# Constraints from High-Precision Measurements of the Cosmic Microwave
Background: The Case of Disintegrating Dark Matter with ${\Lambda}$ or
Dynamical Dark Energy

Wenzhong Liu<EMAIL_ADDRESS>Department of Physics, Liaoning Normal
University, Dalian, 116029, P. R. China  Luis A. Anchordoqui
<EMAIL_ADDRESS>Department of Physics and Astronomy, Lehman
College, City University of New York, NY 10468, USA  Department of Physics,
Graduate Center, City University of New York, NY 10016, USA  Department of
Astrophysics, American Museum of Natural History, NY 10024, USA  Eleonora Di
Valentino<EMAIL_ADDRESS>Institute for Particle Physics
Phenomenology, Department of Physics, Durham University, Durham DH1 3LE, UK
Supriya Pan<EMAIL_ADDRESS>Department of Mathematics,
Presidency University, 86/1 College Street, Kolkata 700073, India  Yabo Wu
<EMAIL_ADDRESS>Department of Physics, Liaoning Normal University, Dalian,
116029, P. R. China  Weiqiang Yang<EMAIL_ADDRESS>Department of Physics,
Liaoning Normal University, Dalian, 116029, P. R. China

###### Abstract

In recent years discrepancies have emerged in measurements of the present-day
rate of expansion of the universe $H_{0}$ and in estimates of the clustering
of matter $S_{8}$. Using the most recent cosmological observations we
reexamine a novel model proposed to address these tensions, in which cold dark
matter disintegrates into dark radiation. The disintegration process is
controlled by its rate $\Gamma=\alpha\mathcal{H}$, where $\alpha$ is a
(constant) dimensionless parameter quantifying the strength of the
disintegration mechanism and $\mathcal{H}$ is the conformal Hubble rate in the
spatially flat Friedmann-Lemaître-Robertson-Walker universe. We constrain this
model with the latest 2018 Planck temperature and polarization data, showing
that there is no evidence for $\alpha\neq 0$ and that it cannot solve the
$H_{0}$ tension below $3\sigma$, clashing with the result obtained by
analyzing the Planck 2015 temperature data. We also investigate two possible
extensions of the model in which the dark energy equation-of-state parameter
$w\neq-1$. In this case it is possible to combine Planck data with the SH0ES
measurement, and we demonstrate that in both these models the $H_{0}$ tension
is resolved at the $1\sigma$ level, but the condition $w\neq-1$ exacerbates
the $S_{8}$ tension. We also demonstrate that the addition of intermediate-
redshift data (from the Pantheon supernova type Ia dataset and baryon acoustic
oscillations) weakens the effectiveness of all these models to address the
$H_{0}$ and $S_{8}$ tensions.

##  I Introduction

Over the last decade or so, the successful operation of WMAP Hinshaw  _et al._
(2013) and Planck Aghanim  _et al._ (2020a) satellites together with large-
scale galaxy surveys and observations from the Hubble space telescope have
provided an avalanche of data, leaving no doubt that a new era is beginning
for “precision cosmology”. Currently, the concordance model of cosmology is
$\Lambda$CDM. Within this model the expansion of the universe today is
dominated by the cosmological constant $\Lambda$ and cold dark matter (CDM).
Even though $\Lambda$CDM has become established as a well tested model,
various discrepancies have emerged, including the more than $4\sigma$ tension
between the observed Riess  _et al._ (2019, 2021) and inferred Aghanim  _et
al._ (2020b) values of the Hubble constant $H_{0}\equiv 100h~{}{\rm km/s/Mpc}$
Di Valentino  _et al._ (2020a), as well as the discrepancy between the
cosmological and local determination of
$S_{8}\equiv\sigma_{8}\sqrt{\Omega_{m}/3}$ Di Valentino  _et al._ (2020b) that
quantifies the rms density fluctuations when smoothed with a top-hat filter of
radius $8h^{-1}/{\rm Mpc}(\equiv\sigma_{8})$ as a function of the present day
value of the non-relativistic matter density parameter $\Omega_{m}$ Di
Valentino  _et al._ (2020a, b). Assuming a flat $\Lambda$CDM model the best-
fit to extract cosmological parameters by the Planck Collaboration leads to
$H_{0}=67.27\pm 0.60~{}{\rm km/s/Mpc}$ at 68% CL Aghanim  _et al._ (2020b),
whereas the SH0ES Collaboration finds a larger value $H_{0}=73.2\pm 1.3~{}{\rm
km/s/Mpc}$ Riess  _et al._ (2021). On the assumption of $\Lambda$CDM the
Planck Collaboration reported $S_{8}=0.830\pm 0.013$ Aghanim  _et al._
(2020b), which is in $3\sigma$ tension with KiDS-1000 data
($S_{8}=0.766^{+0.020}_{-0.014}$) Asgari  _et al._ (2021) and $3.4\sigma$
tension with a combination of BOSS and KV450 data ($S_{8}=0.728\pm 0.026$)
Tröster  _et al._ (2020). Systematic effects do not seem to be responsible for
these discrepancies Verde  _et al._ (2019); Riess (2019); Di Valentino (2021)
and thus a plethora of new cosmological models have been proposed to
accommodate the data Di Valentino  _et al._ (2021a); Perivolaropoulos and
Skara (2021).

