# On the dynamics of a dark sector coupling

Weiqiang Yang<EMAIL_ADDRESS>Department of Physics, Liaoning Normal
University, Dalian, 116029, P. R. China  Supriya Pan
<EMAIL_ADDRESS>Department of Mathematics, Presidency
University, 86/1 College Street, Kolkata 700073, India  Olga Mena
<EMAIL_ADDRESS>Instituto de Física Corpuscular (IFIC), University of
Valencia-CSIC, Parc Científic UV, c/ Catedrático José Beltrán 2, E-46980
Paterna, Spain  Eleonora Di Valentino<EMAIL_ADDRESS>School of
Mathematics and Statistics, University of Sheffield, Hounsfield Road,
Sheffield S3 7RH, United Kingdom

###### Abstract

Interacting dark energy models may play a crucial role in explaining several
important observational issues in modern cosmology and also may provide a
solution to current cosmological tensions. Since the phenomenology of the dark
sector could be extremely rich, one should not restrict the interacting models
to have a coupling parameter which is constant in cosmic time, rather allow
for its dynamical behavior, as it is common practice in the literature when
dealing with other dark energy properties, as the dark energy equation of
state. We present here a compendium of the current cosmological constraints on
a large variety of interacting models, investigating scenarios where the
coupling parameter of the interaction function and the dark energy equation of
state can be either constant or dynamical. For the most general schemes, in
which both the coupling parameter of the interaction function and the dark
energy equation of state are dynamical, we find $95\%$ CL evidence for a dark
energy component at early times and slightly milder evidence for a dynamical
dark coupling for the most complete observational data set exploited here,
which includes CMB, BAO and Supernova Ia measurements. Interestingly, there
are some cases where a dark energy component different from the cosmological
constant case at early times together with a coupling different from zero
today, can alleviate both the $H_{0}$ and $S_{8}$ tension for the full dataset
combination considered here. Due to the energy exchange among the dark
sectors, the current values of the matter energy density and of the clustering
parameter $\sigma_{8}$ are shifted from their $\Lambda$CDM-like values. This
fact makes future surveys, especially those focused on weak lensing
measurements, unique tools to test the nature and the couplings of the dark
energy sector.

Dark matter, dark energy, interacting cosmologies, cosmological observations

###### Contents

  1. I Introduction
  2. II Interacting dark sector scenarios
    1. II.1 Constant coupling
    2. II.2 Dynamical coupling
  3. III Methodology
  4. IV Numerical analyses
    1. IV.1 Constant coupling $\xi_{0}$
      1. IV.1.1 Constant dark energy equation of state
      2. IV.1.2 Dynamical dark energy equation of state
    2. IV.2 Dynamical coupling $\xi(a)$
      1. IV.2.1 Constant dark energy equation of state
      2. IV.2.2 Dynamical dark energy equation of state
  5. V Discussion and Summary
  6. VI Acknowledgments

##  I Introduction

Interacting dark energy cosmologies Amendola (2000) are very appealing
scenarios where to alleviate current cosmological problems, such as the cosmic
coincidence (i.e. the so-called _why now?_ problem) Wetterich (1995); Huey and
Wandelt (2006); Cai and Wang (2005); Pavon and Zimdahl (2005); Berger and
Shojaei (2006); del Campo  _et al._ (2006, 2008, 2009) and cosmological
tensions, such as the one between CMB estimates Aghanim  _et al._ (2020a);
Aiola  _et al._ (2020); Dutcher  _et al._ (2021) and SH0ES (Supernovae and
$H_{0}$ for the Equation of State of dark energy) measurements of the Hubble
constant Riess  _et al._ (2021, 2022), with a significance of $\sim 5.3\sigma$
(see also Refs. Verde  _et al._ (2019); Knox and Millea (2020); Di Valentino
_et al._ (2021a); Jedamzik  _et al._ (2021); Riess (2019); Di Valentino
(2021); Di Valentino  _et al._ (2021b); Perivolaropoulos and Skara (2022);
Schöneberg  _et al._ (2022); Shah  _et al._ (2021); Di Valentino (2022);
Abdalla  _et al._ (2022)). Therefore, over the last several years, the
intriguing possibility of an interaction between the dark matter and dark
energy fluids has been thoroughly investigated using different available
cosmological observations Barrow and Clifton (2006); Valiviita  _et al._
(2008); Gavela  _et al._ (2009, 2010); Li and Zhang (2014); Salvatelli  _et
al._ (2014); Wang  _et al._ (2014); Casas  _et al._ (2016); van de Bruck  _et
al._ (2016); Yang  _et al._ (2016); Kumar and Nunes (2016); van de Bruck  _et
al._ (2017); Pan and Sharov (2017); Yang  _et al._ (2017a); Sharov  _et al._
(2017); Kumar and Nunes (2017); Mifsud and Van De Bruck (2017); Van De Bruck
and Mifsud (2018); Yang  _et al._ (2017b, 2018a); Pan  _et al._ (2018); Yang
_et al._ (2019a, b, 2018b, 2018c); Li  _et al._ (2018); Yang  _et al._ (2018d,
2019c, 2020a); Li  _et al._ (2020); Di Valentino  _et al._ (2020a); Oikonomou
(2019); Di Valentino  _et al._ (2021c); Martinelli  _et al._ (2019); Kumar
_et al._ (2019); Cheng  _et al._ (2020); Lucca and Hooper (2020); Cid  _et
al._ (2021); Beltrán Jiménez  _et al._ (2021); Thipaksorn  _et al._ (2022);
Wang  _et al._ (2022); Kumar (2021); Nunes and Di Valentino (2021); Di
Valentino  _et al._ (2017, 2020b); Yang  _et al._ (2019d, 2020b); Di Valentino
and Mena (2020); Gómez-Valent  _et al._ (2020); Yang  _et al._ (2020c); Pan
_et al._ (2020a); Yao and Meng (2020a, b); Mifsud and van de Bruck (2019); Pan
_et al._ (2019); Paliathanasis  _et al._ (2019); Yang  _et al._ (2019e,
2020d); Kang (2021); da Fonseca  _et al._ (2022); Yang  _et al._ (2021); Lucca
(2021a, b); Gariazzo  _et al._ (2022); Paliathanasis  _et al._ (2021); Mawas
_et al._ (2021); Harko  _et al._ (2022); Nunes  _et al._ (2022); Yengejeh  _et
al._ (2022); Landim (2022); Yao and Meng (2022); Gómez-Valent  _et al._ (2022)
(also see Bolotin  _et al._ (2015); Wang  _et al._ (2016), two review articles
on interacting dark energy models).

The basic underlying idea in these theories relies on the possible non-
gravitational interaction between dark matter (DM) and dark energy (DE). Such
an interaction can be characterized by a continuous flow of energy and/or
momentum between these dark sectors. This energy flow modifies the expansion
history of the universe both at the background and perturbation levels.

The interaction function, also known as the coupling function, is the main
feature of interacting dark energy theories: once the interaction function is
prescribed, the dynamics of the universe can be determined either analytically
or numerically. Despite the fact that in the literature it is usually assumed
a pure phenomenological approach for the dark sectors coupling, a class of
interaction functions can be derived from the field theory perspective van de
Bruck and Morrice (2015); Boehmer  _et al._ (2015a, b); Gleyzes  _et al._
(2015); D’Amico  _et al._ (2016); Pan  _et al._ (2020b); Chatzidakis  _et al._
(2022). Thus, theoretically, the interaction functions can also be well-
motivated.

The most exploited parameterization of the interaction function is $Q=3H\xi
f(\rho_{\rm DM},\rho_{\rm DE})$ Gavela  _et al._ (2009), where $\xi$ is the
coupling parameter that characterizes the strength of the interaction
function, $H$ is the Hubble parameter of the FLRW universe and $f(\rho_{\rm
DM},\rho_{\rm DE})$ is any continuous function of $\rho_{\rm DM}$, $\rho_{\rm
DE}$, i.e. of the energy densities of dark matter and dark energy. For
simplicity, the coupling parameter, $\xi$, is commonly assumed to be
independent of the cosmic time. This economical assumption has been however
questioned in Refs. Li and Zhang (2011); Wang  _et al._ (2014); Yang  _et al._
(2019e, 2020d), where it has been argued that there is no fundamental symmetry
in nature forcing that coupling to be zero or constant. The assumption of a
constant coupling parameter, together with a constant equation of state for
the dark energy component provide only a minimal picture of a (possibly) very
rich dark sector dynamics. A mandatory exercise is therefore to understand the
quality of the fit of the different interacting scenarios to the observational
constraints, assessing the preference for a constant _versus_ a dynamical
coupling parameter in scenarios where the dark energy equation of state is
either constant or dynamical. We follow a bottom up approach here, increasing
step by step the complexity of the dark sectors, starting from the simplest
scenario to arrive to the most general one. The main purpose is to scrutinize
the quality of the fitting of the possible scenarios to the observational
data, devoting special attention to their parameter degeneracies and their
ability in solving the present cosmological tensions.

The study is presented as follows. Sec. II presents the basic equations of
interacting scenarios. Sec. III contains the description of the observational
data and the fitting methodology applied to constrain the interacting models.
In Sec. IV we discuss the results in the different scenarios, ordered from
less to more complexity. Finally, we conclude the study in Sec. V, discussing
some remarks relevant for future work in interacting dark sector cosmologies.

##  II Interacting dark sector scenarios

We consider a homogeneous and isotropic description of our universe which is
well described by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric:

$\displaystyle d{\rm
s}^{2}=-dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\right]~{},$

expressed in terms of the comoving coordinates $(t,r,\theta,\phi)$ where
$a(t)$ refers to the expansion scale factor of the universe and $k$ is the
curvature scalar. The curvature scalar may take three distinct values,
$k=0,~{}+1,~{}-1$, to represent three different geometries of the Universe,
spatially flat, closed and open, respectively. In the following, we shall work
in a flat universe with an interaction between the pressure-less dark
matter/cold dark matter (abbreviated as CDM) and the dark energy components,
governed by the conservation equation:

$\displaystyle\nabla^{\mu}(T^{\rm CDM}_{\mu\nu}+T^{\rm DE}_{\mu\nu})=0~{},$
(1)

which can be decoupled into two equations with the introduction of an
interaction function $Q(t)$ as follows

$\displaystyle\dot{\rho}_{\rm CDM}+3H\rho_{\rm CDM}=-Q(t)~{},$ (2)
$\displaystyle\dot{\rho}_{\rm DE}+3H(1+w)\rho_{\rm DE}=Q(t)~{};$ (3)

where $H\equiv\dot{a}/a$ is the Hubble rate of a FLRW flat universe and $w$
denotes the equation of state of the dark energy component. Let us note that
for $Q(t)>0$, the energy transfer occurs from CDM to DE, while $Q<0$ indicates
the transfer of energy in the reverse direction, i.e. from DE to CDM. In this
work we shall consider the following two well-known interaction functions:

$\displaystyle{\rm Model\;1:}\;Q_{A}=3H\xi\rho_{\rm DE}~{},$ (4)
$\displaystyle{\rm Model\;2:}\;Q_{B}=3H\xi\left(\frac{\rho_{\rm CDM}\rho_{\rm
DE}}{\rho_{\rm CDM}+\rho_{\rm DE}}\right)~{};$ (5)

where $\xi$ is a coupling parameter of the interaction functions and it could
be either time-dependent or time-independent. Now, in agreement with the sign
convention of the interaction function, $\xi>0$ (equivalently, $Q>0$) means an
energy transfer from CDM to DE and $\xi<0$ means that the energy flow is from
DE to CDM. Notice that for some interaction models the energy density of dark
matter and/or of dark energy could be negative Pan  _et al._ (2020c). In this
regard we do not impose any further constraints on the fluids and let the data
discriminate between the most observationally favoured scenarios. We believe
this approach is appropriate because it avoids unwanted biases and also
considers the rising interest in the community in the putative presence of a
negative cosmological constant Delubac  _et al._ (2015); Poulin  _et al._
(2018); Wang  _et al._ (2018); Visinelli  _et al._ (2019); Calderón  _et al._
(2021). We therefore extract the constraints on the cosmological parameters,
assuming both the choices of the interaction functions, namely, Model 1 of Eq.
(4) and Model 2 of Eq. (5), where the coupling parameter of these two
interaction models can be either constant or dynamical. Notice that, as the
dynamics of the interacting cosmologies will also be controlled by the dark
energy equation of state parameter $w$, different choices of the dark energy
equation of state parameter will result into different constraints for the
model parameters. Therefore, for completeness, it is essential to allow for
freedom in the dark energy equation of state parameter. In what follows we
shall specify the models that we will study in this work.

###  II.1 Constant coupling

The first scenario is the one where a vacuum energy component, characterized
by the equation of state $w=-1$, interacts with cold dark matter via the
interaction functions $Q_{A}$ and $Q_{B}$, see Eqs. (4) and (5) respectively,
where $\xi$, the coupling parameter, is constant in cosmic time (redshift). We
refer to these interactions as “$\xi_{0}$IVS$Q_{A}$” and
“$\xi_{0}$IVS$Q_{B}$”.

The next interacting scenario is the one where the dark energy component, with
a constant equation of state $w$ other than $w=-1$, interacts with cold dark
matter through the interaction functions $Q_{A}$ or $Q_{B}$. We refer to these
scenarios as “$\xi_{0}w$IDE$Q_{A,B}$”. Due to the stability criteria of
interacting cosmologies Gavela  _et al._ (2009), we have divided the models
into two regions, accordingly to the values of the dark energy equation of
state: quintessence ($w>-1$) and phantom ($w<-1$) models. We therefore divide
the corresponding interacting scenarios as “$\xi_{0}w_{q}$IDE$Q_{A,B}$” and
“$\xi_{0}w_{p}$IDE$Q_{A,B}$”, where the subscripts $q$ and $p$ of $w$ denote
that the dark energy equation of state is either in the quintessence
($w_{q}>-1$) or in the phantom ($w_{p}<-1$) regimes.

The most generalized interacting scenario in this category is the one where
the dark energy equation of state is dynamical. We shall assume the well-known
CPL dark energy parametrization $w(a)=w_{0}+w_{a}(1-a)$ Chevallier and
Polarski (2001); Linder (2003), which we reformulate as Valiviita  _et al._
(2010):

$\displaystyle w(a)=w_{0}\;a+w_{e}(1-a),$ (6)

in which it is straightforward to identify $w_{e}=w_{0}+w_{a}$, providing the
value of $w(a)$ in the early times, and $w_{0}$ giving the current value of
$w(a)$. In order to ensure the stability of the interaction models, we shall
consider, in the following, two different cases: (i)
$\xi_{0}>0,w_{0}>-1,w_{e}>-1$ (that is, where $w(a)$ remains in the
quintessence regime); and (ii) $\xi_{0}<0,w_{0}<-1,w_{e}<-1$ (i.e. where
$w(a)$ remains in the phantom regime). Thus, we divide the corresponding
interacting scenarios as “$\xi_{0}w_{0}^{q}w_{e}^{q}$IDE$Q_{A,B}$” and
“$\xi_{0}w_{0}^{p}w_{e}^{p}$IDE$Q_{A,B}$” where the subscripts $q$ and $p$ of
$w$ denote that the dark energy equation of state remains within the
quintessence ($w_{q}>-1$) or the phantom ($w_{p}<-1$) regime.

###  II.2 Dynamical coupling

In this study we shall also consider interacting scenarios described by the
interaction functions $Q_{A}$, Eq. (4), and $Q_{B}$, Eq. (5), where the
coupling parameter $\xi$ is dynamical and takes the following simple form Yang
_et al._ (2019e):

$\displaystyle\xi(a)=\xi_{0}+\xi_{a}\;(1-a)~{},$ (7)

where $\xi_{0}$ and $\xi_{a}$ are both constant parameters.

In the very same spirit than in the constant coupling case, we shall explore
one scenario in which a vacuum energy component with $w=-1$ interacts with
cold dark matter (labelled as “$\xi_{0}\xi_{a}$IVS$Q_{A,B}$”) and another one
in which a dark energy fluid with a constant equation of state $w$ other than
$w=-1$, interacts with the cold dark matter through the interaction functions
$Q_{A}$ and $Q_{B}$. Following the stability criteria, we shall consider the
following cases: (i) $\xi_{0}>0,\xi_{a}>0$ and $w>-1$; and (ii)
$\xi_{0}<0,\xi_{a}<0$ and $w<-1$. We divide the corresponding different
parameter spaces as “$\xi_{0}\xi_{a}w_{q,p}$IDE$Q_{A}$” and
“$\xi_{0}\xi_{a}w_{q,p}$IDE$Q_{B}$”, where the subscripts $q$ and $p$ of $w$
denote that the dark energy equation of state is lying in the quintessence
($w_{q}>-1$) or in the phantom ($w_{p}<-1$) regimes.

Finally, we consider the most general interacting scenario, for which both the
interaction rate $\xi(a)$ and the dark energy equation of state $w(a)$ are
dynamical. We shall use the CPL parameterization for $w(a)$ before described.

The stability criteria also defines in this case certain allowed regions in
the parameter space: (i) $\xi_{0}>0,\xi_{a}>0;w_{0}>-1,w_{e}>-1$ (i.e. where
$w_{0}$ and $w_{e}$ remain in the quintessence regimes); and (ii)
$\xi_{0}<0,\xi_{a}<0;w_{0}<-1,w_{e}<-1$ (i.e. where $w_{0}$ and $w_{e}$ remain
in the phantom regimes). The interacting scenarios will be defined as
“$w_{0}^{q,p}w_{e}^{q,p}\xi_{0}\xi_{a}$IDE$Q_{A,B}$” where the subscripts $q$
and $p$ of $w$ refer to either the quintessence ($w_{q}>-1$) or in the phantom
($w_{p}<-1$) regimes and $A,B$ to the interacting model $Q_{A}$ or $Q_{B}$,
respectively.

##  III Methodology

In the following we shall briefly describe the observational datasets that are
used to constrain all the interacting scenarios previously detailed.

  * •

Cosmic Microwave Background (CMB): we shall consider the CMB measurements from
Planck 2018 final release Aghanim  _et al._ (2020a, b).

  * •

Baryon Acoustic Oscillation (BAO) distance measurements: measurements of BAO
data from a number of large scale surveys are also considered in our analyses,
namely, those from 6dFGS Beutler  _et al._ (2011), SDSS-MGS Ross  _et al._
(2015), and BOSS DR12 Alam  _et al._ (2017) (as used by the Planck 2018
collaboration Aghanim  _et al._ (2020a)).

  * •

Supernovae Type Ia (Pantheon): the Pantheon sample of the Supernovae Type Ia
comprising 1048 data points in the redshift interval $z\in[0.01,2.3]$ Scolnic
_et al._ (2018) have also been included in the analyses.

To derive cosmological constraints on the different interacting scenarios
above detailed, we make use of a modified version of the publicly available
Monte-Carlo Markov Chain package CosmoMC Lewis and Bridle (2002). The CosmoMC
package is furnished with a convergence diagnostic based on the Gelman and
Rubin statistic Gelman and Rubin (1992). In Table 1 we have shown the flat
priors on various free parameters that we have considered during the
statistical analyses.

Parameter | IVS | Quintessence | Phantom
---|---|---|---
$\Omega_{\rm b}h^{2}$ | $[0.005\,,\,0.1]$ | $[0.005\,,\,0.1]$ | $[0.005\,,\,0.1]$
$\Omega_{\rm c}h^{2}$ | $[0.001\,,\,0.99]$ | $[0.001\,,\,0.99]$ | $[0.001\,,\,0.99]$
$100\,\theta_{\rm{MC}}$ | $[0.5\,,\,10]$ | $[0.5\,,\,10]$ | $[0.5\,,\,10]$
$\tau$ | $[0.01\,,\,0.8]$ | $[0.01\,,\,0.8]$ | $[0.01\,,\,0.8]$
$\log(10^{10}A_{\rm s})$ | $[1.61\,,\,3.91]$ | $[1.61\,,\,3.91]$ | $[1.61\,,\,3.91]$
$n_{s}$ | $[0.8\,,\,1.2]$ | $[0.8\,,\,1.2]$ | $[0.8\,,\,1.2]$
$w_{0}$ | - | $[-1\,,\,1]$ | $[-3\,,\,-1]$
$w_{e}$ | - | $[-1\,,\,1]$ | $[-3\,,\,-1]$
$\xi_{0}$ | $[-1\,,\,1]$ | $[-1\,,\,0]$ | $[0\,,\,1]$
$\xi_{a}$ | $[-1\,,\,1]$ | $[-1\,,\,0]$ | $[0\,,\,1]$
Table 1: List of the parameters varied and the external flat priors assumed.

##  IV Numerical analyses

###  IV.1 Constant coupling $\xi_{0}$

This section covers the observational constraints extracted out of all the
interacting scenarios when the coupling parameter is constant.

Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $<0.093\,<0.134$ | $0.100_{-0.016-0.038}^{+0.022+0.035}$ | $0.108_{-0.011-0.023}^{+0.013+0.022}$ | $0.1121_{-0.0070-0.015}^{+0.0080+0.015}$
$\Omega_{b}h^{2}$ | $0.02230_{-0.00015-0.00029}^{+0.00014+0.00030}$ | $0.02233_{-0.00014-0.00027}^{+0.00014+0.00028}$ | $0.02230_{-0.00014-0.00029}^{+0.00015+0.00028}$ | $0.02236_{-0.00015-0.00027}^{+0.00014+0.00028}$
$100\theta_{MC}$ | $1.0441_{-0.0041-0.0049}^{+0.0026+0.0055}$ | $1.0419_{-0.0013-0.0021}^{+0.0009+0.0023}$ | $1.04136_{-0.00080-0.0014}^{+0.00066+0.0015}$ | $1.04132_{-0.00076-0.0014}^{+0.00059+0.0014}$
$\tau$ | $0.0542_{-0.0079-0.015}^{+0.0075+0.015}$ | $0.0555_{-0.0083-0.016}^{+0.0076+0.016}$ | $0.0542_{-0.0079-0.016}^{+0.0083+0.016}$ | $0.0558_{-0.0078-0.015}^{+0.0077+0.016}$
$n_{s}$ | $0.9723_{-0.0044-0.0081}^{+0.0043+0.0083}$ | $0.9734_{-0.0040-0.0078}^{+0.0040+0.0079}$ | $0.9720_{-0.0043-0.0085}^{+0.0042+0.0084}$ | $0.9738_{-0.0043-0.0075}^{+0.0038+0.0082}$
${\rm{ln}}(10^{10}A_{s})$ | $3.054_{-0.016-0.030}^{+0.015+0.031}$ | $3.056_{-0.017-0.032}^{+0.016+0.032}$ | $3.055_{-0.016-0.031}^{+0.016+0.033}$ | $3.057_{-0.016-0.032}^{+0.016+0.032}$
$\xi_{0}$ | $0.13_{-0.08-0.20}^{+0.14+0.17}$ | $0.059_{-0.061-0.10}^{+0.053+0.11}$ | $0.039_{-0.036-0.069}^{+0.035+0.070}$ | $0.032_{-0.035-0.069}^{+0.034+0.071}$
$\Omega_{m0}$ | $0.19_{-0.14-0.17}^{+0.08+0.19}$ | $0.261_{-0.046-0.099}^{+0.056+0.095}$ | $0.283_{-0.033-0.064}^{+0.033+0.062}$ | $0.285_{-0.028-0.060}^{+0.032+0.060}$
$\sigma_{8}$ | $1.6_{-1.1-1.3}^{+0.3+2.2}$ | $0.97_{-0.20-0.28}^{+0.09+0.36}$ | $0.895_{-0.098-0.15}^{+0.063+0.17}$ | $0.878_{-0.092-0.15}^{+0.056+0.17}$
$H_{0}$ [km/s/Mpc] | $70.8_{-2.5-5.9}^{+4.3+5.3}$ | $68.8_{-1.5-2.6}^{+1.3+2.78}$ | $68.0_{-1.0-1.9}^{+1.0+2.0}$ | $68.17_{-0.92-1.6}^{+0.81+1.7}$
$S_{8}$ | $1.06_{-0.30-0.35}^{+0.10+0.54}$ | $0.881_{-0.071-0.10}^{+0.035+0.13}$ | $0.864_{-0.038-0.063}^{+0.028+0.067}$ | $0.851_{-0.038-0.062}^{+0.025+0.068}$
$r_{\rm{drag}}$ [Mpc] | $147.08_{-0.30-0.59}^{+0.31+0.61}$ | $147.17_{-0.28-0.52}^{+0.27+0.55}$ | $147.08_{-0.30-0.58}^{+0.29+0.59}$ | $147.23_{-0.27-0.56}^{+0.27+0.53}$
Table 2: Observational constraints on the interacting vacuum scenario “$\xi_{0}$IVS$Q_{A}$” obtained from several observational datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are presented.  Figure 1: One-dimensional marginalized posterior distributions and two-dimensional joint contours for the most relevant parameters of the interacting vacuum scenario “$\xi_{0}$IVS$Q_{A}$” for several datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon.  Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.120_{-0.033-0.078}^{+0.057+0.070}$ | $0.111_{-0.010-0.023}^{+0.012+0.020}$ | $0.1125_{-0.0086-0.018}^{+0.0093+0.017}$ | $0.1121_{-0.0070-0.015}^{+0.0080+0.015}$
$\Omega_{b}h^{2}$ | $0.02230_{-0.00015-0.00031}^{+0.00015+0.00031}$ | $0.02234_{-0.00015-0.00029}^{+0.00015+0.00030}$ | $0.02230_{-0.00015-0.00030}^{+0.00015+0.00029}$ | $0.02235_{-0.00014-0.00028}^{+0.00014+0.00028}$
$100\theta_{MC}$ | $1.0408_{-0.0032-0.0039}^{+0.0017+0.0047}$ | $1.04120_{-0.00071-0.0012}^{+0.00060+0.0013}$ | $1.04107_{-0.00057-0.0011}^{+0.00056+0.0012}$ | $1.04115_{-0.00051-0.00097}^{+0.00045+0.00099}$
$\tau$ | $0.0544_{-0.0083-0.016}^{+0.0077+0.016}$ | $0.0554_{-0.0082-0.015}^{+0.0075+0.017}$ | $0.0543_{-0.0081-0.015}^{+0.0075+0.016}$ | $0.0553_{-0.0082-0.016}^{+0.0076+0.016}$
$n_{s}$ | $0.9721_{-0.0042-0.0084}^{+0.0042+0.0084}$ | $0.9734_{-0.0042-0.0080}^{+0.0042+0.0082}$ | $0.9723_{-0.0044-0.0085}^{+0.0045+0.0091}$ | $0.9734_{-0.0042-0.0079}^{+0.0042+0.0081}$
${\rm{ln}}(10^{10}A_{s})$ | $3.055_{-0.017-0.032}^{+0.016+0.032}$ | $3.056_{-0.017-0.032}^{+0.016+0.033}$ | $3.055_{-0.015-0.032}^{+0.015+0.032}$ | $3.056_{-0.016-0.033}^{+0.016+0.03}$
$\xi_{0}$ | $0.01_{-0.41-0.54}^{+0.24+0.63}$ | $0.062_{-0.093-0.15}^{+0.069+0.17}$ | $0.055_{-0.070-0.12}^{+0.058+0.13}$ | $0.052_{-0.061-0.11}^{+0.050+0.11}$
$\Omega_{m0}$ | $0.35_{-0.22-0.27}^{+0.12+0.31}$ | $0.288_{-0.034-0.065}^{+0.034+0.065}$ | $0.294_{-0.028-0.054}^{+0.027+0.056}$ | $0.291_{-0.022-0.044}^{+0.022+0.044}$
$\sigma_{8}$ | $0.82_{-0.21-0.26}^{+0.12+0.30}$ | $0.843_{-0.054-0.086}^{+0.043+0.098}$ | $0.841_{-0.041-0.072}^{+0.034+0.075}$ | $0.838_{-0.036-0.066}^{+0.031+0.067}$
$H_{0}$ [km/s/Mpc] | $66.3_{-6.1-11}^{+6.9+11}$ | $68.3_{-1.3-2.3}^{+1.2+2.5}$ | $67.9_{-1.1-2.2}^{+1.1+2.2}$ | $68.19_{-0.87-1.6}^{+0.79+1.6}$
$S_{8}$ | $0.823_{-0.021-0.12}^{+0.060+0.08}$ | $0.823_{-0.013-0.026}^{+0.013+0.026}$ | $0.830_{-0.015-0.030}^{+0.016+0.030}$ | $0.824_{-0.013-0.024}^{+0.013+0.024}$
$r_{\rm{drag}}$ [Mpc] | $147.08_{-0.30-0.59}^{+0.30+0.61}$ | $147.19_{-0.29-0.56}^{+0.29+0.57}$ | $147.09_{-0.30-0.59}^{+0.30+0.60}$ | $147.20_{-0.28-0.55}^{+0.28+0.54}$
Table 3: Observational constraints on the interacting vacuum scenario “$\xi_{0}$IVS$Q_{B}$” obtained from several observational datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are presented.  Figure 2:  One-dimensional marginalized posterior distributions and two-dimensional joint contours for the most relevant parameters of the interacting vacuum scenario “$\xi_{0}$IVS$Q_{B}$” for several datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon.  Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.134_{-0.013-0.015}^{+0.007+0.017}$ | $0.1352_{-0.0098-0.014}^{+0.0099+0.014}$ | $0.135_{-0.011-0.014}^{+0.009+0.015}$ | $0.1350_{-0.0098-0.014}^{+0.0098+0.014}$
$\Omega_{b}h^{2}$ | $0.02239_{-0.00015-0.00030}^{+0.00015+0.00030}$ | $0.02239_{-0.00014-0.00028}^{+0.00014+0.00028}$ | $0.02236_{-0.00015-0.00029}^{+0.00015+0.00029}$ | $0.02239_{-0.00014-0.00027}^{+0.00014+0.00028}$
$100\theta_{MC}$ | $1.04018_{-0.00054-0.0011}^{+0.00065+0.0011}$ | $1.04013_{-0.00055-0.0010}^{+0.00055+0.0010}$ | $1.04010_{-0.00055-0.00099}^{+0.00055+0.00098}$ | $1.04015_{-0.00055-0.00099}^{+0.00054+0.00099}$
$\tau$ | $0.0540_{-0.0074-0.015}^{+0.0074+0.015}$ | $0.0548_{-0.0079-0.015}^{+0.0073+0.016}$ | $0.0543_{-0.0075-0.015}^{+0.0075+0.016}$ | $0.0549_{-0.0075-0.015}^{+0.0075+0.016}$
$n_{s}$ | $0.9652_{-0.0044-0.0086}^{+0.0043+0.0086}$ | $0.9657_{-0.0042-0.0082}^{+0.0042+0.0081}$ | $0.9647_{-0.0044-0.0084}^{+0.0043+0.0086}$ | $0.9659_{-0.0040-0.0077}^{+0.0040+0.0078}$
${\rm{ln}}(10^{10}A_{s})$ | $3.043_{-0.015-0.031}^{+0.015+0.029}$ | $3.045_{-0.016-0.030}^{+0.015+0.032}$ | $3.045_{-0.015-0.031}^{+0.015+0.032}$ | $3.045_{-0.015-0.031}^{+0.015+0.032}$
$w_{p}$ | $-1.58_{-0.34-0.44}^{+0.21+0.49}$ | $-1.094_{-0.040}^{+0.070}\;>-1.193$ | $-1.087_{-0.041-0.076}^{+0.049+0.085}$ | $-1.080_{-0.038-0.072}^{+0.047+0.079}$
$\xi_{0}$ | $>-0.051>-0.090$ | $-0.052_{-0.023}^{+0.046}\;>-0.100$ | $-0.051_{-0.021}^{+0.047}\;>-0.101$ | $-0.052_{-0.025}^{+0.044}\;>-0.101$
$\Omega_{m0}$ | $0.226_{-0.072-0.09}^{+0.031+0.11}$ | $0.336_{-0.025-0.042}^{+0.025+0.044}$ | $0.339_{-0.024-0.040}^{+0.024+0.041}$ | $0.339_{-0.023-0.036}^{+0.023+0.037}$
$\sigma_{8}$ | $0.886_{-0.089-0.16}^{+0.086+0.15}$ | $0.760_{-0.047-0.070}^{+0.038+0.074}$ | $0.761_{-0.046-0.064}^{+0.035+0.068}$ | $0.756_{-0.046-0.063}^{+0.034+0.067}$
$H_{0}$ [km/s/Mpc] | $85_{-7-17}^{+13+16}$ | $68.7_{-1.5-2.5}^{+1.1+2.7}$ | $68.3_{-1.0-1.9}^{+1.0+2.0}$ | $68.32_{-0.78-1.5}^{+0.77+1.6}$
$S_{8}$ | $0.756_{-0.034-0.063}^{+0.034+0.064}$ | $0.802_{-0.020-0.036}^{+0.020+0.036}$ | $0.807_{-0.021-0.039}^{+0.021+0.040}$ | $0.802_{-0.019-0.035}^{+0.019+0.036}$
$r_{\rm{drag}}$ [Mpc] | $147.08_{-0.29-0.58}^{+0.29+0.57}$ | $147.11_{-0.28-0.54}^{+0.27+0.54}$ | $147.05_{-0.29-0.56}^{+0.29+0.58}$ | $147.14_{-0.27-0.52}^{+0.27+0.52]}$
Table 4: Observational constraints on the interacting scenario “$\xi_{0}w_{p}$IDE$Q_{A}$” obtained from several observational datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are presented.  Figure 3: One dimensional marginalized posterior distributions and two-dimensional joint contours for the most relevant parameters of the interacting scenario “$\xi_{0}w_{p}$IDE$Q_{A}$” for several datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon.  Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $<0.083\;<0.115$ | $0.076_{-0.018-0.058}^{+0.037+0.046}$ | $0.074_{-0.019-0.057}^{+0.038+0.046}$ | $0.077_{-0.016-0.058}^{+0.036+0.044}$
$\Omega_{b}h^{2}$ | $0.02236_{-0.00015-0.00029}^{+0.00015+0.00029}$ | $0.02239_{-0.00014-0.00027}^{+0.00014+0.00029}$ | $0.02234_{-0.00015-0.00029}^{+0.00015+0.00029}$ | $0.02238_{-0.00014-0.00027}^{+0.00014+0.00028}$
$100\theta_{MC}$ | $1.0448_{-0.0037-0.0043}^{+0.0019+0.0050}$ | $1.0438_{-0.0025-0.0031}^{+0.0010+0.0043}$ | $1.0438_{-0.0026-0.0032}^{+0.0011+0.0043}$ | $1.0437_{-0.0024-0.0031}^{+0.0009+0.0043}$
$\tau$ | $0.0544_{-0.0074-0.015}^{+0.0071+0.016}$ | $0.0550_{-0.0079-0.015}^{+0.0077+0.017}$ | $0.0538_{-0.0075-0.015}^{+0.0074+0.016}$ | $0.0547_{-0.0073-0.015}^{+0.0074+0.016}$
$n_{s}$ | $0.9650_{-0.0042-0.0084}^{+0.0042+0.0084}$ | $0.9658_{-0.0041-0.0084}^{+0.0042+0.0082}$ | $0.9643_{-0.0043-0.0082}^{+0.0043+0.0085}$ | $0.9660_{-0.0040-0.0079}^{+0.0040+0.0078}$
${\rm{ln}}(10^{10}A_{s})$ | $3.045_{-0.015-0.031}^{+0.015+0.031}$ | $3.045_{-0.016-0.032}^{+0.016+0.034}$ | $3.044_{-0.015-0.032}^{+0.016+0.033}$ | $3.044_{-0.015-0.031}^{+0.015+0.033}$
$w_{q}$ | $<-0.897\;<-0.768$ | $<-0.892\;<-0.801$ | $<-0.879\;<-0.793$ | $<-0.887\;<-0.792$
$\xi_{0}$ | $0.15_{-0.07}^{+0.11}\;<0.27$ | $0.119_{-0.090}^{+0.054}\;<0.242$ | $0.125_{-0.090-0.12}^{+0.056+0.13}$ | $0.115_{-0.087-0.12}^{+0.047+0.12}$
$\Omega_{m0}$ | $0.18_{-0.12-0.15}^{+0.07+0.16}$ | $0.212_{-0.047-0.13}^{+0.083+0.11}$ | $0.210_{-0.047-0.13}^{+0.084+0.11}$ | $0.215_{-0.038-0.13}^{+0.080+0.10}$
$\sigma_{8}$ | $1.6_{-1.0-1.2}^{+0.2+2.1}$ | $1.25_{-0.50-0.7}^{+0.09+1.1}$ | $1.27_{-0.51-0.6}^{+0.09+1.1}$ | $1.22_{-0.46-0.6}^{+0.07+1.1}$
$H_{0}$ [km/s/Mpc] | $69.3_{-2.8-6.5}^{+3.9+6.2}$ | $68.4_{-1.4-2.5}^{+1.3+2.7}$ | $68.1_{-1.0-2.0}^{+1.0+2.1}$ | $68.2_{-0.8-1.5}^{+0.8+1.6}$
$S_{8}$ | $1.07_{-0.27-0.32}^{+0.08+0.52}$ | $0.97_{-0.16-0.21}^{+0.04+0.35}$ | $0.99_{-0.17-0.21}^{+0.04+0.35}$ | $0.97_{-0.15-0.20}^{+0.03+0.33}$
$r_{\rm{drag}}$ [Mpc] | $147.04_{-0.29-0.58}^{+0.30+0.59}$ | $147.12_{-0.27-0.52}^{+0.27+0.55}$ | $147.03_{-0.29-0.58}^{+0.29+0.58}$ | $147.15_{-0.27-0.54}^{+0.26+0.52}$
Table 5: Observational constraints on the interacting scenario “$\xi_{0}w_{q}$IDE$Q_{A}$” obtained from several observational datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are presented.  Figure 4: One dimensional marginalized posterior distributions and two-dimensional joint contours for the most relevant parameters of the interacting scenario “$\xi_{0}w_{q}$IDE$Q_{A}$” for several datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon.  Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.137_{-0.018-0.022}^{+0.007+0.032}$ | $0.136_{-0.016-0.019}^{+0.006+0.025}$ | $0.136_{-0.016-0.019}^{+0.006+0.027}$ | $0.135_{-0.014-0.018}^{+0.005+0.024}$
$\Omega_{b}h^{2}$ | $0.02238_{-0.00014-0.00029}^{+0.00014+0.00028}$ | $0.02236_{-0.00014-0.00029}^{+0.00014+0.00029}$ | $0.02234_{-0.00015-0.00029}^{+0.00015+0.00029}$ | $0.02237_{-0.00014-0.00027}^{+0.00014+0.00028}$
$100\theta_{MC}$ | $1.0400_{-0.0005-0.0017}^{+0.0010+0.0014}$ | $1.04008_{-0.00050-0.0015}^{+0.00084+0.0012}$ | $1.04003_{-0.00050-0.0015}^{+0.00087+0.0013}$ | $1.04016_{-0.00046-0.0014}^{+0.00080+0.0012}$
$\tau$ | $0.0544_{-0.0076-0.015}^{+0.0078+0.016}$ | $0.0552_{-0.0075-0.015}^{+0.0075+0.015}$ | $0.0548_{-0.0084-0.015}^{+0.0074+0.016}$ | $0.0555_{-0.0077-0.015}^{+0.0077+0.016}$
$n_{s}$ | $0.9645_{-0.0044-0.0083}^{+0.0044+0.0086}$ | $0.9644_{-0.0041-0.0081}^{+0.0043+0.0082}$ | $0.9636_{-0.0044-0.0085}^{+0.0043+0.0083}$ | $0.9649_{-0.0039-0.0077}^{+0.0039+0.0078}$
${\rm{ln}}(10^{10}A_{s})$ | $3.045_{-0.015-0.032}^{+0.015+0.032}$ | $3.046_{-0.015-0.031}^{+0.016+0.032}$ | $3.047_{-0.017-0.031}^{+0.016+0.033}$ | $3.047_{-0.016-0.031}^{+0.016+0.032}$
$w_{p}$ | $-1.67_{-0.33-0.41}^{+0.16+0.52}$ | $-1.111_{-0.044}^{+0.089}\;>-1.237$ | $-1.095_{-0.035}^{+0.070}\;>-1.20$ | $-1.084_{-0.032}^{+0.064}\;>-1.18$
$\xi_{0}$ | $>-0.368\;>-0.766$ | $>-0.309\;>-0.619$ | $>-0.306\;>-0.657$ | $>-0.277\;>-0.591$
$\Omega_{m0}$ | $0.216_{-0.064-0.08}^{+0.026+0.11}$ | $0.334_{-0.036-0.052}^{+0.020+0.061}$ | $0.342_{-0.037-0.050}^{+0.018+0.064}$ | $0.337_{-0.033-0.044}^{+0.015+0.056}$
$\sigma_{8}$ | $0.92_{-0.07-0.16}^{+0.10+0.15}$ | $0.774_{-0.032-0.092}^{+0.052+0.078}$ | $0.769_{-0.025-0.090}^{+0.053+0.070}$ | $0.771_{-0.024-0.084}^{+0.050+0.067}$
$H_{0}$ [km/s/Mpc] | $88_{-6-18}^{+12+14}$ | $69.0_{-1.5-2.7}^{+1.2+2.8}$ | $68.3_{-1.0-2.0}^{+1.0+2.0}$ | $68.37_{-0.85-1.5}^{+0.78+1.6}$
$S_{8}$ | $0.764_{-0.035-0.053}^{+0.024+0.060}$ | $0.814_{-0.015-0.036}^{+0.019+0.033}$ | $0.819_{-0.017-0.038}^{+0.020+0.037}$ | $0.816_{-0.014-0.036}^{+0.019+0.033}$
$r_{\rm{drag}}$ [Mpc] | $147.04_{-0.32-0.57}^{+0.30+0.59}$ | $147.07_{-0.28-0.54}^{+0.28+0.53}$ | $147.00_{-0.29-0.57}^{+0.29+0.56}$ | $147.10_{-0.26-0.52}^{+0.26+0.51}$
Table 6: Observational constraints on the interacting scenario “$\xi_{0}w_{p}$IDE$Q_{B}$” obtained from several observational datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are presented.  Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.1153_{-0.0026-0.0090}^{+0.0053+0.0072}$ | $0.1129_{-0.0028-0.0090}^{+0.0057+0.0074}$ | $0.1130_{-0.0033-0.0090}^{+0.0056+0.0079}$ | $0.1125_{-0.0031-0.0084}^{+0.0054+0.0074}$
$\Omega_{b}h^{2}$ | $0.02236_{-0.00015-0.00029}^{+0.00015+0.00030}$ | $0.02244_{-0.00014-0.00028}^{+0.00015+0.00027}$ | $0.02240_{-0.00014-0.00029}^{+0.00014+0.00029}$ | $0.02244_{-0.00014-0.00027}^{+0.00014+0.00027}$
$100\theta_{MC}$ | $1.04117_{-0.00042-0.00073}^{+0.00036+0.00080}$ | $1.04139_{-0.00040-0.00070}^{+0.00035+0.00078}$ | $1.04134_{-0.00041-0.00075}^{+0.00037+0.00078}$ | $1.04140_{-0.00037-0.00070}^{+0.00037+0.00075}$
$\tau$ | $0.0538_{-0.0074-0.016}^{+0.0074+0.016}$ | $0.0554_{-0.0075-0.014}^{+0.0074+0.016}$ | $0.0543_{-0.0076-0.016}^{+0.0078+0.016}$ | $0.0550_{-0.0083-0.015}^{+0.0073+0.016}$
$n_{s}$ | $0.9650_{-0.0043-0.0085}^{+0.0043+0.0087}$ | $0.9680_{-0.0040-0.0075}^{+0.0039+0.0081}$ | $0.9664_{-0.0043-0.0084}^{+0.0043+0.0084}$ | $0.9678_{-0.0038-0.0076}^{+0.0038+0.0077}$
${\rm{ln}}(10^{10}A_{s})$ | $3.044_{-0.015-0.032}^{+0.016+0.031}$ | $3.044_{-0.015-0.030}^{+0.015+0.032}$ | $3.043_{-0.016-0.032}^{+0.016+0.032}$ | $3.044_{-0.017-0.031}^{+0.016+0.033}$
$w_{q}$ | $<-0.894\;<-0.759$ | $<-0.955\;<-0.908$ | $<-0.972\;<-0.939$ | $<-0.973\;<-0.944$
$\xi_{0}$ | $<0.097\;<0.203$ | $<0.121\;<0.238$ | $<0.137\;<0.244$ | $<0.136\;<0.239$
$\Omega_{m0}$ | $0.327_{-0.033-0.058}^{+0.022+0.066}$ | $0.301_{-0.012-0.030}^{+0.017+0.027}$ | $0.299_{-0.013-0.031}^{+0.016+0.027}$ | $0.296_{-0.011-0.026}^{+0.015+0.024}$
$\sigma_{8}$ | $0.809_{-0.028-0.066}^{+0.031+0.061}$ | $0.826_{-0.027-0.044}^{+0.018+0.050}$ | $0.834_{-0.027-0.042}^{+0.017+0.046}$ | $0.832_{-0.027-0.040}^{+0.017+0.045}$
$H_{0}$ [km/s/Mpc] | $65.2_{-1.5-5.1}^{+2.9+4.1}$ | $67.30_{-0.76-1.8}^{+0.99+1.7}$ | $67.45_{-0.75-1.6}^{+0.84+1.5}$ | $67.72_{-0.61-1.2}^{+0.63+1.2}$
$S_{8}$ | $0.842_{-0.017-0.033}^{+0.018+0.034}$ | $0.826_{-0.014-0.027}^{+0.014+0.027}$ | $0.833_{-0.016-0.030}^{+0.016+0.032}$ | $0.826_{-0.013-0.026}^{+0.013+0.027}$
$r_{\rm{drag}}$ [Mpc] | $147.06_{-0.30-0.58}^{+0.29+0.58}$ | $147.28_{-0.26-0.53}^{+0.27+0.51}$ | $147.15_{-0.29-0.56}^{+0.28+0.55}$ | $147.26_{-0.25-0.50}^{+0.25+0.50}$
Table 7: Observational constraints on the interacting scenario
“$\xi_{0}w_{q}$IDE$Q_{B}$” obtained from several observational datasets,
namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are presented.

