# Testing the growth rate in homogeneous and inhomogeneous interacting vacuum
models

H. A. Borges  C. Pigozzo  P. Hepp  L. O. Baraúna  M. Benetti

###### Abstract

In this work we consider a class of interacting vacuum corresponding to a
generalised Chaplygin gas (gCg) cosmology. In particular we analyse two
different scenarios at perturbation level for the same background interaction
characterised by the parameter $\alpha$: (i) matter that follows geodesics,
corresponding to homogeneous vacuum, and (ii) a covariant ansatz for vacuum
density perturbations. In the latter case, we show that the vacuum
perturbations are very tiny as compared to matter perturbations on sub-horizon
scales. In spite of that, depending on the value of the Chaplygin gas
parameter $\alpha$, vacuum perturbations suppress or enhance the matter growth
rate as compared to the case (i). We use Cosmic Microwave Background (CMB),
type Ia supernovae (SNe) and Redshift Space Distortion (RSD) measurements to
test the observational viability of the model. We found that the mean value of
our joint analysis clearly favours a positive interaction, i.e., an energy
flux from dark matter to dark energy, with $\alpha\approx 0.143$ in both
cases, while the cosmological standard model, recovered for $\alpha$=0, is
ruled out by 3$\sigma$ confidence level. Noteworthy, the positive value of
interaction can alleviate both the $H_{0}$ and $S_{8}$ tension for the dataset
considered here.

##  1 Introduction

The accelerated expansion of the universe detected by the precision
measurements of type Ia supernovae [1, 2, 3], anisotropies in the cosmic
microwave background radiation [4, 5, 6, 7] and baryon acoustic oscillations
[8, 9] indicates that about $96\%$ of the total energy of our universe is in
the form of unknown dark fluid components, namely, dark energy [10] and dark
matter, and the remaining components are in the form of baryonic matter and
radiation. The clustering dark matter with zero pressure (cold dark matter) is
concentrated in the local structures and plays crucial role for forming
galaxies and clusters of galaxies while dark energy is featured by a negative
pressure that drives the recent accelerated expansion. The simplest model to
describe dark energy is a constant vacuum energy density $\rho_{V}$
characterised by the equation of state parameter $w=-1$, equivalent to a
cosmological constant in Einstein gravity [11, 12, 13]. The cosmological model
that incorporates a constant vacuum energy plus cold dark matter is known as
$\Lambda$CDM. Despite its relative success when tested against the most
precise observations, one of its problems is the large discrepancy between the
current vacuum energy density, $\rho_{V}\sim 10^{-29}g/cm^{3}$, and the
theoretical value predicted by quantum field theories [13].

Among of many alternatives to circumvent the problem of the cosmological
constant we can consider the generalised Chaplygin gas (gCg), a unified dark
sector whose equation of state is given by [14, 15, 16, 17, 18]

$p=-\frac{A}{\rho^{\alpha}},$ (1.1)

where A is a constant, $\rho$ is the energy density and $\alpha$ is a free
parameter. This gCg interpolates between a cold dark matter dominating at
early times and a dark energy component dominating at late times. However, due
to a non-zero adiabatic sound speed, the perturbations exhibit strong
instabilities and oscillations affecting radically the matter power spectrum,
unless the parameter $\alpha$ does not differ too much from zero. An
alternative to avoid this problem is to include a non-adiabatic pressure
contribution in order to make the effective sound speed vanish [22]. Another
possibility is to split the gCg fluid into two interacting components, a
pressuless cold dark matter with energy density $\rho_{m}$ and a vacuum term
with equation of state $p_{V}=-\rho_{V}$ [23, 24, 25]. In this way, the sound
speed comes only from the vacuum energy perturbations, that need to be zero or
negligible as compared to matter perturbations.

The above condition is reached if we assume that dark matter follows
geodesics, with energy transfer proportional to its $4$-velocity, which
implies that energy and momentum exchange between vacuum and CDM is absent at
perturbative level [26, 27]. However, due to the dynamical nature of vacuum,
it is important to consider the possibility of inhomogeneities in its energy
density and verify explicitly if their fluctuations are in fact negligible at
sub-horizon scales. This was done in a model equivalent to the generalised
Chaplygin gas with $\alpha=$-1/2 [28], but here we extend the analysis for any
value of $\alpha$.

Since we are interested to explore the impact of vacuum energy fluctuations on
the evolution of the dark matter growth rate in a decomposed gCg model with
interacting dark matter and vacuum energy, for the sake of simplicity, we
neglect the contributions of the baryonic component at first moment. Under the
assumption of a covariant ansatz for the vacuum energy density, we are able to
compute the energy-momentum transfer up to first order using a gauge invariant
perturbative approach. A scale-dependent second order differential equation
for the matter density contrast is obtained, which allows us to follow the
evolution of the linear dark matter growth rate, defined by

$f=\frac{\dot{\delta}_{m}}{H\delta_{m}}.$ (1.2)

We show explicitly that the vacuum perturbations are smaller than the matter
fluctuations by several orders of magnitude on sub-horizon scales. However,
due to the corrections in the second order differential equation for the
matter density contrast we show that the growth rate evolution at late time is
not the same evolution as found in the geodesic scenario.

Since the observed matter power spectrum $P(k)$ and the redshift space
distortion (RSD) is a measure of the galaxy distribution, the conserved
baryons will be included in the perturbative dynamics.

In light of observations, previous analysis considering Cosmic Microwave
Background measurements (CMB) from Planck 2015 with the SNeIa data from the
JLA sample are consistent with values very close to the $\Lambda$CDM(i.e. zero
$\alpha$), with $|\alpha|\leq 0.05$ [32]. On the other hand, considering the
lensing amplitude, $\mathcal{A}_{l}$, and the effective number of relativistic
species, $N_{eff}$, as a free parameters, the data favour negative values of
$\alpha$, reducing the discrepancy between the local and CMB $H_{0}$
measurements [35]. The update results using CMB measurements from Planck 2018,
SNeIa data and SH0ES prior, assuming a spatially flat universe, favours a
positive interaction with $\alpha\approx$ 0.2, ruled out $\Lambda$CDM model by
more than 3$\sigma$ confidence level. The influences of non-zero spatial
curvature and SH0ES prior on $H_{0}$ in the joint analyses are also considered
[36].

Motivated by such a previous results, in this work we take a step forward and
we generalize the analysis done in [28] not assuming $\alpha$ at a specific
value but leaving it free to vary, including the dynamics of the baryonic
component to be able to compute RSD and the observed matter power spectrum. We
also perform an updated analysis of [32] and [36] to take into account RSD
data.

This paper is organised as follows. In section II we present the background
evolution of the decomposed gCg model. In section III we perform a
perturbative analysis of the evolution of the dark matter growth rate sourced
by vacuum energy perturbations. A comparison is then made with the case of
geodesic scenario. In section IV we present the observational tests and the
results. Finally, we conclude the study in Sec. V.