The above mentioned discrepancies have become a common test-ground to uncover
properties of the dark sector. New cosmological models modifying the dark
sector now may include a period of early dark energy Poulin  _et al._ (2019);
Karwal and Kamionkowski (2016); Sakstein and Trodden (2020); Agrawal  _et al._
(2019); Niedermann and Sloth (2021, 2020), phantom dark energy Yang  _et al._
(2019a); Di Valentino  _et al._ (2020c, 2021b); Yang  _et al._ (2021a); Di
Valentino  _et al._ (2021c), interacting dark energy Kumar and Nunes (2016);
Di Valentino  _et al._ (2020d); Kumar  _et al._ (2019); Lucca and Hooper
(2020); Yang  _et al._ (2020a); Di Valentino  _et al._ (2017); Yang  _et al._
(2020b, 2018); Anchordoqui  _et al._ (2020); Pan  _et al._ (2020a, 2019a);
Yang  _et al._ (2019b); Pan  _et al._ (2019b); Anchordoqui  _et al._ (2021a);
Allali  _et al._ (2021); Nunes and Di Valentino (2021), emergent dark energy
Li and Shafieloo (2019); Pan  _et al._ (2020b); Li and Shafieloo (2020);
Hernández-Almada  _et al._ (2020); Benaoum  _et al._ (2020); Yang  _et al._
(2021b), scattering-induced disintegrating dark matter Bjaelde  _et al._
(2012); Bringmann  _et al._ (2018); Pandey  _et al._ (2020) and decaying dark
matter Berezhiani  _et al._ (2015); Benisty (2019); Vattis  _et al._ (2019);
Desai  _et al._ (2020); Anchordoqui (2021); Blinov  _et al._ (2020); Nygaard
_et al._ (2021); Chen  _et al._ (2020). It would be engaging and at the same
time intriguing if both the $H_{0}$ and $S_{8}$ discrepancies were to be
resolved simultaneously, but as yet model building of the dark sector on this
front has not been done to a satisfactory degree Schöneberg  _et al._ (2021);
Anchordoqui  _et al._ (2021b).

In this paper we reexamine the idea that scattering-induced disintegration of
dark matter into dark radiation (dr), with an interaction rate proportional to
the Hubble parameter, could re-accelerate the expansion rate to accommodate
the $H_{0}$ tension. The relativistic degrees of freedom are generally
parametrized via $N_{\rm eff}$ Steigman  _et al._ (1977) and constrained by
the cosmic microwave background (CMB). In the Standard Model, we have $N_{\rm
eff}^{\rm SM}=3.046$ Mangano  _et al._ (2005); de Salas and Pastor (2016);
Akita and Yamaguchi (2020); Froustey  _et al._ (2020); Bennett  _et al._
(2021), and so the disintegrating dark matter (ddm) model would produce
$\Delta N_{\rm eff}\equiv N_{\rm eff}-N_{\rm eff}^{\rm SM}>0$. At the same
time the ddm would reduce the amount of CDM relaxing the $S_{8}$ tension.
Indeed, in Ref. Pandey  _et al._ (2020) a likelihood analysis was carried out
considering only 2015 Planck TT CMB data and the local SH0ES prior on $H_{0}$.
This study leads to a 1% upper bound of CDM disintegration decreasing the
$S_{8}$ tension down to $0.3\sigma$ while simultaneously increasing the
central value of $H_{0}$ by a factor of about $1.023$ km/s/Mpc. However, the
addition of intermediate-redshift data from supernova and baryon acoustic
oscillations (BAO) weakens the effectiveness of the ddm model, because when
these data sets are included the upper bound on ddm reduces to about 0.5%
(bringing the $S_{8}$ tension to roughly $1.5\sigma$) and the increment in the
mean value of $H_{0}$ is only a factor of $1.005$ km/s/Mpc. Herein we combine
the ddm idea with various dark energy sectors including the dynamical dark
energy as a generalized candidate of the dark energy.