####  IV.1.1 Constant dark energy equation of state

The observational constraints on the interacting scenarios when the dark
energy represents the vacuum energy (i.e. $w=-1$) are shown in Tabs. 2 and 3
and Figs. 1 and 2.

The constraints arising from CMB, CMB + BAO, CMB + Pantheon, and CMB + BAO +
Pantheon observations on the $\xi_{0}$IVS$Q_{A}$ model are shown in Tab. 2.
Even if for CMB alone there is a very mild evidence for a non-zero coupling
($\xi_{0}=0.132_{-0.076}^{+0.142}$ at 68% CL), this evidence gets completely
diluted when we include BAO measurements ($\xi_{0}=0.059_{-0.061}^{+0.053}$ at
68% CL for CMB+BAO). A similar situation happens with $\Omega_{m0}$,
$\sigma_{8}$ and $H_{0}$. The large shifts in the mean values of these
parameters obtained in the case of CMB data alone disappear when adding BAO
information. Namely, the lower value of $\Omega_{m0}$ (due to the energy flow
from the dark matter sector to the dark energy one), the higher value of
$\sigma_{8}$ (to compensate for the lower value of the matter energy density)
and the higher value of the Hubble constant (to keep $\Omega_{m0}h^{2}$
unchanged) from CMB data alone analyses are restored to their $\Lambda$CDM
values after the addition of BAO in the data analyses, see Fig. 1. For the
CMB+Pantheon case, instead we observe a mild preference for a non-zero
coupling ($\xi_{0}=0.039_{-0.036}^{+0.035}$ at 68% CL) but with the inclusion
of BAO to CMB+Pantheon the coupling parameter is back in agreement with zero
($\xi_{0}=0.032_{-0.035}^{+0.034}$ at 68% CL).

Table 3 summarizes the observational constraints on various parameters of the
interacting scenario $\xi_{0}$IVS$Q_{B}$ for various datasets, namely, CMB
alone, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon. For all the datasets, the
coupling parameter is statistically consistent with a vanishing value, i.e.
with a non-interacting model within $1\sigma$. Notice that the shift in the
parameters is the opposite of what we found for $Q_{A}$, see Fig. 2. In this
case, the energy flow is such that the amount of dark matter today is higher,
as the energy transfer is from the dark energy sector to the dark matter one.
Consequently, the mean value of $\Omega_{m0}$ is higher and those of
$\sigma_{8}$ and $H_{0}$ are lower compared to the interacting scenario with
$Q_{A}$. Nevertheless the addition of BAO measurements restores their usual
(i.e. canonical, $\Lambda$CDM-like) values, as in the case of $Q_{A}$.

Table 4 and Fig. 3 shows the observational constraints for the model
$\xi_{0}w_{p}$IDE$Q_{A}$, characterized by a parameter space given by
$w_{p}<-1$, $\xi_{0}<0$ and the coupling function $Q_{A}$, see Eq. (4). Note
that, following section II.1, we use $w_{p}$ to represent the phantom dark
energy equation of state and $w_{q}$ representing the quintessence dark energy
equation of state. From the CMB data, we do not find any evidence for a non-
zero coupling. However, what we observe for CMB data alone is a preference for
phantom dark energy at more than $95\%$ CL ($w_{p}=-1.58_{-0.44}^{+0.49}$ at
95% CL) and a much larger value of the Hubble constant, as expected in these
scenarios Di Valentino  _et al._ (2016) and also noticed from the results
depicted in Fig. 3. Once BAO observations are also considered, a very mild
evidence for a non-vanishing coupling parameter is found
($\xi_{0}=-0.052^{+0.046}_{-0.023}$ at 68% CL for CMB+BAO). Also, the mean
values of the parameters are shifted closer to their $\Lambda$CDM-like values
and the preference for $w_{p}<-1$ is
decreased($w_{p}=-1.094_{-0.040}^{+0.070}$ at 68% CL for CMB+BAO). However,
the value of $\Omega_{m0}$ ($\sigma_{8}$) is still mildly higher (lower) than
in the canonical $\Lambda$CDM picture. The inclusion of Pantheon data leads to
very similar results to the CMB+BAO case except for $w_{p}$, which, in this
case, remains in the phantom regime, $w_{p}=-1.087_{-0.076}^{+0.085}$ at 95%
CL for CMB+Pantheon. Finally, the constraints from CMB+BAO+Pantheon are very
similar to those obtained with CMB+Pantheon, i.e. show an indication for a
phantom interacting scenario at $1\sigma$ level. However, we observe that this
full dataset prefers a smaller value of $S_{8}$ compared to the Planck’s
estimation Aghanim  _et al._ (2020a) (within the canonical $\Lambda$CDM
picture), improving the consistency with weak lensing measurements Heymans
_et al._ (2021); Tröster  _et al._ (2021); Secco  _et al._ (2022); Abbott  _et
al._ (2022).

We present in Tab. 5 and Fig. 4 the constraints for the model
$\xi_{0}w_{q}$IDE$Q_{A}$, i.e. ($w_{q}>-1$, $\xi_{0}>0$ and the coupling
function $Q_{A}$, see Eq. (4)) for CMB and CMB+BAO, CMB+Pantheon and
CMB+BAO+Pantheon measurements. For CMB alone we find a mild preference of a
non-zero coupling at 68% CL ($\xi_{0}=0.15_{-0.07}^{+0.11}$ at 68% CL). When
BAO measurements are added to the CMB, the behavior of the coupling parameter
remains the same, that means, we find a mild evidence of the coupling
parameter at 68% CL ($\xi_{0}=0.119_{-0.090}^{+0.054}$ at 68% CL).
Interestingly, for this case, the addition of BAO does not restore the values
of the parameters (other than $w_{q}$) to their $\Lambda$CDM-like ones.
Namely, the value of the matter density is much lower (due to the energy flow)
and consequently, to leave unchanged the overall power spectra normalization,
the clustering parameter $\sigma_{8}$ is required to be much higher than in
the standard paradigm, and its value is much relaxed, with very large error
bars. The addition of Pantheon to CMB does not offer anything new compared to
the CMB+BAO constraints except for some mild shifts in the mean values of the
parameters and a 95% CL significance for a coupling different from zero.
Finally, the results for the combined analysis CMB+BAO+Pantheon are also
similar to those of CMB+BAO.

Table 6 presents the observational constraints of the model
$\xi_{0}w_{p}$IDE$Q_{B}$, characterized by a parameter space given by
$w_{p}<-1$, $\xi_{0}<0$ and the coupling function $Q_{B}$, see Eq. (5) for
CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon. Focusing on the CMB
constraints, we do not find any evidence for a non-zero value of the coupling
parameter $\xi_{0}$ but the dark energy equation of state shows its phantom
behaviour at more than 95% CL ($w_{p}=-1.67_{-0.41}^{+0.52}$ at 95% CL).
Notice that a much larger value of the Hubble constant is also obtained, as
expected in these phantom scenarios Di Valentino  _et al._ (2016). When the
BAO data are added to CMB, the mean values of some of the parameters are
shifted to their $\Lambda$CDM-like values, and the preference for $w_{p}<-1$
is somehow decreased ($w_{p}=-1.111_{-0.044}^{+0.089}$ at 68% CL for CMB+BAO).
However, the value of $\Omega_{m0}$ ($\sigma_{8}$) is still mildly higher
(lower) than in the $\Lambda$CDM model. For the remaining two combined
datasets, namely, CMB+Pantheon and CMB+BAO+Pantheon, the results do not
significantly differ compared to the constraints obtained with the CMB+BAO
combination. This means that we do not find any evidence for a non-zero
coupling in the dark sector. The phantom nature of the dark energy equation of
state is evident at more than 68% CL and for both the datasets the value of
$\Omega_{m0}$ ($\sigma_{8}$) is still mildly higher (lower) than in the
$\Lambda$CDM model.

To conclude with the constant coupling and constant dark energy equation of
state framework, we show in Tab. 7 the constraints on the model
$\xi_{0}w_{q}$IDE$Q_{B}$, characterized by a parameter space given by
$w_{q}>-1$, $\xi_{0}>0$ and the coupling function $Q_{B}$, see Eq. (5), for
the observational datasets CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon.
Starting with the CMB constraints, we note that the model does not lead to any
non-zero value of the coupling parameter and it is very close to the non-
interacting scenario. The Hubble constant takes a slightly lower value
($H_{0}=65.2_{-1.5}^{+2.9}$ km/s/Mpc at 68% CL) compared to the $\Lambda$CDM
value due to a slightly higher value of the matter density parameter and also
to the fact that the dark energy equation of state is required to lie within
the quintessence region. When the BAO observations are added to CMB, we find
that the values of the parameters are very similar to their $\Lambda$CDM-like
ones. Our findings do not change for the remaining two cases with CMB+Pantheon
and CMB+BAO+Pantheon datasets.

Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.137_{-0.017-0.019}^{+0.008+0.024}$ | $0.137_{-0.008-0.016}^{+0.011+0.015}$ | $0.138_{-0.007-0.016}^{+0.012+0.015}$ | $0.136_{-0.007-0.015}^{+0.011+0.014}$
$\Omega_{b}h^{2}$ | $0.02241_{-0.00015-0.00029}^{+0.00015+0.00030}$ | $0.02237_{-0.00014-0.00028}^{+0.00014+0.00028}$ | $0.02234_{-0.00015-0.00029}^{+0.00015+0.00029}$ | $0.02238_{-0.00014-0.00028}^{+0.00014+0.00029}$
$100\theta_{MC}$ | $1.04007_{-0.00061-0.0013}^{+0.00082+0.0011}$ | $1.04001_{-0.00054-0.00094}^{+0.00054+0.00098}$ | $1.03993_{-0.00063-0.0010}^{+0.00050+0.0010}$ | $1.04005_{-0.00056-0.00094}^{+0.00051+0.00098}$
$\tau$ | $0.0535_{-0.0075-0.015}^{+0.0073+0.016}$ | $0.0542_{-0.0074-0.015}^{+0.0072+0.015}$ | $0.0538_{-0.0075-0.015}^{+0.0076+0.015}$ | $0.0540_{-0.0072-0.014}^{+0.0070+0.016}$
$n_{s}$ | $0.9656_{-0.0044-0.0083}^{+0.0044+0.0087}$ | $0.9652_{-0.0043-0.0082}^{+0.0043+0.0081}$ | $0.9645_{-0.0044-0.0085}^{+0.0044+0.0085}$ | $0.9653_{-0.0042-0.0080}^{+0.0042+0.0082}$
${\rm{ln}}(10^{10}A_{s})$ | $3.042_{-0.015-0.032}^{+0.015+0.031}$ | $3.044_{-0.015-0.031}^{+0.015+0.031}$ | $3.044_{-0.015-0.031}^{+0.016+0.031}$ | $3.044_{-0.016-0.032}^{+0.015+0.032}$
$w_{0}^{p}$ | $-1.84_{-0.27}^{+0.84}\;uncons.$ | $>-1.205\;>-1.456$ | $>-1.142\;>-1.27$ | $>-1.101\;>-1.19$
$w_{e}^{p}$ | $>-1.79\;>-2.64$ | $-1.11_{-0.04}^{+0.10}\;>-1.24$ | $>-1.135\;>-1.25$ | $-1.108_{-0.047}^{+0.082}\;>-1.22$
$\xi_{0}$ | $>-0.052\;>-0.090$ | $>-0.095\;>-0.103$ | $-0.060_{-0.042-0.048}^{+0.025+0.058}$ | $-0.056_{-0.038-0.048}^{+0.028+0.055}$
$\Omega_{m0}$ | $0.211_{-0.058-0.071}^{+0.024+0.093}$ | $0.331_{-0.025-0.044}^{+0.024+0.044}$ | $0.344_{-0.026-0.044}^{+0.026+0.045}$ | $0.340_{-0.019-0.038}^{+0.025+0.035}$
$\sigma_{8}$ | $0.910_{-0.080-0.14}^{+0.079+0.14}$ | $0.764_{-0.049-0.070}^{+0.034+0.075}$ | $0.754_{-0.049-0.063}^{+0.029+0.075}$ | $0.754_{-0.047-0.061}^{+0.030+0.071}$
$H_{0}$ [km/s/Mpc] | $88_{-5-14}^{+11+13}$ | $69.7_{-2.0-3.1}^{+1.2+3.6}$ | $68.5_{-1.2-2.2}^{+1.1+2.3}$ | $68.52_{-0.85-1.4}^{+0.74+1.5}$
$S_{8}$ | $0.754_{-0.032-0.066}^{+0.033+0.061}$ | $0.801_{-0.020-0.034}^{+0.018+0.037}$ | $0.805_{-0.022-0.042}^{+0.022+0.041}$ | $0.801_{-0.020-0.035}^{+0.019+0.037}$
$r_{\rm{drag}}$ [Mpc] | $147.09_{-0.29-0.57}^{+0.29+0.58}$ | $147.08_{-0.28-0.55}^{+0.28+0.55}$ | $147.02_{-0.29-0.59}^{+0.29+0.60}$ | $147.09_{-0.28-0.55}^{+0.27+0.55}$
Table 8: Observational constraints on the interacting scenario
“$\xi_{0}w_{0}^{p}w_{e}^{p}$IDE$Q_{A}$” obtained from several observational
datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are
presented.

Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $<0.072\;<0.110$ | $0.059_{-0.030-0.056}^{+0.035+0.050}$ | $<0.054\;<0.095$ | $0.055_{-0.034}^{+0.033}\;<0.101$
$\Omega_{b}h^{2}$ | $0.02240_{-0.00016-0.00030}^{+0.00016+0.00031}$ | $0.02243_{-0.00015-0.00030}^{+0.00015+0.00029}$ | $0.02238_{-0.00015-0.00028}^{+0.00015+0.00028}$ | $0.02242_{-0.00014-0.00028}^{+0.00014+0.00029}$
$100\theta_{MC}$ | $1.0453_{-0.0032-0.0043}^{+0.0026+0.0043}$ | $1.0449_{-0.0029-0.0038}^{+0.0017+0.0044}$ | $1.0462_{-0.0021-0.0045}^{+0.0032+0.0042}$ | $1.0453_{-0.0031-0.0038}^{+0.0019+0.0044}$
$\tau$ | $0.0525_{-0.0076-0.016}^{+0.0075+0.016}$ | $0.0555_{-0.0088-0.016}^{+0.0076+0.017}$ | $0.0540_{-0.0080-0.015}^{+0.0076+0.016}$ | $0.0551_{-0.0083-0.014}^{+0.0070+0.017}$
$n_{s}$ | $0.9664_{-0.0046-0.0090}^{+0.0046+0.0092}$ | $0.9678_{-0.0042-0.0086}^{+0.0043+0.0083}$ | $0.9656_{-0.0051-0.0089}^{+0.0047+0.0093}$ | $0.9671_{-0.0041-0.0082}^{+0.0041+0.0082}$
${\rm{ln}}(10^{10}A_{s})$ | $3.040_{-0.016-0.033}^{+0.016+0.033}$ | $3.045_{-0.018-0.032}^{+0.016+0.035}$ | $3.044_{-0.016-0.031}^{+0.016+0.034}$ | $3.045_{-0.018-0.031}^{+0.015+0.035}$
$w_{0}^{q}$ | $-0.59_{-0.29}^{+0.20}\;<-0.190$ | $-0.82_{-0.09}^{+0.10}\;<-0.687$ | $-0.80_{-0.06-0.18}^{+0.11+0.13}$ | $-0.839_{-0.065-0.16}^{+0.099+0.12}$
$w_{e}^{q}$ | $<-0.897\;<-0.769$ | $<-0.921\;<-0.830$ | $<-0.931\;<-0.843$ | $<-0.910\;<-0.814$
$\xi_{0}$ | $0.25_{-0.14}^{+0.13}\;<0.458$ | $0.176_{-0.098-0.15}^{+0.087+0.15}$ | $0.24_{-0.07-0.18}^{+0.13+0.16}$ | $0.185_{-0.093-0.15}^{+0.093+0.15}$
$\Omega_{m0}$ | $0.20_{-0.13-0.16}^{+0.07+0.18}$ | $0.181_{-0.074-0.13}^{+0.073+0.12}$ | $0.142_{-0.089-0.11}^{+0.043+0.13}$ | $0.168_{-0.073-0.12}^{+0.071+0.12}$
$\sigma_{8}$ | $1.7_{-1.1-1.2}^{+0.3+2.0}$ | $1.55_{-0.80-1.0}^{+0.20+1.7}$ | $2.1_{-1.3-1.5}^{+0.5+2.1}$ | $1.66_{-0.90-1.1}^{+0.24+1.8}$
$H_{0}$ [km/s/Mpc] | $62.7_{-4.8-8.7}^{+4.7+8.6}$ | $67.6_{-1.5-3.0}^{+1.45+3.0}$ | $67.5_{-1.0-2.2}^{+1.1+2.1}$ | $68.06_{-0.81-1.6}^{+0.80+1.6}$
$S_{8}$ | $1.20_{-0.38-0.43}^{+0.14+0.62}$ | $1.08_{-0.26-0.32}^{+0.08+0.49}$ | $1.25_{-0.40-0.48}^{+0.19+0.60}$ | $1.11_{-0.29-0.34}^{+0.10+0.52}$
$r_{\rm{drag}}$ [Mpc] | $147.11_{-0.30-0.59}^{+0.30+0.59}$ | $147.24_{-0.28-0.56}^{+0.29+0.59}$ | $147.07_{-0.31-0.58}^{+0.31+0.62}$ | $147.21_{-0.27-0.53}^{+0.28+0.52}$
Table 9: Observational constraints on the interacting scenario
“$\xi_{0}w_{0}^{q}w_{e}^{q}$IDE$Q_{A}$” obtained from several observational
datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are
presented.

Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.144_{-0.026-0.029}^{+0.012+0.038}$ | $0.139_{-0.017-0.021}^{+0.008+0.027}$ | $0.142_{-0.020-0.024}^{+0.010+0.029}$ | $0.137_{-0.015-0.019}^{+0.007+0.024}$
$\Omega_{b}h^{2}$ | $0.02238_{-0.00017-0.00031}^{+0.00016+0.00030}$ | $0.02234_{-0.00014-0.00028}^{+0.00014+0.00029}$ | $0.02232_{-0.00015-0.00030}^{+0.00015+0.00030}$ | $0.02235_{-0.00014-0.00028}^{+0.00014+0.00028}$
$100\theta_{MC}$ | $1.0397_{-0.0008-0.0020}^{+0.0013+0.0017}$ | $1.03992_{-0.00055-0.0015}^{+0.00088+0.0013}$ | $1.0398_{-0.0007-0.0016}^{+0.0010+0.0014}$ | $1.04000_{-0.00051-0.0014}^{+0.00080+0.0012}$
$\tau$ | $0.0538_{-0.0077-0.015}^{+0.0078+0.017}$ | $0.0547_{-0.0077-0.015}^{+0.0076+0.016}$ | $0.0550_{-0.0084-0.015}^{+0.0073+0.016}$ | $0.0550_{-0.0077-0.014}^{+0.0071+0.015}$
$n_{s}$ | $0.9648_{-0.0042-0.0086}^{+0.0042+0.0086}$ | $0.9640_{-0.0042-0.0082}^{+0.0042+0.0084}$ | $0.9633_{-0.0041-0.0080}^{+0.0042+0.0080}$ | $0.9642_{-0.0041-0.0082}^{+0.0041+0.0082}$
${\rm{ln}}(10^{10}A_{s})$ | $3.044_{-0.016-0.032}^{+0.016+0.033}$ | $3.046_{-0.016-0.031}^{+0.016+0.033}$ | $3.047_{-0.017-0.030}^{+0.015+0.032}$ | $3.047_{-0.016-0.030}^{+0.015+0.030}$
$w_{0}^{p}$ | $-1.96_{-0.53}^{+0.73}\;uncons.$ | $>-1.21\;>-1.50$ | $>-1.17\;>-1.43$ | $>-1.10\;>-1.21$
$w_{e}^{p}$ | $-1.69_{-0.18}^{+0.68}\;>-2.61$ | $-1.13_{-0.05}^{+0.11}\;>-1.28$ | $>-1.15\;>-1.26$ | $-1.124_{-0.059}^{+0.094}\;>-1.25$
$\xi_{0}$ | $>-0.48\;>-0.85$ | $>-0.36\;>-0.67$ | $-0.33_{-0.10}^{+0.32}\;>-0.73$ | $-0.27_{-0.07}^{+0.26}\;>-0.62$
$\Omega_{m0}$ | $0.218_{-0.066-0.09}^{+0.025+0.12}$ | $0.333_{-0.038-0.054}^{+0.024+0.062}$ | $0.354_{-0.051-0.064}^{+0.025+0.084}$ | $0.340_{-0.034-0.047}^{+0.017+0.056}$
$\sigma_{8}$ | $0.91_{-0.07-0.18}^{+0.11+0.16}$ | $0.774_{-0.037-0.094}^{+0.058+0.082}$ | $0.755_{-0.037-0.10}^{+0.066+0.08}$ | $0.767_{-0.029-0.084}^{+0.052+0.074}$
$H_{0}$ [km/s/Mpc] | $89_{-5-16}^{+11+13}$ | $69.7_{-2.0-3.2}^{+1.3+3.6}$ | $68.3_{-1.2-2.5}^{+1.2+2.5}$ | $68.60_{-0.76-1.5}^{+0.76+1.5}$
$S_{8}$ | $0.761_{-0.032-0.052}^{+0.024+0.056}$ | $0.813_{-0.015-0.040}^{+0.022+0.036}$ | $0.816_{-0.017-0.038}^{+0.021+0.037}$ | $0.815_{-0.015-0.038}^{+0.019+0.033}$
$r_{\rm{drag}}$ [Mpc] | $147.06_{-0.28-0.58}^{+0.29+0.57}$ | $147.03_{-0.29-0.56}^{+0.29+0.56}$ | $146.98_{-0.28-0.55}^{+0.28+0.56}$ | $147.03_{-0.28-0.56}^{+0.28+0.54}$
Table 10: Observational constraints on the interacting scenario
“$\xi_{0}w_{0}^{p}w_{e}^{p}$IDE$Q_{B}$” obtained from several observational
datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are
presented.

Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.1164_{-0.0021-0.0075}^{+0.0042+0.0060}$ | $0.1132_{-0.0028-0.0080}^{+0.0049+0.0068}$ | $0.1128_{-0.0035-0.0086}^{+0.0054+0.0079}$ | $0.1121_{-0.0033-0.0083}^{+0.0053+0.0075}$
$\Omega_{b}h^{2}$ | $0.02235_{-0.00014-0.00028}^{+0.00016+0.00026}$ | $0.02247_{-0.00014-0.00027}^{+0.00014+0.00029}$ | $0.02240_{-0.00014-0.00027}^{+0.00014+0.00030}$ | $0.02245_{-0.00014-0.00026}^{+0.00014+0.00027}$
$100\theta_{MC}$ | $1.04110_{-0.00036-0.00070}^{+0.00036+0.00071}$ | $1.04140_{-0.00036-0.00067}^{+0.00036+0.00070}$ | $1.04135_{-0.00037-0.00072}^{+0.00037+0.00078}$ | $1.04142_{-0.00039-0.00071}^{+0.00036+0.00076}$
$\tau$ | $0.0543_{-0.0073-0.016}^{+0.0075+0.016}$ | $0.0559_{-0.0086-0.015}^{+0.0074+0.016}$ | $0.0544_{-0.0081-0.015}^{+0.0075+0.016}$ | $0.0552_{-0.0073-0.015}^{+0.0072+0.016}$
$n_{s}$ | $0.9650_{-0.0041-0.0086}^{+0.0046+0.0083}$ | $0.9688_{-0.0040-0.0077}^{+0.0039+0.0079}$ | $0.9666_{-0.0041-0.0084}^{+0.0041+0.0085}$ | $0.9686_{-0.0038-0.0076}^{+0.0039+0.0075}$
${\rm{ln}}(10^{10}A_{s})$ | $3.045_{-0.015-0.031}^{+0.015+0.032}$ | $3.044_{-0.018-0.030}^{+0.015+0.032}$ | $3.043_{-0.015-0.032}^{+0.016+0.032}$ | $3.044_{-0.017-0.031}^{+0.015+0.033}$
$w_{0}^{q}$ | $<-0.722\;<-0.359$ | $<-0.888\;<-0.750$ | $<-0.963\;<-0.917$ | $<-0.957\;<-0.907$
$w_{e}^{q}$ | $<-0.868\;<-0.714$ | $<-0.952\;<-0.892$ | $<-0.941\;<-0.861$ | $<-0.957\;<-0.903$
$\xi_{0}$ | $<0.080\;<0.17$ | $<0.112\;<0.21$ | $<0.145\;<0.23$ | $0.108_{-0.103}^{+0.034}\;<0.23$
$\Omega_{m0}$ | $0.356_{-0.048-0.071}^{+0.026+0.089}$ | $0.309_{-0.016-0.032}^{+0.015+0.032}$ | $0.303_{-0.014-0.029}^{+0.016+0.027}$ | $0.297_{-0.012-0.026}^{+0.014+0.024}$
$\sigma_{8}$ | $0.780_{-0.028-0.081}^{+0.042+0.068}$ | $0.812_{-0.026-0.045}^{+0.022+0.050}$ | $0.829_{-0.025-0.039}^{+0.017+0.042}$ | $0.828_{-0.025-0.039}^{+0.017+0.043}$
$H_{0}$ [km/s/Mpc] | $62.8_{-2.1-6.5}^{+3.9+5.3}$ | $66.4_{-0.9-2.5}^{+1.4+2.2}$ | $67.02_{-0.75-1.7}^{+0.96+1.5}$ | $67.4_{-0.63-1.3}^{+0.64+1.3}$
$S_{8}$ | $0.847_{-0.018-0.034}^{+0.018+0.034}$ | $0.824_{-0.014-0.026}^{+0.013+0.027}$ | $0.832_{-0.015-0.030}^{+0.015+0.032}$ | $0.824_{-0.013-0.026}^{+0.013+0.026}$
$r_{\rm{drag}}$ [Mpc] | $147.05_{-0.32-0.57}^{+0.29+0.58}$ | $147.34_{-0.26-0.51}^{+0.26+0.50}$ | $147.18_{-0.28-0.55}^{+0.29+0.55}$ | $147.32_{-0.25-0.49}^{+0.25+0.50}$
Table 11: Observational constraints on the interacting scenario
“$\xi_{0}w_{0}^{q}w_{e}^{q}$IDE$Q_{B}$” obtained from several observational
datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are
presented.

####  IV.1.2 Dynamical dark energy equation of state

We extend in this section the previous scenarios by allowing the dark energy
equation of state to be dynamical, adopting the most common choice in the
literature, see Eq. (6). The stability of these interacting scenarios strongly
relies on the parameter space. We shall follow here the analysis of Ref.
(Gavela  _et al._ , 2009), dividing the parameter space into two distinct
regions, namely, $\xi_{0}<0,w_{0}<-1,w_{e}<-1$ and
$\xi_{0}>0,w_{0}>-1,w_{e}>-1$, where the cosmological evolution will be
stable. In the following, we shall describe the constraints for all the
interacting scenarios.

We start by presenting in Tab. 8 the constraints on the interacting scenario
$\xi_{0}w_{0}^{p}w_{e}^{p}$IDE$Q_{A}$, characterized by the parameter space
$\xi_{0}<0,w_{0}^{p}<-1,w_{e}^{p}<-1$ and the interaction function $Q_{A}$,
Eq. (4), by making use of the CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon
datasets. For CMB alone, we do not find any indication for a non-zero coupling
in the dark sector and we also do not get any evidence for phantom dark energy
at the present time: indeed, $w_{0}^{p}$ remains unconstrained at 95% CL. On
the other hand, the evidence for a dark energy component at early times
(different from pure vacuum energy) is also not present. The strong degeneracy
between the dark energy equation of state parameter and the Hubble constant
leads to a much larger value of $H_{0}$, and consequently, to a much lower
value of $\Omega_{m0}$, to leave unchanged $\Omega_{m}h^{2}$. When the BAO
observations are added to CMB measurements, we still do not find any evidence
of a non-zero coupling parameter but the mean values of some of the parameters
are shifted closer to their $\Lambda$CDM-like values, leaving a signal for a
dark energy component at early times (different from the minimal cosmological
constant) at 68% CL ($w_{e}^{p}=-1.11^{+0.10}_{-0.04}$ at 68% CL for CMB+BAO)
which is of phantom nature, and solving the Hubble constant tension $H_{0}$
within $2\sigma$. Interestingly, the addition of Pantheon to CMB data shows an
evidence for a non-zero coupling in the dark sector at more than 95% CL
($\xi_{0}=-0.060_{-0.048}^{+0.058}$ at 95% CL for CMB+Pantheon) and the mean
values of some of the parameters are shifted closer to their $\Lambda$CDM-like
values. The inclusion of both BAO and Pantheon observations to CMB does not
change the overall picture except for an evidence of early dark energy
(different from a cosmological constant at early times) at 68% CL compared to
the CMB+Pantheon constraints. Similarly to the CMB+Pantheon case, the coupling
parameter is found to be non-zero at more than 95% CL
($\xi_{0}=-0.056_{-0.048}^{+0.055}$ at 95% CL for CMB+BAO+Pantheon). It is
important to note that for this full dataset combination the Hubble constant
tension $H_{0}$ is alleviated at $2.5\sigma$, and we also obtain a lower value
for $S_{8}$, improving the consistency with weak lensing data Heymans  _et
al._ (2021); Tröster  _et al._ (2021); Secco  _et al._ (2022); Abbott  _et
al._ (2022).

Table 9 corresponds to the constraints on the interacting scenario
$\xi_{0}w_{0}^{q}w_{e}^{q}$IDE$Q_{A}$ characterized by the parameter space
$\xi_{0}>0,w_{0}^{q}>-1,w_{e}^{q}>-1$ and the interaction function $Q_{A}$,
given in Eq. (4), for CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon
datasets. We find that CMB alone shows a preference of a non-zero coupling at
68% CL ($\xi_{0}=0.25_{-0.14}^{+0.13}$ at 68% CL) while this preference is
diluted within 95% CL. When BAO data are added to CMB alone, the preference of
a non-zero coupling is pronounced as we can see from its 95% CL constraints:
$\xi_{0}=0.176_{-0.15}^{+0.15}$ at 95% CL for CMB+BAO. The preference for a
non-zero coupling at more than 95% CL is also found for CMB+Pantheon and
CMB+BAO+Pantheon. Notice the extreme low values of $\Omega_{c}h^{2}$ (and,
consequently, the very high values of $\sigma_{8}$), which are a consequence
of two effects. On one hand, the values of $\Omega_{m0}$ are very small, due
to the energy flow from the dark matter sector to the dark energy one. On the
other hand, since the equation of state parameter is required to lie within
the quintessence region, the values of $H_{0}$ are lower. The product of these
two parameters (i.e. $\Omega_{m0}$ and $H_{0}$) is consequently much lower.

In Tables 10 and 11 we have summarized the constraints for the interaction
scenarios driven by the interaction function $Q_{B}$, Eq. (5), for different
parameter spaces. More concretely, Tab. 10 corresponds to the interacting
scenario $\xi_{0}w_{0}^{p}w_{e}^{p}$IDE$Q_{B}$ characterized by the parameter
space $\xi_{0}<0,w_{0}^{p}<-1,w_{e}^{p}<-1$. Focusing on the CMB constraints,
notice that we do not find any evidence for a non-zero coupling in the dark
sector. Further, the present value of the dark energy equation of state lies
in the phantom regime at 68% CL ($w_{0}^{p}=-1.96_{-0.53}^{+0.73}$ at 68% CL
for CMB alone) while it remains unconstrained at 95% CL. Due to the strong
degeneracy between the dark energy equation of state and the Hubble constant,
a very high value of $H_{0}$ is also found. A mild evidence for an early dark
energy component (which is not vacuum energy at early times) for CMB data
alone appears ($w_{e}^{p}=-1.69_{-0.18}^{+0.68}$ at 68% CL). When BAO data are
added to CMB observations, the mean values of some of the parameters are
shifted to their $\Lambda$CDM values, except for the mild preference for a
dark energy component different from the standard cosmological constant case
at early times ($w_{e}\neq-1$), which still persists
($w_{e}^{p}=-1.13_{-0.05}^{+0.11}$ at 68% CL, CMB+BAO), and the value of
$\Omega_{m0}$ ($\sigma_{8}$), which are higher (lower) than in the
$\Lambda$CDM picture. Interestingly, the inclusion of Pantheon data to CMB
measurements leads a non-null value of the coupling parameter
($\xi_{0}=-0.33_{-0.10}^{+0.32}$ at 68% CL, CMB+Pantheon). Again we see that
for CMB+Pantheon dataset, the value of $\Omega_{m0}$ ($\sigma_{8}$) is still
mildly higher (lower) than in the $\Lambda$CDM picture. Finally, for
CMB+BAO+Pantheon, we have the same indication for a non-zero coupling at
$1\sigma$ together with an early dark energy parameter
($w_{e}^{p}=-1.124_{-0.059}^{+0.094}$ at 68% CL), which shows a mild
preference for early dark energy instead than for vacuum energy, compared to
the constraints from CMB+Pantheon.

In Table 11 we summarize the constraints on the scenario
$\xi_{0}w_{0}^{q}w_{e}^{q}$IDE$Q_{B}$ characterized by the parameter space
$\xi_{0}>0,w_{0}^{q}>-1,w_{e}^{q}>-1$ arising from CMB, CMB+BAO, CMB+Pantheon
and CMB+BAO+Pantheon datasets. For CMB alone, we do not find any indication
for a non-zero coupling in the dark sector. The matter density parameter takes
a very high value due to the energy flow and also by the fact that $w(a)>-1$.
As a consequence, the Hubble constant assumes a relatively small value, to
leave $\Omega_{c}h^{2}$ unchanged. When the BAO observations are added to CMB,
the mean values of the parameters are shifted toward their $\Lambda$CDM values
and again no evidence for a non vanishing coupling parameter is found, and
similar conclusions hold for CMB+Pantheon. For the remaining CMB+BAO+Pantheon,
we obtain instead very similar results to those obtained with CMB+BAO and
CMB+Pantheon, with the exception of $1\sigma$ preference for a coupling
different from zero.

Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $<0.092\;<0.137$ | $0.119_{-0.014-0.040}^{+0.026+0.034}$ | $0.110_{-0.017-0.037}^{+0.021+0.036}$ | $0.117_{-0.014-0.031}^{+0.018+0.031}$
$\Omega_{b}h^{2}$ | $0.02231_{-0.00016-0.00028}^{+0.00014+0.00030}$ | $0.02232_{-0.00014-0.00030}^{+0.00014+0.00027}$ | $0.02228_{-0.00015-0.00030}^{+0.00015+0.00028}$ | $0.02232_{-0.00014-0.00029}^{+0.00015+0.00028}$
$100\theta_{MC}$ | $1.0441_{-0.0036-0.0052}^{+0.0030+0.0054}$ | $1.0408_{-0.0015-0.0020}^{+0.0008+0.00234}$ | $1.0412_{-0.0013-0.0022}^{+0.0010+0.0022}$ | $1.0409_{-0.0011-0.0019}^{+0.0009+0.0019}$
$\tau$ | $0.0547_{-0.0075-0.016}^{+0.0077+0.015}$ | $0.0549_{-0.0088-0.015}^{+0.0077+0.016}$ | $0.0545_{-0.0086-0.015}^{+0.0079+0.015}$ | $0.0552_{-0.0074-0.016}^{+0.0077+0.016}$
$n_{s}$ | $0.9724_{-0.0043-0.0082}^{+0.0044+0.0085}$ | $0.9732_{-0.0039-0.0080}^{+0.0041+0.0078}$ | $0.9718_{-0.0042-0.0081}^{+0.0042+0.0086}$ | $0.9728_{-0.0042-0.0081}^{+0.0041+0.0082}$
${\rm{ln}}(10^{10}A_{s})$ | $3.056_{-0.015-0.031}^{+0.015+0.032}$ | $3.055_{-0.018-0.031}^{+0.016+0.034}$ | $3.056_{-0.016-0.032}^{+0.016+0.031}$ | $3.056_{-0.015-0.032}^{+0.015+0.030}$
$\xi_{0}$ | $0.09_{-0.15-0.27}^{+0.15+0.28}$ | $-0.05_{-0.14-0.21}^{+0.10+0.23}$ | $0.01_{-0.13-0.25}^{+0.14+0.24}$ | $-0.04_{-0.10-0.20}^{+0.11+0.20}$
$\xi_{a}$ | $0.19_{-0.47}^{+0.38}\;>-0.421$ | $0.21_{-0.27-0.51}^{+0.27+0.52}$ | $0.10_{-0.38-0.63}^{+0.30+0.68}$ | $0.19_{-0.29-0.50}^{+0.23+0.53}$
$\Omega_{m0}$ | $0.19_{-0.13-0.16}^{+0.07+0.20}$ | $0.306_{-0.039-0.10}^{+0.063+0.09}$ | $0.286_{-0.038-0.087}^{+0.048+0.082}$ | $0.300_{-0.035-0.075}^{+0.043+0.073}$
$\sigma_{8}$ | $1.6_{-1.1-1.3}^{+0.3+2.1}$ | $0.83_{-0.16-0.21}^{+0.06+0.28}$ | $0.89_{-0.16-0.23}^{+0.08+0.28}$ | $0.84_{-0.12-0.18}^{+0.06+0.21}$
$H_{0}$ [km/s/Mpc] | $71.8_{-3.2-7.2}^{+4.3+6.2}$ | $68.2_{-1.2-2.2}^{+1.2+2.4}$ | $68.2_{-1.1-2.3}^{+1.1+2.2}$ | $68.30_{-0.90-1.6}^{+0.82+1.7}$
$S_{8}$ | $1.04_{-0.29-0.35}^{+0.10+0.53}$ | $0.828_{-0.065-0.09}^{+0.032+0.12}$ | $0.860_{-0.076-0.12}^{+0.045+0.13}$ | $0.831_{-0.052-0.085}^{+0.034+0.094}$
$r_{\rm{drag}}$ [Mpc] | $147.08_{-0.30-0.58}^{+0.30+0.59}$ | $147.15_{-0.28-0.58}^{+0.29+0.55}$ | $147.04_{-0.29-0.60}^{+0.30+0.61}$ | $147.14_{-0.29-0.56}^{+0.28+0.55}$
Table 12: Observational constraints on the interacting scenario
“$\xi_{0}\xi_{a}$IVS$Q_{A}$” obtained from several observational datasets,
namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are presented.

Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.118_{-0.036-0.054}^{+0.028+0.056}$ | $0.123_{-0.017-0.035}^{+0.018+0.034}$ | $0.117_{-0.011-0.024}^{+0.013+0.022}$ | $0.122_{-0.011-0.021}^{+0.011+0.021}$
$\Omega_{b}h^{2}$ | $0.02230_{-0.00014-0.00028}^{+0.00014+0.00027}$ | $0.02233_{-0.00015-0.00030}^{+0.00015+0.00029}$ | $0.02229_{-0.00015-0.00029}^{+0.00015+0.00031}$ | $0.02232_{-0.00014-0.00029}^{+0.00014+0.00029}$
$100\theta_{MC}$ | $1.0408_{-0.0017-0.0029}^{+0.0018+0.0030}$ | $1.0405_{-0.0010-0.0018}^{+0.0009+0.0020}$ | $1.04080_{-0.00071-0.0013}^{+0.00070+0.0014}$ | $1.04058_{-0.00068-0.0012}^{+0.00066+0.0013}$
$\tau$ | $0.0548_{-0.0082-0.015}^{+0.0074+0.016}$ | $0.0550_{-0.0075-0.015}^{+0.0073+0.016}$ | $0.0548_{-0.0080-0.016}^{+0.0079+0.017}$ | $0.0551_{-0.0075-0.016}^{+0.0076+0.017}$
$n_{s}$ | $0.9723_{-0.0042-0.0082}^{+0.0043+0.0084}$ | $0.9732_{-0.0042-0.0080}^{+0.0041+0.0082}$ | $0.9721_{-0.0044-0.0084}^{+0.0044+0.0086}$ | $0.9730_{-0.0041-0.0079}^{+0.0040+0.0082}$
${\rm{ln}}(10^{10}A_{s})$ | $3.056_{-0.016-0.032}^{+0.016+0.034}$ | $3.055_{-0.015-0.031}^{+0.015+0.033}$ | $3.056_{-0.017-0.032}^{+0.016+0.034}$ | $3.056_{-0.016-0.032}^{+0.015+0.033}$
$\xi_{0}$ | $-0.04_{-0.31-0.45}^{+0.19+0.50}$ | $-0.14_{-0.23-0.39}^{+0.19+0.41}$ | $-0.07_{-0.21-0.34}^{+0.16+0.35}$ | $-0.13_{-0.18-0.31}^{+0.16+0.32}$
$\xi_{a}$ | $0.20_{-0.34-0.59}^{+0.26+0.68}$ | $0.37_{-0.33-0.56}^{+0.41+0.63}$ | $0.30_{-0.38-0.62}^{+0.41+0.70}$ | $0.38_{-0.34-0.54}^{+0.37+0.62}$
$\Omega_{m0}$ | $0.32_{-0.13-0.17}^{+0.07+0.22}$ | $0.319_{-0.047-0.094}^{+0.046+0.093}$ | $0.301_{-0.032-0.062}^{+0.031+0.063}$ | $0.311_{-0.030-0.055}^{+0.028+0.058}$
$\sigma_{8}$ | $0.83_{-0.11-0.20}^{+0.12+0.20}$ | $0.813_{-0.063-0.12}^{+0.054+0.12}$ | $0.837_{-0.043-0.074}^{+0.038+0.083}$ | $0.820_{-0.035-0.070}^{+0.035+0.072}$
$H_{0}$ [km/s/Mpc] | $67.7_{-3.3-7.9}^{+4.9+6.9}$ | $67.9_{-1.3-2.7}^{+1.3+2.6}$ | $68.3_{-1.1-2.3}^{+1.2+2.3}$ | $68.17_{-0.81-1.6}^{+0.80+1.6}$
$S_{8}$ | $0.828_{-0.022-0.068}^{+0.038+0.054}$ | $0.831_{-0.015-0.032}^{+0.017+0.029}$ | $0.835_{-0.017-0.034}^{+0.017+0.033}$ | $0.833_{-0.015-0.030}^{+0.015+0.029}$
$r_{\rm{drag}}$ [Mpc] | $147.09_{-0.29-0.59}^{+0.28+0.60}$ | $147.17_{-0.28-0.55}^{+0.27+0.56}$ | $147.07_{-0.30-0.59}^{+0.30+0.59}$ | $147.17_{-0.28-0.54}^{+0.28+0.56}$
Table 13: Observational constraints on the interacting scenario
“$\xi_{0}\xi_{a}$IVS$Q_{B}$” obtained from several observational datasets,
namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are presented.

###  IV.2 Dynamical coupling $\xi(a)$

In the following sections we shall present the interacting cosmologies for the
two interaction functions $Q_{A}$ and $Q_{B}$ of Eqs. (4) and (5) assuming
that a time-dependent coupling parameter $\xi(a)$, see Eq. (7).

Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.149_{-0.008-0.021}^{+0.012+0.018}$ | $0.141_{-0.006-0.015}^{+0.010+0.013}$ | $0.1444_{-0.0042-0.015}^{+0.0092+0.012}$ | $0.139_{-0.007-0.015}^{+0.010+0.014}$
$\Omega_{b}h^{2}$ | $0.02246_{-0.00016-0.00030}^{+0.00016+0.00031}$ | $0.02246_{-0.00015-0.00029}^{+0.00015+0.00030}$ | $0.02244_{-0.00015-0.00030}^{+0.00016+0.00030}$ | $0.02246_{-0.00015-0.00028}^{+0.00014+0.00030}$
$100\theta_{MC}$ | $1.03949_{-0.00069-0.0010}^{+0.00042+0.0012}$ | $1.03989_{-0.00054-0.00087}^{+0.00046+0.00096}$ | $1.03966_{-0.00054-0.0009}^{+0.00043+0.0010}$ | $1.04001_{-0.00048-0.00086}^{+0.00048+0.00089}$
$\tau$ | $0.0522_{-0.0074-0.016}^{+0.0079+0.015}$ | $0.0544_{-0.0080-0.015}^{+0.0073+0.016}$ | $0.0527_{-0.0081-0.015}^{+0.0076+0.016}$ | $0.0552_{-0.0083-0.015}^{+0.0076+0.016}$
$n_{s}$ | $0.9672_{-0.0041-0.0086}^{+0.0044+0.0087}$ | $0.9676_{-0.0045-0.0086}^{+0.0043+0.0085}$ | $0.9666_{-0.0044-0.0087}^{+0.0048+0.0083}$ | $0.9680_{-0.0044-0.0081}^{+0.0044+0.0087}$
${\rm{ln}}(10^{10}A_{s})$ | $3.038_{-0.015-0.031}^{+0.017+0.029}$ | $3.042_{-0.016-0.031}^{+0.015+0.033}$ | $3.039_{-0.017-0.032}^{+0.016+0.032}$ | $3.044_{-0.016-0.031}^{+0.016+0.033}$
$w_{p}$ | $-1.83_{-0.38}^{+0.54}\;>-2.63$ | $-1.21_{-0.08-0.22}^{+0.14+0.20}$ | $-1.203_{-0.082-0.16}^{+0.092+0.16}$ | $-1.131_{-0.052-0.11}^{+0.065+0.11}$
$\xi_{0}$ | $>-0.030\;>-0.062$ | $>-0.048\;>-0.082$ | $>-0.035\;>-0.070$ | $>-0.055\;>-0.087$
$\xi_{a}$ | $>-0.106\;>-0.162$ | $>-0.051\;>-0.095$ | $-0.071_{-0.040}^{+0.049}\;>-0.136$ | $>-0.066\;>-0.073$
$\Omega_{m0}$ | $0.266_{-0.091-0.12}^{+0.051+0.14}$ | $0.340_{-0.021-0.040}^{+0.021+0.039}$ | $0.372_{-0.020-0.049}^{+0.028+0.045}$ | $0.348_{-0.018-0.036}^{+0.021+0.034}$
$\sigma_{8}$ | $0.853_{-0.088-0.14}^{+0.075+0.15}$ | $0.763_{-0.035-0.058}^{+0.030+0.060}$ | $0.748_{-0.027-0.043}^{+0.019+0.048}$ | $0.751_{-0.035-0.054}^{+0.028+0.057}$
$H_{0}$ [km/s/Mpc] | $82_{-11-19}^{+12+19}$ | $69.4_{-1.9-3.2}^{+1.5+3.4}$ | $67.1_{-1.1-2.2}^{+1.1+2.2}$ | $68.19_{-0.78-1.5}^{+0.78+1.6}$
$S_{8}$ | $0.789_{-0.042-0.067}^{+0.038+0.070}$ | $0.811_{-0.016-0.033}^{+0.018+0.032}$ | $0.832_{-0.020-0.044}^{+0.024+0.043}$ | $0.808_{-0.018-0.036}^{+0.018+0.034}$
$r_{\rm{drag}}$ [Mpc] | $147.16_{-0.29-0.59}^{+0.29+0.57}$ | $147.25_{-0.29-0.54}^{+0.29+0.56}$ | $147.14_{-0.29-0.58}^{+0.30+0.59}$ | $147.28_{-0.28-0.54}^{+0.28+0.58}$
Table 14: Observational constraints on the interacting scenario “$\xi_{0}\xi_{a}w_{p}$IDE$Q_{A}$” obtained from several observational datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are presented.  Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $<0.034\;<0.110$ | $0.079_{-0.014-0.059}^{+0.034+0.043}$ | $0.079_{-0.015-0.054}^{+0.034+0.042}$ | $0.075_{-0.017-0.058}^{+0.037+0.0445}$
$\Omega_{b}h^{2}$ | $0.02232_{-0.00016-0.00031}^{+0.00015+0.00030}$ | $0.02233_{-0.00015-0.00027}^{+0.00014+0.00029}$ | $0.02232_{-0.00015-0.00030}^{+0.00017+0.00029}$ | $0.02234_{-0.00015-0.00029}^{+0.00015+0.00029}$
$100\theta_{MC}$ | $1.0450_{-0.0036-0.0042}^{+0.0020+0.0049}$ | $1.0435_{-0.0023-0.0030}^{+0.0008+0.0044}$ | $1.0435_{-0.0023-0.0029}^{+0.0010+0.0040}$ | $1.0437_{-0.0025-0.0031}^{+0.0010+0.0043}$
$\tau$ | $0.0549_{-0.0075-0.014}^{+0.0072+0.015}$ | $0.0549_{-0.0075-0.015}^{+0.0076+0.016}$ | $0.0547_{-0.0076-0.016}^{+0.0077+0.016}$ | $0.0546_{-0.0073-0.015}^{+0.0076+0.015}$
$n_{s}$ | $0.9639_{-0.0044-0.0088}^{+0.0048+0.0089}$ | $0.9645_{-0.0046-0.0082}^{+0.0042+0.0085}$ | $0.9640_{-0.0048-0.0084}^{+0.0043+0.0090}$ | $0.9647_{-0.0041-0.0083}^{+0.0042+0.0080}$
${\rm{ln}}(10^{10}A_{s})$ | $3.046_{-0.016-0.030}^{+0.015+0.029}$ | $3.046_{-0.016-0.032}^{+0.016+0.033}$ | $3.046_{-0.016-0.033}^{+0.015+0.035}$ | $3.045_{-0.016-0.031}^{+0.016+0.032}$
$w_{q}$ | $<-0.901\;<-0.783$ | $<-0.872\;<-0.779$ | $-0.905_{-0.088}^{+0.033}\;<-0.790$ | $-0.892_{-0.097}^{+0.040}\;<-0.774$
$\xi_{0}$ | $0.136_{-0.088}^{+0.085}\;<0.254$ | $<0.117\;<0.220$ | $<0.125<0.212$ | $<0.132<0.225$
$\xi_{a}$ | $<0.022\;<0.050$ | $<0.021\;<0.044$ | $<0.020<0.044$ | $<0.020<0.042$
$\Omega_{m0}$ | $0.17_{-0.11-0.15}^{+0.07+0.15}$ | $0.220_{-0.036-0.13}^{+0.077+0.10}$ | $0.217_{-0.039-0.12}^{+0.075+0.10}$ | $0.210_{-0.039-0.13}^{+0.079+0.10}$
$\sigma_{8}$ | $1.7_{-1.0-1.2}^{+0.2+2.1}$ | $1.18_{-0.43-0.6}^{+0.07+1.0}$ | $1.17_{-0.40-0.53}^{+0.09+0.86}$ | $1.23_{-0.47-0.63}^{+0.08+1.07}$
$H_{0}$ [km/s/Mpc] | $70.4_{-3.1-7.2}^{+4.2+6.8}$ | $68.3_{-1.2-2.4}^{+1.2+2.5}$ | $68.5_{-1.1-2.1}^{+1.1+2.2}$ | $68.40_{-0.81-1.5}^{+0.79+1.6}$
$S_{8}$ | $1.06_{-0.27-0.32}^{+0.08+0.50}$ | $0.94_{-0.14-0.19}^{+0.03+0.32}$ | $0.94_{-0.14-0.18}^{+0.04+0.27}$ | $0.96_{-0.15-0.20}^{+0.04+0.33}$
$r_{\rm{drag}}$ [Mpc] | $147.01_{-0.30-0.59}^{+0.30+0.59}$ | $147.07_{-0.29-0.56}^{+0.28+0.57}$ | $147.01_{-0.29-0.59}^{+0.29+0.59}$ | $147.06_{-0.27-0.54}^{+0.27+0.52}$
Table 15: Observational constraints on the interacting scenario “$\xi_{0}\xi_{a}w_{q}$IDE$Q_{A}$” obtained from several observational datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are presented.  Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.165_{-0.012-0.028}^{+0.018+0.027}$ | $0.145_{-0.016-0.023}^{+0.009+0.028}$ | $0.155_{-0.011-0.024}^{+0.014+0.023}$ | $0.140_{-0.015-0.020}^{+0.007+0.026}$
$\Omega_{b}h^{2}$ | $0.02246_{-0.00015-0.00030}^{+0.00015+0.00031}$ | $0.02244_{-0.00016-0.00030}^{+0.00015+0.00030}$ | $0.02244_{-0.00015-0.00030}^{+0.00015+0.00030}$ | $0.02244_{-0.00016-0.00028}^{+0.00014+0.00030}$
$100\theta_{MC}$ | $1.03869_{-0.00085-0.0014}^{+0.00066+0.0015}$ | $1.03965_{-0.00058-0.0015}^{+0.00086+0.0014}$ | $1.03911_{-0.00078-0.0013}^{+0.00059+0.0014}$ | $1.03993_{-0.00048-0.0014}^{+0.00081+0.0012}$
$\tau$ | $0.0530_{-0.0077-0.016}^{+0.0081+0.016}$ | $0.0551_{-0.0086-0.015}^{+0.0078+0.016}$ | $0.0543_{-0.0076-0.016}^{+0.0076+0.016}$ | $0.0561_{-0.0076-0.016}^{+0.0077+0.016}$
$n_{s}$ | $0.9668_{-0.0046-0.0090}^{+0.0045+0.0091}$ | $0.9664_{-0.0043-0.0085}^{+0.0044+0.0082}$ | $0.9660_{-0.0045-0.0086}^{+0.0045+0.0086}$ | $0.9668_{-0.0041-0.0079}^{+0.0042+0.0083}$
${\rm{ln}}(10^{10}A_{s})$ | $3.040_{-0.016-0.032}^{+0.016+0.032}$ | $3.044_{-0.016-0.033}^{+0.016+0.033}$ | $3.044_{-0.016-0.034}^{+0.016+0.033}$ | $3.046_{-0.016-0.031}^{+0.016+0.033}$
$w_{p}$ | $-2.10_{-0.58-0.84}^{+0.40+0.86}$ | $-1.26_{-0.10-0.27}^{+0.16+0.24}$ | $-1.32_{-0.14-0.24}^{+0.14+0.25}$ | $-1.143_{-0.053-0.15}^{+0.089+0.13}$
$\xi_{0}$ | $>-0.478\;>-0.765$ | $>-0.289\;>-0.641$ | $>-0.242>-0.531$ | $>-0.252>-0.604$
$\xi_{a}$ |  $<-0.47$ unconstr. | $-0.21_{-0.08}^{+0.18}\;>-0.44$ |  $-0.50_{-0.26}^{+0.26}\;$ unconstr. | $>-0.17>-0.31$
$\Omega_{m0}$ | $0.28_{-0.10-0.13}^{+0.04+0.18}$ | $0.344_{-0.031-0.050}^{+0.022+0.056}$ | $0.405_{-0.031-0.073}^{+0.045+0.066}$ | $0.351_{-0.033-0.048}^{+0.018+0.058}$
$\sigma_{8}$ | $0.823_{-0.086-0.17}^{+0.086+0.16}$ | $0.759_{-0.034-0.085}^{+0.050+0.077}$ | $0.719_{-0.038-0.074}^{+0.038+0.076}$ | $0.755_{-0.026-0.085}^{+0.048+0.069}$
$H_{0}$ [km/s/Mpc] | $85_{-7-22}^{+15+18}$ | $69.9_{-2.0-3.4}^{+1.7+3.5}$ | $66.4_{-1.4-2.4}^{+1.2+2.6}$ | $68.21_{-0.82-1.6}^{+0.82+1.6}$
$S_{8}$ | $0.770_{-0.048-0.069}^{+0.029+0.081}$ | $0.811_{-0.015-0.041}^{+0.023+0.036}$ | $0.833_{-0.020-0.039}^{+0.022+0.037}$ | $0.814_{-0.014-0.038}^{+0.020+0.034}$
$r_{\rm{drag}}$ [Mpc] | $147.15_{-0.30-0.58}^{+0.30+0.61}$ | $147.18_{-0.28-0.54}^{+0.28+0.57}$ | $147.10_{-0.30-0.58}^{+0.30+0.59}$ | $147.23_{-0.28-0.54}^{+0.27+0.54}$
Table 16: Observational constraints on the interacting scenario “$\xi_{0}\xi_{a}w_{p}$IDE$Q_{B}$” obtained from several observational datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are presented.  Figure 5: One dimensional marginalized posterior distributions and the two-dimensional joint contours for the most relevant parameters of the interacting scenario “$\xi_{0}\xi_{a}w_{p}$IDE$Q_{B}$” for several datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon.  Figure 6: One dimensional marginalized posterior distributions and the two-dimensional joint contours for the most relevant parameters of the interacting scenario “$\xi_{0}\xi_{a}w_{q}$IDE$Q_{B}$” for several datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon.  Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.1109_{-0.0044-0.011}^{+0.0065+0.010}$ | $0.1098_{-0.0039-0.0089}^{+0.0053+0.0084}$ | $0.1095_{-0.0043-0.0097}^{+0.0057+0.0093}$ | $0.1096_{-0.0037-0.0085}^{+0.0050+0.0080}$
$\Omega_{b}h^{2}$ | $0.02234_{-0.00015-0.00029}^{+0.00015+0.00030}$ | $0.02239_{-0.00014-0.00028}^{+0.00014+0.00029}$ | $0.02237_{-0.00015-0.00029}^{+0.00015+0.00030}$ | $0.02238_{-0.00014-0.00028}^{+0.00015+0.00028}$
$100\theta_{MC}$ | $1.04140_{-0.00043-0.00086}^{+0.00042+0.00087}$ | $1.04152_{-0.00040-0.00070}^{+0.00036+0.00075}$ | $1.04152_{-0.00042-0.00075}^{+0.00038+0.00079}$ | $1.04151_{-0.00040-0.00070}^{+0.00036+0.00076}$
$\tau$ | $0.0537_{-0.0078-0.016}^{+0.0078+0.016}$ | $0.0546_{-0.0083-0.015}^{+0.0074+0.016}$ | $0.0541_{-0.0081-0.015}^{+0.0076+0.016}$ | $0.0545_{-0.0076-0.014}^{+0.0077+0.016}$
$n_{s}$ | $0.9648_{-0.0044-0.0091}^{+0.0044+0.0087}$ | $0.9664_{-0.0040-0.0075}^{+0.0039+0.0080}$ | $0.9654_{-0.0043-0.0090}^{+0.0044+0.0087}$ | $0.9662_{-0.0040-0.0081}^{+0.0040+0.0080}$
${\rm{ln}}(10^{10}A_{s})$ | $3.043_{-0.016-0.032}^{+0.016+0.033}$ | $3.044_{-0.017-0.031}^{+0.015+0.033}$ | $3.044_{-0.016-0.032}^{+0.016+0.032}$ | $3.044_{-0.017-0.030}^{+0.015+0.032}$
$w_{q}$ | $<-0.900\;<-0.762$ | $<-0.938\;<-0.879$ | $<-0.958<-0.916$ | $<-0.956<-0.912$
$\xi_{0}$ | $<0.082\;<0.173$ | $<0.088\;<0.183$ | $<0.096<0.194$ | $<0.098<0.198$
$\xi_{a}$ | $<0.100\;<0.234$ | $<0.108<0.213$ | $<0.116<0.238$ | $<0.105<0.205$
$\Omega_{m0}$ | $0.305_{-0.036-0.066}^{+0.028+0.070}$ | $0.290_{-0.014-0.029}^{+0.014+0.028}$ | $0.285_{-0.017-0.035}^{+0.019+0.034}$ | $0.286_{-0.013-0.026}^{+0.013+0.025}$
$\sigma_{8}$ | $0.828_{-0.034-0.073}^{+0.034+0.069}$ | $0.838_{-0.025-0.044}^{+0.021+0.047}$ | $0.848_{-0.026-0.043}^{+0.020+0.047}$ | $0.844_{-0.024-0.042}^{+0.020+0.044}$
$H_{0}$ [km/s/Mpc] | $66.4_{-2.0-5.9}^{+3.1+5.0}$ | $67.7_{-0.9-2.1}^{+1.1+2.0}$ | $68.2_{-1.2-2.1}^{+1.0+2.2}$ | $68.10_{-0.72-1.4}^{+0.72+1.4}$
$S_{8}$ | $0.833_{-0.019-0.039}^{+0.019+0.038}$ | $0.823_{-0.014-0.026}^{+0.014+0.028}$ | $0.826_{-0.016-0.033}^{+0.017+0.033}$ | $0.824_{-0.013-0.026}^{+0.013+0.026}$
$r_{\rm{drag}}$ [Mpc] | $147.05_{-0.30-0.62}^{+0.30+0.59}$ | $147.17_{-0.27-0.54}^{+0.27+0.51}$ | $147.09_{-0.30-0.59}^{+0.31+0.58}$ | $147.15_{-0.26-0.53}^{+0.27+0.51}$
Table 17: Observational constraints on the interacting scenario “$\xi_{0}\xi_{a}w_{q}$IDE$Q_{B}$” obtained from several observational datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are presented.  Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.149_{-0.007-0.019}^{+0.013+0.017}$ | $0.1431_{-0.0044-0.014}^{+0.0084+0.011}$ | $0.1461_{-0.0034-0.012}^{+0.0070+0.009}$ | $0.1429_{-0.0038-0.013}^{+0.0084+0.010}$
$\Omega_{b}h^{2}$ | $0.02247_{-0.00016-0.00030}^{+0.00015+0.00030}$ | $0.02247_{-0.00015-0.00029}^{+0.00015+0.00029}$ | $0.02244_{-0.00016-0.00030}^{+0.00015+0.00030}$ | $0.02245_{-0.00015-0.00030}^{+0.00015+0.00031}$
$100\theta_{MC}$ | $1.03944_{-0.00065-0.0010}^{+0.00048+0.0011}$ | $1.03977_{-0.00049-0.00078}^{+0.00039+0.00090}$ | $1.03958_{-0.00044-0.00073}^{+0.00035+0.00086}$ | $1.03975_{-0.00047-0.00079}^{+0.00041+0.00088}$
$\tau$ | $0.0526_{-0.0074-0.016}^{+0.0075+0.016}$ | $0.0541_{-0.0082-0.015}^{+0.0074+0.016}$ | $0.0532_{-0.0076-0.015}^{+0.0076+0.016}$ | $0.0540_{-0.0079-0.016}^{+0.0078+0.016}$
$n_{s}$ | $0.9677_{-0.0044-0.0087}^{+0.0044+0.0089}$ | $0.9675_{-0.0044-0.0086}^{+0.0044+0.0089}$ | $0.9668_{-0.0044-0.0086}^{+0.0043+0.0088}$ | $0.9675_{-0.0044-0.0088}^{+0.0045+0.0089}$
${\rm{ln}}(10^{10}A_{s})$ | $3.039_{-0.015-0.034}^{+0.016+0.033}$ | $3.042_{-0.016-0.030}^{+0.016+0.032}$ | $3.041_{-0.016-0.032}^{+0.016+0.033}$ | $3.041_{-0.016-0.032}^{+0.015+0.033}$
$w_{0}^{p}$ |  $-2.00_{-0.62}^{+0.62}$ unconstr. | $>-1.20\;>-1.46$ | $>-1.24>-1.45$ | $>-1.11>-1.20$
$w_{e}^{p}$ |  $>-2.07$ unconstr. | $-1.34_{-0.14-0.32}^{+0.21+0.31}$ | $-1.27_{-0.12}^{+0.22}\;>-1.56$ | $-1.32_{-0.13-0.30}^{+0.19+0.28}$
$\xi_{0}$ | $>-0.033\;>-0.072$ | $>-0.040\;>-0.075$ | $>-0.033>-0.067$ | $>-0.042>-0.078$
$\xi_{a}$ | $-0.073_{-0.031}^{+0.059}\;>-0.154$ | $-0.064_{-0.029}^{+0.052}\;>-0.131$ | $-0.084_{-0.040-0.067}^{+0.040+0.071}$ | $-0.065_{-0.028}^{+0.052}\;>-0.136$
$\Omega_{m0}$ | $0.256_{-0.080-0.10}^{+0.043+0.12}$ | $0.345_{-0.019-0.042}^{+0.024+0.039}$ | $0.375_{-0.023-0.044}^{+0.023+0.044}$ | $0.356_{-0.014-0.034}^{+0.020+0.030}$
$\sigma_{8}$ | $0.863_{-0.078-0.12}^{+0.065+0.13}$ | $0.760_{-0.032-0.050}^{+0.025+0.056}$ | $0.747_{-0.024-0.039}^{+0.017+0.043}$ | $0.749_{-0.027-0.047}^{+0.022+0.049}$
$H_{0}$ [km/s/Mpc] | $84_{-9.4-16}^{+10+17}$ | $69.4_{-1.9-3.1}^{+1.2+3.5}$ | $67.2_{-1.4-2.7}^{+1.4+2.7}$ | $68.26_{-0.81-1.6}^{+0.82+1.7}$
$S_{8}$ | $0.786_{-0.040-0.078}^{+0.040+0.080}$ | $0.815_{-0.016-0.034}^{+0.018+0.031}$ | $0.834_{-0.020-0.044}^{+0.023+0.040}$ | $0.816_{-0.016-0.036}^{+0.020+0.034}$
$r_{\rm{drag}}$ [Mpc] | $147.18_{-0.30-0.58}^{+0.30+0.60}$ | $147.20_{-0.30-0.56}^{+0.30+0.58}$ | $147.13_{-0.30-0.60}^{+0.30+0.58}$ | $147.23_{-0.29-0.56}^{+0.29+0.56}$
Table 18: Observational constraints of the interacting scenario “$\xi_{0}\xi_{a}w_{0}^{p}w_{e}^{p}$IDE$Q_{A}$” obtained from several observational datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are presented.  Figure 7: One-dimensional marginalized posterior distributions and two-dimensional joint contours for the most relevant parameters of the interacting scenario “$\xi_{0}\xi_{a}w_{0}^{p}w_{e}^{p}$IDE$Q_{A}$” for several datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon.  Figure 8: One-dimensional marginalized posterior distributions and two-dimensional joint contours for the most relevant parameters of the interacting scenario “$\xi_{0}\xi_{a}w_{0}^{q}w_{e}^{q}$IDE$Q_{A}$” for several datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon.  Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $<0.050\;<0.091$ | $<0.060\;<0.095$ | $<0.046\;<0.090$ | $<0.054\;<0.095$
$\Omega_{b}h^{2}$ | $0.02239_{-0.00018-0.00032}^{+0.00016+0.00032}$ | $0.02238_{-0.00015-0.00030}^{+0.00015+0.00030}$ | $0.02236_{-0.00015-0.00030}^{+0.00015+0.00029}$ | $0.02238_{-0.00015-0.00031}^{+0.00015+0.00029}$
$100\theta_{MC}$ | $1.0465_{-0.0020-0.0044}^{+0.0030+0.0040}$ | $1.0460_{-0.0027-0.0042}^{+0.0027+0.0043}$ | $1.0467_{-0.0015-0.0045}^{+0.0031+0.0037}$ | $1.0463_{-0.0019-0.0044}^{+0.0033+0.0040}$
$\tau$ | $0.0523_{-0.0075-0.016}^{+0.0076+0.016}$ | $0.0549_{-0.0072-0.015}^{+0.0074+0.015}$ | $0.0538_{-0.0077-0.015}^{+0.0076+0.016}$ | $0.0542_{-0.0076-0.014}^{+0.0075+0.016}$
$n_{s}$ | $0.9662_{-0.0047-0.0096}^{+0.0051+0.0093}$ | $0.9660_{-0.0045-0.0084}^{+0.0044+0.0091}$ | $0.9651_{-0.0045-0.0090}^{+0.0044+0.0086}$ | $0.9660_{-0.0046-0.0090}^{+0.0045+0.0089}$
${\rm{ln}}(10^{10}A_{s})$ | $3.040_{-0.015-0.033}^{+0.015+0.031}$ | $3.045_{-0.015-0.030}^{+0.015+0.031}$ | $3.044_{-0.016-0.032}^{+0.016+0.033}$ | $3.043_{-0.016-0.029}^{+0.015+0.032}$
$w_{0}^{p}$ | $-0.42_{-0.20}^{+0.28}\;>-0.82$ | $-0.68_{-0.16-0.27}^{+0.13+0.28}$ | $-0.766_{-0.055-0.17}^{+0.097+0.14}$ | $-0.770_{-0.07-0.19}^{+0.10+0.15}$
$w_{e}^{q}$ | $<-0.91\;<-0.80$ | $<-0.92\;<-0.834$ | $<-0.93<-0.85$ | $<-0.92<-0.82$
$\xi_{0}$ | $0.26_{-0.11-0.23}^{+0.11+0.21}$ | $0.17_{-0.07}^{+0.10}\;<0.30$ | $0.22_{-0.06-0.17}^{+0.11+0.14}$ | $0.20_{-0.06-0.16}^{+0.10+0.14}$
$\xi_{a}$ | $<0.090\;<0.219$ | $<0.068\;<0.160$ | $<0.036<0.079$ | $<0.036<0.080$
$\Omega_{m0}$ | $0.17_{-0.11-0.13}^{+0.05+0.17}$ | $0.155_{-0.095-0.11}^{+0.049+0.13}$ | $0.127_{-0.078-0.09}^{+0.031+0.12}$ | $0.139_{-0.088-0.10}^{+0.040+0.13}$
$\sigma_{8}$ | $1.9_{-1.2-1.4}^{+0.5+1.9}$ | $1.8_{-1.0-1.2}^{+0.4+1.8}$ | $2.2_{-1.3-1.5}^{+0.6+1.9}$ | $2.0_{-1.2-1.3}^{+0.5+1.9}$
$H_{0}$ [km/s/Mpc] | $61.3_{-5.6-7.8}^{+3.3+9.1}$ | $66.6_{-1.7-3.5}^{+1.7+3.4}$ | $68.1_{-1.1-2.3}^{+1.2+2.3}$ | $68.03_{-0.86-1.7}^{+0.87+1.7}$
$S_{8}$ | $1.27_{-0.38-0.47}^{+0.19+0.58}$ | $1.13_{-0.31-0.36}^{+0.13+0.49}$ | $1.26_{-0.38-0.45}^{+0.21+0.52}$ | $1.19_{-0.36-0.41}^{+0.17+0.50}$
$r_{\rm{drag}}$ [Mpc] | $147.09_{-0.30-0.59}^{+0.30+0.60}$ | $147.16_{-0.30-0.58}^{+0.29+0.58}$ | $147.05_{-0.30-0.59}^{+0.30+0.59}$ | $147.12_{-0.29-0.57}^{+0.29+0.58}$
Table 19: Observational constraints on the interacting scenario “$\xi_{0}\xi_{a}w_{0}^{q}w_{e}^{q}$IDE$Q_{A}$” obtained from several observational datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are presented.  Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.157_{-0.015-0.027}^{+0.018+0.026}$ | $0.151_{-0.013-0.023}^{+0.013+0.023}$ | $0.158_{-0.009-0.020}^{+0.011+0.019}$ | $0.152_{-0.013-0.024}^{+0.014+0.023}$
$\Omega_{b}h^{2}$ | $0.02244_{-0.00017-0.00032}^{+0.00016+0.00033}$ | $0.02244_{-0.00016-0.00029}^{+0.00016+0.00030}$ | $0.02242_{-0.00016-0.00030}^{+0.00015+0.00032}$ | $0.02245_{-0.00016-0.00030}^{+0.00016+0.00030}$
$100\theta_{MC}$ | $1.03906_{-0.00094-0.0013}^{+0.00070+0.0015}$ | $1.03937_{-0.00070-0.0013}^{+0.00067+0.0013}$ | $1.03897_{-0.00061-0.0011}^{+0.00052+0.0011}$ | $1.03930_{-0.00075-0.0012}^{+0.00064+0.0013}$
$\tau$ | $0.0529_{-0.0072-0.014}^{+0.0075+0.014}$ | $0.0548_{-0.0086-0.015}^{+0.0071+0.016}$ | $0.0538_{-0.0076-0.016}^{+0.0077+0.016}$ | $0.0558_{-0.0076-0.016}^{+0.0076+0.015}$
$n_{s}$ | $0.9664_{-0.0050-0.0092}^{+0.0044+0.0092}$ | $0.9666_{-0.0044-0.0085}^{+0.0044+0.0088}$ | $0.9657_{-0.0046-0.0090}^{+0.0046+0.0091}$ | $0.9668_{-0.0040-0.0082}^{+0.0042+0.0079}$
${\rm{ln}}(10^{10}A_{s})$ | $3.040_{-0.014-0.030}^{+0.016+0.028}$ | $3.044_{-0.017-0.030}^{+0.015+0.032}$ | $3.042_{-0.016-0.034}^{+0.016+0.033}$ | $3.045_{-0.015-0.031}^{+0.015+0.032}$
$w_{0}^{p}$ |  $-2.0_{-0.54}^{+0.63}$ unconstr. | $>-1.19\;>-1.44$ | $>-1.31>-1.60$ | $>-1.10>-2.0$
$w_{e}^{p}$ |  $-1.97_{-0.54}^{+0.64}$ unconstr. | $-1.48_{-0.21-0.45}^{+0.30+0.43}$ | $-1.48_{-0.24-0.39}^{+0.22+0.40}$ | $-1.47_{-0.21-0.39}^{+0.28+0.37}$
$\xi_{0}$ | $>-0.405\;>-0.712$ | $>-0.264\;>-0.507$ | $>-0.244>-0.484$ | $>-0.272>-0.521$
$\xi_{a}$ |  $-0.44_{-0.19}^{+0.38}$ unconstr. |  $-0.44_{-0.18}^{+0.38}$, unconstr. | $-0.61_{-0.38}^{+0.13}\;<0.19$ |  $-0.47_{-0.21}^{+0.39}$, unconstr.
$\Omega_{m0}$ | $0.261_{-0.087-0.11}^{+0.040+0.14}$ | $0.364_{-0.035-0.058}^{+0.031+0.062}$ | $0.405_{-0.032-0.064}^{+0.036+0.065}$ | $0.377_{-0.030-0.058}^{+0.034+0.056}$
$\sigma_{8}$ | $0.845_{-0.091-0.15}^{+0.078+0.16}$ | $0.741_{-0.041-0.080}^{+0.045+0.076}$ | $0.715_{-0.033-0.066}^{+0.033+0.066}$ | $0.728_{-0.044-0.075}^{+0.038+0.076}$
$H_{0}$ [km/s/Mpc] | $85_{-7-19}^{+13+16}$ | $69.3_{-1.9-3.0}^{+1.2+3.4}$ | $66.9_{-1.5-3.0}^{+1.5+3.0}$ | $68.16_{-0.81-1.6}^{+0.81+1.5}$
$S_{8}$ | $0.774_{-0.046-0.068}^{+0.028+0.078}$ | $0.813_{-0.017-0.036}^{+0.019+0.033}$ | $0.829_{-0.020-0.040}^{+0.021+0.039}$ | $0.814_{-0.016-0.036}^{+0.019+0.032}$
$r_{\rm{drag}}$ [Mpc] | $147.15_{-0.30-0.60}^{+0.29+0.58}$ | $147.18_{-0.29-0.59}^{+0.30+0.57}$ | $147.10_{-0.30-0.60}^{+0.30+0.60}$ | $147.20_{-0.29-0.55}^{+0.29+0.56}$
Table 20: Observational constraints on the interacting scenario “$\xi_{0}\xi_{a}w_{0}^{p}w_{e}^{p}$IDE$Q_{B}$” obtained from several observational datasets, namely, CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon are presented.  Parameters | CMB | CMB+BAO | CMB+Pantheon | CMB+BAO+Pantheon
---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.1121_{-0.0036-0.011}^{+0.0065+0.009}$ | $0.1086_{-0.0041-0.0096}^{+0.0055+0.0090}$ | $0.1085_{-0.0046-0.0101}^{+0.0057+0.0096}$ | $0.1085_{-0.0040-0.0095}^{+0.0054+0.0087}$
$\Omega_{b}h^{2}$ | $0.02233_{-0.00015-0.00028}^{+0.00015+0.00030}$ | $0.02239_{-0.00015-0.00030}^{+0.00015+0.00029}$ | $0.02236_{-0.00016-0.00029}^{+0.00015+0.00030}$ | $0.02238_{-0.00014-0.00028}^{+0.00014+0.00028}$
$100\theta_{MC}$ | $1.04134_{-0.00048-0.00086}^{+0.00039+0.00088}$ | $1.04160_{-0.00042-0.00071}^{+0.00036+0.00078}$ | $1.04158_{-0.00042-0.00080}^{+0.00041+0.00084}$ | $1.04159_{-0.00040-0.00071}^{+0.00035+0.00077}$
$\tau$ | $0.0533_{-0.0074-0.015}^{+0.0075+0.015}$ | $0.0552_{-0.0082-0.015}^{+0.0073+0.016}$ | $0.0542_{-0.0081-0.015}^{+0.0072+0.016}$ | $0.0547_{-0.0078-0.015}^{+0.0078+0.016}$
$n_{s}$ | $0.9648_{-0.0047-0.0083}^{+0.0042+0.0089}$ | $0.9670_{-0.0042-0.0083}^{+0.0042+0.0082}$ | $0.9657_{-0.0044-0.0088}^{+0.0044+0.0086}$ | $0.9664_{-0.0040-0.0082}^{+0.0039+0.0080}$
${\rm{ln}}(10^{10}A_{s})$ | $3.043_{-0.015-0.032}^{+0.016+0.031}$ | $3.045_{-0.015-0.031}^{+0.015+0.032}$ | $3.044_{-0.016-0.031}^{+0.016+0.032}$ | $3.044_{-0.017-0.031}^{+0.016+0.033}$
$w_{0}^{p}$ | $<-0.706\;<-0.333$ | $<-0.846\;<-0.704$ | $<-0.944<-0.884$ | $<-0.938<-0.877$
$w_{e}^{q}$ | $<-0.879\;<-0.731$ | $<-0.933\;<-0.856$ | $<-0.929<-0.847$ | $<-0.933<-0.846$
$\xi_{0}$ | $<0.070<0.158$ | $<0.082<0.168$ | $<0.094<0.188$ | $<0.089<0.183$
$\xi_{a}$ | $<0.096<0.228$ | $<0.140<0.266$ | $<0.135<0.259$ | $<0.132<0.243$
$\Omega_{m0}$ | $0.338_{-0.054-0.088}^{+0.037+0.102}$ | $0.296_{-0.018-0.034}^{+0.018+0.036}$ | $0.286_{-0.017-0.035}^{+0.020+0.034}$ | $0.286_{-0.013-0.028}^{+0.015+0.027}$
$\sigma_{8}$ | $0.795_{-0.041-0.094}^{+0.049+0.084}$ | $0.828_{-0.025-0.050}^{+0.024+0.053}$ | $0.846_{-0.026-0.044}^{+0.021+0.046}$ | $0.843_{-0.025-0.041}^{+0.019+0.043}$
$H_{0}$ [km/s/Mpc] | $63.6_{-2.9-7.5}^{+4.2+6.4}$ | $66.8_{-1.2-2.9}^{+1.6+2.6}$ | $67.9_{-1.2-1.9}^{+1.0+2.2}$ | $67.87_{-0.74-1.5}^{+0.72+1.5}$
$S_{8}$ | $0.839_{-0.020-0.040}^{+0.020+0.039}$ | $0.822_{-0.013-0.027}^{+0.013+0.027}$ | $0.825_{-0.017-0.032}^{+0.017+0.032}$ | $0.822_{-0.014-0.025}^{+0.013+0.026}$
$r_{\rm{drag}}$ [Mpc] | $147.04_{-0.31-0.59}^{+0.31+0.62}$ | $147.21_{-0.28-0.55}^{+0.28+0.56}$ | $147.11_{-0.29-0.58}^{+0.29+0.58}$ | $147.16_{-0.27-0.54}^{+0.28+0.51}$
Table 21: Observational constraints on the interacting scenario
“$\xi_{0}\xi_{a}w_{0}^{q}w_{e}^{q}$IDE$Q_{B}$” obtained from several
observational datasets, namely, CMB, CMB+BAO, CMB+Pantheon and
CMB+BAO+Pantheon are presented.

####  IV.2.1 Constant dark energy equation of state

The simplest interacting cosmological scenario in this framework is the one
where the dark energy equation of state mimics the vacuum energy,
corresponding to $w=-1$. In Tabs. 12 and 13 we present the constraints for the
two interacting scenarios driven by the functions $Q_{A}$ and $Q_{B}$,
respectively.

