Observational constraints on scale-dependent cosmology Pedro D. Alvarez1 2Benjamin Koch3 4Cristobal Laporte5andAngel Rinc on6 1Centro de Estudios Cient ıficos CECs Arturo Prat 514 Valdivia Chile

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Observational constraints on scale-dependent cosmology

Pedro D. Alvarez,1, 2, ∗Benjamin Koch,3, 4, †Cristobal Laporte,5, ‡and ´

Angel Rinc´on6, §

1Centro de Estudios Cient´ıﬁcos (CECs), Arturo Prat 514, Valdivia, Chile

2Universidad San Sebasti´an, General Lagos 1163, Valdivia, Chile

3Institut f¨ur Theoretische Physik, Technische Universit¨at Wien, Wiedner Hauptstrasse 8-10,

A-1040 Vienna, Austria and Atominstitut, Stadionalle 2, A-1020 Vienna, Austria

4Instituto de F´ısica, Pontiﬁcia Cat´olica Universidad de Chile, Av. Vicu˜na Mackenna 4860, Santiago, Chile

5Institute for Mathematics, Astrophysics and Particle Physics (IMAPP),

Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen,The Netherlands

6Departamento de Fsica Aplicada, Universidad de Alicante,

Campus de San Vicente del Raspeig, E-03690 Alicante, Spain

This paper examines a cosmological model of scale-dependent gravity. The gravitational action

is taken to be the Einstein-Hilbert term supplemented with a cosmological constant, where the

couplings, Gkand Λk, run with the energy scale k. Also, notice that, by construction, our formalism

recovers general relativity when in the limit of constant Newton’s coupling. Two sub-models based

on the scale-dependent cosmological model are confronted with recent observational data from: i)

the Hubble parameter H(z), ii) distance modulus µ(z), and iii) baryon acoustic scale evolution as

functions of redshift (BAO). The viability of the model is discussed, obtaining the best-ﬁt parameters

and the maximum likelihood contours for these observables. Finally, a joint analysis is performed

for H(z)+µ(z)+BAO.

∗pedro.alvarez@uss.cl

†benjamin.koch@tuwien.ac.at

‡cristobal.laportemunoz@ru.nl

§angel.rincon@ua.es

arXiv:2210.11853v2 [gr-qc] 28 May 2024

CONTENTS

I. Introduction 3

II. Framework: the scale-dependent formalism 3

A. Scale-dependent gravity with a null energy condition 4

B. The scale-dependent cosmological model 5

C. Why this model? 6

D. Quantum gravity? 7

III. Observational constraints 7

A. Constraints from H(z) data 8

B. Constraints from µ(z) data 9

C. Constraints from BAO data 10

D. Joint analysis 11

IV. Discussion and conclusion 12

Acknowlegements 12

References 12

I. INTRODUCTION

Current observational evidence is compatible with a spatially ﬂat Universe dominated by dark matter and dark

energy [1]. The standard cosmological model (ΛCDM hereafter) is good enough to give highly accurate predictions

of the features of the Cosmic Microwave Background (CMB), primordial abundances of light elements, and large-

scale structures. Although we have made signiﬁcant progress, the nature of the dark sector remains an open question.

ΛCDM is based on General Relativity (GR), a theory consistent with a wealth of observational data from astrophysics

and cosmology. However, GR cannot be the ultimate gravity theory since some problems naturally emerge. To mention

a few, the theory has the several issues: i) the existence of singularities (suggesting that GR is still incomplete), ii) the

non-renormalizability, and iii) the ﬁne-tuning problem (i.e., the issue in which fundamental constants seem to require

precise values of parameters to be compatible with the observable universe).

Apart from the issues of GR mentioned above, there are a couple of tensions within the ΛCDM model itself that

have been made clear by the high precision measurements of recent years. The ﬁrst one to be mentioned is the weak

gravitational lensing at low redshifts (see, for instance [2,3]) and the second one the tension between measurements

of the Hubble constant H0(deﬁned as H(t)≡˙a/a at the present-day (z= 0)) (see, for instance [4–9] and references

therein). To be more precise, it is important to clarify that the Hubble tension arises not from a contrast between high-

redshift and low-redshift data, but rather from discrepancies between predictions of H0within the ΛCDM model based

on high-redshift data and measurements derived from a model-independent approach utilizing low-redshift data. The

value of the Hubble constant H0[km s−1Mpc−1] = 67.3±1.1 (at 68%CL, Planck TT,TE,EE+lowE+lensing+BAO)

[10] is now at ∼5σtension with the most recent result found by local measurements, H0[km s−1Mpc−1] = 73.1±1.1

(at 68 CL) [11].

