Holographic energy density dark energy sound speed and tensions in cosmological parameters H0andS8

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Holographic energy density, dark

energy sound speed, and tensions in

cosmological parameters: H0and S8

Wilmar Cardona,a,1M. A. Sabogal,b

aICTP South American Institute for Fundamental Research & Instituto de F´ısica Te´orica,

Universidade Estadual Paulista, 01140-070, S˜ao Paulo, Brazil

bPrograma de F´ısica, Universidad del Atl´antico, Carrera 30 N´umero 8-49, Puerto Colombia-

Atl´antico, Colombia

E-mail: wilmar.cardona@unesp.br,msabogal@est.uniatlantico.edu.co

Abstract. Interesting discrepancies in cosmological parameters are challenging the success

of the ΛCDM model. Direct measurements of the Hubble constant H0using Cepheid variables

and supernovae turn out to be higher than inferred from the Cosmic Microwave Background

(CMB). Weak galaxy lensing surveys consistently report values of the strength of matter

clustering σ8lower than values derived from the CMB in the context of ΛCDM. In this

paper we address these discrepancies in cosmological parameters by considering Dark Energy

(DE) as a ﬂuid with evolving equation of state wde(z), constant sound speed squared ˆc2

s, and

vanishing anisotropic stress σ. Our wde(z) is derived from the Holographic Principle and

can consecutively exhibit radiation-like, matter-like, and DE-like behaviour, thus aﬀecting

the sound horizon and the comoving angular diameter distance, hence H0. Here we show

DE sound speed plays a part in the matter clustering behaviour through its eﬀect on the

evolution of the gravitational potential. We compute cosmological constraints using several

data set combinations including primary CMB, CMB lensing, redshift-space-distortions, local

distance-ladder, supernovae, and baryon acoustic oscillations. In our analysis we marginalise

over ˆc2

sand ﬁnd ˆc2

s= 1 is excluded at &3σ. For our baseline result including the whole

data set we found H0and σ8in good agreement (within ≈2σ) with low redshift probes. Our

constraint for the baryon energy density ωbis however in ≈3σtension with BBN constraints.

We conclude evolving DE also having non-standard clustering properties [e.g., ˆc2

s(z, k)] might

be relevant for the solution of current discrepancies in cosmological parameters.

1Corresponding author.

arXiv:2210.13335v3 [astro-ph.CO] 22 Feb 2023

Contents

1 Introduction 1

2 Theoretical framework and holographic dark energy model 2

2.1 Background 3

2.2 First order perturbations 6

2.2.1 Matter dominance 7

2.2.2 Implementation in Boltzmann solver 9

3 Data and Methodology 11

4 Results and discussion 13

5 Conclusions 21

1 Introduction

While the concordance model provides a reasonable, good phenomenological description of

most astrophysical measurements [1–4], it also becomes clear that our ignorance about the

nature of Dark Matter as well as the so-called cosmological constant problem represent major

drawbacks in the model. In addition, over the past years we have seen the emergence of pretty

interesting discrepancies (e.g., the Hubble constant H0, the strenght of matter clustering σ8)

in cosmological parameters whose understanding could reveal new physics disregarded in the

standard cosmological model [5–25].

Although Bayesian analyses show that the standard cosmological model ΛCDM per-

forms better than its simplest alternatives [26], there exists the possibility that more elabo-

rate models could explain the shortcomings ΛCDM is facing. Dynamical Dark Energy and

Modiﬁed Gravity (MG) have become the two leading approaches when trying to explain the

late-time accelerating universe [27–30]. There is however no conclusive evidence for new Dark

Energy (DE) ﬁelds or deviations from General Relativity [31–33].

Within the wide spectrum of proposals to address the DE problem, there is a hypothesis

known as the Holographic Principle (HP). Roughly speaking, the HP asserts that everything

inside a region of space can be described by bits of information conﬁned to the boundary

[34–39]. This non-extensive scaling would suggest that quantum ﬁeld theory ceases to be

valid in a large volume. Nevertheless, it is also true that the performance of local quantum

ﬁeld theory at describing particle phenomenology is quite remarkable. It turns out that a

relationship between ultraviolet (UV) and infrared (IR) cut-oﬀs of an eﬀective quantum ﬁeld

theory could make these regimes compatible with each other [40]. If ρis the quantum zero-

point energy density associated to a UV cut-oﬀ, the total energy in a region of size Lshould

not exceed the mass of a black hole of the same size, namely,

L3ρ≤LM2

p,(1.1)

where Mpis the reduced Planck mass. The largest, allowed IR cut-oﬀ LIR saturates the

inequality (1.1) so that the maximum energy density in the eﬀective theory is given by

ρ= 3γ2M2

pL−2

IR ,(1.2)

– 1 –

where γis an arbitrary parameter. The UV/IR relationship (1.2) is a consequence of recog-

nising that quantum ﬁeld theory overestimates states. Moreover, it oﬀers a possible way of

understanding the cosmological constant problem [41–43], one of the main shortcomings of

the standard cosmological model ΛCDM.