Table 12 summarizes the constraints of the interacting scenario
$\xi_{0}\xi_{a}$IVS$Q_{A}$ using the different observational datasets, namely,
CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon. A common feature which
remains valid for all the observational datasets is that we do not find any
preference for ($\xi_{0},\xi_{a}$) $\neq\;(0,0)$, i.e. we do not find any
evidence for a non-zero coupling in the dark sector. Nevertheless there are
large differences in the cosmological parameter constraints arising from CMB
alone and from CMB plus other external datasets, such as BAO or Pantheon. For
CMB alone, we see a very low value of the matter density parameter which is
due to the energy flow from the dark matter sector to the dark energy one,
which is always associated to a larger value of the Hubble constant. However,
when BAO observations are added to CMB, the mean values of all the parameters
shift to closer values to the $\Lambda$CDM framework. The matter density
parameter increases significantly (compared to the CMB alone case) because in
this case the one probability distribution of $\xi_{0}$ is shifted towards
negative values of this parameter, changing the energy flow at present times
(i.e. from the dark energy to the dark matter one). That means the transfer of
energy is reversed after the inclusion of BAO data to CMB, fact that is clear
from the mean value of $\xi_{0}$: $\xi_{0}=-0.05_{-0.14}^{+0.10}$ at 68% CL
for CMB+BAO, while $\xi_{0}=0.09_{-0.15}^{+0.15}$ at 68% CL for CMB alone.
When Pantheon SNIa survey measurements are added to CMB observations, no
significant changes appear except a mild reduction of the matter density
parameter, and a slight increase in $\sigma_{8}$ parameter compared to the
CMB+BAO case. The final analysis with CMB+BAO+Pantheon leads to very similar
constraints to the CMB+BAO dataset.

Table 13 summarizes the constraints of the interacting scenario
$\xi_{0}\xi_{a}$IVS$Q_{B}$ using the observational datasets, namely, CMB,
CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon. For CMB alone, we do not find any
evidence of $(\xi_{0},\xi_{a})\neq(0,0)$ and therefore this scenario is very
close to the non-interacting one. When the BAO observations are added to CMB,
the mean values of some of the parameters slightly change, and the most
interesting change appears in $\xi_{a}$ which prefers a non-zero value at 68%
CL ($\xi_{a}=0.37_{-0.33}^{+0.41}$ at 68% CL, CMB+BAO) and hence a mild
evidence of a dynamical $\xi(a)$ is suggested for CMB+BAO. The inclusion of
Pantheon to CMB slightly changes the mean values of the parameters, however,
we do not find any indication for a non-zero dynamical coupling in the dark
sector. Finally, for the combined dataset CMB+BAO+Pantheon we find, again, a
mild evidence of a non-zero $\xi_{a}$ ($=0.38_{-0.34}^{+0.37}$) at 68% CL,
that means, a dynamical coupling in the dark sector is mildly preferred also
for this dataset. This feature is mainly driven by the BAO data.

The consideration of $w$ other than $-1$ but constant generalizes the previous
interacting scenarios (for which $w=-1$). Here we explore the constraints on
the interacting scenarios with a dynamical coupling parameter when the dark
energy equation of state is constant but it is other than $-1$. As already
mentioned, the nature of $w$ may affect the stability of the interacting
scenario. Therefore, following the doom factor analysis of Ref. Gavela  _et
al._ (2009), one can select accordingly the parameter space where the
interacting scenario will be free of instabilities.

In Tabs. 14 and 15 we have summarized the constraints on the interacting
scenarios driven by the interaction function $Q_{A}$ of Eq. (4) assuming two
different regions of the parameter space.

Table 14 presents the observational constraints of the model
$\xi_{0}\xi_{a}w_{p}$IDE$Q_{A}$ characterized by the parameter space
$\xi_{0}<0$, $\xi_{a}<0$, $w_{p}<-1$ for CMB, CMB+BAO, CMB+Pantheon and
CMB+BAO+Pantheon. For CMB alone we do not find any preference for a non-zero
and dynamical coupling parameters, however, we observe a preference for the
phantom dark energy at 68% CL ($w_{p}=-1.83_{-0.38}^{+0.54}$ at 68% CL) and a
higher value of the Hubble constant, due to the strong correlation between
these two parameters. Additionally, we find a lower (higher) value of
$\Omega_{m0}$ ($\sigma_{8}$) compared to its estimation within the canonical
$\Lambda$CDM picture, due to the fact that the energy flows from the dark
matter sector to the dark energy one. When BAO observations are added to CMB
data we have no evidence for a non-zero dynamical coupling in the dark sector
but in this case the preference for $w_{p}<-1$ is much more significant, as
$w_{p}$ remains in the phantom region at more than 95% CL
($w_{p}=-1.21_{-0.22}^{+0.20}$ at 95% CL). The inclusion of Pantheon data to
CMB observations leads to a non-zero value of $\xi_{a}$
($=-0.071_{-0.040}^{+0.049}$ at 68% CL, CMB+Pantheon) although the present day
value of $\xi(a)$, i.e. $\xi_{0}$, does not show any preference for a non-
vanishing value. The dark energy equation of state remains in the phantom
regime at more than 95% CL ($w_{p}=-1.20_{-0.16}^{+0.16}$ at 95% CL). We also
find in this case that $\Omega_{m0}$ and $\sigma_{8}$ take larger and smaller
values compared to their values in the canonical $\Lambda$CDM picture,
respectively. Similar results are obtained for the combined dataset
CMB+BAO+Pantheon except in the constraints on $H_{0}$ and $S_{8}$: there is no
indication for a non-zero dynamical coupling in the dark sector, and the dark
energy equation of state remains again in the phantom regime at more than 95%
CL ($w_{p}=-1.13_{-0.11}^{+0.11}$ at 95% CL). As observed in the earlier cases
with CMB plus either BAO or Pantheon data, in this case, $\Omega_{m0}$
($\sigma_{8}$) adopts a higher (lower) value compared to the $\Lambda$CDM
picture (however, it should be noted that $\Omega_{m0}$ takes relatively a
lower value compared to the CMB+Pantheon case), due to the energy flow.
Lastly, we notice that $H_{0}$ is increased slightly, alleviating the Hubble
tension within $3\sigma$, and additionally we obtain a smaller value of
$S_{8}$ compared to the Planck’s estimation (within the canonical $\Lambda$CDM
picture Aghanim  _et al._ (2020a)), indicating a better consistency with the
weak lensing measurements Heymans  _et al._ (2021); Tröster  _et al._ (2021);
Secco  _et al._ (2022); Abbott  _et al._ (2022).

Table 15 presents the observational constraints of the model
$\xi_{0}\xi_{a}w_{q}$IDE$Q_{A}$, characterized by the parameter space
$\xi_{0}>0$, $\xi_{a}>0$, $w_{q}>-1$ using the observational datasets CMB,
CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon. Our observations for CMB alone are
as follows. We find no evidence of dynamical coupling in the dark sector, but
an indication for a non-zero coupling today at $1\sigma$
($\xi_{0}=0.136^{+0.085}_{-0.088}$) at 68% CL for CMB alone. The matter
density parameter takes a very low value due to the exchange of energy from
the dark matter to dark energy and also due to the fact that the dark energy
equation of state is required to lie within the quintessence region. Despite
the very strong anti-correlation between $w_{q}$ and $H_{0}$, a higher value
of $H_{0}$ compared to the $\Lambda$CDM picture is obtained to increase the
$\Omega_{m}h^{2}$ parameter, governing the angular location of the CMB
acoustic peaks. When BAO observations are added to CMB data, we find no
preference for either a non-zero or a dynamical coupling in the dark sector.
Even if the value of the matter density parameter increases,
$\Omega_{m0}=0.220_{-0.036}^{+0.077}$ at 68% CL, the value is low if compared
to its estimated figure within the $\Lambda$CDM picture. The inclusion of the
Pantheon catalogue to CMB (i.e. CMB+Pantheon) does not lead to any significant
changes in the constraints (the same is true for CMB+BAO+Pantheon dataset).

Tables 16 and 17 together with Figs. 5 and 6 summarize the observational
constraints on the interacting scenarios driven by the interaction function
$Q_{B}$, Eq. (5), for two different regions of the parameter space.

Firstly, Tab. 16 presents the constraints on the model
$\xi_{0}\xi_{a}w_{p}$IDE$Q_{B}$, characterized by the parameter space
$\xi_{0}<0$, $\xi_{a}<0$, $w_{p}<-1$ for CMB, CMB+BAO, CMB+Pantheon and
CMB+BAO+Pantheon. For CMB alone, we find no indication for a non-zero
$\xi_{0}$ and a non-zero $\xi_{a}$, but the dark energy equation of state
deeply lies in the phantom region, $w_{p}=-2.10^{+0.86}_{-0.84}$ at more than
95% CL. Due to the phantom nature of $w_{p}$, this scenario leads to a high
value of the Hubble constant, see Fig. 5. The addition of BAO observations to
CMB data shifts the parameter values. First of all, we get an evidence for a
non-zero $\xi_{a}$ at more than 68% CL ($\xi_{a}=-0.21_{-0.08}^{+0.18}$ at 68%
CL) while within 95% CL, we do not have any evidence for $\xi_{a}\neq 0$. Even
though the dark energy equation of state remains in the phantom regime at more
than 95% CL ($w_{p}=-1.26_{-0.27}^{+0.24}$ at 95% CL, CMB+BAO), the indication
for a phantom nature is slightly decreased (compared to what it is observed in
the CMB data alone case). Additionally, we find a very high (low) value of
$\Omega_{m0}$ ($\sigma_{8}$), and the requirement of a lower value of $H_{0}$,
see the one-dimensional probability distribution for $\Omega_{m0}$ in Fig. 5.
The inclusion of Pantheon measurements to CMB also offers almost similar
results to the CMB+BAO case: we find $\xi_{a}\neq 0$ at 68% CL but it is found
to be unconstrained within our prior at 95%; $w_{p}$ remains in the phantom
region at more than 95% CL but also the preference for a phantom nature has
decreased compared to the CMB alone case. The final combination
CMB+BAO+Pantheon does not exhibit any indication for $\xi_{0}\neq 0$ and
$\xi_{a}\neq 0$, but the dark energy equation of state behaves such as
$w_{p}<-1$ (at more than 95% CL). The other parameter values are very close to
the CMB+BAO constraints.

Secondly, Tab. 17 shows the constraints on the model
$\xi_{0}\xi_{a}w_{q}$IDE$Q_{B}$ characterized by the parameter space
$\xi_{0}>0$, $\xi_{a}>0$, $w_{q}>-1$ for CMB, CMB+BAO, CMB+Pantheon and
CMB+BAO+Pantheon. For all the four datasets, we find no indication for a non-
zero and dynamical coupling in the dark sector. And the remaining parameters
take values very similar to those in the $\Lambda$CDM cosmology: in practise,
this scenario is perfectly consistent with the non-interacting standard
cosmological scenario, see also the one-dimensional probability distributions
in Fig. 6.

####  IV.2.2 Dynamical dark energy equation of state

In the following, we shall present the constraints on the most general
scenario so far discussed in this work. Despite the dimension of the parameter
space, we restrict ourselves to the regions allowed by the doom factor
analyses provided in Ref. Gavela  _et al._ (2009). We have therefore explored
the interacting scenarios driven by Eqs. (4) and (5) dividing the entire
parameter space into two disjoint regions.

Namely, Tab. 18, together with Fig. 7, summarize the observational constraints
on the interacting model $\xi_{0}\xi_{a}w_{0}^{p}w_{e}^{p}$IDE$Q_{A}$,
characterized by the parameter space
$w_{0}^{p}<-1,\;w_{e}^{p}<-1,\;\xi_{0}<0,\;\xi_{a}<0$ for the CMB, CMB+BAO,
CMB+Pantheon and CMB+BAO+Pantheon datasets. For CMB alone, we find a
preference for a non-zero $\xi_{a}$ at 68% CL
($\xi_{a}=-0.073_{-0.031}^{+0.059}$) which is diluted at 95% CL. The dark
energy remains in the phantom region at 68% CL
($w_{0}^{p}=-2.00_{-0.62}^{+0.62}$), and it is unconstrained at 95% CL. The
Hubble constant assumes a higher value which is, as usual, driven by the
phantom dark energy. As a consequence, a relatively small value of the matter
density parameter compared to what we see in the $\Lambda$CDM picture is
required (see Fig. 7). When the BAOs are added to CMB, we find no evidence for
$w_{0}^{p}<-1$ but an indication for an early dark energy component different
from a cosmological constant at early times is suggested at more than 95% CL
($w_{e}^{p}=-1.34_{-0.32}^{+0.31}$ at 95% CL, CMB+BAO). Interestingly, we
notice that $\xi_{a}$ is non-zero at 68% CL
($\xi_{a}=-0.064_{-0.029}^{+0.052}$ at 68% CL) but this indication is diluted
at a higher significance. Notice that the value of $\Omega_{m0}$
($\sigma_{8}$) is still mildly higher (lower) than in the $\Lambda$CDM model.
When the Pantheon catalogue is added to CMB, no evidence for $w_{0}^{p}<-1$ is
found, as already observed in CMB+BAO case, and the evidence for an early dark
energy fluid and not for the simplest vacuum energy scenario is reduced down
to 68% CL ($w_{e}^{p}=-1.27_{-0.12}^{+0.22}$ at 68% CL for CMB+Pantheon).
Interestingly, we find a strong evidence for a dynamical $\xi_{a}$ at more
than 95% CL ($=-0.084_{-0.067}^{+0.071}$ at 95% CL, CMB+Pantheon) and a very
large (small) value of $\Omega_{m0}$ ($\sigma_{8}$) (while the remaining
parameters take almost similar results than the $\Lambda$CDM values). For the
final combination, i.e. CMB+BAO+Pantheon, our results are similar to the
CMB+BAO analysis, that means, no preference for $w_{0}^{p}<-1$ and $\xi_{a}<0$
at 68% CL are obtained, but an indication for an early dark energy component
different from the cosmological constant picture is suggested with a
significance above 95% CL, see the one-dimensional distribution for
$w_{e}^{p}$ in Fig. 7.

Table 19 and Fig. 8 show the constraints on the interacting scenario
$\xi_{0}\xi_{a}w_{0}^{q}w_{e}^{q}$IDE$Q_{A}$, characterized by the parameter
space $w_{0}^{q}>-1,\;w_{e}^{q}>-1,\;\xi_{0}>0,\;\xi_{a}>0$ for CMB, CMB+BAO,
CMB+Pantheon and CMB+BAO+Pantheon data analyses. For CMB alone, we find a
strong evidence for a non-zero interaction ($\xi_{0}=0.26_{-0.23}^{+0.21}$ at
95% CL) while $\xi_{a}$ is consistent with a null value, i.e. there is no
evidence for a dynamical coupling for this dataset. The matter density
parameter takes a very low value due to the values of both $w_{0}^{q}$ an
$\xi_{0}$, see the parameter degeneracies illustrated in the two-dimensional
contours in Fig. 8. As a result, this leads to a very high value of the
$\sigma_{8}$ parameter. When BAO are added to CMB, the mean values of the
parameters are shifted: namely, the evidence for a non-zero $\xi_{0}$ is
reduced down to 68% CL ($\xi_{0}=0.17_{-0.07}^{+0.10}$ at 68% CL). For
CMB+Pantheon instead we find a strong evidence for a non-zero $\xi_{0}$ at
more than 95% CL while no indication for a non-zero $\xi_{a}$ is observed.
Finally, for the entire combined dataset, i.e. for CMB+BAO+Pantheon, an
evidence for a non-zero $\xi_{0}$ is found at more than 95% CL, and the same
significance is present for a quintessence dark energy today, while the
behaviours of the other parameters are similar to the previous cases,
including we a very small value of the matter density parameter and a very
large value of $\sigma_{8}$.

Tables 20 and 21 summarize the observational constraints on the interacting
scenarios driven by the interaction function $Q_{B}$ of Eq. (5) for two
disjoint parameter spaces, ensuring the stability of the scenarios at the
level of perturbations.

Table 20 corresponds to the constraints on the interacting scenario
$\xi_{0}\xi_{a}w_{0}^{p}w_{e}^{p}$IDE$Q_{B}$ characterized by the parameter
space $w_{0}^{p}<-1,\;w_{e}^{p}<-1,\;\xi_{0}<0,\;\xi_{a}<0$ for the
observational datasets CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon. For
CMB alone we find an evidence for a dynamical coupling at 68% CL:
$\xi_{a}=-0.44_{-0.19}^{+0.38}$ at 68% CL (even though at 95% CL, this
parameter is found to be unconstrained) together with an evidence for a
phantom dark energy at present ($w_{0}^{p}<-1$ at 68% CL) and also for an
early dark energy component ($w_{e}^{p}<-1$). Due to the phantom nature of
$w_{0}$, we find a higher value of the Hubble constant. When BAO are added to
CMB, the evidence for a dynamical coupling is still found at more than 68% CL
(although it remains unconstrained at 95% CL) together with an early dark
energy fluid instead of the minimal cosmological constant, present at more
than 95% CL ($w_{e}^{p}=-1.48_{-0.45}^{+0.43}$ at 95% CL, CMB+BAO).
Additionally, a higher (lower) value of $\Omega_{m0}$ ($\sigma_{8}$) is found,
due to both the energy flow and the parameter space of $w(a)$. The inclusion
of the Pantheon catalogue to CMB shifts the mean values of the parameters a
bit. Notice that, similarly to the CMB+BAO case, we find an evidence for an
early dark energy component at more than 95% CL together with an evidence for
a dynamical coupling in the dark sector at 68% CL, which is diluted at 95% CL
(unlike to the previous cases, where $\xi_{a}$ has been found to be
unconstrained). The final dataset combination, i.e. CMB+BAO+Pantheon, leads to
very similar results to the CMB+BAO case, except for (very mild) changes in
the mean parameter values.

To conclude, Tab. 21 corresponds to the constraints on the interacting
scenario $\xi_{0}\xi_{a}w_{0}^{q}w_{e}^{q}$IDE$Q_{B}$, characterized by the
parameter space $w_{0}^{q}>-1,\;w_{e}^{q}>-1,\;\xi_{0}>0,\;\xi_{a}>0$, for the
observational datasets CMB, CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon. For
all four cases, the results agree completely with the $\Lambda$CDM scenario:
we find no evidence for $\xi_{0}\neq 0$, $\xi_{a}\neq 0$, while statistically
we cannot exclude the possibility for a non-zero coupling, and for a dark
energy instead of the cosmological constant. Further, for CMB data alone we
obtain a relatively higher value of $\Omega_{m0}$ (caused due to the exchange
of energy from the dark energy sector to the dark matter one), and
consequently, a lower value of the Hubble constant is found. For the remaining
three analyses, i.e. CMB+BAO, CMB+Pantheon and CMB+BAO+Pantheon, we notice
that both free and derived parameters, adopt very similar values to the
$\Lambda$CDM case.

##  V Discussion and Summary

Interacting dark energy models are very appealing cosmologies which play a
crucial role in explaining several important observational issues in modern
cosmology. Ranging from the ‘why now?’ problem in cosmology, where interacting
theories have shown their ability to offer a possible explanation Amendola
(2000); Huey and Wandelt (2006); Cai and Wang (2005); Pavon and Zimdahl
(2005); Berger and Shojaei (2006); del Campo  _et al._ (2006, 2008, 2009) to
current cosmological tensions Kumar and Nunes (2017); Di Valentino  _et al._
(2017); Yang  _et al._ (2018c, b); Pan  _et al._ (2019, 2020a); Di Valentino
_et al._ (2021b), these particular theories have a very rich phenomenology.
The underlying mechanism of interaction between dark matter and dark energy is
driven by an energy and/or momentum transfer between these dark fluids and
this is usually quantified by an interaction function, $Q$. The function $Q$
modifies the expansion history of the universe both at the background and the
perturbation levels, and, as a consequence, cosmological parameters are
directly affected. Nevertheless, and for the shake of simplicity, almost _all_
works in the literature have assumed a constant coupling parameter.
Considering this issue, the possibility of dynamical nature in the coupling
parameter has been raised and investigated by a few authors in a few articles
Li and Zhang (2011); Guo  _et al._ (2018); Wang and Zhao (2018); Yang  _et
al._ (2019e, 2020d). Note that interacting scenarios with dynamical coupling
parameter offer a much more general picture of the universe in which a
constant coupling parameter would be a special case of the former. On the
other hand, there is no reason to exclude the possibility of the dynamical
coupling parameter in the interaction functions. In this article we have
investigated this issue with a special focus on the dark energy sector, making
the analyses as general as possible. We have therefore considered three
different types of dark energy, quantified through their equation-of-state,
namely, vacuum energy (characterized by $w=-1$), a dark energy fluid with
constant $w\neq-1$, and a dynamical $w(a)$ with $w(a)=w_{0}a+w_{e}(1-a)$, in
which $w_{0}$ refers to its present value and $w_{e}$ refers to its value in
the early time. Additionally, following Refs. Yang  _et al._ (2019e, 2020d),
we have assumed that the coupling parameter takes the following expression,
$\xi(a)=\xi_{0}+\xi_{a}(1-a)$, which as one can recognizes quickly, is the
first two terms of the Taylor expansion of $\xi(a)$ around the present value
of the scale factor, i.e. unity. We note that the above choice of the coupling
parameter is the most natural one in order to detect the deviation, if any, of
the coupling parameter from $\xi(a)=\xi_{0}=$ constant. Finally, we further
note that we have considered two interaction functions, $Q_{A}$ and $Q_{B}$,
given by Eqs. (4)) and (5)) respectively.