In this paper we focus on the following hypothesis: perhaps the assumption of time independence of the Newton

constant and the vacuum energy density is no longer valid at cosmological scales and therefore an appropriate de-

scription can be given in terms of an eﬀective action. This could mean that an improvement to the ΛCDM model

could be made by including the scale dependence of the coupling constants.

Many other approaches are used to study the nature of dark energy. A phenomenological description of dark energy

could use a perfect ﬂuid characterized by a time-varying equation of state (EOS) denoted w(a), where arepresents

the scale factor. However, many other proposals based on fundamental principles do exist and can be classiﬁed

into two broad categories: i) Dynamical dark energy models, where a new dynamical ﬁeld is introduced to induce

the acceleration of the universe [12], and ii) Geometric dark energy models, where a new gravitational theory is

postulated to modify Einstein’s general relativity on cosmic scales [13,14]. We refer to [15,16] for comprehensive

reviews. Recently, the study of cosmological models in which the cosmological constant is assumed to be a function of

a certain energy scale has attracted attention. The reason is that a time-dependent vacuum energy makes possible to

alleviate the tension in measurements of H0[17,18] (see also e.g. [19–22] and references therein). The cosmological

equations in the most popular Λ-varying scenarios are generalizations of the Friedmann equations, and they oﬀer a

richer phenomenology compared to the ΛCDM model [23]. There are also some works on cosmological models with a

variable Newton’s constant, see e.g. [24–26].

The present work focuses on a scale-dependent (SD) gravity model that, when used for cosmology, implies generalized

Friedmann equations with time-dependent quantities. The paper is organized as follows: Section II introduces the

framework of the SD gravity model used in this paper. In subsection II A we discuss the Null Energy Condition (NEC)

to complement the gravitational equations, while subsection II B provides the fundamental dynamical equations of

cosmological SD model. Then, in section III, we performed a likelihood analysis using H(z) data (subsection III A),

µ(z) data (subsection III B), BAO/CMB data (subsection III C), and a joint likelihood analysis (subsection III D). To

ﬁnish the paper, we summarize our main ﬁndings, and provide a short discussion in section IV.

II. FRAMEWORK: THE SCALE-DEPENDENT FORMALISM

The eﬀective action, Γ[φ, k], where φrepresents the relevant ﬁelds in the theory and kthe renormalization energy

scale, is a powerful tool commonly used in quantum ﬁeld theory. The eﬀective action encompasses quantum features

absent in its classical counterpart that are relevant for an accurate description of the phenomena. Moreover, the

eﬀective action has an extensive range of applications in fundamental particle physics, condensed matter physics,

quantum cosmology, and quantum gravity. For a comprehensive review see [27]. In this formalism the coupling

constants of the classical theory become SD quantities i.e.,

{A0, B0, C0,··· , Z0} −→ {Ak, Bk, Ck,··· , Zk},(1)

where the left-hand side corresponds to the classical set and the right-hand side corresponds to the quantum-corrected

SD set of couplings.

Inspired by ideas of eﬀective ﬁeld theory, eﬀective models of gravity have been proposed. The implementation of

eﬀective ﬁeld theory for gravity or in the presence of gravity, is a complicated subject and, therefore, diﬀerent methods

and models have been proposed (see [28–38] and references therein). We will study the eﬀective action

Γ[gµν , k] = Zd4x√−g"1

16πGkR−2Λk+LM#,(2)

where Λkand Gkare the scale-dependent cosmological and Newton couplings, respectively. A great advantage of

working with eﬀective quantum actions like (2) is that they are capable of incorporating some eﬀects of quantum

