$, as suggested by the non-trivial ultraviolet fixed point scenario for Einstein gravity with a cosmological constant term. Here, we examine what possible effects a scale-dependent coupling might have on large-scale cosmological density perturbations. Starting from a set of manifestly covariant effective field equations, we develop the linear theory of density perturbations for a non-relativistic perfect fluid. The result is a modified equation for the matter density contrast, which can be solved and thus provides an estimate for the corrections to the growth index parameter, γ.

## 1. Introduction

Recent years have seen the development of a bewildering variety of alternative theories of gravity, in addition to the more traditional ones, such as scalar-tensor, higher derivative and dilaton gravities, just to mention a few examples. Many of these theories eventually predict some level of deviation from classical gravity, which is often parametrized either by a suitable set of post-Newtonian parameters or, more recently, by the introduction of a slip function [1,2,3,4]. The latter has been quite useful in describing deviations from classical General Relativity (GR) and specifically from the standard ΛCDM (Lambda Cold Dark Matter) model, when analyzing the latest cosmological CMB (Cosmic Microwave Background), weak lensing, supernovae and galaxy clustering data.

In this work, we will focus on the systematic analysis of departures from GR in the growth history of matter perturbations arising from a quantum running of G, within the narrow context of the non-trivial ultraviolet fixed point scenario for Einstein gravity with a cosmological term. Thus, instead of looking at deviations from GR at very short distances, due to new interactions, such as the ones suggested by string theories [5,6], we will be considering here infrared effects, which could therefore become manifest at very large distances. We will argue here that such effects are in principle calculable and could therefore be confronted with present and future astrophysical observations. The classical theory of small density perturbations is by now well developed in standard textbooks, and the resulting theoretical predictions for the growth exponents are simple to state and well understood. Except possibly on the very largest scales, where the data so far is still rather limited, the predictions agree quite well with current astrophysical observations. Here, we will be interested in computing and predicting possible small deviations in the growth history of matter perturbations and specifically in the values of the growth exponents, arising from a very specific scenario, namely a weakly scale-dependent gravitational coupling, whose value very gradually increases with distance. The specific nature of the scenario we will be investigating here is motivated by the treatment of field-theoretic models of quantum gravity, based on the Einstein action with a bare cosmological term. Its long-distance scaling properties are derived from the existence of a non-trivial ultraviolet fixed point of the renormalization group in Newton’s constant, G [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21].

The first step in analyzing the consequences of a running of G is thus to re-write the expression for $G\left({k}^{2}\right)$ in a coordinate-independent way, either by the use of a non-local Vilkovisky-type effective gravity action [22,23,24,25] or by the use of a set of consistent effective field equations. In going from momentum to position space, one employs ${k}^{2}\to -\square $, which gives for the quantum-mechanical running of the gravitational coupling the replacement $G\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\to \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}G(\square )$. One then finds that the running of G is given, in the vicinity of the Ultra Violet (UV) fixed point, by:

Within the quantum-field-theoretic renormalization group treatment, the quantity, ξ, arises as an integration constant of the Callan–Symanzik renormalization group equations. One challenging issue, and of great relevance to the physical interpretation of the results, is a correct identification of the renormalization group invariant scale, ξ. A number of arguments can be given in support of the suggestion that the infrared scale, ξ (analogous to the ${\Lambda}_{\overline{MS}}$ of QCD), can, in fact, be very large, even cosmological, in the gravity case (see, for example, [8] and references therein). From these arguments, one would then first infer that the constant, ${G}_{0}$, can, to a very close approximation, be identified with the laboratory value of Newton’s constant, $\sqrt{{G}_{0}}\sim 1.6\times {10}^{-33}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{cm}$. The appearance of the d’Alembertian □ in the running of G then naturally leads to a set of non-local field equations; instead of the ordinary Einstein field equations with constant G, one is now led to consider the modified effective field equations:

As stated above, physically, it would seem at first, based on the perturbative treatment alone [7,18], that the non-perturbative scale, ξ, could take any value (including perhaps a very small one), which could then possibly preclude any observable quantum effects in the foreseeable future. In perturbation theory, the reason for this is that the non-perturbative scale, ξ, appears, as in gauge theories, as an integration constant of the renormalization group equations and is therefore not fixed by perturbation theory alone. However, a number of recent non-perturbative results for the gravitational Wilson loop on the Euclidean lattice at strong coupling, giving an area law, and their subsequent interpretation in light of the observed large-scale semiclassical curvature [8,44,45,46], would suggest otherwise: namely that the non-perturbative scale, ξ, appears, in fact, to be related to macroscopic curvature. From astrophysical observation, the average curvature on very large scales, or, stated in somewhat better terms, the measured cosmological constant, λ, is very small. This would then suggest that the new scale, ξ, can be very large, even cosmological, and comparable to the Hubble scale, $1/{\xi}^{2}\phantom{\rule{0.166667em}{0ex}}\simeq \phantom{\rule{0.166667em}{0ex}}\lambda /3$. This would then give a more concrete semi-quantitative estimate for the scale in the $G(\square )$ of Equation (1), namely $\xi \sim 1/\sqrt{\lambda /3}\sim 1.51\times {10}^{28}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{cm}$. Note that in common astronomical units (Mpcs), the reference scale appearing in $G(\square )$ is then of the order of $\xi \simeq 4890$ Mpc.

A scale-dependent Newton’s constant is already expected to lead to small modifications of the standard cosmological solutions to the Einstein field equations. The starting point is the quantum effective field equations of Equation (2), with $G(\square )$ defined in Equation (1). In the Friedmann–Lemaître–Robertson–Walker (FLRW) framework, these are applied to the standard homogeneous isotropic metric. In the following, we will mainly consider the case $k=0$ (a spatially flat universe). The next step, therefore, is a systematic examination of the nature of the solutions to the full effective field equations, with $G(\square )$ involving the relevant covariant d’Alembertian operator $\square \phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}{g}^{\mu \nu}\phantom{\rule{0.166667em}{0ex}}{\nabla}_{\mu}{\nabla}_{\nu}$. To start the process, one assumes, for example, that the matter, ${T}_{\mu \nu}$, has the perfect fluid form:

A more general calculation [47] shows that a non-vanishing pressure contribution is generated in the effective field equations, even if one initially assumes a pressureless fluid, $p\left(t\right)=0$. After a somewhat lengthy derivation, one obtains, for a universe filled with non-relativistic matter ($p=0$), a set of effective Friedmann equations incorporating the running of G. It was also noted in [47] that the effective field equations with a running G can be recast in an equivalent, but slightly more appealing, form by defining a vacuum polarization pressure, ${p}_{vac}$, and density, ${\rho}_{vac}$, such that, for the FLRW background, one has:

Then, the source term in the effective $tt$ field equation can be regarded as a combination of the two density terms $\rho \left(t\right)$ + ${\rho}_{vac}\left(t\right)$, while the effective $rr$ equation involves the new vacuum polarization pressure term, ${p}_{vac}\left(t\right)$. Just as one introduces the parameter, w, describing the matter equation of state, $p\left(t\right)=w\phantom{\rule{0.166667em}{0ex}}\rho \left(t\right)$, with $w=0$ for non-relativistic matter, one can do the same for the remaining contribution by setting ${p}_{vac}\left(t\right)={w}_{vac}\phantom{\rule{0.277778em}{0ex}}{\rho}_{vac}\left(t\right)$. This more compact notation allows one to finally re-write the field equations for the FLRW background (and $k=0$) as:

## 2. Relativistic Treatment of Matter Density Perturbations

Besides the modified cosmic scale factor evolution just discussed, the running of $G(\square )$ given in Equation (1) also affects the nature of matter density perturbations on very large scales. In computing these effects, it is customary to introduce a perturbed metric of the form:

The first equation one obtains is the zeroth (in the fluctuations) order energy conservation in the presence of $G(\square )$, which reads:

The zeroth order field equations with the running of G included were already given in Equation (6). The next step consists in obtaining the equations that govern the effects of small field perturbations. These equations will involve, apart from the metric perturbation, ${h}_{ij}$, the matter and vacuum polarization contributions. The latter arise from:

As far as the magnitude of the correction, ${c}_{h}$, in Equation (22), one can argue that from Equation (23), one can relate the combination $(\dot{h}/h)(a/\dot{a})$ to the growth index, $f\left(a\right)$,

Finally, one can do the same analysis in the opposite, but less physical, limit $\omega \ll \Gamma $ or $\dot{h}/h\ll \dot{a}/a$. However, this second limit is, in our opinion, less physical, because of the fact that, now, the background is assumed to be varying more rapidly in time than the metric perturbation itself, $\dot{a}/a\gg \dot{h}/h$. Furthermore, one disturbing, but not entirely surprising, general aspect of the whole calculation in this second $\omega \ll \Gamma $ limit is its extreme sensitivity as far as magnitudes and signs of the results are concerned, to the set of assumptions initially made about the time development of the background. For the reasons mentioned, in the following, we will no longer consider this limit of rapid background fluctuations any further.