##  2 Background model

In this section we present the background evolution of the decomposed gCg
model as the interacting model. For this aim, let us consider a flat FLRW
cosmology in which the cosmic interacting fluid is described by the balance
equations

$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=Q,$ (2.1)
$\displaystyle\dot{\rho}_{V}=-Q,$ (2.2)

where $\rho_{m}$ and $\rho_{V}$ are respectively the densities of pressureless
matter and vacuum. The energy transfer between the components is given by

$Q=6\alpha
H_{0}(1-\Omega_{m0})\bigg{(}\frac{H}{H_{0}}\bigg{)}^{-(2\alpha+1)}\dot{H}.$
(2.3)

This model corresponds to the decomposed gCg [14, 15, 16, 17, 18, 19, 20]
whose pressure is given by (1.1). The sign of $Q$ depends on the sign of the
gCg parameter $\alpha$, since $\dot{H}<0$. If $Q>0$ the vacuum energy decays
into matter, and in the opposite case matter decays into vacuum energy.
Therefore, we see that, for $\alpha<0$, the vacuum energy density decays along
the expansion, while matter is created in the process, whereas for $\alpha>0$
matter is annihilated.

On the other hand, for $\alpha=0$ we re-obtain the standard model with a
cosmological constant and conserved matter.

The solutions for matter and vacuum densities are, respectively

$\rho_{m}=3H^{2}-3H_{0}^{2(1+\alpha)}(1-\Omega_{m0})H^{-2\alpha},$ (2.4)
$\rho_{V}=\rho_{V0}\bigg{(}\frac{H}{H_{0}}\bigg{)}^{-2\alpha},$ (2.5)

where a subindex 0 indicates the present value of the corresponding quantities
and $\Omega_{m0}=\rho_{m0}/3H_{0}^{2}$ is the present matter density
parameter.

The Hubble expansion rate is governed by the Friedmann equation [21]

$H=H_{0}\sqrt{\bigg{[}1-\Omega_{m0}+\frac{\Omega_{m0}}{a^{3(1+\alpha)}}\bigg{]}^{\frac{1}{1+\alpha}}+\frac{\Omega_{r0}}{a^{4}}}.$
(2.6)

###  2.1 The covariant energy transfer

We can explicitly write the energy transfer term (2.3) in a covariant manner
for a perfect fluid described by two interacting components. For this purpose,
let us assume a covariant form for the vacuum energy density (2.5) [31, 32],

$\rho_{V}=\rho_{V0}\bigg{(}\frac{\Theta}{3H_{0}}\bigg{)}^{-2\alpha},$ (2.7)

where we use the scalar expansion $\Theta=u^{\mu}_{;\mu}$ with $u^{\mu}$ being
the four velocity of the fluid. In the background universe the scalar
expansion is $\Theta=3H$.

The energy-momentum conservation equations for each component are given by

${T^{\nu\mu}_{m}}_{;\mu}=Q^{\nu},$ (2.8) ${T^{\nu\mu}_{V}}_{;\mu}=-Q^{\nu},$
(2.9)

where

$T^{\mu\nu}_{A}=\rho_{A}u^{\mu}u^{\nu}+p_{A}h^{\mu\nu}$ (2.10)

is the energy-momentum tensor of each component,
$h^{\mu\nu}=g^{\mu\nu}+u^{\mu}u^{\nu}$ is the projector tensor and $Q^{\mu}$
is the energy-momentum transfer between dark matter and vacuum. The latter can
be decomposed parallel and perpendicular to the four velocity $u^{\mu}$ as

$Q^{\mu}=u^{\mu}Q+\bar{Q}^{\mu},$ (2.11)

with $Q=-u_{\mu}Q^{\mu}$, $\bar{Q}^{\mu}=h^{\mu}_{\nu}Q^{\nu}$,
$u_{\mu}\bar{Q}^{\mu}=0$ and $u_{\mu}u^{\mu}=-1$.

Projecting equations $(\ref{asd})$ and $(\ref{asdw})$ parallel and
perpendicular to $u^{\mu}$, we find the energy conservation equations

$\rho_{m,\mu}u^{\mu}+\Theta\rho_{m}=-u_{\mu}Q^{\mu},$ (2.12)
$\rho_{V,\mu}u^{\mu}=u_{\mu}Q^{\mu},$ (2.13)

and the momentum conservation equations

$\rho_{m}{u^{\mu}}_{;\nu}u^{\nu}=\bar{Q}^{\mu},$ (2.14)
$\rho_{V,\nu}h^{\nu\mu}=-\bar{Q}^{\mu}.$ (2.15)

Using the ansatz $(\ref{pls})$ in the last equation, we find explicitly the
covariant energy transfer source term

$Q=\frac{2}{3}\alpha(1-\Omega_{m0})(3H_{0})^{2(\alpha+1)}\Theta^{-(2\alpha+1)}{\Theta}_{,\mu}u^{\mu}.$
(2.16)

We complete our system of equations with the Raychaudhuri equation

$\Theta_{,\mu}u^{\mu}=-\frac{1}{3}\Theta^{2}+(u^{\mu}_{;\nu}u^{\nu})_{;{\mu}}-\frac{1}{2}(\rho_{m}-2\rho_{V}),$
(2.17)

where we have neglected the shear and vorticity contributions. In the comoving
frame, where the components of the four-velocity are $u_{0}=-1$, $u^{0}=1$ and
$u_{i}=0=u^{i}$, one has $\bar{Q}^{\mu}=0$, which shows that there is no
momentum transfer in the homogeneous and isotropic background and the
expression (2.3) is recovered.

##  3 The perturbed equations

Now let us focus our attention to linear perturbations around a spatially flat
FLRW universe. Let us start with the most general line element for scalar
perturbations [29],

$ds^{2}=-(1+2\phi)dt^{2}+2a^{2}B_{,i}dtdx^{i}+a^{2}[(1-2\psi)\delta_{ij}dx^{i}dx^{j}+2E_{,ij}dx^{i}dx^{j}].$
(3.1)

We follow the approach presented in the reference [28] whose analysis is
gauge-invariant, here we generalise the results for any value of $\alpha$. At
the moment we neglect the perturbation dynamics for the baryon component. The
fluid velocity potencial $v$ can be defined by perturbing the four velocity
$u_{\mu}=g_{\mu\nu}u^{\nu}$, which results in

$\delta u_{j}=a^{2}\delta u^{j}+a^{2}B_{j}=v_{,j},$ (3.2)

assuming that $v$ is irrotational. We postulate that it coincides with the
matter velocity potential $v_{m}$, since we cannot properly define the four
velocity for the vacuum component. The time component of the perturbed four-
velocity is related to the perturbed metric through

$\delta u_{0}=\delta u^{0}=-\phi.$ (3.3)

The next step is to obtain the conservation equations for each interacting
component, namely matter and vacuum energy. In order to provide a set of basic
equations to calculate the matter density perturbation
$\delta_{m}=\delta\rho_{m}/\rho_{m}$, we start by considering the equations
for the vacuum. The perturbation of the momentum equation $(\ref{huma})$
yields, in the comoving gauge, the result

$\partial^{i}\delta
p_{V}^{c}=-\partial^{i}\delta\rho_{V}^{c}=-\delta\bar{Q}^{ic}.$ (3.4)

So, a non-zero momentum transfer $\delta\bar{Q}^{i}$ is related to the
presence of spatial variations of vacuum perturbations. Here the gauge
invariant scalar quantities
$\delta\mathcal{A}^{c}=\delta\mathcal{A}+\dot{\mathcal{A}}v$ that characterise
perturbations on comoving hypersurfaces were introduced.

