Perturbations in the interacting vacuum Albert Munyeshyaka1Joseph Ntahompagaze2Tom Mutabazi1 Manasse.R Mbonye234 Abra- ham Ayirwanda2Fidele Twagirayezu2and Amare Abebe56

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Perturbations in the interacting vacuum

Albert Munyeshyaka1,Joseph Ntahompagaze2,Tom Mutabazi1, Manasse.R Mbonye2,3,4, Abra-

ham Ayirwanda2,Fidele Twagirayezu 2and Amare Abebe5,6

1Department of Physics, Mbarara University of Science and Technology, Mbarara, Uganda

2Department of Physics, College of Science and Technology, University of Rwanda, Rwanda

3International Center for Theoretical Physics (ICTP)-East African Institute for Fundamental

Research, University of Rwanda, Kigali, Rwanda

4Rochester Institute of Technology, NY, USA

5Centre for Space Research, North-West University, Mahikeng 2745, South Africa

6National Institute for Theoretical and Computational Sciences (NITheCS), South Africa

Correspondence:munalph@gmail.com

Abstract

In this study, we present the evolution of cosmological perturbations in a universe consisting of

standard matter and interacting vacuum. We use the 1 + 3 covariant formalism in perturba-

tion framework and consider two diﬀerent models for the interacting vacuum; namely, a linear

interacting model and interaction with creation pressure model. For both models, we derive

the evolution equations governing the growth of linear perturbations for both radiation- and

dust-dominated Universe. We ﬁnd numerical solutions in appropriate limits, namely long and

short wavelengths. For both models, the perturbations grow with time (decay with redshift),

showing that structure formation is possible in an accelerated cosmic background. The pertur-

bation amplitudes – and their relative scalings with those of ΛCDM – depend on the values of

the interaction parameters considered, and in a way that can be used to constrain the models

using existing and future large-scale structure data. In the vanishing limits of the coupling

parameters of the interaction, we show that standard ΛCDM cosmology, both background and

perturbed, is recovered.

keywords: Cosmology— dark matter— dark energy— interacting vacuum—cosmological constant—

covariant formalism— Cosmological perturbations.

PACS numbers: 04.50.Kd, 98.80.-k, 95.36.+x, 98.80.Cq; MSC numbers: 83F05, 83D05

This manuscript was accepted for publication in International Journal of Geometric Methods

in Modern Physics.

1 Introduction

Recent astronomical data show that the expansion of the Universe is accelerating [1,2]. Diﬀer-

ent hypotheses propose diﬀerent mechanisms for such a cosmic acceleration, the most prominent

proposal being a certain ﬂuid with negative pressure known as dark energy. Among the widely

explored candidates for dark energy is the cosmological constant Λ [3,4,5], which has now

led to the development of the concordance cosmological model known as ΛCDM dominated by

dark energy. The CDM part of ΛCDM represents [cold] dark matter, another sub-dominant

component of the universe which is invisible in the electromagnetic spectrum but whose eﬀects

can be detected via gravitational interactions. The cosmological constant as the energy of free

arXiv:2210.01184v1 [gr-qc] 3 Oct 2022

space (vacuum) suﬀers from the cosmological constant problem [6,7,8] and the coincidence

problem [9], serious theoretical and observational issues that cannot and should not be ignored.

Since the standard model of cosmology is based on General Relativity (GR), recent studies

tried to propose diﬀerent modiﬁed gravity theories [10].

One possible avenue to addressing the above issues related to the cosmological constant is

treating Λ as some limiting manifestation of an interacting vacuum [11,12,13,14,15]. For

instance in the work done in [14], it was found that one can produce nonsingular cosmology

under the consideration of the interacting vacuum energy, and the cosmological constant ap-

pears as a late-time limiting case of the vacuum energy. In the work done by [15], it was found

that through the consideration of the creation pressure in the continuity equation led to the

situation where the universe experiences quasi-periodic acceleration phases. On the other side,

one ﬁnds a detailed exploration of interacting vacuum energy using Planck Data [13].

