Observational constraints on interacting vacuum energy with linear interactions

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Observational constraints on

interacting vacuum energy with

linear interactions

Chakkrit Kaeonikhom,aHooshyar Assadullahi,a,b Jascha

Schewtschenko,aDavid Wandsa

aInstitute of Cosmology and Gravitation, University of Portsmouth,

Dennis Sciama Building, Burnaby Road, Portsmouth, PO1 3FX, UK

bSchool of Mathematics and Physics, University of Portsmouth,

Lion Gate Building, Lion Terrace, Portsmouth, PO1 3HF, UK

E-mail: chakkrit.kaeonikhom@port.ac.uk,hooshyar.assadullahi@port.ac.uk,

jascha.schewtschenko@port.ac.uk,david.wands@port.ac.uk

Abstract. We explore the bounds that can be placed on interactions between cold dark

matter and vacuum energy, with equation of state w=−1, using state-of-the-art cosmo-

logical observations. We consider linear perturbations about a simple background model

where the energy transfer per Hubble time, Q/H, is a general linear function of the

dark matter density, ρc, and vacuum energy, V. We explain the parameter degeneracies

found when ﬁtting cosmic microwave background (CMB) anisotropies alone, and show

how these are broken by the addition of supernovae data, baryon acoustic oscillations

(BAO) and redshift-space distortions (RSD). In particular, care must be taken when

relating redshift-space distortions to the growth of structure in the presence of non-zero

energy transfer. Interactions in the dark sector can alleviate the tensions between low-

redshift measurements of the Hubble parameter, H0, or weak-lensing, S8, and the values

inferred from CMB data. However these tensions return when we include constraints

from supernova and BAO-RSD datasets. In the general linear interaction model we show

that, while it is possible to relax both the Hubble and weak-lensing tensions simultane-

ously, the reduction in these tensions is modest (reduced to less slightly than 4σand 2σ

respectively).

arXiv:2210.05363v2 [astro-ph.CO] 7 Feb 2023

Contents

1 Introduction 1

2 Interacting vacuum energy models 3

2.1 Covariant equations 3

2.2 FLRW cosmology 3

2.3 Model: Q=αHρc4

2.4 Model: Q=βHV 5

2.5 Model: Q=αHρc+βHV 5

3 Linear perturbations 6

3.1 Energy continuity and Euler equations 7

3.2 Einstein equations 8

3.3 Geodesic model 8

4 Cosmic microwave background anisotropies 10

4.1 Model: Q=αHρc12

4.2 Model: Q=βHV 14

5 Redshift-space distortions 17

6 Observational constraints 19

6.1 Likelihoods 19

6.2 Implementation 20

6.3 Results 23

7 Discussion 29

8 Conclusion 30

1 Introduction

There is now strong evidence that the expansion of the Universe is accelerating from

a variety of observations, since early measurements of the magnitude-redshift relation

for type-Ia Supernovae (SNe Ia) [1,2], to detailed observations of cosmic microwave

background (CMB) anisotropies made by Planck satellite [3], as well as galaxy surveys [4–

7]. Such an acceleration suggests that the Universe contains some kind of dark energy,

homogeneously distributed exerting a negative pressure, in addition to both ordinary,

baryonic matter and non-relativistic cold dark matter (CDM) which freely falls under

gravity. The simplest model for dark energy is a cosmological constant, Λ, with an

eﬀective equation of state w=−1, which is equivalent to a non-zero vacuum energy as

predicted by quantum ﬁeld theory. However, the huge discrepancy between the vacuum

– 1 –

energy inferred from observations and that expected from theoretical predictions gives

rise to the cosmological constant problem [8,9].