Before proceeding, we pause to note that since a fraction of dark matter
disintegrates into dr per Hubble time, the effect is amplified near the onset
of matter domination and therefore becomes constrained by CMB data. Now, the
Planck 2015 dataset only contains information on the temperature spectrum,
featuring a large correlation between $N_{\rm eff}$ and $H_{0}$. This implies
that by accommodating a $\Delta N_{\rm eff}>0$ it is possible to naturally
increase $H_{0}$. To improve the determination of $N_{\rm eff}$, in our study
we consider the latest Planck 2018 data sample which contains both temperature
and polarization measurements, together with a new optical depth $\tau$
estimate which strongly correlates with this parameter, and shifts the $N_{\rm
eff}$ best fit towards lower values. Thereby, although $N_{\rm eff}$ affects
the temperature spectrum modifying the smoothing in the damping tail and
increasing the early Integrated Sachs-Wolfe (eISW) effect Bowen  _et al._
(2002); Hou  _et al._ (2013), the polarization is not affected by the eISW
effect, breaking the degeneracy while yielding a very robust measurement of dr
Di Valentino  _et al._ (2018).111The well-known eISW effect encodes the
contribution to CMB anisotropies originating in time-varying gravitational
potentials at early times, shortly after recombination, when the universe was
not entirely matter-dominated Sachs and Wolfe (1967); Rees and Sciama (1968).
The dr measurement is so robust, that even adding additional data (BAO or
Pantheon) the central value does not shift from $3.046$.

The paper has been organized as follows. In Sec. II we present the background
and perturbations equations of the ddm scenarios. After that in Sec. III we
describe the observational datasets and the statistical methodology to
constrain all the cosmological scenarios described in this article. Then, in
Sec. IV we discuss the observational constraints extracted out of all the
scenarios considered. Finally, in Sec. V we close this article with a brief
summary of the results.

##  II Disintegrating Dark matter: Background and Perturbations

We consider a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) line
element which provides a good description of our homogeneous and isotropic
universe:

$ds^{2}=-dt^{2}+a^{2}(t)\,\delta_{ij}\,dx^{i}dx^{j}\,,$ (1)

where $a(t)$ is the expansion scale factor of the universe. We expand the
metric tensor into spatial average and small linear perturbations. We work in
the synchronous gauge in conformal time $d\eta=dt/a$, for which the scalar
component of the inhomogeneous line element reduces to,

$ds^{2}=a^{2}(\eta)[-d\eta^{2}+(\delta_{ij}+h_{ij})\ dx^{i}dx^{j}]\,,$ (2)

with $h_{ij}$ a rank-2 symmetric tensor field associated to the six components
of the perturbations of the spatial part. Following Ma and Bertschinger
(1995), $h_{ij}$ is parametrized in Fourier space $k$. We assume that the
gravitational sector of the universe is perfectly described by the General
Relativity and within the matter sector not all the components are
independently conserved. Here we explore the cosmology of the disintegration
mechanism between CDM and dr but the remaining components, namely baryons and
dark energy are independently conserved. In the scattering-induced ddm model,
the CDM (with equation of state parameter $w_{\rm ddm}=0$) disintegrates into
dr (with $w_{\rm dr}=1/3$), and hence the continuity equations read:

$\displaystyle\rho^{\prime}_{\rm ddm}+3\mathcal{H}\rho_{\rm ddm}$
$\displaystyle=$ $\displaystyle-\Gamma\ \rho_{\rm ddm},$ (3)
$\displaystyle\rho^{\prime}_{\rm dr}+4\mathcal{H}\rho_{\rm dr}$
$\displaystyle=$ $\displaystyle\Gamma\ \rho_{\rm ddm}\,,$ (4)

where $\rho_{\rm ddm}$ and $\rho_{\rm dr}$ are respectively the densities of
ddm and dr, the prime denotes the derivative with respect to the conformal
time $\eta$, $\Gamma=\alpha\mathcal{H}$ is the interaction rate, $\mathcal{H}$
is the conformal Hubble rate, and $\alpha$ is the coupling parameter that
quantifies the strength of the CDM disintegration into dark radiation.