To summarize, the results for a scalar box and for a very slowly varying background, $\dot{h}/h\gg \dot{a}/a$, give the $O\left(h\right)$ corrected expression for $\delta {\rho}_{vac}\left(t\right)$ in Equation (21) and $\delta {p}_{vac}\left(t\right)={w}_{vac}\phantom{\rule{0.166667em}{0ex}}\delta {\rho}_{vac}\left(t\right)$ with ${c}_{h}\simeq +2.162$, while the tensor box calculation, under essentially the same assumptions, gives the somewhat larger result ${c}_{h}\simeq +7.927$. From now on, these will be the only two choices we shall consider here.

The next step in the analysis involves the derivation of the energy-momentum conservation to first order in the fluctuations and a derivation of the relevant field equations to the same order. After that, energy conservation is used to eliminate the h field entirely and, thus, obtain a single equation for the matter density fluctuation, δ. First, we will look here at the implications of energy-momentum conservation, ${\nabla}^{\mu}\left({T}_{\mu \nu}+{T}_{\mu \nu}^{vac}\right)=0$, to first order in the fluctuations. After defining the matter density contrast, $\delta \left(t\right)$, as the ratio $\delta \left(t\right)\equiv \delta \rho \left(t\right)/\overline{\rho}\left(t\right)$, the energy conservation equation to first order in the perturbations is found to be:

Furthermore, to first order in the perturbations, the $tt$ and $ii$ effective field equations become, respectively,

To obtain an equation for the matter density contrast $\delta \left(t\right)=\delta \rho \left(t\right)/\overline{\rho}\left(t\right)$, one needs to eliminate the metric trace field, $h\left(t\right)$, from the field equations. This is first done by taking a suitable linear combination of the two field equations in Equations (25) and (26), to get the equivalent equation:

The above equation can now be compared with the corresponding, much simpler, equation obtained for constant G, i.e., for $G\to {G}_{0}$ and still $w=0$ (see, for example, [50,51]):

It is common practice at this point to write an equation for the density contrast, $\delta \left(a\right)$, as a function not of t, but of the scale factor, $a\left(t\right)$. This is done by utilizing simple derivative identities to relate derivatives with respect to t to derivatives with respect to $a\left(t\right)$, with $H\equiv \dot{a}\left(t\right)/a\left(t\right)$ the Hubble constant. This last quantity can be obtained from the zeroth order $tt$ field equation, sometimes written in terms of current density fractions:

It is convenient at this stage to introduce a parameter, θ, describing the cosmological constant fraction as measured today, $\theta \equiv {\Omega}_{\lambda}/\Omega $. While the following discussion will continue with some level of generality, in practice, one is mostly interested in the observationally favored case of a current matter fraction $\Omega \approx 0.25$, for which $\theta \approx 3$. In terms of the parameter, θ, the growing solution to the differential equation for the density contrast, $\delta \left(a\right)$, for constant G is:

After the various substitutions and insertions have been performed, one obtains, after expanding to linear order in ${a}_{0}$, a second order linear differential equation for the correction, $\mathcal{F}\left(a\right)$ to $\delta \left(a\right)$, as defined in Equation (33). Since this equation looks rather complicated for general $\delta G\left(a\right)$, it will not be recorded here, but it is easily obtained from Equation (28) by a sequence of straightforward substitutions and expansions. The resulting equation can then be solved for $\mathcal{F}\left(a\right)$, giving the desired density contrast, $\delta \left(a\right)$, as a function of the parameter, Ω.

To obtain an explicit solution to the $\delta \left(a\right)$ equation, one needs to know the coefficient, ${c}_{a}$, and the exponent, ${\gamma}_{\nu}$, in Equation (9), whose likely values are discussed above and right after the quoted expression for $G\left(a\right)$. For the exponent, ν, one has $\nu \simeq \frac{1}{3}$, whereas for the value for ${c}_{h}$, one finds, according to the discussion in the previous section, ${c}_{h}\simeq 7.927$ for the tensor box case. Furthermore, one needs at some point to insert a value for the matter density fraction parameter, θ, which, based on current observation, is close to $\theta =(1-\Omega )/\Omega \simeq 3$.

