Growth of structures and redshift-space distortion data in scale-dependent gravity

Current data seem to suggest that we live in a spatially flat Universe dominated by dark matter and dark energy turner . Although as of today the origin and nature of the dark sector still remains unknown, we have come up with the standard cosmological model, which is based on cold dark matter combined with a positive cosmological constant ($\Lambda$CDM) within Einstein’s General Relativity (GR) GR . This model, despite its simplicity, is overall in a very good agreement with a wealth of observational data coming from Astrophysics and Cosmology. As economic and successful as it may be, the concordance cosmological model unfortunately does not come without problems.

To begin with, as far as the theory itself is concerned, i) it contains singularities that indicate that the theory is incomplete, and ii) it is a non-renormalizable theory. In addition to those, $\Lambda$CDM, too, comes with its own problems. In particular, apart from the well-known fine-tuning problem, there are a couple of tensions related to $\Lambda$CDM discussed in the literature over the last years. To be more precise, there is nowadays a tension regarding the value of the Hubble constant, $H_{0}$, between high red-shift CMB data and low red-shift data, see e.g. tension ; tension1 ; tension2 ; tension3 . The value of the Hubble constant extracted by the Planck Collaboration planck1 ; planck2 , $H_{0}=(67-68)~\text{km/(Mpc sec)}$, is found to be lower than the value obtained by local measurements, $H_{0}=(73-74)~\text{km/(Mpc sec)}$ hubble ; recent . What is more, regarding large scale structure formation data, the growth rate from red-shift space distortion measurements has been found to be lower than expected from Planck eriksen ; basilakos .

Regarding the aforementioned problems and possible alternatives to the $\Lambda$CDM model, either a modified theory of gravity is assumed, providing correction terms to GR at cosmological scales, or a new dynamical degree of freedom with an equation-of-state (EOS) parameter $w<-1/3$ must be introduced. In the first class of models (geometrical DE) one finds for instance $f(R)$ theories of gravity mod1 ; mod2 ; HS ; starobinsky (see also NewRef1 ; NewRef2 for a more general class of theories, $f(R,L)$, that allow for more general couplings between matter and curvature), brane-world models langlois ; maartens ; dgp and Scalar-Tensor theories of gravity BD1 ; BD2 ; leandros1 , while in the second class (dynamical DE) one finds models such as quintessence DE1 , phantom DE2 , quintom DE3 , tachyonic DE4 or k-essence DE5 . What is more, people have proposed and analysed the implications of cosmological models with a varying cosmological constant, see e.g. Sola1 ; Sola2 ; Sola3 ; Sola4 and references therein. The effective cosmological equations in $\Lambda$-varying scenarios can be computed without additional assumptions, they are a natural generalization of the more standard concordance model, and they offer a richer phenomenology compared to the $\Lambda$CDM model Oztas:2018jsu . There are also some works on cosmological models with a variable Newton’s constant, see e.g. VarG1 ; VarG2 ; VarG3 .

Over the last years scale-dependent gravity has emerged as an interesting framework with a few appealing properties. As it is inspired by the renormalization group approach, it naturally allows for a varying cosmological constant and a varying Newton’s constant at the same time. Its impact on black hole physics has been investigated in detail SD1 ; SD0 ; SD2 ; SD3 ; SD4 ; SD5 ; SD6 ; SD7 ; SD8 , while recently some astrophysical and cosmological implications have been studied as well astro ; cosmo1 ; cosmo2 . Recently, in cosmo2 the authors considered current cosmic acceleration within scale-dependent gravity, analyzing in particular the evolution of the Universe at the background level. In the present work we propose to study for the first time the evolution of linear matter perturbations, and in particular the predictions for growth rate from red-shift space distortions. Since scale-dependent gravity applied to cosmological contexts implies a varying Newton´s constant, $G_{\textrm{eff}}$, which is lower than ordinary Newton’s constant, $G$, at red-shift $z\sim(1-2)$, it would be both natural and interesting to investigate the predictions of scale dependent gravity on the $A$ parameter, see the discussion below in section 4.

The paper is organized as follows: In the next section we briefly describe the formalism in the framework of scale-dependent gravity. It is then applied to Cosmology in section 3, where we present the cosmological equations for classical and scale-dependent background evolution. In the fourth section, where linear cosmological perturbation theory is discussed, we study the evolution of density perturbations for pressure-less matter, and we compute the combination $A=f\sigma_{8}$. Finally, we finish with some concluding remarks in section 5.