The evolution of the ddm and dr components at the background level can be
completely solved from Eqs. (3) and (4), leading to:

$\displaystyle\rho_{\rm ddm}$ $\displaystyle=$ $\displaystyle\rho_{{\rm
ddm},0}\ a^{-(3+\alpha)},$ (5) $\displaystyle\rho_{\rm dr}$ $\displaystyle=$
$\displaystyle=\beta\ a^{-4}+\frac{\alpha}{1-\alpha}\rho_{\rm ddm,0}\
a^{-(3+\alpha)}\,,$ (6)

where $\beta$ is a constant. For simplicity, herein we assume that the initial
abundance of CDM $\equiv$ ddm is established at some early time $t_{\rm
prod}\ll t_{\rm LS}$ and that $\Delta N_{\rm eff}\ll 1$ at $t_{\rm prod}$,
where $t_{\rm LS}$ denotes the time of last scattering. All through the paper
we adopt the subscript $0$ to indicate the quantities are evaluated today,
with $a_{0}=1$. Note that the first term in the right-hand-side of Eq. (6)
behaves like a standard radiation density whereas the second term behaves like
a fluid with an equation of state $\alpha/3$. Note also that both $\rho_{\rm
ddm}$ and $\rho_{\rm dr}$ will be taken as independent parameters in the
likelihood analysis and independently normalized to data at and after
recombination. Therefore, the relation $\rho_{\rm
ddm,prod}\leftrightharpoons\rho_{\rm dr,prod}$ remains unconstrained and,
without loss of generality, we can set $\beta=0$. With this in mind, the
fraction of dark matter that disintegrates into dark radiation,

$\displaystyle f_{\rm ddm}=\frac{\rho_{{\rm dr},0}}{\rho_{{\rm
ddm},0}}=\frac{\alpha}{1-\alpha},$ (7)

remains constant over the time. Thus, the system of equations describing the
evolution of CDM and dr are simplified to:

$\displaystyle\rho_{\rm ddm}$ $\displaystyle=$ $\displaystyle\rho_{\rm ddm,0}\
a^{-(3+\alpha)},$ (8) $\displaystyle\rho_{\rm dr}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{1-\alpha}\rho_{\rm ddm,0}\ a^{-(3+\alpha)}\,.$ (9)

Now, for the total energy density of the universe given by $\rho_{\rm
total}=\rho_{b}+\rho_{\rm ddm}+\rho_{\rm dr}+\rho_{\rm de}$, we can write down
the evolution of the Hubble expansion as follows

$\displaystyle\frac{H^{2}}{H_{0}^{2}}=\Omega_{b}\ a^{-3}+\Omega_{\rm ddm}\
a^{-(3+\alpha)}+\frac{\alpha}{1-\alpha}\ \Omega_{\rm ddm}\
a^{-(3+\alpha)}+\Omega_{\rm de}\ \exp\left(3\int^{1}_{a}\frac{1+w_{\rm
de}(a^{\prime})}{a^{\prime}}da^{\prime}\right),$ (10)

where $\Omega_{i}=\rho_{i,0}/\rho_{\rm crit,0}$ denotes the present-day
density parameters, with $\rho_{\rm crit,0}=3H_{0}^{2}/(8\pi G)$ the present-
day value of the critical density, and where $w_{\rm de}(a)$ is the barotropic
equation of state of the dark energy, which could be either time independent
or time dependent.222Because we are always referring to the present day
density parameters, we omit the subscript $0$ in this case. In this work we
consider the two possibilities, with $w_{\rm de}(a)$ defined as follows:

  * •

Non-dynamical dark energy: We consider the cosmological constant $\Lambda$
(i.e. $w_{\rm de}(a)=-1$) as the canonical example of non-dynamical dark
energy. The Hubble expansion in this case becomes,

$\displaystyle\frac{H^{2}}{H_{0}^{2}}=\Omega_{b}\ a^{-3}+\Omega_{\rm ddm}\
a^{-(3+\alpha)}+\frac{\alpha}{1-\alpha}\Omega_{\rm ddm}\
a^{-(3+\alpha)}+\Omega_{\Lambda}\;,$ (11)

and we label this cosmological scenario as $\Lambda$ddm.

  * •

Dynamical dark energy: In this category we assume two different dark energy
candidates as follows

    * –

We consider the simplest dynamical dark energy model characterized by the
constant equation of state $w_{\rm de}(a)=w_{0}\neq-1$. The Hubble expansion
in this case takes the form

$\displaystyle\frac{H^{2}}{H_{0}^{2}}=\Omega_{b}\ a^{-3}+\Omega_{\rm ddm}\
a^{-(3+\alpha)}+\frac{\alpha}{1-\alpha}\Omega_{\rm ddm}\
a^{-(3+\alpha)}+\Omega_{\rm de}\ a^{-3(1+w_{0})}\;,$ (12)

and we label this scenario as $w_{0}$ddm.