## 3. Relativistic Growth Index with $\mathbf{G}(\square )$

When discussing the growth of density perturbations in classical General Relativity, it is customary at this point to introduce a scale factor-dependent growth index, $f\left(a\right)$, defined as:

It is known that in the absence of a running Newton’s constant G ($G\to {G}_{0}$, thus ${c}_{a}=0$), one has $f(a={a}_{0})=0.4625$ and $\gamma =0.5562$ for the standard ΛCDM scenario with $\Omega =0.25$ [50]. On the other hand, when the running of $G(\square )$ is taken into account, one finds from the solution to Equation (28) for the growth index parameter, γ, at $\Omega =0.25$ the following set of results. For γ, one has:

To quantitatively estimate the actual size of the correction in the above expressions for the growth index parameter, γ, and to make some preliminary comparison to astrophysical observations, some additional information is needed. The first item is the coefficient ${c}_{0}\approx 8.02\pm 0.55$ in Equation (1), as obtained from lattice gravity calculations of invariant correlation functions at a fixed geodesic distance [52]. We have re-analyzed and extended the results of [52], which involve rather large uncertainties for this particular quantity; nevertheless, it would seem very difficult to accommodate values for ${c}_{0}$ that are more than an order of magnitude smaller than the quoted value.

The next item that is needed here is a quantitative estimate for the magnitude of the coefficient, ${c}_{a}$, in Equation (9) in terms of ${c}_{t}$ in Equation (5) and, therefore, in terms of ${c}_{0}$ in the original Equation (1). First of all, one has ${c}_{a}={c}_{t}\phantom{\rule{0.166667em}{0ex}}{({t}_{0}/\xi )}^{3}$, where ${t}_{0}$ represents the current age of the universe (about 13.82 Gyrs), giving ${c}_{a}=0.646\phantom{\rule{0.166667em}{0ex}}{c}_{t}$. The factor ${({t}_{0}/\xi )}^{3}\approx 0.646$ accounts for the fact that the two time scales, ${t}_{0}$ (“today”) and $\xi /c$, do not coincide exactly and differ instead by a rather small, but relevant, amount. Regarding the numerical value of the coefficient, ${c}_{t}$, itself, it was found in [47] that in Equation (5), ${c}_{t}\simeq 0.785\phantom{\rule{0.166667em}{0ex}}{c}_{0}$ in the scalar box case and ${c}_{t}\simeq 0.450\phantom{\rule{0.166667em}{0ex}}{c}_{0}$ in the tensor box case. In both cases, these estimates refer to values obtained from the zeroth order covariant effective field equations. In the following, we will take for concreteness the tensor box value, thus ${c}_{t}\approx 0.450\phantom{\rule{0.166667em}{0ex}}{c}_{0}$; then, for all three covariant calculations recorded above ${c}_{a}\approx 0.450\times 8.02\times {({t}_{0}/\xi )}^{3}\approx 2.33$, a rather large coefficient. From all of these considerations, one would tend to get estimates for the growth parameter, γ, with rather large corrections! For example, in the tensor box case, the corrections would add up to $-199.20\phantom{\rule{0.166667em}{0ex}}{c}_{a}\approx -464.1$.

Nevertheless, it would seem that one should account somewhere for the fact that the largest galaxy clusters and superclusters studied today up to redshifts $z\simeq 1$ extend for only about, at the very most, $1/20$ the overall size of the visible universe. This would suggest then that the corresponding scale for the running coupling, $G\left(t\right)$ or $G\left(a\right)$, in Equations (5) and (9), respectively, should be reduced by a suitable ratio of the two relevant length scales, one for the largest observed galaxy clusters or superclusters and the second for the very large, cosmological scale $\xi \sim 1/\sqrt{\lambda /3}\sim 1.51\times {10}^{28}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{cm}$, entering the expression for $\delta G(\square )$ in Equations (1) and (2). This would then reduce the overall magnitude of the quantum correction by a factor of ${(\mathrm{scale}/\xi )}^{3}$ on slightly smaller scales, with the correction factor attaining unity only on the largest accessible scales, comparable to ξ. Note that this cubic scaling is already implicit in the form of $G\left(k\right)$ or $G(\square )$ given in Equation (1) with $\nu =\frac{1}{3}$. When this suppression is taken properly into account, one finds that the quantum correction to the growth exponent, γ, drops to about 10% on scales of 240 Mpc, to 5% on scales of 190 Mpc and to a rather small 1% effect on scales of about 110 Mpc.

It should also be emphasized here that, so far, all of the above results have been obtained by solving the differential equation for $\delta \left(a\right)$ with $G\left(a\right)$ given in Equation (9) and exponent ${\gamma}_{\nu}=3/2\nu \simeq 9/2$ relevant for a matter dominated background universe. However, it is this last choice that needs to be critically analyzed, as it might give rise to a definite bias. Our value for ${\gamma}_{\nu}$, so far, reflects our choice of a matter dominated background. More general choices, such as an “effective” ${\gamma}_{\nu}=3(1+w)/2\nu $ with an “effective” w, are, in principle, possible. Then, although Equation (28) for $\delta \left(t\right)$ remains unchanged, the differential equation for $\delta \left(a\right)$ needs to be solved with new parameters. Therefore, we will discuss here a number of options that should allow one to refine the accuracy of the above result and, in particular, correct the possible shortcomings coming so far from the specific choice of the exponent, ${\gamma}_{\nu}$.