The perturbation of equation $(\ref{lapis})$ allows us to compute the energy
transfer between the components,

$\delta Q^{c}=\dot{\rho}_{V}(\dot{v}+\phi)-\delta\dot{\rho}^{c}_{V}.$ (3.5)

For the matter component, the energy balance $(\ref{lopq})$ and the momentum
balance $(\ref{hum})$ can be written, up to first order, respectively as

$\dot{\delta}_{m}^{c}+\frac{Q}{\rho_{m}}\delta_{m}^{c}+\delta\Theta^{c}=\frac{\delta
Q^{c}}{\rho_{m}}+\bigg{(}\frac{Q}{\rho_{m}}-3H\bigg{)}(\dot{v}+\phi),$ (3.6)
$(\dot{v}+\phi)_{,j}=\frac{\delta\bar{Q}_{j}^{c}}{\rho_{m}}.$ (3.7)

The latter shows that, if the momentum transfer $\delta\bar{Q}_{j}^{c}$ is non
zero, the matter particles are forced to deviate from their geodesic motions.
This means that the evolution of the matter perturbation $\delta_{m}^{c}$
should be affected by the background evolution and the source terms owing to
vacuum inhomogeneities.

To complete our system of equations, the Raychaudhuri equation for the
expansion is obtained from the perturbation of $(\ref{sabe})$,

$\delta(\Theta_{,\mu}u^{\mu})+\frac{2}{3}\Theta\delta\Theta-\frac{\nabla^{2}}{a^{2}}(\dot{v}+\phi)+\frac{1}{2}(\delta\rho_{m}-2\delta\rho_{V})=0,$
(3.8)

or in terms of gauge invariant quantities

$\delta\dot{\Theta}^{c}+\frac{2}{3}\Theta\delta\Theta^{c}+\frac{1}{2}\rho_{m}\delta_{m}^{c}=\delta\rho_{V}^{c}+\bigg{(}\frac{\nabla^{2}}{a^{2}}+\dot{\Theta}\bigg{)}(\dot{v}+\phi).$
(3.9)

For investigating the possibility of non-zero vacuum perturbations
$\delta\rho^{c}_{V}$ and how they affect structure formation, we need now to
specify a precise form for the energy and momentum transfer.

###  3.1 Homogeneous vacuum density - geodesic

Firstly, we assume that the energy transfer between the components follows the
matter velocity, $Q^{\mu}=Qu^{\mu}$. In this case the momentum transfer
$\bar{Q}^{\mu}$ is zero at the background and perturbative levels, which
implies that the dynamical vacuum is homogeneous, with $\delta\rho^{c}_{V}=0$
according to $(\ref{momentum})$. Consequently, matter particles follow
geodesics in a comoving frame. Furthermore, there is no energy transfer at
first order, as we can see from $(\ref{vai})$ and $(\ref{sei})$. We refer to
this homogeneous interaction as geodesic scenario [26]. So, the basic
equations that describe the dynamics of the matter perturbations and scalar
expansion are given by $(\ref{mn})$ and $(\ref{opai})$ in the absence of
source terms,

$\dot{\delta}_{m}^{c}+\frac{Q}{\rho_{m}}\delta_{m}^{c}+\delta\Theta^{c}=0,$
(3.10)
$\delta\dot{\Theta}^{c}+\frac{2}{3}\Theta\delta\Theta^{c}+\frac{1}{2}\rho_{m}\delta_{m}^{c}=0.$
(3.11)

These equations are the same obtained in the synchronous comoving gauge [33],
and a simpler second order differential equation for the density contrast can
be found. To do that, we differentiate the continuity equation $(\ref{eq1})$
with respect to time and eliminate $\delta\Theta^{c}$ and
$\delta\dot{\Theta}^{c}$ by using $(\ref{eq1})$ and $(\ref{opai1})$, to obtain
the following equation

$\ddot{\delta}_{m}^{c}+\bigg{[}\frac{Q}{\rho_{m}}+2H\bigg{]}\dot{\delta}_{m}^{c}+\bigg{[}\frac{d}{dt}\bigg{(}\frac{Q}{\rho_{m}}\bigg{)}+2H\frac{Q}{\rho_{m}}-\frac{1}{2}\rho_{m}\bigg{]}\delta_{m}^{c}=0.$
(3.12)

###  3.2 Inhomogeneous vacuum density - non geodesic

An alternative choice is to explicitly consider inhomogeneities in the vacuum,
since neglecting them may lead to false interpretations of the observations
[37]. The natural manner for calculating the energy-momentum transfer at the
perturbative level is to assume the covariant ansatz $({\ref{pls}})$ for the
vacuum energy density, such that the perturbation of this quantity up to first
order is related to the scalar expansion through the expression

$\delta\rho_{V}^{c}=\frac{2Q}{3\rho_{m}}\delta\Theta^{c}.$ (3.13)

The relation above can be used into $(\ref{momentum})$ to obtain the right-
hand side of the momentum equation $(\ref{sei})$, given by

$\delta\bar{Q}_{j}={\delta\rho_{V}^{c}}_{,j}.$ (3.14)

Perturbing $(\ref{werb})$ and using the Raychaudhuri equation $(\ref{opai})$
and the relations $(\ref{paises4})$ and $(\ref{sei23})$, it is possible to
write the energy transfer function in the Fourier space as

$\frac{\delta
Q^{c}}{\rho_{m}}=\frac{Q}{3\rho_{m}}\delta_{m}^{c}+\bigg{[}2H-\frac{2Q}{3\rho_{m}}+\frac{2QH^{2}}{3\rho_{m}^{2}}\bigg{(}\frac{k}{aH}\bigg{)}^{2}-\frac{(2\alpha+1)\rho_{m}}{2H}\bigg{]}\frac{\delta\rho_{V}^{c}}{\rho_{m}},$
(3.15)

where $k$ is the comoving wave number. The scale dependence that appears in
the third term into the brackets is due to the momentum transfer between the
dark components, owing to the presence of vacuum perturbations. We refer to
this scenario as non geodesic. The amplitude of these perturbations compared
to the matter perturbations can be evaluated by using $(\ref{mn})$ together
with $(\ref{sei})$, $(\ref{paises4})$ and $(\ref{open45})$, leading to

$\frac{\delta\rho_{V}^{c}}{\delta\rho_{m}^{c}}=-\frac{2Q}{3\rho_{m}^{2}K}\bigg{[}Hf+\frac{2Q}{3\rho_{m}}\bigg{]}.$
(3.16)

Here we have defined the scale dependent function

$K(a,k)=1-\frac{2Q}{3\rho_{m}^{2}}\bigg{[}A-H-\frac{(2\alpha+1)\rho_{m}}{2H}\bigg{]},$
(3.17)

where

$A(a,k)=\frac{Q}{3\rho_{m}}+\frac{2QH^{2}}{3\rho_{m}^{2}}\bigg{(}\frac{k}{aH}\bigg{)}^{2}.$
(3.18)

The Raychaudhuri equation $(\ref{opai})$ can be written as

${\delta\dot{\Theta}}^{c}=-\frac{1}{2}\rho_{m}\delta_{m}-2H\delta\Theta^{c}-\bigg{[}\frac{H^{2}}{\rho_{m}}\bigg{(}\frac{k}{aH}\bigg{)}^{2}+\frac{1}{2}\bigg{]}\delta\rho_{V}^{c}.$
(3.19)