We summarize below the most relevant results of our analyses, focusing on the
key issue of the article, namely, the constant coupling parameter versus the
dynamical coupling parameter.

  * •

Constant dark energy - dark matter coupling:

The simplest interacting scenario in this class of models is the one where the
dark energy equation of state $w$ is equal to $-1$. We find only mild evidence
for a coupling parameter (quantified through $\xi_{a}\neq 0$) at 68% CL for
model A, that is completely diluted when considering BAO observations. For the
case in which $w\neq-1$, we have divided the parameter space into two regions,
to satisfy the stability requirements. For model A and $w<-1$, i.e. for the
interacting scenario $\xi_{0}w_{p}$IDE$Q_{A}$, $\xi_{0}$ is required to be
such that $\xi_{0}<-1$. In this case, there is a strong preference for $w<-1$
together with a very large value of $H_{0}$ without BAO data. Once BAO
measurements are considered, the data suggest a phantom-interacting scenario
at $1\sigma$, that persists and increases with the addition of the Pantheon
data. For model A and dark energy within the quintessence region,
$\xi_{0}w_{q}$IDE$Q_{A}$, there is a preference for a (constant) dark coupling
that is not diluted after including BAO and Pantheon measurements, but rather
exacerbated at $2\sigma$. Model B gives no indication for a coupling for all
the combination of datasets considered, contrarily to model A, in the phantom
region, while in the quintessence region it is very consistent with a
$\Lambda$CDM non-interacting scenario. Finally, we have also explored the case
of a time-varying dark energy equation of state governed by two parameters,
$w_{0}$, which states its current value, and $w_{e}$, which refers to its
component at early times. For model A and $\xi_{0}<0$ (and therefore both
$w_{0}$ and $w_{e}$ in the phantom region), for the most complete combination
of data sets, i.e. CMB+BAO+Pantheon, the coupling parameter is found to be
non-zero at $95\%$ CL, and a slight ($68\%$ CL) evidence for an early dark
energy component (i.e. a dark energy fluid different from a cosmological
constant) is also found. In this case the Hubble tension is alleviated at
$2.5\sigma$ and the $S_{8}$ tension is relaxed. For the very same model but
for $\xi_{0}>0$ (and therefore both $w_{0}$ and $w_{e}$ in the quintessence
region) and also for the most complete combination of data sets, i.e.
CMB+BAO+Pantheon, there is evidence for a quintessence interacting scenario
today at more than 95% CL, but the value of $\Omega_{m0}$ is much lower than
its $\Lambda$CDM-like value. This case exacerbates the so-called $S_{8}$
tension, as the value of the clustering parameter $\sigma_{8}$ is much higher
than in the canonical $\Lambda$CDM scheme. For model B and $\xi_{0}<0$ (and
therefore both $w_{0}$ and $w_{e}$ in the phantom region), we only find slight
evidence for a non-zero coupling and for an early dark energy component
different from a cosmological constant at $1\sigma$ for the full combination
CMB+BAO+Pantheon, while if we restrict the model to the quintessence region
and $\xi>0$, we just note a $1\sigma$ indication for $\xi\neq 0$ for the most
complete data set.

  * •

Dynamical dark energy - dark matter coupling:

In the following, we shall summarize our main results for a dynamical coupling
of the form $\xi_{0}+\xi_{a}\;(1-a)$, see Eq. (7). For vacuum dark energy and
model A, the parameter values are shifted accordingly to the sign of
$\xi_{0}$: if $\xi_{0}<0$ ($\xi_{0}>0$), the energy flows from the dark energy
(matter) sector to the dark matter (energy) one, resulting in higher (lower)
values of $\Omega_{m0}$ and lower (higher) values of $\sigma_{8}$. However,
all the datasets are in agreement with no interaction in the dark sector. For
model B, BAO data seems to prefer $\xi_{a}\neq 0$ at 68% CL, and therefore a
mild evidence for a dynamical coupling arises when considering the former
observations. If instead $w\neq-1$, we have as usual the phantom (associated
to $\xi_{0}<0$ and $\xi_{a}<0$) and quintessence (associated to $\xi_{0}>0$
and $\xi_{a}>0$) regions. For model A in the phantom region $w>-1$ at more
than $95\%$ CL and there is mild evidence for $\xi_{a}\neq 0$ when Pantheon
observations are added to CMB data. For this very same model but within the
quintessence region, the dark matter energy density is very low due to the
energy flow and no evidence for a dynamical dark sector is found. For Model B,
we find again strong evidence for $w<-1$ within the phantom region
($\xi_{0}\xi_{a}w_{0}^{p}w_{e}^{p}$IDE$Q_{B}$) and a mild evidence for
$\xi_{a}\neq 0$ in some parameter combinations. The quintessence region
($\xi_{0}\xi_{a}w_{0}^{q}w_{e}^{p}$IDE$Q_{B}$) is extremely in agreement to a
non-interacting model.

Finally, we consider the most generalized interacting scenarios in which both
the coupling and the dark energy equation of state are dynamical. Following
the stability criteria Gavela  _et al._ (2009) to have divided the parameter
space. For model A in its phantom region ($w_{0}^{p}<-1$ and $w_{e}^{p}<-1$,
i.e. $\xi_{0}\xi_{a}w_{0}^{p}w_{e}^{p}$IDE$Q_{A}$), the coupling parameters
are required to be negative, and we find evidence at $95\%$ CL ($68\%$ CL)
significance for an early dark energy component (a dynamical coupling) for
CMB+BAO+Pantheon. In the case of the model
$\xi_{0}\xi_{a}w_{0}^{q}w_{e}^{q}$IDE$Q_{A}$, the energy flow is reversed and
therefore $\Omega_{m0}$ ($\sigma_{8}$) takes much lower (higher) values than
within the minimal $\Lambda$CDM scenario. For CMB+BAO+Pantheon a non-zero
value of $\xi_{0}$ and $w_{0}^{q}$ are found at more than $95\%$ CL, or an
indication for a quintessence interacting model. Concerning model B within its
phantom region ($\xi_{0}\xi_{a}w_{0}^{p}w_{e}^{p}$IDE$Q_{B}$) the results are
very similar to model A in this very same region, finding evidence at $95\%$
CL ($68\%$ CL) significance for an early dark energy component (a dynamical
coupling) for CMB+BAO+Pantheon. Model
$\xi_{0}\xi_{a}w_{0}^{q}w_{e}^{q}$IDE$Q_{B}$ instead provides results in
agreement with the canonical non-interacting case except for slightly larger
(smaller) values of $\Omega_{m0}$ ($\sigma_{8}$) for the CMB alone data, due
to the energy exchange among the dark sectors.

Our summary above clearly states that interacting cosmologies have a very rich
phenomenology. In particular, really interesting are those cases, where a dark
energy component different from the cosmological constant at early times, and
a coupling different from zero today are explored, that for the full
combination CMB+BAO+Pantheon can alleviate both the $H_{0}$ and $S_{8}$
tensions, as for example for the “$\xi_{0}w_{0}^{p}w_{e}^{p}$IDE$Q_{A}$”
model. Scenarios in which the coupling parameter is time dependent are highly
promising and are expected to be strongly constrained with upcoming
cosmological observations.

##  VI Acknowledgments

WY was supported by the National Natural Science Foundation of China under
Grants No. 12175096 and No. 11705079, and Liaoning Revitalization Talents
Program under Grant no. XLYC1907098. SP acknowledges the financial support
from the Department of Science and Technology (DST), Govt. of India under the
Scheme “Fund for Improvement of S&T Infrastructure (FIST)” (File No.
SR/FST/MS-I/2019/41). OM is supported by the Spanish grants
PID2020-113644GB-I00, PROMETEO/2019/083 and by the European ITN project HIDDeN
(H2020-MSCA-ITN-2019//860881-HIDDeN). EDV is supported by a Royal Society
Dorothy Hodgkin Research Fellowship.

## References

  * Amendola (2000) L. Amendola, Phys. Rev. D 62, 043511 (2000), arXiv:astro-ph/9908023 .
  * Wetterich (1995) C. Wetterich, Astron. Astrophys. 301, 321 (1995), arXiv:hep-th/9408025 .
  * Huey and Wandelt (2006) G. Huey and B. D. Wandelt, Phys. Rev. D 74, 023519 (2006), arXiv:astro-ph/0407196 .
  * Cai and Wang (2005) R.-G. Cai and A. Wang, JCAP 03, 002 (2005), arXiv:hep-th/0411025 .
  * Pavon and Zimdahl (2005) D. Pavon and W. Zimdahl, Phys. Lett. B 628, 206 (2005), arXiv:gr-qc/0505020 .
  * Berger and Shojaei (2006) M. S. Berger and H. Shojaei, Phys. Rev. D 73, 083528 (2006), arXiv:gr-qc/0601086 .
  * del Campo  _et al._ (2006) S. del Campo, R. Herrera, G. Olivares, and D. Pavon, Phys. Rev. D 74, 023501 (2006), arXiv:astro-ph/0606520 .
  * del Campo  _et al._ (2008) S. del Campo, R. Herrera, and D. Pavon, Phys. Rev. D 78, 021302 (2008), arXiv:0806.2116 [astro-ph] .
  * del Campo  _et al._ (2009) S. del Campo, R. Herrera, and D. Pavon, JCAP 01, 020 (2009), arXiv:0812.2210 [gr-qc] .
  * Aghanim  _et al._ (2020a) N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A6 (2020a), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Aiola  _et al._ (2020) S. Aiola _et al._ (ACT), JCAP 12, 047 (2020), arXiv:2007.07288 [astro-ph.CO] .
  * Dutcher  _et al._ (2021) D. Dutcher _et al._ (SPT-3G), Phys. Rev. D 104, 022003 (2021), arXiv:2101.01684 [astro-ph.CO] .
  * Riess  _et al._ (2021) A. G. Riess _et al._ , (2021), arXiv:2112.04510 [astro-ph.CO] .
  * Riess  _et al._ (2022) A. G. Riess, L. Breuval, W. Yuan, S. Casertano, L. M. Macri, D. Scolnic, T. Cantat-Gaudin, R. I. Anderson, and M. C. Reyes, (2022), arXiv:2208.01045 [astro-ph.CO] .
  * Verde  _et al._ (2019) L. Verde, T. Treu, and A. G. Riess,  _Nature Astronomy 2019_ ,  (2019), 10.1038/s41550-019-0902-0, arXiv:1907.10625 [astro-ph.CO] .
  * Knox and Millea (2020) L. Knox and M. Millea, Phys. Rev. D 101, 043533 (2020), arXiv:1908.03663 [astro-ph.CO] .
  * Di Valentino  _et al._ (2021a) E. Di Valentino _et al._ , Astropart. Phys. 131, 102605 (2021a), arXiv:2008.11284 [astro-ph.CO] .
  * Jedamzik  _et al._ (2021) K. Jedamzik, L. Pogosian, and G.-B. Zhao, Commun. in Phys. 4, 123 (2021), arXiv:2010.04158 [astro-ph.CO] .
  * Riess (2019) A. G. Riess, Nature Rev. Phys. 2, 10 (2019), arXiv:2001.03624 [astro-ph.CO] .
  * Di Valentino (2021) E. Di Valentino, Mon. Not. Roy. Astron. Soc. 502, 2065 (2021), arXiv:2011.00246 [astro-ph.CO] .
  * Di Valentino  _et al._ (2021b) E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, Class. Quant. Grav. 38, 153001 (2021b), arXiv:2103.01183 [astro-ph.CO] .
  * Perivolaropoulos and Skara (2022) L. Perivolaropoulos and F. Skara, New Astron. Rev. 95 (2022), 10.1016/j.newar.2022.101659, arXiv:2105.05208 [astro-ph.CO] .
  * Schöneberg  _et al._ (2022) N. Schöneberg, G. Franco Abellán, A. Pérez Sánchez, S. J. Witte, V. Poulin, and J. Lesgourgues, Phys. Rept. 984, 1 (2022), arXiv:2107.10291 [astro-ph.CO] .
  * Shah  _et al._ (2021) P. Shah, P. Lemos, and O. Lahav, Astron. Astrophys. Rev. 29, 9 (2021), arXiv:2109.01161 [astro-ph.CO] .
  * Di Valentino (2022) E. Di Valentino, Universe 8, 399 (2022).
  * Abdalla  _et al._ (2022) E. Abdalla _et al._ , JHEAp 34, 49 (2022), arXiv:2203.06142 [astro-ph.CO] .
  * Barrow and Clifton (2006) J. D. Barrow and T. Clifton, Phys. Rev. D 73, 103520 (2006), arXiv:gr-qc/0604063 .
  * Valiviita  _et al._ (2008) J. Valiviita, E. Majerotto, and R. Maartens, JCAP 07, 020 (2008), arXiv:0804.0232 [astro-ph] .
  * Gavela  _et al._ (2009) M. B. Gavela, D. Hernandez, L. Lopez Honorez, O. Mena, and S. Rigolin, JCAP 07, 034 (2009), [Erratum: JCAP 05, E01 (2010)], arXiv:0901.1611 [astro-ph.CO] .
  * Gavela  _et al._ (2010) M. B. Gavela, L. Lopez Honorez, O. Mena, and S. Rigolin, JCAP 11, 044 (2010), arXiv:1005.0295 [astro-ph.CO] .
  * Li and Zhang (2014) Y.-H. Li and X. Zhang, Phys. Rev. D 89, 083009 (2014), arXiv:1312.6328 [astro-ph.CO] .
  * Salvatelli  _et al._ (2014) V. Salvatelli, N. Said, M. Bruni, A. Melchiorri, and D. Wands, Phys. Rev. Lett. 113, 181301 (2014), arXiv:1406.7297 [astro-ph.CO] .
  * Wang  _et al._ (2014) Y. Wang, D. Wands, G.-B. Zhao, and L. Xu, Phys. Rev. D 90, 023502 (2014), arXiv:1404.5706 [astro-ph.CO] .
  * Casas  _et al._ (2016) S. Casas, L. Amendola, M. Baldi, V. Pettorino, and A. Vollmer, JCAP 01, 045 (2016), arXiv:1508.07208 [astro-ph.CO] .
  * van de Bruck  _et al._ (2016) C. van de Bruck, J. Mifsud, J. P. Mimoso, and N. J. Nunes, JCAP 11, 031 (2016), arXiv:1605.03834 [gr-qc] .
  * Yang  _et al._ (2016) W. Yang, H. Li, Y. Wu, and J. Lu, JCAP 10, 007 (2016), arXiv:1608.07039 [astro-ph.CO] .
  * Kumar and Nunes (2016) S. Kumar and R. C. Nunes, Phys. Rev. D 94, 123511 (2016), arXiv:1608.02454 [astro-ph.CO] .
  * van de Bruck  _et al._ (2017) C. van de Bruck, J. Mifsud, and J. Morrice, Phys. Rev. D 95, 043513 (2017), arXiv:1609.09855 [astro-ph.CO] .
  * Pan and Sharov (2017) S. Pan and G. Sharov, Mon. Not. Roy. Astron. Soc. 472, 4736 (2017), arXiv:1609.02287 [gr-qc] .
  * Yang  _et al._ (2017a) W. Yang, N. Banerjee, and S. Pan, Phys. Rev. D 95, 123527 (2017a), arXiv:1705.09278 [astro-ph.CO] .
  * Sharov  _et al._ (2017) G. S. Sharov, S. Bhattacharya, S. Pan, R. C. Nunes, and S. Chakraborty, Mon. Not. Roy. Astron. Soc. 466, 3497 (2017), arXiv:1701.00780 [gr-qc] .
  * Kumar and Nunes (2017) S. Kumar and R. C. Nunes, Phys. Rev. D96, 103511 (2017), arXiv:1702.02143 [astro-ph.CO] .
  * Mifsud and Van De Bruck (2017) J. Mifsud and C. Van De Bruck, JCAP 11, 001 (2017), arXiv:1707.07667 [astro-ph.CO] .
  * Van De Bruck and Mifsud (2018) C. Van De Bruck and J. Mifsud, Phys. Rev. D 97, 023506 (2018), arXiv:1709.04882 [astro-ph.CO] .
  * Yang  _et al._ (2017b) W. Yang, S. Pan, and D. F. Mota, Phys. Rev. D 96, 123508 (2017b), arXiv:1709.00006 [astro-ph.CO] .
  * Yang  _et al._ (2018a) W. Yang, S. Pan, and J. D. Barrow, Phys. Rev. D 97, 043529 (2018a), arXiv:1706.04953 [astro-ph.CO] .
  * Pan  _et al._ (2018) S. Pan, A. Mukherjee, and N. Banerjee, Mon. Not. Roy. Astron. Soc. 477, 1189 (2018), arXiv:1710.03725 [astro-ph.CO] .
  * Yang  _et al._ (2019a) W. Yang, S. Pan, and A. Paliathanasis, Mon. Not. Roy. Astron. Soc. 482, 1007 (2019a), arXiv:1804.08558 [gr-qc] .
  * Yang  _et al._ (2019b) W. Yang, S. Pan, L. Xu, and D. F. Mota, Mon. Not. Roy. Astron. Soc. 482, 1858 (2019b), arXiv:1804.08455 [astro-ph.CO] .
  * Yang  _et al._ (2018b) W. Yang, A. Mukherjee, E. Di Valentino, and S. Pan, Phys. Rev. D98, 123527 (2018b), arXiv:1809.06883 [astro-ph.CO] .
  * Yang  _et al._ (2018c) W. Yang, S. Pan, E. Di Valentino, R. C. Nunes, S. Vagnozzi, and D. F. Mota, JCAP 09, 019 (2018c), arXiv:1805.08252 [astro-ph.CO] .
  * Li  _et al._ (2018) H. Li, W. Yang, Y. Wu, and Y. Jiang, Phys. Dark Univ. 20, 78 (2018).
  * Yang  _et al._ (2018d) W. Yang, S. Pan, R. Herrera, and S. Chakraborty, Phys. Rev. D 98, 043517 (2018d), arXiv:1808.01669 [gr-qc] .
  * Yang  _et al._ (2019c) W. Yang, N. Banerjee, A. Paliathanasis, and S. Pan, Phys. Dark Univ. 26, 100383 (2019c), arXiv:1812.06854 [astro-ph.CO] .
  * Yang  _et al._ (2020a) W. Yang, S. Pan, R. C. Nunes, and D. F. Mota, JCAP 04, 008 (2020a), arXiv:1910.08821 [astro-ph.CO] .
  * Li  _et al._ (2020) H.-L. Li, D.-Z. He, J.-F. Zhang, and X. Zhang, JCAP 06, 038 (2020), arXiv:1908.03098 [astro-ph.CO] .
  * Di Valentino  _et al._ (2020a) E. Di Valentino, A. Melchiorri, O. Mena, and S. Vagnozzi, Phys. Dark Univ. 30, 100666 (2020a), arXiv:1908.04281 [astro-ph.CO] .
  * Oikonomou (2019) V. K. Oikonomou, Phys. Rev. D 99, 104042 (2019), arXiv:1905.00826 [gr-qc] .
  * Di Valentino  _et al._ (2021c) E. Di Valentino, A. Melchiorri, O. Mena, S. Pan, and W. Yang, Mon. Not. Roy. Astron. Soc. 502, L23 (2021c), arXiv:2011.00283 [astro-ph.CO] .
  * Martinelli  _et al._ (2019) M. Martinelli, N. B. Hogg, S. Peirone, M. Bruni, and D. Wands, Mon. Not. Roy. Astron. Soc. 488, 3423 (2019), arXiv:1902.10694 [astro-ph.CO] .
  * Kumar  _et al._ (2019) S. Kumar, R. C. Nunes, and S. K. Yadav, Eur. Phys. J. C79, 576 (2019), arXiv:1903.04865 [astro-ph.CO] .
  * Cheng  _et al._ (2020) G. Cheng, Y.-Z. Ma, F. Wu, J. Zhang, and X. Chen, Phys. Rev. D 102, 043517 (2020), arXiv:1911.04520 [astro-ph.CO] .
  * Lucca and Hooper (2020) M. Lucca and D. C. Hooper, Phys. Rev. D 102, 123502 (2020), arXiv:2002.06127 [astro-ph.CO] .
  * Cid  _et al._ (2021) A. Cid, C. Rodriguez-Benites, M. Cataldo, and G. Casanova, Eur. Phys. J. C 81, 31 (2021), arXiv:2005.07664 [astro-ph.CO] .
  * Beltrán Jiménez  _et al._ (2021) J. Beltrán Jiménez, D. Bettoni, D. Figueruelo, F. A. Teppa Pannia, and S. Tsujikawa, JCAP 03, 085 (2021), arXiv:2012.12204 [astro-ph.CO] .
  * Thipaksorn  _et al._ (2022) W. Thipaksorn, S. Sapa, and K. Karwan, Phys. Rev. D 105, 063527 (2022), arXiv:2201.03261 [gr-qc] .