    * –

As a general dynamical dark energy model, we assume the most well known
dynamical equation of state parametrization, namely the Chevallier-Polarski-
Linder (CPL) parametrization $w_{\rm de}(a)=w_{0}+w_{a}(1-a)$, where $w_{0}$
and $w_{a}=dw_{\rm de}(a)/da$ are the free parameters. The Hubble expansion
here becomes,

$\displaystyle\frac{H^{2}}{H_{0}^{2}}=\Omega_{b}\ a^{-3}+\Omega_{\rm ddm}\
a^{-(3+\alpha)}+\frac{\alpha}{1-\alpha}\Omega_{\rm ddm}\
a^{-(3+\alpha)}+\Omega_{\rm de}\ a^{-3(1+w_{0}+w_{a})}\exp[-3w_{a}(1-a)]\;,$
(13)

and we label this scenario as $w_{0}w_{a}$ddm.

All in all, for the ddm scenario described by Eqs. (3) and (4), we can
explicitly solve its evolution at the background level for three specific dark
energy equation of state parameters. Now, to address the impact of the ddm
cosmological models on CMB and large-scale-structure (LSS) observables, we
must not only account for the modified evolution of the background densities,
but also include the effect of perturbations. Since dark energy does not
interfere with the dark matter disintegration mechanism, the density and
velocity perturbations for the dark energy fluid will be exactly the same as
we have seen in the non-interacting cosmological models where dark energy is a
component. For the perturbations in the cosmological fluid, the dimensionless
density contrast $\delta_{i}=\delta\rho_{i}/\bar{\rho}_{i}$ conveniently
describes the fluctuations in the energy density field of a given cosmological
species $i$ and $\theta_{i}$ the velocity divergence of the fluid with respect
to the expansion, where the bar denotes background quantities. This means that
the density perturbations and the velocity perturbations for the dark energy
fluid assuming the synchronous gauge will respectively be given by the
following set of equations Ma and Bertschinger (1995):

$\displaystyle\delta^{\prime}_{\rm de}$ $\displaystyle=$
$\displaystyle-(1+w_{\rm de})\left(\theta_{\rm
de}+\frac{h^{\prime}}{2}\right)-3\mathcal{H}w^{\prime}_{\rm
de}\frac{\theta_{\rm de}}{k^{2}}-3\mathcal{H}\left(c^{2}_{s}-w_{\rm
de}\right)\left[\delta_{\rm de}+3\mathcal{H}(1+w_{\rm de})\frac{\theta_{\rm
de}}{k^{2}}\right],$ (14) $\displaystyle\theta^{\prime}_{\rm de}$
$\displaystyle=$ $\displaystyle-\mathcal{H}(1-3c^{2}_{s})\theta_{\rm
de}+\frac{c^{2}_{s}}{1+w_{\rm de}}k^{2}\delta_{\rm de}-k^{2}\sigma_{\rm de}$
(15)

where $h\equiv h_{ii}$ denotes the trace part of the metric perturbation and
$c^{2}_{s}$ is the physical sound speed of the dark energy in the rest frame.
We have taken the usual assumption in which $c^{2}_{s}=1$. Under this
assumption we are considering that the dark energy does not cluster in the
sub-Hubble scale. However, one can also consider $c_{s}^{2}$ to be a free
parameter, but this parameter has been found to be unconstrained, see for
instance Bean and Dore (2004); Hannestad (2005); Corasaniti  _et al._ (2005);
Xia  _et al._ (2008). Thus, the assumption of free $c_{s}^{2}$ does not offer
any interesting physics. In what follows the shear perturbation of dark energy
is taken to be $\sigma_{\rm de}$ = 0. The most vital changes that appear in
the perturbations equations are due to the disintegration mechanism between
CDM and dr.