Before we used $a\left(t\right)\sim {a}_{0}{(t/{t}_{0})}^{2/3}$ in relating $G\left(a\right)$ in Equation (9) to $G\left(t\right)$ in Equation (5). In general, if w is not small, one should use the more general equation relating the variable, t, to $a\left(t\right)$. The problem here is that, loosely speaking, for $w\ne 0$, at least two ws are involved, $w=0$ (matter) and $w=-1$ (λ term). This issue considerably complicates the problem of relating $\delta G\left(t\right)$ to $\delta G\left(a\right)$ and, therefore, the solution to the resulting differential equation for $\delta \left(a\right)$ [53,54,55]. As a tractable approximation, we will set here instead $a\left(t\right)\sim {a}_{0}{(t/{t}_{0})}^{2/3(1+w)}$ and, then, use an “effective” value of $w\approx -7/9$, which would seem more appropriate for the final target value of a matter fraction $\Omega \approx 0.25$. For this choice, one then obtains a significantly reduced power in Equation (9), namely ${\gamma}_{\nu}=3(1+w)/2\nu =1$. Furthermore, the resulting differential equation for $\delta \left(a\right)$ is still relatively easy to solve, by the same methods used in the previous section.

One then obtains for the growth index parameter, γ, at $\Omega =0.25$ the following set of results:

As a practical example, consider the galaxy clusters studied recently in [56,57,58,59,60], which typically involve comoving radii of ∼8.5 Mpc and viral radii of ∼1.4 Mpc. For these, one would obtain an approximate overall scale reduction factor of ${(8.5/4890)}^{3}\approx 5.3\times {10}^{-9}$. This would give for the tensor box (${c}_{h}=7.927$) correction to the growth index, γ, in Equation (39) the still rather small order of magnitude estimate, ∼${10}^{-6}$. To clearly see the corrections, larger clusters will be needed. Nevertheless this last case is suggestive of a trend, quite independent of the specific value of ${c}_{h}$ and, therefore, of the overall numerical coefficient of the correction in Equation (39): namely, that the correction to the growth index parameter is expected to be negative and will gradually and quite rapidly increase with the size of the cluster. In view of these results, it would seem rather worthwhile to further investigate, from an observational point of view, what happens on very large astrophysical scales, comparable to either ${H}_{0}^{-1}$ or ξ.

## 4. Conclusions

In this work, we have presented some preliminary results on the evolution of matter density perturbations in quantum-gravity motivated models, where Newton’s constant, $G(\square )$, is scale dependent, due to strong infrared quantum radiative corrections. Such a scale dependence arises naturally in the non-perturbative treatment of Euclidean quantum gravity via the covariant path integral approach, in close analogy to what happens in other perturbatively non-renormalizable theories, such as the non-linear sigma model. The relevant scaling violation scale, ξ, similar to the ${\Lambda}_{\overline{MS}}$ of QCD, is seen here to be naturally related to the infrared cutoff, ${H}_{0}^{-1}$, or perhaps, more appropriately (since the latter is time dependent), to the scaled cosmological constant, λ, via $\xi \sim 1/\sqrt{\lambda /3}$. As in non-abelian gauge theories and QCD, the infrared cutoff scale is, in turn, expected to be connected, by scaling considerations, to a non-perturbative gravitational condensate:

From all of the above considerations, it should be clear that the results presented in this paper are completely different from what one would expect based on a naive perturbative treatment of quantum gravity.

## Acknowledgments

One of the authors (Herbert W. Hamber) wishes to thank Thibault Damour and Gabriele Veneziano for inspiration and discussions leading to the present work and Alexey Vikhlinin for correspondence regarding the astrophysical measurements of growth indices. He also wishes to thank Hermann Nicolai and the Max Planck Institut für Gravitationsphysik (Albert Einstein Institut) in Potsdam for warm hospitality. The work of Herbert W. Hamber was supported in part by the Max Planck Gesellschaft zur Förderung der Wissenschaften and by the University of California. The work of Reiko Toriumi was supported in part by a Department of Education fellowship.

## Author Contributions

Each author has contributed in a significant way to the reported research and to the writing of the paper.

## Conflicts of Interest

The authors declare no conflict of interest.

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