The standard ΛCDM cosmology can successfully describe all current observations

with suitable choices for the cosmological parameters. However the parameter values

inferred from CMB anisotropies appear to be in tension with other determinations of

the Hubble constant H0based distance-ladder measurements at lower redshifts. In par-

ticular, the SH0ES collaboration estimates H0= 73.04 ±1.04 km s−1Mpc−1based on

observations of SNe calibrated by a sample of Cepheid variable stars [10]. This gives rise

to about 4.8σstatistically signiﬁcant diﬀerence with respect to the Planck baseline CMB

constraint, given by H0= 67.36 ±0.54 km s−1Mpc−1[3] in ΛCDM. If this discrepancy

is not due to undiscovered systematic errors in either of the analyses, the only expla-

nation for this Hubble tension appears to be new physics beyond the standard ΛCDM

cosmology. At the same time there is growing evidence of another tension between the

amplitude of matter density ﬂuctuations measured by weak-lensing and other surveys of

large-scale structure, denoted by σ8or S8≡σ8(Ωm/0.3)1/2, compared with the ampli-

tude inferred from CMB anisotropies in ΛCDM. The most common approach to address

these tensions by exploring cosmologies beyond ΛCDM is to consider a dynamical form

of dark energy by introducing additional degrees of freedom in the form of a ﬁeld or ﬂuid

description, leading to an eﬀective equation of state w6= 1 [11]. However most attempts

to resolve the Hubble tension tend to exacerbate the discrepancy in S8[12].

Here we investigate an alternative, less frequently studied extension of ΛCDM where

the equation of state of dark energy remains ﬁxed to be that of a vacuum energy, w=−1,

but where we allow the vacuum to exchange energy-momentum with dark matter [13–

18]. For example, one might consider vacuum decay due to particle creation in an

expanding universe [19,20]. Cosmological probes allow us to constrain the form of the

interaction [21,22] while studying the eﬀect of the interaction on other cosmological

parameters such as H0and S8.

In this paper we consider interacting vacuum cosmologies where the energy transfer

per Hubble time, Q/H =αρc+βV , is a general linear function of the cold dark matter

density, ρc, and/or vacuum energy, V, updating and extending previous studies [23–

27]. This model generalises time-dependent vacuum models with Q=βHV [28,29], or

time-dependent dark matter models with Q=αHρc[30–34], which are closely related

to running vacuum models [18]. It also generalises the decomposed Chaplygin gas mod-

els [35–38], where Q=γHρcV/(ρc+V), and we have Q/H →γV at early times where

V/ρc→0, and Q/H →γρcat late times where ρc/V →0.

This paper is organised as follows. In Section 2we introduce the general covariant

equations for the vacuum-dark matter interaction and give analytic solutions for the

background cosmological models for speciﬁc and general linear interaction models. In

Section 3we present the dynamical equations for scalar cosmological perturbations,

focusing on the geodesic model for inhomogeneous perturbations where dark matter

follows geodesics and hence clusters while the vacuum energy remains homogeneous in

the synchronous, comoving frame [17]. The next Section 4investigates the eﬀect of

interaction parameters on the CMB power spectra and in Section 5we consider the

– 2 –

interpretation of redshift-space distortions in this model. We present constraints on the

model parameters from current observations of CMB anisotropies, type-Ia supernovae

and large-scale structure survey data in Section 6and discuss the impact on H0and S8

tensions in Section 7. We present our conclusions in Section 8.

2 Interacting vacuum energy models

2.1 Covariant equations

The energy-momentum tensor of the vacuum energy Vcan be deﬁned as proportional

to the metric tensor ˇ

Tµ

ν=−V gµ

ν(2.1)

with the energy density ρV=Vand pressure PV=−V. The equation of state parameter

thus corresponds to that of a cosmological constant, w=PV/ρV=−1. Comparing

Eq. (2.1) with the energy-momentum tensor for a perfect ﬂuid,

Tµ

ν=P gµ

ν+ (ρ+P)uµuν,(2.2)

we can see that the four-velocity for the vacuum, ˇuµ, is not deﬁned since ρV+PV= 0

and the 4-momentum is identically zero [17].