Interestingly, the UV/IR relation (1.2) has been widely applied in cosmology as an al-

ternative to the cosmological constant causing the late-time accelerating expansion in the

concordance model. These kinds of cosmological models are now known as Holographic Dark

Energy (HDE) models (see [44] for a review). In this context, the IR cut-oﬀ LIR has a cosmo-

logical origin and various choices are found in the literature [45–51]. Despite being appealing

as an alternative to ΛCDM, the HDE models investigated here are not derived from a La-

grangian which is a disadvantage when studying the evolution of cosmological perturbations:

since HDE models do not have a Lagrangian, we cannot derive equations of motion for linear

order perturbations.1Nevertheless, fairly general theories relying on scalar and vector ﬁelds

(e.g., scalar-vector-tensor theories [53]) could provide background phenomenology matching

HDE models while allowing the investigation of cosmological perturbations. Here we will

adopt a phenomenological approach and assume the existence of a DE ﬂuid having an evolv-

ing equation of state wde(a) derived from the UV/IR relation (1.2). As for the description of

DE perturbations, we opt for a constant sound speed in the ﬂuid rest-frame ˆc2

sand vanishing

anisotropic stress π= 0.

In this work we want to determine whether or not HDE is viable given current astro-

physical measurements. Although cosmological constraints have been computed for HDE

models (see, for instance, [54–68]), a few details have been overlooked. Firstly, while HDE

models usually feature an evolving wde(a) which might cross the phantom divide wde =−1,

this behaviour is not properly addressed in the literature when also considering the evolu-

tion of perturbations. Here we will take it into consideration by using the Parameterized

Post-Friedmann (PPF) formalism [69]. Secondly, when modelling DE perturbations, studies

exist which a priori set ˆc2

sto a constant value. However, this choice could bias cosmolog-

ical constraints as it directly aﬀects the clustering properties of DE. In our investigation

we marginalise over ˆc2

sand inquire about its phenomenological signatures in the context of

HDE. Thirdly, with regard to cosmological constraints of HDE models, most studies focus on

the background evolution and use only low red-shift data to constrain the parameter space

fully disregarding the impact on earlier stages of the Universe. Here we ﬁll this gap in the

literature by also studying the impact of HDE on linear order perturbations.

The manuscript is organised as follows. In Section 2we set our notation, discuss the

particular HDE model and explain its background phenomenology as well as the behaviour

of linear order perturbations. In Sections 3-4we present and discuss results for cosmological

constraints. Finally, in Section 5we give our conclusions.

2 Theoretical framework and holographic dark energy model

The Einstein-Hilbert action reads

S=Zd4x√−gR

2κ+Lm,(2.1)

where Lmdenotes the Lagrangian for any matter ﬁelds appearing in the theory, gis the

determinant of the metric gµν ,Ris the Ricci scalar and κ≡8πG is a constant with Gbeing

1However, see Ref. [52] for a relation between HDE and massive gravity theory that could provide a

framework for investigating perturbations.

– 2 –

the bare Newton’s constant. By applying the Principle of Least Action we can derive the

well known Einstein ﬁeld equations

Rµν −1

2Rgµν =κTµν ,(2.2)

where Rµν is the Ricci tensor and Tµν is the energy-momentum tensor of matter ﬁelds.2

Since observations and simulations indicate that on large enough scales the Universe is sta-

tistically homogeneous and isotropic also having vanishing curvature [1,70–72], here we will

assume a ﬂat, linearly perturbed Friedmann-Lemaˆıtre-Robertson-Walker metric (FLRW). In

the conformal Newtonian gauge [73]

ds2=a(τ)2−(1 + 2ψ(~x, τ ))dτ 2+ (1 −2φ(~x, τ))d~x2,(2.3)

where a(τ) is the scale factor, and ψ,φdenote the gravitational potentials. As usual we will

consider the material content as described by a perfect ﬂuid with energy-momentum tensor

Tµ

ν=Pﬂdδµ

ν+ (ρﬂd +Pﬂd)Uµ,(2.4)

where ρﬂd,Pﬂd, and Uµrespectively denote the energy density, pressure, and four-velocity

vector of the ﬂuid. At ﬁrst order the four-velocity vector is given by Uµ=a(τ)−1(1 −ψ, ~u),

which satisﬁes UµUµ=−1, with ~u =˙

~x. Taking into account linear perturbations, the

elements of the energy-momentum tensor are given by

0=−(¯ρﬂd +δρﬂd),(2.5)

i= (¯ρﬂd +¯

Pﬂd)ui,(2.6)

j= ( ¯

Pﬂd +δPﬂd)δi

j+ Σi

j,(2.7)

where ¯ρﬂd,¯

Pﬂd are background quantities and only depend on time. The perturbations

δρﬂd,δPﬂd, Σi

jdepend on (~x, τ ). The anisotropic stress tensor of the ﬂuid is deﬁned as

Σi

j≡Ti

j−δi

jTk

k/3.