Diﬀerent formalism such as metric and the 1 + 3 covariant formalisms can be used to study

cosmological perturbations. The work done in [16] investigated the matter density perturbation

and matter power spectrum in the running vacuum model using metric formalism. In addition

to that, the work done in [17] studied linear scalar perturbations using metric formalism where

they assumed the running vacuum as the sum of independent contributions associated with each

of the matter species. On the other hand, the work done in [18] studied the growth of matter

perturbations in an interacting dark energy scenario emerging from the metric-scalar-torsion

couplings using metric formalism and obtained appropriate ﬁtting formula for the growth in-

dex in terms of the coupling function and the matter density parameters. The works done in

[19,20,21,22] explored the perturbation aspects of interacting vacuum energy. In the work

done by [20,23], the consideration of decaying vacuum was done focusing on the homogeneous

interactions of both matter and dark energy. On the other hand the treatment of generalised

Chaplygin gas models as inhomogeneous interacting dark energy with matter was done in [21]

focusing on cosmic microwave background (CMB) anisotropies. In [22], the authors investi-

gated perturbations of interacting vacuum focusing on the contributions of gravitational waves

data, where they found a signiﬁcant improvement in the CMB measurements.

The present paper aims to apply the 1 + 3 covariant and gauge-invariant perturbations for-

malism to study large-scale structure formation scenarios for two interacting vacuum models

treated in the works by Bruni et al [14] and Mbonye [15]. The 1 + 3 covariant gauge-invariant

formalism is used to study cosmological perturbations for both GR and modiﬁed gravity theo-

ries such as f(R), f(T) and f(G). In the 1 + 3 covariant formalism, the perturbation variables

deﬁned describe true physical degrees of freedom and no unphysical modes exist.

The rest of this paper is organised as follows: in Sec. 2we give a covariant description and

the general linearised ﬁeld equations involving the interacting dynamical vacuum. In Sec. 3we

deﬁne the covariant perturbation variables, derive their evolution equations and analyse their

solutions. Finally in Sec. 4we discuss the results and give conclusions.

Natural units in which c= 8πG = 1 will be used throughout this paper, indices like a,b...

run from 1 to 3 and Greek indices run from 0 to 3. The symbols ∇,˜

∇and the overdot .rep-

resent the usual covariant derivative, the spatial covariant derivative, and diﬀerentiation with

respect to cosmic time, respectively. We use the (−+ ++) spacetime signature.

2 Background Field Equations

The standard GR gravitational action with a matter ﬁeld contribution to the Lagrangian, Lm,

is given by

A=1

2Zd4x√−g[R+ 2Lm].(1)

Using the variational principle of least action with respect to the metric gab, the generalised

Einstein Field Equations (EFEs) can be given in a compact form as

Gab =Tab ,(2)

with the ﬁrst (geometric) term represented by the Einstein tensor, and energy-momentum

tensor of matter ﬂuid forms given by

Tab =µuaub+phab +qaub+qbua+πab ,(3)

where µ,p,qaand πab are the energy density, isotropic pressure, heat ﬂux and anisotropic

pressure of the ﬂuid, respectively. Here ua≡dxa

dt is the 4-velocity of fundamental observers

comoving with the ﬂuid. In a multi-component ﬂuid universe ﬁlled with standard matter ﬁelds

(dust, radiation, etc) and vacuum contributions, the total energy density, isotropic pressures

and heat ﬂux are given, respectively, by ρ=ρm+ρv,p=pm+pvand qa=qa

m+qa

v, where

mand vspecify matter and vacuum respectively. The vacuum equation of state parameter w

is given by w=−1. An arbitrary energy transfer Qcan reproduce an arbitrary background

cosmology with energy density [11,24]

ρ=ρm+V , (4)

p=−V , (5)

which reduces to ΛCDM when Q= 0 and we have a constant V= Λ. The 4-vector Qµcan in

general be decomposed into parallel and orthogonal parts to the 4-velocity of the ﬂuid

Qa=Qua+qa,(6)

where qahere is due to momentum exchange between matter and vacuum. In this paper, we

will assume an homogeneous isotropic model in which Qais parallel to the matter 4-velocity,

Qa=Qua. We will consider the case where the interactions reduce to pure energy exchange

so that qa= 0 [13,20,25]. Moreover for each interacting ﬂuid, the following conservation

equations considered in [11,19,26] hold:

˙ρ+ 3(1 + ω)Hρ =−Q , (7)

V=Q . (8)

The equation of state for the standard matter (such as dust and radiation) component is

presented as p=wρ, where wis constant, and the total thermodynamic quantities p=pm+

pvand ρ=ρm+ρvwhere the subscripts mand vstand for standard matter and vacuum

contributions. We deﬁne two diﬀerent covariant choices for Qas follows: the ﬁrst one considered

by Bruni et al [14], hereafter referred to as Case 1, and the second by Mbonye [15], hereafter

referred to as Case 2:

Q1= [ξ(VΛ−V) + σρ]θ , (9)

Q2=πcθ , (10)

with πc=K[(3γ−2)ρm−2ρv]. ξ,σ,Kand γare dimensionless coupling parameters and VΛ

plays the role of an eﬀective cosmological constant, and γ= 1 + w. Note that throughout the

remainder of the paper, we refer to Case 1 when using the interaction form Q1and Case 2 when

using Q2.

Consider a spatially ﬂat FRW universe with the metric

ds2=−dt2+a(t)2(dx2+dy2+dz2),(11)

where the Friedman and the Raychaudhuri equations for ﬂat spacetime are given respectively

H2≡˙a

a2

3(ρ+V),(12)

H=−H2−1

6[(1 + 3w)ρ−2V],(13)

where θ= 3H,H≡˙a

ais the Hubble expansion rate, and a(t) is the cosmological scale factor.

We dedicate the next subsections to analyse some of the cosmological solutions (in terms of

the solutions for Hubble expansion and the deceleration parameters) to help us explain cosmic

history of the late-time background due to the presence of the interacting vacuum, compared

against the ΛCDM model.

2.1 Background expansion for Case 1

We now consider linear models [14] presented by:

ρ=E1aα1+E2aα2,(14)

V=VΛ+λ1aα1+λ2aα2,(15)

where E1,E2,λ1and λ2are energy contributions of the universe. Equation 12 can be repre-

sented as

H2=1

3(E1aα1+E2aα2+VΛ+λ1aα1+λ2aα2).(16)

We can make change of variable and use redshift transformation as the redshift is a physically

measurable quantity. Using a=1

1+z, here we assume a present value of scale factor to be 1, eq.

16 is transformed as

h1(z) = p(Ω1+ Ω3)(1 + z)−α1+ (Ω2+ Ω4)(1 + z)−α2+ ΩΛ,(17)

where Ω1=E1

3H2

0, Ω2=E2

3H2

0, Ω3=λ1

3H2

0, Ω4=λ2

3H2

0, ΩΛ=VΛ

3H2

0and h=H

H0.

The total fractional density parameter for a ﬂat universe today is given by

Ω1+ Ω2+ Ω3+ Ω4+ ΩΛ= 1.(18)

The deceleration parameter qis deﬁned as

q=−¨aa

˙a2,(19)

Figure 1: A plot of the normalised Hubble parameter, h(z) versus redshift, zfor eq. 17.

which can be re-written (in redshift space) as

q(z) = −1 + (1 + z)h0

h,(20)

where prime means derivative with respect to redshift z. The deceleration parameter in our

case is presented as

q1(z) = −1 + (Ω1+ Ω3)α1(1 + z)−α1+ (Ω2+ Ω4)α2(1 + z)−α2

2h2

1.(21)

Numerical results of normalised Hubble parameter (eq. 17) and deceleration parameter (eq.

21) are presented in Figs. 1and 2, respectively.

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PerturbationsintheinteractingvacuumAlbertMunyeshyaka1,JosephNtahompagaze2,TomMutabazi1,Manasse.RMbonye2;3;4,Abra-hamAyirwanda2,FideleTwagirayezu2andAmareAbebe5;61DepartmentofPhysics,MbararaUniversityofScienceandTechnology,Mbarara,Uganda2DepartmentofPhysics,CollegeofScienceandTechnology,UniversityofR...

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Perturbations in the interacting vacuum Albert Munyeshyaka1Joseph Ntahompagaze2Tom Mutabazi1 Manasse.R Mbonye234 Abra- ham Ayirwanda2Fidele Twagirayezu2and Amare Abebe56

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