Now we can differentiate $(\ref{fui})$, eliminate $\delta\dot{\Theta}^{c}$
through the perturbed Raychaudhuri equation $(\ref{opai3})$ and
$\delta\Theta^{c}$ through $(\ref{fui})$ and $(\ref{sei23})$, to obtain a
second order differential equation for the evolution of the matter contrast,

$\ddot{\delta}_{m}^{c}+\bigg{[}\frac{2Q}{3\rho_{m}}+2H+\bigg{(}A-\frac{\dot{K}}{K}\bigg{)}\bigg{]}\dot{\delta}_{m}^{c}+\bigg{[}\frac{d}{dt}\bigg{(}\frac{2Q}{3\rho_{m}}\bigg{)}+2H\bigg{(}\frac{2Q}{3\rho_{m}}\bigg{)}-\frac{1}{2}\rho_{m}K+\frac{2Q}{3\rho_{m}}\bigg{(}A-\frac{\dot{K}}{K}\bigg{)}\bigg{]}\delta_{m}^{c}=0.$
(3.20)

We see that differences arises as compared to the geodesic scenario
$(\ref{sin})$, namely, a reduction by a factor $2/3$ in the
creation/annihilation rate and a change in the evolution of the dark matter
contrast through the scale-dependent function $K$. Furthermore, a new function
$A-\frac{\dot{K}}{K}$ appears in the coefficients of $\dot{\delta}_{m}$ and
$\delta_{m}$. The standard $\Lambda$CDM model is recovered if we choose
$\alpha=0$. For the case of constant matter creation, which corresponds to the
value of $\alpha=-0.5$, the results presented in [28] are recovered.

To estimate the importance of vacuum energy perturbations relative to matter
as given by expression $(\ref{fui})$, we start by looking for their values in
the deeper matter dominated phase ($z\gg 1$). Since $H\gg Q/\rho_{m}$ at high
redshifts, from $(\ref{poise})$ we have $K\approx 1$, and hence the density
contrast is proportional to the scale factor, $\delta_{m}\propto a$, resulting
in the standard growth rate $f=1$. So, the expression $(\ref{fui})$ assumes
the scale-independent form

$\frac{\delta\rho_{V}^{c}}{\delta\rho_{m}^{c}}\approx\frac{2Q}{3\rho_{m}}\Omega_{m0}^{-\frac{1}{2(1+\alpha)}}z^{-3/2}.$
(3.21)

This ratio is very tiny and depends essentially on the interaction rate and
the present value of the matter density. For comparison purposes, if we assume
the model with $\alpha=-0.5$ and $\Omega_{m0}=0.45$, corresponding to a
constant interaction rate, we found
$\frac{\delta\rho_{V}^{c}}{\delta\rho_{m}^{c}}\sim 10^{-5}$ at $z_{i}=1000$.
On the other hand, at the same redshift, for $\alpha=-0.1$ and
$\Omega_{m0}=0.3$ we have $\frac{\delta\rho_{V}^{c}}{\delta\rho_{m}^{c}}\sim
10^{-9}$, which is different in several orders of magnitude.

Figure 1: Relative difference between the matter contrasts for the scale
$k=0.2$ and for the scale $k=0.01$, as a function of the redshift for
different values of the gCg parameter.

Figure 2: Matter contrast (left panel) and growth rate (right panel) for the
following models: $\alpha=-0.5$ (red), $\alpha=-0.2$ (yellow), $\alpha=-0.1$
(green), $\alpha=0.1$ (blue) and $\alpha=0.2$ (gray). We have used the value
$\Omega_{m0}=0.45$ for the model with $\alpha=-0.5$. For all the other models
we have used $\Omega_{m0}=0.3$. Solid curves correspond to the geodesic and
dotted curves to the non geodesic.

On the other hand, at late times the vacuum perturbations depend on the scale.
The observational data of the linear power spectrum lie in the comoving wave
number range $0.01$ Mpc${}^{-1}<k<0.2$ Mpc-1. In this range, taking the gCg
parameter in the interval $-0.5\leq\alpha\leq 0.5$, we find the ratio between
the vacuum and dark matter perturbations at $z=0$ in the interval

$10^{-6}<\frac{\delta\rho_{V}^{c}}{\delta\rho_{m}^{c}}<10^{-4},$ (3.22)

where the upper value corresponds to $k=0.01$, and the lower value to $k=0.2$.
Therefore, vacuum perturbations are strongly suppressed inside the Hubble
horizon respect to matter perturbations for the class of interacting models
considered here. The smaller the scale value, the stronger the suppression.
These results are shown in Fig. 1, where we plot the ratio between matter
perturbations for the scale $k$=0.2 and for the scale $k$=0.01 as a function
of the redshift, which is less than $0.4\%$ for $|\alpha|<0.5$.

To obtain the evolution of $f$ we need to solve the second order equation
$(\ref{sin})$ and (3.20) by fixing the same initial amplitude
$\delta_{m}(z_{i})$ for all models at $z_{i}=1000$ such that in the matter
dominated era we have the standard value $f(z_{i})=1$, since the matter
contrast is proportional to the scale factor, $\delta_{m}\propto a$. Fig. 2
shows the matter density contrast and growth rate when we use the geodesic
(solid curves) and non geodesic (dotted curves) scenarios. The differences
increase for large values of $|\alpha|$. We see an enhancement in the curves
as compared to the geodesic for $\alpha=-0.1,-0.2$, and a suppression for
$\alpha=0.1,0.2$. For the case $\alpha=-0.5$, corresponding to matter creation
at constant rate, a large suppression appears in the matter growth rate.
Therefore, the correction terms introduced in equation $(\ref{bo})$ by the
vacuum inhomogeneities should be taken into account when we analyse the growth
rate evolution.

###  3.3 Newtonian gauge

Let us show that another way to determine the evolution of the growth rate is
to calculate the scalar expansion up to first order, through the covariant
derivative of the 4-velocity. In the Newtonian gauge it results in [32]

$\delta\Theta=-\frac{3}{a}(\psi^{\prime}+\mathcal{H}\phi)+\frac{\nabla^{2}v}{a},$
(3.23)

where the prime represents a derivative with respect to the conformal time. We
can show that the above expression satisfies the Raychaudhury equation
$(\ref{eq2})$, which we have used to find the second order differential
equation (3.20). To do that, write the first term of $(\ref{eq2})$ as

$\delta(\Theta_{,\mu}u^{\mu})=\frac{1}{a}(\delta\Theta^{\prime}-\Theta^{\prime}\phi),$
(3.24)

where $u^{0}=1/a$, $\delta u^{0}=-\phi/a$ and $\mathcal{H}=aH$ in the
conformal time. The derivative of $(\ref{W1})$ is given by

$\delta\Theta^{\prime}=-\frac{3}{a}(\psi^{\prime\prime}+\mathcal{H}\phi^{\prime}+\mathcal{H}^{\prime}\phi)+\frac{\nabla^{2}v^{\prime}}{a}-\mathcal{H}\delta\Theta.$
(3.25)