ﬂuctuations at the level of equations of motion. To obtain background solutions for this eﬀective action one has to

derive the corresponding equation of motion. Varying the eﬀective action with respect to the inverse metric ﬁeld, one

obtains the corresponding Einstein ﬁeld equations [35]

Gµν =Teﬀec

µν =−Λkgµν −∆tµν ,(3)

where

∆tµν =Gkhgµν ∇α∇α− ∇µ∇νiG−1

k.(4)

To get physical information out of those equations one has to set the renormalization scale in terms of the physical

variables of the system under consideration k→k(x, . . . ). This process is called “scale-setting”. The connection

between kand the system’s physical variables is not uniquely deﬁned. Thus, by imposing a particular relation

between kand {t, r, θ, ϕ}, one makes a choice that aﬀects the physical observables and their interpretation of this

system. Thus, if one wishes to bypass such a disadvantage, one then should close the system following an alternative

way. This can, for example, be achieved by taking variations with respect to the renormalization scale, i.e.,

dΓ[gµν , k]

dkk=kopt

= 0.(5)

The solution of the above equation is called optimal scale k=kopt, since small variations in this scale do, by

construction not alter the value of the eﬀective action Γ. Albeit possible, the implementation of such an equation

turns out to be cumbersome. Finally, in order to close the system of equations in a consistent way, one can use the

contraction Teﬀec

µν ℓµℓν= 0, where ℓµis a null vector. This condition can be understood as an eﬀective null energy

condition, in the sense of the eﬀective action. In the following three sub-sections we will elaborate on the motivation,

implementation, advantages and limitations of this condition. Independent of which scale-setting condition is chosen,

the symmetry of the physical system under consideration will also be applied for the scale-setting. E.g. in cosmology

with a homogeneous space-time, the dynamical variable is time, which implies that k=k(t), G =G(t),Λ = Λ(t), . . . .

A. Scale-dependent gravity with a null energy condition

Energy conditions play an essential role in many applications of general relativity, from cosmology to black-hole

physics, and their importance in formulating singularity theorems. These energy conditions consist of restrictions on

the stress-energy tensor. Their purpose is three-fold. Firstly, energy conditions allow us to get a sense of “normal

matter” since they capture standard features of a diﬀerent kind of matter. Secondly, all the properties of matter ﬁelds

are contained in the Einstein’s equations and many of their modiﬁcations (including the present work) through the

stress-energy tensor. Therefore, one can analyze the resulting dynamical system without recurring to the complex

behavior of the ﬁeld content of the theory. Thirdly, energy conditions enable a conceptual simpliﬁcation for bypassing

complicated computation, as shown in the singularity theorems [39,40]

This last point is one of the strongest criticisms about the range of validity of the energy conditions. Pointwise

energy conditions on the stress-energy tensor are generally considered as over-simpliﬁcation that are not able to

capture all the features of the systems under scrutiny. An example is their application to quantum ﬁelds, where the

violation of all pointwise energy conditions motivates the introduction of the quantum energy inequalities [41,42],

which allows a ﬁnite, possibly negative, lower bound. An intermediate step consists in the averaged energy conditions

that average the components of the stress-energy tensor along suitable causal curves while preserving lower bounds

to zero. It is noteworthy to remark that the validity of the energy conditions strongly depends on the contributions

to the total stress-energy tensor.

SD gravity with a pointwise NEC has been used in: i) cosmology [43–47], ii) relativistic stars [48,49] and iii)

black holes [50–54]. In this manuscript, we will continue the ideas presented in [43–45], with the inclusion of the

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Observationalconstraintsonscale-dependentcosmologyPedroD.Alvarez,1,2,∗BenjaminKoch,3,4,†CristobalLaporte,5,‡and´AngelRinc´on6,§1CentrodeEstudiosCient´ıficos(CECs),ArturoPrat514,Valdivia,Chile2UniversidadSanSebasti´an,GeneralLagos1163,Valdivia,Chile3Institutf¨urTheoretischePhysik,TechnischeUniversit¨...

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Observational constraints on scale-dependent cosmology Pedro D. Alvarez1 2Benjamin Koch3 4Cristobal Laporte5andAngel Rinc on6 1Centro de Estudios Cient ıficos CECs Arturo Prat 514 Valdivia Chile

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