The goal of this section is to briefly review the key concepts and ingredients of scale-dependent gravity (see e.g. SD0 ; SD8 and references therein). Scale-dependent (SD) gravity is one of the current alternative scenarios in which the couplings of the underlying theory are allowed to vary. The motivation for studying alternatives scenarios is, roughly speaking, to address a few shortcomings of the $\Lambda$CDM model. In particular, the $H_{0}$ tension as well as the cosmological constant problem maybe alleviated when the couplings of the theory vary with time. Among other formalisms, scale-dependent gravity is one where a soft entanglement between classical and quantum is established. To be more precise, the technical motivation of this approach relies on the idea of quantum gravity Rovelli:2007uwt . Exact Renormalization Groups (ERG) is one of the now many alternatives to quantum gravity, and it serves as an inspiration. Currently, we do not have a favorite approach to combine classical gravity to quantum mechanics and, therefore, a ”consistent and predictive” parameterization of quantum gravity is still missing. Scale-dependent gravity takes advantage of certain ideas of ERG to consistently account for quantum effects. Thus, considering an average effective action with running couplings, a successful theory can be achieved. Also, it is remarkable that the ERG and the SD scenario allow us to obtain the equations for running couplings of an average effective action exactly. Thus, the latter approaches do not need additional expansion of the couplings in powers of some small control parameter.

Scale-dependent gravity is able to extend classical GR solutions via the inclusion of scale-dependent couplings, which account for quantum features. Notice that the corrections are assumed to be small. The idea can also be shown considering the truncated average effective Einstein-Hilbert action with a non-vanishing cosmological constant. Thus, in the absence of matter content, there are two ”running” couplings in the theory, namely: i) Newton’s constant $G_{k}$ and ii) the cosmological constant $\Lambda_{k}$. For convenience, we can define the parameter $\kappa_{k}\equiv 8\pi G_{k}$ as the Einstein’s constant. Besides, apart from the the metric tensor, $g_{\mu\nu}$, an arbitrary renormalization scale $k$ must be included.

The effective action $\Gamma[g_{\mu\nu},k]$ is then written as SD0 ; SD8

where $\mathcal{L}_{M}$ is the Lagrangian density of the matter fields, $g$ is the determinant of the metric tensor $g_{\mu\nu}$, $\Lambda_{k}$ is the scale-dependent cosmological constant (CC), $\mathcal{R}$ is the corresponding Ricci scalar and $\kappa_{k}$ is the scale-dependent gravitational coupling. To obtain the field equations we vary (the average effective action) with respect to the metric tensor. The latter gives the effective Einstein’s field equations, which can be written down as:

where the effective energy-momentum tensor is computed to be

Notice that the effective energy-momentum tensor includes two parts: i) the usual matter content and ii) the non-matter source (provided by the running of the gravitational coupling), which is given by SD9 :

In the present work we propose to investigate the impact of scale-dependent gravity on linear cosmological perturbations including a non-vanishing cosmological constant.

The second step to obtain the full set of equations consists of taking the variation of the average effective action with respect to the additional field $k(x)$. Doing so, we obtain a supplementary equation to close the set of equations. Thus, such a condition can be written down as

The latter is taken as an aposteriori condition towards background independence Stevenson:1981vj ; Reuter:2003ca ; Becker:2014qya ; Dietz:2015owa ; Labus:2016lkh ; Morris:2016spn ; Ohta:2017dsq . Also, using (5) we are able to link $G_{k}$ with $\Lambda_{k}$. The system is now closed after the inclusion of the aforementioned equation. The auxiliary scalar field $k(x)$ is a real physical scale, identified with the momentum as a function of any space-time point $x^{\mu}$. The way in which the theory depends on $k$ is accounted for via the running of the coupling constants. The dependence of $k$ on the physical coordinates is not unique. Thus, an alternative choice to circumvent such a problem could be the following: First we recognize that $\mathcal{O}(k(x))\rightarrow\mathcal{O}(x)$, and then we can forget about the concrete relation between $k$ and the spatial coordinates.

Finally, it is instructive to include one additional equation Koch:2010nn

which although is not used in practice, it serves as a consistency relation that shows that a running cosmological constant implies a running Newton’s coupling and vice-versa, a fact that has been previously reported in cosmo1 ; Cai:2011kd .

In this section we briefly present the cosmological equations for the background, first the classical one and then its scale-dependent counterpart.

Let us briefly review linear cosmological perturbation theory, see e.g. Mukhanov:2005sc ; other .

The goal is to solve the perturbed Einstein’s field equations

On the one hand, for scalar perturbations relevant in growth of structures, the metric tensor has the form

where $\Psi$ is the metric perturbation, while for the cosmological fluid the perturbed stress-energy tensor has the form

The full set of coupled equations for matter and metric perturbations may be found e.g. in mariam ; pano1 ; pano2 .

The Fourier transform of the density contrast, $\delta_{k}=\delta\rho_{m}/\rho_{m}$, with $k$ being the wave number, for pressure-less matter satisfies the following linear differential equation rogerio ; leandros2 ; review2

at linear level, $\delta_{k}\ll 1$, and for sub-horizon scales, $k/(2\pi a)\gg aH$, when only non-relativistic matter clusters. During matter domination

the matter density contrast grows linearly with the scale factor, $\delta_{k}(a)\sim a$.

The equation for $\delta$ may be take equivalently the following form

where for simplicity we drop the sub-index $k$, a prime denotes differentiation with respect to the scale factor, while the dimensionless Hubble rate corresponding to $\Lambda$CDM can be written down as follows

neglecting radiation at low red-shift, $1+z=a_{0}/a$, of order one or so.