Putting all this together the density and the velocity perturbation equations
for the ddm and dr sectors are found to be

$\delta^{\prime}_{\rm ddm}=\theta_{\rm ddm}-\frac{h^{\prime}}{2}\,,$ (16)
$\theta^{\prime}_{\rm ddm}=-\mathcal{H}\ \theta_{\rm ddm}\,,$ (17)
$\delta^{\prime}_{\rm dr}=-\frac{4}{3}\left(\theta_{\rm
dr}+\frac{h^{\prime}}{2}\right)+\alpha\mathcal{H}\frac{\rho_{\rm
ddm}}{\rho_{\rm dr}}(\delta_{\rm ddm}-\delta_{\rm dr})\,,$ (18)
$\theta^{\prime}_{\rm dr}=\frac{k^{2}}{4}\delta_{\rm
dr}-\alpha\mathcal{H}\frac{3\rho_{\rm ddm}}{4\rho_{\rm
dr}}\left(\frac{4}{3}\theta_{\rm dr}-\theta_{\rm ddm}\right)-k^{2}\sigma_{\rm
dr}\,.$ (19)

Note that one can set $\theta_{\rm ddm}=0$ in the ddm comoving frame and with
this consideration (16) and (17) simplify to: $\delta^{\prime}_{\rm
ddm}=-h^{\prime}/2$ and $\theta^{\prime}_{\rm ddm}=0$. Hereafter, the shear
perturbation of dark radiation is also taken to be $\sigma_{\rm dr}=0$.

In Figs. 1, 2 and 3, we show the effects of the $\alpha$ and $w_{0}$
parameters in modifying the CMB power spectrum. We can see that while $w_{0}$
mainly changes the amplitude of the low-$\ell$ multipoles, $\alpha$ completely
modifies the CMB peak structure, changing the peak amplitudes and positions.

Figure 1: CMB TT power spectrum obtained by varying $\alpha$ (upper panel) and
by varying $w_{0}$ (bottom panel) for the $w_{0}$ddm scenario. Figure 2: CMB
TE power spectrum obtained by varying $\alpha$ for the $w_{0}$ddm scenario.
Figure 3: CMB EE power spectrum obtained by varying $\alpha$ for the
$w_{0}$ddm scenario.

##  III Observational data and statistical methodology

In this section we provide a description of the observational datasets and the
statistical methodology. To constrain the free parameters of the ddm models we
make use of a series of cosmological probes:

  * •

CMB: the Planck 2018 legacy release temperature and polarization CMB
measurements Aghanim  _et al._ (2020b, c).

  * •

lensing: the Planck 2018 CMB lensing reconstruction likelihood Aghanim  _et
al._ (2020d).

  * •

BAO: the BAO measurements from 6dFGS Beutler  _et al._ (2011), SDSS-MGS Ross
_et al._ (2015), and BOSS DR12 Alam  _et al._ (2017).

  * •

Pantheon: the Pantheon sample of 1048 Supernovae Type Ia distributed in the
redshift interval $z\in[0.01,2.3]$ Scolnic  _et al._ (2018).

  * •

R20: a gaussian prior on the Hubble constant in agreement with the SH0ES
collaboration measurement in Riess  _et al._ (2021).

To study the impact of the ddm models on the CMB and LSS observables, we
modify the Boltzmann code CAMB Lewis  _et al._ (2000). More concretely, we
implement the non-standard time evolution of ddm and dr energy densities
according to Eqs. (16) and (19). To investigate and constrain the imprints of
ddm models on the CMB and LSS we adopt Markov Chain Monte Carlo (MCMC) methods
to sample the posterior distribution of the cosmological parameters. We used a
modified version of the MCMC cosmological package CosmoMC Lewis and Bridle
(2002); Lewis (2013), publicly available at http://cosmologist.info/cosmomc/,
that is equipped with a convergence diagnostic based on the Gelman-Rubin
criterion Gelman and Rubin (1992) and supports the Planck 2018 likelihood
Aghanim  _et al._ (2020c). We monitor the convergence of the generated MCMC
chains using the standard $R$ parameter, requiring $R-1<0.02$ for the MCMC
chains to be considered as converged. We consider the models outlined in Sec.
II, which span three different parameter spaces; namely, a 7-dimensional space
for $\Lambda$ddm, 8-dimensional for $w_{0}$ddm, and 9-dimensional for
$w_{0}w_{a}$ddm. The free parameters in these models are