Now consider the set of Einstein’s equations in the Newtonian gauge [34],

$-\nabla^{2}\psi+3\mathcal{H}(\psi^{\prime}+\mathcal{H}\phi)=-\frac{a^{2}}{2}(\delta\rho_{m}+\delta\rho_{V}),$
(3.26)
$\psi^{\prime\prime}+2\mathcal{H}\psi^{\prime}+\mathcal{H}\phi^{\prime}+(2\mathcal{H}^{\prime}+\mathcal{H}^{2})\phi=-\frac{a^{2}}{2}\delta\rho_{V}+\frac{a^{2}}{3}\nabla^{2}\pi,$
(3.27) $\psi^{\prime}+\mathcal{H}\phi=-\frac{a^{2}}{2}\rho_{m}v,$ (3.28)
$\psi-\phi=a^{2}\pi,$ (3.29)

where $\pi$ is the anisotropic stress perturbation. Using $(\ref{hi1})$,
$(\ref{hi2})$ and $(\ref{W1})$ in Eq. $(\ref{ra})$ it is not difficult to show
that $(\ref{ya})$ is given by

$\delta(\Theta_{,\mu}u^{\mu})+\frac{2}{3}\Theta\delta\Theta-\frac{\nabla^{2}}{a^{2}}(v^{\prime}+\psi)+\frac{1}{2}(\delta\rho_{m}-2\delta\rho_{V})=\nabla^{2}\bigg{(}\frac{\mathcal{H}}{a^{2}}v-\pi\bigg{)}.$
(3.30)

Finally, we use Eq. $(\ref{hi4})$, make the substitution $v\rightarrow v/a$
and change from conformal time to cosmological time to obtain the Raychaudhury
equation

$\delta(\Theta_{,\mu}u^{\mu})+\frac{2}{3}\Theta\delta\Theta-\frac{\nabla^{2}}{a^{2}}(\dot{v}+\phi)+\frac{1}{2}(\delta\rho_{m}-2\delta\rho_{V})=0.$
(3.31)

We can also obtain the evolution of coupled equations for total matter density
contrast $\delta_{m}$, matter velocity defined by $\theta_{m}=-k^{2}v$ and
gravitational potential $\phi$ in Newtonian gauge. We consider the anisotropic
stress perturbations equal to zero, which implies $\phi=\psi$.

In the limit of small scales $k\gg aH$, after some calculations using (3.23),
(3.15), (3.14), (3.13) and (3.7) into (3.6), it is straightforward to show
that the energy balance assumes the form

$\delta^{\prime}_{m}+\theta_{m}+\frac{aQ}{\rho_{m}}\delta_{m}=\frac{aQ}{3\rho_{m}}\delta_{m}+\frac{4Q^{2}H^{2}}{9\rho_{m}^{4}}\bigg{(}\frac{k}{aH}\bigg{)}^{2}\theta_{m}.$
(3.32)

In the sub-horizon limit $k\gg aH$, the Poisson equation (3.26) is given by

$-k^{2}\phi=\frac{a^{2}}{2}\rho_{m}\delta_{m},$ (3.33)

where we neglected the vacuum energy perturbation $\delta\rho_{V}$ as compared
with matter density perturbation $\delta\rho_{m}$ and the second term on the
right of equation. Finally, it is easy to show that the momentum balance (3.7)
becomes

$\theta^{\prime}_{m}+aH\theta_{m}-k^{2}\phi=-\frac{2aQH^{2}}{3\rho_{m}^{2}}\bigg{(}\frac{k}{aH}\bigg{)}^{2}\theta_{m}.$
(3.34)

For the geodesic case in the sub-horizon limit, the resulting equations are
given in the reference [35].

###  3.4 Perturbation of baryons

In this section the perturbation of baryons will be included in the dynamics
to account the effect of Redshift-Space Distortion (RSD) on large scale
structures (LSS) data. The assumption is that baryons reflect the motion of
galaxies, since what we directly observe is their emitted light.

To derive the evolution equation for the baryonic contrast $\delta_{b}$ in the
Newtonian gauge we assume that vacuum does not interact with baryons. Thus,
the resulting continuity and Euler equations in the sub-horizon limit, are
given, respectively, by [38]

$\delta^{\prime}_{b}+\theta_{b}=0,$ (3.35)
$\theta^{\prime}_{b}+aH\theta_{b}-k^{2}\phi=0.$ (3.36)

For $\Lambda$CDM model where the energy transfer is absent ($Q$=0)
$\delta_{m}$ and $\delta_{b}$ have exactly the same evolution, however the
suppression or enhanced in the matter density contrast for models with
$\alpha\neq 0$ affect the baryon density contrast through Poisson equation
(3.33).

Let us define the linear matter power spectrum in Fourier space at a redshift
$z$, as

$P_{m,b}(k,z)=P(k,z_{i})(\delta_{m,b}(z))^{2},$ (3.37)

where the suffices $m$ and $b$ refer to the baryon and total matter and
$P(k,z_{i})$ is the matter power spectrum defined at matter dominated era.

The amplitude of baryon and total matter density fluctuation within sphere of
comoving radius 8 $h^{-1}$Mpc at redshift $z$ is given by

$\sigma_{8m,b}(z)=\bigg{[}\frac{1}{2\pi^{2}}\int
k^{2}dkW(kR)P(k,z_{i})\bigg{]}^{1/2}\delta_{m,b}(z),$ (3.38)

with the window function

$W(kR)=\bigg{[}\frac{sin(kR)}{(kR)^{3}}-\frac{cos(kR)}{(kR)^{2}}\bigg{]}.$
(3.39)

Now, using (3.38) we can take the relation for each component

$\sigma_{8m,b}(z)=\sigma_{8m,b}(0)\frac{\delta_{m,b}(z)}{\delta_{m,b}(0)}.$
(3.40)

Similarly, we can derive $\sigma_{8m}$ from $\sigma_{8b}$ through the relation

$\sigma_{8m}(0)=\sigma_{8b}(0)\frac{\delta_{m}(0)}{\delta_{b}(0)}.$ (3.41)

For $\Lambda$CDM model we have $\sigma_{8m}(0)=\sigma_{8b}(0)$ due to the
absence of interactions. However, for interaction models the today’s amplitude
of baryons and total matter density fluctuations are, in fact, different by
the above relationship. From (3.37) and (3.41) it is possible to find the
baryon power spectrum in terms of the total matter power spectrum at $z$=0

$P_{b}(k,0)=P_{m}(k,0)\bigg{[}\frac{\sigma_{8b}(0)}{\sigma_{8m}(0)}\bigg{]}^{2}.$
(3.42)

Now we can compute the observed galaxy power spectrum in redshift space as

$P_{g}(k,z)=[b_{1}\sigma_{8m}(z)+\mu^{2}_{k}f_{b}(z)\sigma_{8b}(z)]^{2}\frac{P_{m}(k,0)}{\sigma_{8m}^{2}(z)},$
(3.43)

onde $b_{1}$ is the linear bias and $\mu_{k}$ is the cosine of the angle
between the line of sight and the wavevector $\hat{k}$. Regarding to the RSD,
the important quantity is the bias independent combination
$f_{b}(z)\sigma_{8b}(z)$, where $f_{b}$ is the baryon growth rate and
$\sigma_{8b}$ represents the amplitude of baryon density fluctuation within
sphere of comoving radius 8 Mpc given by $(\ref{sigmas_relation01})$.