2.1 Background

From the time-time component of Eq. (2.2) and using the unperturbed (i.e., ψ=φ= 0)

FLRW metric (2.3), we obtain

H2=κ

3(ρr+ρm+ρde),(2.8)

where the Hubble parameter H≡1

a(t)

dt , and ρde,ρr,ρmrespectively denote DE, radiation,

and matter energy densities. While radiation and matter will be taken into account as in

the standard cosmological model ΛCDM, we will consider DE as a ﬂuid with energy density

given by (1.2). We choose the so-called GO cut-oﬀ [47]

L−2

IR ≡αH2+βdH

dt (2.9)

2Unless stated otherwise, throughout this paper we adopt the following conventions: speed of light c= 1, τ

is the conformal time, ~x denotes conformal comoving coordinates, and the metric signature is (−+ ++). For

a generic function f,df

dτ ≡˙

fand df

da ≡f0. Cosmic time tand conformal time τare related via dτ =dt/a(τ).

– 3 –

where αand βare dimensionless constants. Eqs. (1.2) and (2.9) allow us to deﬁne a HDE

density

ρde =3

καH2+βdH

dt ,(2.10)

where the constant γwas absorbed by αand β. Taking into account Eq. (2.10), we can

rewrite the Friedmann equation (2.8) as3

H2= Ωr,0H2

0a−4+ Ωm,0H2

0a−3+αH2+βa

dH2

da .(2.11)

We deﬁne E2≡H2

0and ﬁnd an analytical solution for the diﬀerential equation (2.11) given

E2(a)=Ωeﬀ

r,0a−4+ Ωeﬀ

m,0a−3+ Ωeﬀ

de,0a

−2(α−1)

β,(2.12)

where

Ωeﬀ

r,0≡1 + (α−2β)

(1 −α+ 2β)Ωr,0,(2.13)

Ωeﬀ

m,0≡1 + (2α−3β)

(2 −2α+ 3β)Ωm,0,(2.14)

Ωeﬀ

de,0≡1−2Ωm,0

(2 −2α+ 3β)−Ωr,0

(1 −α+ 2β),(2.15)

and the eﬀective parameter densities satisfy Ωeﬀ

m,0+ Ωeﬀ

r,0+ Ωeﬀ

de,0= 1. Note that the HDE

parameter density reads

Ωde =α−2β

1−α+ 2βΩr,0a−4+2α−3β

2−2α+ 3βΩm,0a−3+ Ωeﬀ

de,0a

−2(α−1)

β.(2.16)

Assuming a barotropic ﬂuid with Pde =wdeρde, from the condition for energy conservation

dρde

dt + 3Hρde (1 + wde) = 0 (2.17)

and Eqs. (2.10) and (2.12), we can derive the equation of state for our DE ﬂuid

wde(a) = 2α−3β−2

3βΩeﬀ

de,0a

−2(α−1)

β+2β−α

3α−6β−3Ωr,0a−4

2α−3β

2−2α+3βΩm,0a−3+α−2β

1−α+2βΩr,0a−4+ Ωeﬀ

de,0a

−2(α−1)

.(2.18)

Figure 1shows the evolution of parameter densities as well as the HDE equation of

state wde(a) in Eq. (2.18). It becomes clear that when α > 2βthe HDE equation of state

evolves from radiation-like [wde(a)≈1/3] to pressure-less matter-like [wde(a)≈0] until

reaching a DE-like [wde(a)<−1/3] behaviour at late times. Consequently, a non-vanishing

HDE (2.10) can eﬀectively add both pressure-less matter and radiation to the cosmological

model [see Eqs. (2.13)-(2.14)]. While for the case where α= 2βthere is no radiation-like

behaviour of HDE in the early universe, HDE contributes to the eﬀective matter parameter

density in the matter dominated epoch. Since in this work we focus on a possible explanation

3As it is usual, we deﬁne the density parameters Ωi,0≡κ

3H2

ρi,0and use d

dt =aH d

da .

– 4 –

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Holographicenergydensity,darkenergysoundspeed,andtensionsincosmologicalparameters:H0andS8WilmarCardona,a;1M.A.Sabogal,baICTPSouthAmericanInstituteforFundamentalResearch&InstitutodeFsicaTeorica,UniversidadeEstadualPaulista,01140-070,S~aoPaulo,BrazilbProgramadeFsica,UniversidaddelAtlantico,Carre...

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Holographic energy density dark energy sound speed and tensions in cosmological parameters H0andS8

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