$\mathcal{P}_{1}=\\{\Omega_{b}h^{2},\Omega_{\rm
ddm}h^{2},100\theta_{MC},\tau,n_{s},{\rm{ln}}(10^{10}A_{s}),\alpha\\},$
$\mathcal{P}_{2}=\\{\Omega_{b}h^{2},\Omega_{\rm
ddm}h^{2},100\theta_{MC},\tau,n_{s},{\rm{ln}}(10^{10}A_{s}),\alpha,w_{0}\\},$
$\mathcal{P}_{3}=\\{\Omega_{b}h^{2},\Omega_{\rm
ddm}h^{2},100\theta_{MC},\tau,n_{s},{\rm{ln}}(10^{10}A_{s}),\alpha,w_{0},w_{a}\\},$

where $\Omega_{b}$ and $\Omega_{\rm ddm}$ are the baryons and ddm densities
normalized to the critical density, $\theta_{MC}$ is an approximation of the
ratio of sound horizon to the angular diameter distance (which is adopted in
CosmoMC Lewis and Bridle (2002); Lewis (2013) and is based on fitting formulae
given in Hu and Sugiyama (1996)), $\tau$ is the reionization optical depth,
$n_{s}$ is the scalar spectral index, $A_{s}$ is the amplitude of the
primordial scalar power spectrum, $\alpha$ gauges the strength of the ddm
mechanism, and $w_{0}$ and $w_{a}$ are the free parameters describing the dark
energy equation of state. We impose flat priors on the free parameters as
specified in Table 1.

Parameter | prior
---|---
$\Omega_{b}h^{2}$ | $[0.005,0.1]$
$\Omega_{\rm ddm}h^{2}$ |  $[0.001,0.99]$
$100\theta_{MC}$ | $[0.5,10]$
$\tau$ | $[0.01,0.8]$
$n_{\mathrm{s}}$ | $[0.7,1.3]$
${\rm{ln}}(10^{10}A_{s})$ | $[1.7,5.0]$
$\alpha$ | $[0,1]$
$w_{0}$ | $[-2,0.5]$
$w_{a}$ | $[-3,3]$
Table 1: Flat priors on various cosmological parameters of the ddm scenarios.

##  IV Observational Results

In this section we summarize the observational constraints extracted out of
the three distinct ddm scenarios distinguished by the dark energy dependence;
namely, (i) $\Lambda$ddm, (ii) $w_{0}$ddm and  (iii) $w_{0}w_{a}$ddm.

###  IV.1 ddm and non-dynamical dark energy: The $\bm{\Lambda}$ddm model

In Table 2 we display the constraints at 68% CL and at 95% CL on the free
cosmological parameters of the $\Lambda$ddm model (above the horizontal line)
and on the derived parameters (below the horizontal line). In addition, we
display the corresponding 2D contour plots in Fig. 4. Remarkably,
independently of the dataset combinations considered, we obtain an upper limit
for $\alpha$ that is consistent with zero, recovering the constraints on the
cosmological parameters that we have for a standard $\Lambda$CDM model, and
therefore the same level of tension with $H_{0}$ and $S_{8}$. Moreover, the
upper bound on $\alpha$ is robust and does not change among the various
dataset combinations, with the only exception being the CMB+lensing dataset,
for which the bound becomes more restrictive by a factor of $0.8$, and when
the R20 prior is included, as discussed below. In particular, $\alpha<0.0033$
at 95% CL for CMB alone, whereas $\alpha<0.0027$ at 95% CL for CMB+lensing,
while it remains unaltered when BAO or Pantheon are included in the analysis.
For a fair comparison with the previous results present in the literature, we
combine Planck and R20, despite the tension between them is above $3\sigma$
for this model. This combination should be considered therefore with caution,
and this is the reason why it gives completely different results and larger
error bars with respect to the CMB alone case. Actually, the combination of
CMB+R20 can reduce the $H_{0}$ tension down to $1.4\sigma$, while resolving
the $S_{8}$ mismatch at $1\sigma$ level. This however is driven by the $H_{0}$
prior and happens at the price of $1\sigma$ indication for $\alpha\neq 0$,
i.e. $\alpha=0.0038^{+0.0019}_{-0.0023}$ at 68% CL, and a much larger upper
limits at 95% CL than the other dataset combinations. This solution of the
Hubble tension is in contrast with the intermediate-redshift data that reduce
the effectiveness of this model to relax the $H_{0}$ and $S_{8}$ tensions.

From Fig. 4 it is possible to understand the effect of the $\Lambda$ddm model
on the $H_{0}$ and $S_{8}$ tensions analyzing the correlation between $\alpha$
and the relevant cosmological parameters. The correlation works in the right
direction to potentially alleviate the current cosmological tensions, being
positively correlated with $H_{0}$ (the Hubble value shifts about $1\sigma$
towards a higher value) and at the same time inversely correlated with the
sound horizon $r_{\rm drag}$ (although mildly in this case), and
anticorrelated with both $S_{8}$ and $\Omega_{m}$. Therefore, we can speculate
that even if the observational datasets do not favor the $\Lambda$ddm scenario
analyzed herein, it is possible that a different functional form for $\Gamma$
could lead to a resolution of the tensions.