In Fig. 3 we plot the evolution of $f_{b}(z)\sigma_{8b}(z)$ on the left panel
for $\alpha=0$, $\alpha=0.2$ and $\alpha=-0.5$ fixing $\sigma_{8b}(0)=0.83$.
The values of the total density parameters are chosen to be $\Omega_{m0}=0.45$
for $\alpha=-0.5$ and $\Omega_{m0}=0.3$ for the other two models. In the right
panel, we show the relative difference between homogeneous and inhomogeneous
vacuum ansatz for the expected combination of RSD for baryons.

Figure 3: Left panel: The evolution of combination $f_{b}(z)\sigma_{8b}(z)$
for $\Lambda$CDM (black curve), for $\alpha=0.2$ (gray curve) and for
$\alpha=-0.5$ (red curve). Solid curves correspond to the geodesic model and
dotted curves to inhomogeneous vacuum. Right panel: The percent relative
difference between homogeneous and inhomogeneous vacuum ansatz.

##  4 Observational tests

Once we have the description of both homogeneous and inhomogeneous vacuum
models, equations (3.12) and (3.20), it is possible to construct the quantity
$f_{b}(z)\sigma_{8b}(z)$ and perform observational tests against available
data. Our goal here is to compare the constraints obtained with redshift space
distortion data alone, and also to check the impact of this analysis when
combining with independent data with bigger sample and more controlled errors.
For that, we used type Ia supernovae samples to combine with
$f_{b}\sigma_{8b}$ data set. In this section we present the data used in our
analysis, the applied statistics and some results.

###  4.1 Data set and Method

In our analysis, we choose to use a joint data set combining several probes,
such as Redshift Space Distortion data, Supernovae Ia samples and Cosmic
Microwave Background likelihoods, in order to cover evolution of the universe
at different scales.

About the RSD probe, we considered $(f\sigma_{8})^{\prime}$ data presented in
Tab. IV of Ref. [39], containing each fiducial cosmology on the last column,
and the prime refers to the value in the fiducial cosmology. Taking into
account corrections due to Alcock-Paczynski effect, the value $(f\sigma_{8})$
for the cosmology of interest is obtained making

$\displaystyle f\sigma_{8}\sim q(f\sigma_{8})^{\prime},$ (4.1)

where

$\displaystyle
q=\frac{H(z|s)D_{A}(z|s)}{H^{\prime}(z|s^{\prime})D^{\prime}_{A}(z|s^{\prime})},$
(4.2)

with $s^{\prime}$ and $s$ representing sets of cosmological parameters in both
fiducial and tested models. In order to not having to evaluate errors and
covariance matrix for the models of interest, avoiding error propagation in
this process, we built the likelihood for RSD data proportional to
$\exp(-\chi^{2}/2)$, where

$\displaystyle\chi^{2}=\sum_{ij}V^{\prime}_{i}(C^{\prime-1}_{ij})V^{\prime}_{j},$
(4.3)

and,

$\displaystyle
V^{\prime}_{i}\equiv(f\sigma_{8})^{\prime}_{i}-\frac{f\sigma_{8}(z_{i}|s)}{q(z_{i}|s,s^{\prime})}.$
(4.4)

Only WiggleZ data present correlations, translated into the following
covariance matrix,

$\displaystyle
C_{ij}^{\prime\text{WiggleZ}}=10^{-3}\left(\begin{array}[]{ccc}6.400&2.570&0.000\\\
2.570&3.969&2.540\\\ 0.000&2.540&5.184\\\ \end{array}\right).$ (4.8)

At the same time, we consider both Supernovae Ia by the Joint Light-curve
(JLA) sample [40], and Pantheon compilation [41]. The first is a sample of 740
data points of the Supernova Legacy Survey (SNLS) and Sloan Digital SkySurvey
(SDSS), covering the redshift range $0.01<z<1.3$. This likelihood is
marginalized over two nuisance parameters, with the advantage of allowing the
light-curve recalibration with the model under consideration, which is an
important issue when testing alternative cosmologies [42, 35]. Instead, the
Pantheon compilation cover the range $0.01<z<2.3$ with 1048 SNe Ia, combining
the subset of 276 newPan-STARRS1 SNe Ia with useful distance estimates of SNIa
from SNLS, SDSS, low-z and Hubble space telescope (HST) samples.

Finally, we also take into account CMB measurements, through the Planck (2018)
data [43], using “TT,TE,EE+lowE" data by combination of temperature power
spectra and cross correlation TE and EE over the range $\ell\in[30,2508]$, the
low-$\ell$ temperature Commander likelihood, and the low-$\ell$ SimAll EE
likelihood. We refer to this data set as “Planck18".

Our analysis are divided into two distinct methodologies. The first is a RSD
and SNe data joint analysis, building chains with the Python MultiNest
algorithm [44, 45]. We set $1500$ livepoints and a sampling efficiency of
$0.5$, giving both parameter and evidence estimations. Those PyMultiNest
parameters were chosen to guarantee stability in the exploration of the space
parameter achieving and acceptable convergence.

Parameter | Priors
---|---
$H_{0}$ (km/s/Mpc) | ${\cal{G}}[74.03,1.42]$
$\Omega_{b}h^{2}$ | ${\cal{G}}[0.02166,0.00026]$
$\Omega_{c}h^{2}$ | $\mathcal{U}[0.01,0.25]$
$\alpha$ | $\mathcal{U}[-0.99,1.00]$
$\sigma_{8}$ | $\mathcal{U}[0.4,1.2]$
$\alpha_{s}$ | $\mathcal{U}[0.0,1.0]$
$\beta_{s}$ | $\mathcal{U}[0.0,4.0]$
$\mathcal{M}_{B}$ | $\mathcal{U}[-22,-16]$
$\Delta_{M}$ | $\mathcal{U}[-1.0,1.0]$
Table 1: Adopted priors for cosmological and supernovae (JLA) nuisance free
parameters on performed analysis. A Gaussian prior for $H_{0}$ inspired on
Riess _et al._ measurements was chosen, once this parameter is not relevant
for RSD-only analysis and for supernovae data fitting, $H_{0}$ is fully
correlated to $\mathcal{M}_{B}$. $\sigma_{8}$, is a derived parameter.

The priors used in our analysis are summarized in Table 1, where Gaussian
priors are indicated for local Hubble-Lemaître parameter, $H_{0}$ [46], and
the physical baryon density, $\Omega_{b}h^{2}$ [47], and uniform priors for
the physical cold dark matter density, $\Omega_{c}h^{2}$, for interaction
parameter, $\alpha$, and for $\sigma_{8b}$. Also, uniform priors were adopted
for nuisance parameters associate with SNeIa light curve calibration, i.e.
{$\alpha_{s}$,$\beta_{s}$,$\mathcal{M}_{B}$,$\Delta_{M}$} for JLA likelihood,
and $\mathcal{M}_{B}$ for Pantheon likelihood.