In general relativity the total energy-momentum tensor of matter plus vacuum

is covariantly conserved ∇µTµ

ν+ˇ

Tµ

ν= 0 ,however an energy-momentum exchange

between these two components is allowed. We deﬁne the energy-momentum transfer to

the vacuum energy

Qν≡ ∇µˇ

Tµ

ν=−∇νV , (2.3)

where the ﬁnal equality can be obtained directly from the deﬁnition of the vacuum

energy-momentum tensor, Eq. (2.1). Hence, from the conservation of the total energy-

momentum, we have

∇µTµ

ν=−Qν.(2.4)

Without loss of generality we can decompose the energy-momentum transfer [39,40]

into an energy transfer Qalong the matter 4-velocity, uµ, and a momentum transfer (or

force), fµorthogonal to uµ:

Qµ=Quµ+fµ.(2.5)

2.2 FLRW cosmology

In a spatially-ﬂat Friedmann-Lemaitre-Robertson-Walker (FLRW) background cosmol-

ogy with vacuum energy V, the Friedmann equation reads

H2=8πG

3(ρr+ρb+ρc+V),(2.6)

with the Hubble rate H= ˙a/a, where ais the scale factor and a dot denotes a time

derivative. The radiation and matter densities are denoted by ρiwhere the subscripts

r,band cstand for radiation, baryons and cold dark matter respectively.

– 3 –

In an FLRW background, all the matter components, including the vacuum energy,

are spatially homogeneous. Hence, the energy transfer Qonly depends on time. In this

case the FLRW symmetry required that the energy transfer four-vector is parallel to the

dark matter four-velocity, Qµ=Quµ. The continuity equations take the simple form:

V=Q ,

˙ρr+ 4Hρr= 0 ,

˙ρb+ 3Hρb= 0 ,

˙ρc+ 3Hρc=−Q ,

(2.7)

and we assume that the interaction between vacuum and matter is restricted to the dark

sector. Radiation and baryons are only gravitationally coupled to the vacuum energy,

giving the standard background solutions

ρr=ρr,0a−4, ρb=ρb,0a−3,(2.8)

In this paper, we consider the simple linear interaction model [23–27]

Q=αHρc+βHV , (2.9)

where αand βare dimensionless parameters controlling the strength of the interaction.

We will study separately the cases β= 0 and α= 0, i.e., the models Q=αHρcand

Q=βHV , before considering the general case.

2.3 Model: Q=αHρc

In this model, Eqs. (2.7) can be integrated to give

ρc=ρc,0a−(3+α),(2.10)

ρV=α

α+ 3 1−a−(3+α)ρc,0+V0,(2.11)

where ρc,0and V0are the present energy densities of cold dark matter and vacuum energy

respectively. The Friedmann equation, Eq. (2.6), can then be rewritten in terms of the

present-day dimensionless density parameters

Ωi≡8πGρi

3H2,(2.12)

H2(z) = H2

0Ωr,0(z+ 1)4+ Ωb,0(z+ 1)3+α+ 3 (z+ 1)α+3

α+ 3 Ωc,0+ ΩV,0.(2.13)

where the redshift z=a−1−1. Note that the total matter density is given by Ωm,0=

Ωb,0+ Ωc,0and, by construction, ΩV,0= 1 −Ωr,0−Ωm,0.

When exploring observational constraints on the expansion history, H(z), and the

dependence on diﬀerent cosmological parameters it will be helpful to eliminate ΩV,0and

– 4 –

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ObservationalconstraintsoninteractingvacuumenergywithlinearinteractionsChakkritKaeonikhom,aHooshyarAssadullahi,a;bJaschaSchewtschenko,aDavidWandsaaInstituteofCosmologyandGravitation,UniversityofPortsmouth,DennisSciamaBuilding,BurnabyRoad,Portsmouth,PO13FX,UKbSchoolofMathematicsandPhysics,Universityo...

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