In theories with a varying effective Newton’s constant, $G_{\textrm{eff}}$, the equation for $\delta$ remains the same, the only difference being that $G$ is replaced by $G_{\textrm{eff}}$ L3 ; L5 . Thus, in scale-dependent gravity the new equation to be solved is the following proDodelson ; Dodelson ; L4 ; L2

where $E=H/H_{0}$, as before, but now the Hubble parameter is the solution of the cosmological equation of scale-dependent gravity. Since the cosmological constant does not cluster, we need to solve a single fluid differential equation for the evolution of matter density contrast, and therefore $\Lambda(a)$ does not directly affects $\delta(a)$. Notice, however, that a varying cosmological constant affects the expansion of the Universe through the generalized Friedmann equations, and therefore it influences the evolution of $\delta(a)$ indirectly via $H(a)$.

In Fig. (3) we show the density contrast of non-relativistic matter versus the scale factor both for $\Lambda$CDM (black curves) and for scale-dependent gravity (red curves). The left panel corresponds to the Gold data set, while the right panel corresponds to the Union2 compilation. At early times, when matter dominates, all curves grow linearly with $a$ and they coincide, as expected. At late times, when dark energy starts to dominate, we observe that i) the $a$ dependence ceases to be linear, and ii) the curves split exhibiting different trends, i.e. in the case of Gold data set the density contrast for scale-dependent gravity lies above the one for $\Lambda$CDM, whereas for the Union2 compilation the opposite holds.

To make contact with observations, although there are several probes for dark energy, see e.g. the recent reviews review1 ; review2 , we shall consider here redshift-space distortions, and therefore we shall compute the combination L5 ; L1

where the first function, $f(a)$, is defined to be

while the second function, $\sigma_{8}(a)$, is given by

with $\sigma_{8}\equiv\sigma_{8}(a=1)$ being the rms density fluctuation on scales of $0.125h^{-1}\>\text{Mpc}$.

The $\sigma_{8}$ values within scale-dependent gravity are obtained minimizing $\chi_{\text{SD}}^{2}$ corresponding to the $A(z)$ parameter

with $\sigma_{A}$ being the error of $A$ in the observational data taken from mariam .

The value of $\sigma_{8}$ for $\Lambda$CDM extracted by the Planck Collaboration is found to be $\sigma_{8}=0.831$ planck1 , whereas upon comparison with the data used here we have obtained the following numerical values

which are found to be lower than the Planck value.

In Fig.  (4) we show the combination $A=f\sigma_{8}$ versus red-shift both for $\Lambda$CDM (black curves) and for scale-dependent gravity (red curves). We also show the observational points, taken from mariam . As $\sigma_{8}$ was found to be lower in the case of the Gold data set, the deviation from the $\Lambda$CDM model is larger in this case.

Before we conclude our work, a couple of final comments are in order. Notice that we have not included more recent data $A(z)$, althought they do exist in the literature, since they are shifted towards higher red-shift, while it turns out that lower red-shift data are more adequate to probe $G$ variations corresponding to a weaker gravity at $z\sim 1$, as pointed out in L4 .

Finally, we have noticed that our findings are similar to the results reported in another closely related approach; the running vacuum scenario. In Gomez-Valent:2018nib , for instance, the authors analyzing a running vacuum model computed the combination $A=f\sigma_{8}$, and they showed that in such a scenario the value of $f\sigma_{8}$ was slightly lower than its classical counterpart, in agreement with our results.

To summarize our work, in this article we have analyzed some implications of scale-dependent gravity to Cosmology. In particular, we have studied the evolution of the density contrast for non-relativistic matter within linear cosmological perturbation theory taking into account a varying Newton’s constant. First, we solved the background cosmological equations using the first Friedmann equation as a constraint, which allowed us to impose the initial conditions properly and obtain self-consistent solutions. An effective gravitational coupling consistent with Primordial Big-Bang Nucleosynthesis was obtained. After that, we solved the generalized version of the equation for matter density contrast on top of a flat FLRW Universe, where Newton’s constant was replaced by a varying effective gravitational coupling. Finally, with an eye on redshift-space distortion data and the $\Lambda$CDM crisis, the combination $A=f\sigma_{8}$ was computed. Next, it was contrasted against available observational data, and the numerical value of $\sigma_{8}$ that best fits the data was obtained. Our findings show that it is computed to be lower than that of $\Lambda$CDM extracted from the Planck Collaboration.

We are grateful to the anonymous reviewer for a careful reading of the manuscript as well as for numerous useful comments and suggestions, which helped us significantly improve the quality of our work. The author G. P. thanks the Fundação para a Ciência e Tecnologia (FCT), Portugal, for the financial support to the Center for Astrophysics and Gravitation-CENTRA, Instituto Superior Técnico, Universidade de Lisboa, through the Project No. UIDB/00099/2020 and grant No. PTDC/FIS-AST/28920/2017. The author A. R. acknowledges Universidad de Tarapacá for financial support.