The second methodology regards the analysis using the CMB data, i.e when the
combined CMB+SNe data set are considered. In this case we modified both the
background and perturbation equations, explained in the previous sections, in
the numerical Cosmic Linear Anisotropy Solving System (CLASS) code [48],
choosing to work with the Newtonian gauge. In this way, we get the theoretical
prediction of cosmological observables, i.e. among others the TT, TE and EE
anisotropy CMB power spectra as well as the matter power spectrum. Then, using
the Monte Python [49] code it is possible to constrain the parameters of the
theory by a Monte Carlo Markov Chains analysis, using the chosen data set.
These chains, are then post-processed using RSD data. In these analysis, we
vary the standard six $\Lambda$CDM cosmological parameters in large and flat
priors. This means that, besides the $H_{0}$, $\Omega_{b}h^{2}$ and
$\Omega_{c}h^{2}$, we also consider the optical depth, $\tau_{reio}$, the
primordial scalar amplitude, $\mathcal{A}_{s}$, the primordial spectral index,
$n_{s}$, in addition to the interaction parameter, $\alpha$. Also, we consider
the CMB nuisance foreground parameters [50]. Once $\sigma_{8}=\sigma_{8m}$ is
given as a derived parameter in MontePython analysis, with the help of (3.41)
it was possible to also derive $\sigma_{8b}$.

Parameter | Data |  $\Lambda$CDM | Geodesic CG | non-Geodesic CG
---|---|---|---|---
$\alpha$ | RSD | 0 | $-0.011^{+0.695}_{-0.494}$ | $-0.14^{+0.336}_{-0.343}$
RSD+JLA | 0 | $-0.112^{+0.382}_{-0.316}$ | $-0.111^{+0.289}_{-0.266}$
RSD+Pantheon | 0 | $-0.062^{+0.366}_{-0.298}$ | $-0.053^{+0.249}_{-0.269}$
$\Omega_{m0}$ | RSD | $0.28^{+0.056}_{-0.051}$ | $0.294^{+0.163}_{-0.156}$ | $0.333^{+0.132}_{-0.097}$
RSD+JLA | $0.286^{+0.042}_{-0.04}$ | $0.313^{+0.092}_{-0.088}$ | $0.314^{+0.079}_{-0.068}$
RSD+Pantheon | $0.291^{+0.034}_{-0.032}$ | $0.305\pm 0.071$ | $0.305^{+0.059}_{-0.051}$
$\sigma_{8b}$ | RSD | $0.779^{+0.036}_{-0.034}$ | $0.794^{+0.142}_{-0.094}$ | $0.803^{+0.103}_{-0.060}$
RSD+JLA | $0.775^{+0.032}_{-0.031}$ | $0.807^{+0.11}_{-0.08}$ | $0.796^{+0.077}_{-0.051}$
RSD+Pantheon | $0.773\pm 0.029$ | $0.792^{+0.096}_{-0.072}$ | $0.785^{+0.069}_{-0.043}$
$\sigma_{8}$ | RSD | $0.779^{+0.036}_{-0.034}$ | $0.805^{+0.582}_{-0.254}$ | $0.69^{+0.143}_{-0.196}$
RSD+JLA | $0.775^{+0.032}_{-0.031}$ | $0.723^{+0.220}_{-0.148}$ | $0.709^{+0.122}_{-0.156}$
RSD+Pantheon | $0.773\pm 0.029$ | $0.745^{+0.206}_{-0.144}$ | $0.736^{+0.099}_{-0.149}$
Table 2: Results obtained when constraining the discussed models against RSD
data, and also the joint analysis with both SNeIa samples. Parameters mean
values and $95\%$ credible intervals are presented. Figure 4: RSD and
RSD+SNeIa posterior distributions and 2-D credible regions for $\Omega_{m0}$
and $\sigma_{8}$ in the $\Lambda$CDM scenario. RSD results show a good
agreement with SNeIa, and the joint analysis return tighter constraints on
matter density parameter.

Figure 5: Geodesic Chaplygin gas (left panel) and Non-Geodesic (right panel)
parameters plots of 1D PDF and 2D credible regions (1$\sigma$ and 2$\sigma$)
as results of statistical analysis with RSD and RSD+SNeIa. Those results are
summarized on table 2.

Figure 6: Left: Geodesic Chaplygin gas; Right: non-geodesic. Results obtained
with RSD only and some combinations of datasets. Red curves show the effect of
post-processing Planck18+JLA chains with RSD likelihood. Mean and credible
intervals are presented at table 3.

Figure 7: Comparison of 1-D and 2-D distributions of cosmological parameters for both Geodesic and Non-Geodesic with Planck18, JLA and priors (orange curves) and effect of postprocessing the chains with RSD likelihood (red curves). The overlap of solid gray and dashed black curves shows that RSD postprocessing causes no effect on $\Lambda$CDM parameter constraints. The posteriors of $S_{8}$ parameter shows the agreement of Geodesic results with KiDS-1000 measurements [51] (green shaded area). Parameter | Data |  $\Lambda$CDM | Geodesic CG | non-Geodesic CG
---|---|---|---|---
$\alpha$ | Planck18+JLA | 0 | $0.198^{+0.113}_{-0.118}$ | $0.155^{+0.133}_{-0.131}$
Planck18+JLA +RSD | 0 | $0.143^{+0.061}_{-0.063}$ | $0.143^{+0.117}_{-0.112}$
$\Omega_{m0}$ | Planck18+JLA | $0.309^{+0.012}_{-0.013}$ | $0.226^{+0.045}_{-0.044}$ | $0.242^{+0.049}_{-0.047}$
Planck18+JLA +RSD | $0.309^{+0.012}_{-0.013}$ | $0.247^{+0.025}_{-0.023}$ | $0.247^{+0.055}_{-0.054}$
$\sigma_{8b}$ | Planck18+JLA | $0.81\pm 0.012$ | $0.722^{+0.058}_{-0.063}$ | $0.804^{+0.018}_{-0.015}$
Planck18+JLA +RSD | $0.81\pm 0.012$ | $0.751^{+0.025}_{-0.027}$ | $0.797^{+0.011}_{-0.01}$
$\sigma_{8}$ | Planck18+JLA | $0.81\pm 0.012$ | $0.879^{+0.039}_{-0.041}$ | $0.864^{+0.043}_{-0.041}$
Planck18+JLA +RSD | $0.81\pm 0.012$ | $0.862^{+0.027}_{-0.028}$ | $0.86^{+0.043}_{-0.042}$
$S_{8}$ | Planck18+JLA | $0.823\pm 0.023$ | $0.761^{+0.044}_{-0.048}$ | $0.775^{+0.042}_{-0.046}$
Planck18+JLA +RSD | $0.823\pm 0.023$ | $0.781\pm 0.022$ | $0.779^{+0.042}_{-0.045}$
$H_{0}$ | Planck18+JLA | $67.771^{+0.965}_{-0.899}$ | $70.664^{+1.887}_{-1.932}$ | $70.09^{+1.937}_{-1.92}$
Planck18+JLA +RSD | $67.771^{+0.965}_{-0.899}$ | $69.894^{+1.051}_{-1.146}$ | $69.9^{+1.922}_{-1.948}$
$n_{s}$ | Planck18+JLA | $0.968\pm 0.008$ | $0.963\pm 0.008$ | $0.964\pm 0.008$
Planck18+JLA +RSD | $0.968\pm 0.008$ | $0.964\pm 0.008$ | $0.964\pm 0.008$
$\tau_{reio}$ | Planck18+JLA | $0.057^{+0.016}_{-0.014}$ | $0.056\pm 0.015$ | $0.057^{+0.015}_{-0.014}$
Planck18+JLA +RSD | $0.057^{+0.016}_{-0.014}$ | $0.058^{+0.016}_{-0.015}$ | $0.057^{+0.016}_{-0.014}$
$ln10^{10}A_{s}$ | Planck18+JLA | $3.048^{+0.03}_{-0.027}$ | $3.052^{+0.029}_{-0.03}$ | $3.051^{+0.03}_{-0.027}$
Planck18+JLA +RSD | $3.048^{+0.03}_{-0.027}$ | $3.054^{+0.03}_{-0.029}$ | $3.05^{+0.03}_{-0.027}$
Table 3: Mean values and $2\sigma$ credible intervals for cosmological parameters obtained with MontePython chains for Planck18, JLA SNe Ia Compilation and priors. Also, in the second row of each parameter, the results obtained after postprocessing the chains with RSD likelihood. Data | Geodesic Chaplygin | non-Geodesic Chaplygin
---|---|---
RSD | -0.05 | -0.05
RSD+JLA | -0.46 | -0.46
RSD+Pantheon | -0.18 | -0.04
Planck18+JLA | -9.84 | -10.04
Table 4: The values of $\Delta(\chi^{2}_{min})$ for each dataset, comparing
both Chaplygin models against the $\Lambda$CDM , where $\chi^{2}\equiv-2$
loglikelihood.

###  4.2 Results

The constraints obtained with only RSD data are represented as the black
curves on figure 4 for $\Lambda$CDM model and figure 5 for both cases of the
gCg discussed here. Combined with JLA or Pantheon SNe Ia compilation, tighter
parameter constraints are obtained. It is shown on table 2 that the inclusion
of JLA data does not affect much $\alpha$ mean value in the non-geodesic
scenario, but a considerable change is noted when using Pantheon data. On the
other hand the opposite happens when the latter and the former are included in
the geodesic. Besides RSD data and both SNeIa data used show a preference for
a flux of energy from dark energy to dark matter, Pantheon dataset deviates
the curves towards the $\Lambda$CDM model. No relevant differences between
geodesic and non geodesic models where observed apart from slightly tighter
constraints for the latter, what could be addressed to divergences when using
certain set of parameters, drawing forbidden regions. The preference towards
the standard model when using Pantheon dataset is expected once nuisance
parameters are fixed on $\Lambda$CDM best-fit values, and also a lower set of
free parameters tends to return tigther constraints on the remanent
parameters.

We extend our results by also considering CMB data. In Fig. 6, with orange
curves, we show that Planck+JLA likelihoods favour positive values of $\alpha$
for both the geodesic (left panel) and for the non geodesic (right panel),
leading to lower values of $\Omega_{m0}$.

While our geodesic scenario result is an updated analysis and is compatible
with previous results [35, 52], the non-geodesic case has never been analysed
in the literature. In this case we note a sligthly higher value for the
$\alpha$ parameter with broader errors. Also, the Planck18+JLA chains are
post-processed with RSD likelihood generating the red curves on Figures 6 and
7.

Table 3 summarizes the mean values and $2\sigma$ credible regions for the
parameters of interest. The parameters of the $\Lambda$CDM model are
insensitive to the RSD data when combined with Planck18+JLA. This behavior is
not observed when we analyse the interacting model. The amplitude of the
baryon density fluctuation $\sigma_{8b}$ is slightly larger for the non-
geodesic than for the geodesic. As a consequence, the inclusion of RSD yields
a shift of $\alpha$ to lower values, leading to slightly higher values of
$\Omega_{m0}$. Thus, except for the $\sigma_{8b}$ values, the cosmological
parameters resulting from the joint analysis are practically the same for both
scenarios. Through the equation (3.42) it is possible to find that the baryon
power spectrum for non-geodesic case is $13\%$ greater than the geodesic one.
This difference increases for $28\%$ if only the analysis of Planck18+JLA is
considered. We also note that the $\Lambda$CDM model ($\alpha=0$) is ruled out
with and without RSD, with a clear preference for a positive $\alpha$,
corresponding to annihilated dark matter.

It is quite interesting that positive values of $\alpha$ move matter density
parameter to lower values, what allows a better agreement for the $S_{8}$
$(\equiv\sigma_{8}\sqrt{\Omega_{m0}/0.3})$ parameter, besides the preferred
higher $\sigma_{8}$ values. These behaviour is emphasized on Figure 7, where
the KiDS-1000 measurements of $S_{8}$ parameter [51] are indicated in the
green shaded area.

##  5 Final Remarks

In this work we investigated the evolution of dark matter perturbations in the
context of an interacting dark sector corresponding to a decomposed Chaplygin
gas model. We have considered two distinct scenarios for the covariant energy-
momentum transfer. In the first, the energy transfer follows the dark matter
$4$-velocity. In this simplest case the momentum transfer is zero, which
implies a homogeneous vacuum energy. We recover the scale-independent second
order differential equation for the density contrast, showing that, compared
to the $\Lambda$CDM cosmology, the growth rate is suppressed for $\alpha<0$
due to the homogeneous creation of dark matter, and enhanced when $\alpha>0$.

In the second scenario, the momentum transfer is determined by the gradient of
the vacuum perturbations, which is proportional to the scalar expansion,
$\delta\rho^{c}_{V}\propto\delta\Theta^{c}$. The dynamics in this case is
reduced to a single scale-dependent second order equation. We are able to
evaluate the size of vacuum energy perturbations compared with the dark matter
ones, determining the evolution of the growth rate for diverse gCg background
solutions. The vacuum perturbations show to be negligible on scales inside the
horizon.

To explore the bounds of the cosmological parameters of these two scenarios,
we use the observational datasets from Planck18, Type Ia Supernova and
redshift-space distortions. We used the growth rate of baryons $f_{b}$
combined with the $\sigma_{8b}$ to account redshift-space distortions inferred
from peculiar velocities.

Furthermore, we have updated the results of [32] and [36] using Planck18, JLA
and RSD observational datasets. Our results are shown in table 3. One can note
that the included RSD data used in our analysis do not change the mean value
$H_{0}$ estimates for the $\Lambda$CDM cosmology. The discrepancy on $H_{0}$
between the $\Lambda$CDM prediction $H_{0}=67.771\pm^{0.965}_{0.889}$ with
respect to the latest Hubble constant measurement by Hubble Space Telescope
and the SH0ES Team [53] is about 3.8 $\sigma$.

In the presence of a non-zero energy transfer between dark matter and the
vacuum the uncertainty on $H_{0}$ inferred from the combination of
Planck+JLA+RSD datasets decreased. For the geodesic scenario the bounds on
$H_{0}$ reduce the tension to 2.0 $\sigma$ while for the non geodesic, where
the error is about twice as large, one finds 1.4 $\sigma$. In both scenarios
the data prefer a positive values of $\alpha=0.143$ and a lower values of
$S_{8}$ with good agreement with KiDS-1000.

In conclusion, our analysis shows an interesting agreement between the
analysed models and the data, as well as a reduction in the cosmological
tensions, that are one of the most heated debates of current cosmology.
Finally, we note a clear indication for an energy flux from dark matter to
dark energy, being $\alpha$ positive and also different from zero at
3$\sigma$.

## Acknowledgements

The authors are thankful to Saulo Carneiro and Winfried Zimdahl for a critical
reading. M.B. acknowledge Istituto Nazionale di Fisica Nucleare (INFN),
sezione di Napoli, iniziativa specifica QGSKY. We also acknowledge the use of
Monte Python package. This work was developed thanks to the use of the
National Observatory Data Center (CPDON).

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