Parameters | CMB | CMB+lensing | CMB+Pantheon | CMB+BAO | CMB+lensing+Pantheon+BAO | CMB+R20
---|---|---|---|---|---|---
$\Omega_{\rm ddm}h^{2}$ | $0.1194_{-0.0015-0.0032}^{+0.0016+0.0030}$ | $0.1195_{-0.0013-0.0025}^{+0.0013+0.0025}$ | $0.1191_{-0.0014-0.0029}^{+0.0015+0.0027}$ | $0.1189_{-0.0011-0.0021}^{+0.0011+0.0021}$ | $0.11889_{-0.00094-0.0019}^{+0.00094+0.0019}$ | $0.1159_{-0.0016-0.0033}^{+0.0016+0.0031}$
$\Omega_{b}h^{2}$ | $0.02231_{-0.00015-0.00031}^{+0.00015+0.00030}$ | $0.02233_{-0.00014-0.00028}^{+0.00014+0.00028}$ | $0.02232_{-0.00015-0.00030}^{+0.00015+0.00030}$ | $0.02234_{-0.00015-0.00030}^{+0.00015+0.00030}$ | $0.02235_{-0.00014-0.00029}^{+0.00015+0.00029}$ | $0.02235_{-0.00017-0.00036}^{+0.00017+0.00035}$
$100\theta_{MC}$ | $1.04092_{-0.00031-0.00061}^{+0.00031+0.00061}$ | $1.04091_{-0.00030-0.00060}^{+0.00030+0.00060}$ | $1.04093_{-0.00030-0.00060}^{+0.00031+0.00061}$ | $1.04097_{-0.00029-0.00057}^{+0.00029+0.00058}$ | $1.04097_{-0.00029-0.00058}^{+0.00029+0.00058}$ | $1.04118_{-0.00030-0.00059}^{+0.00030+0.00060}$
$\tau$ | $0.0546_{-0.0081-0.015}^{+0.0074+0.016}$ | $0.0550_{-0.0076-0.015}^{+0.0072+0.015}$ | $0.0550_{-0.0076-0.015}^{+0.0077+0.016}$ | $0.0552_{-0.0080-0.015}^{+0.0073+0.016}$ | $0.0565_{-0.0070-0.014}^{+0.0070+0.015}$ | $0.0578_{-0.0083-0.015}^{+0.0076+0.017}$
$n_{s}$ | $0.9653_{-0.0045-0.0087}^{+0.0045+0.0089}$ | $0.9652_{-0.0041-0.0079}^{+0.0041+0.0080}$ | $0.9658_{-0.0041-0.0082}^{+0.0041+0.0082}$ | $0.9664_{-0.0038-0.0077}^{+0.0040+0.0076}$ | $0.9665_{-0.0037-0.0073}^{+0.0038+0.0073}$ | $0.9708_{-0.0041-0.0081}^{+0.0042+0.0080}$
${\rm{ln}}(10^{10}A_{s})$ | $3.046_{-0.015-0.031}^{+0.016+0.032}$ | $3.047_{-0.015-0.029}^{+0.014+0.029}$ | $3.047_{-0.016-0.032}^{+0.016+0.033}$ | $3.046_{-0.016-0.030}^{+0.016+0.032}$ | $3.049_{-0.014-0.028}^{+0.014+0.029}$ | $3.051_{-0.016-0.031}^{+0.016+0.033}$
$\alpha$ | $<0.0013\,<0.0033$ | $<0.0011\,<0.0027$ | $<0.0015\,<0.0033$ | $<0.0015\,<0.0031$ | $<0.0013\,<0.0029$ | $0.0038^{+0.0019}_{-0.0023}\,<0.0073$
$\Omega_{m}$ | $0.310_{-0.010-0.021}^{+0.011+0.020}$ | $0.3105_{-0.0081-0.017}^{+0.0089+0.016}$ | $0.308_{-0.009-0.020}^{+0.010+0.017}$ | $0.3066_{-0.0071-0.014}^{+0.0071+0.013}$ | $0.3066_{-0.0061-0.013}^{+0.0062+0.012}$ | $0.284_{-0.011-0.022}^{+0.011+0.021}$
$\sigma_{8}$ | $0.8149_{-0.0077-0.015}^{+0.0077+0.016}$ | $0.8147_{-0.0071-0.013}^{+0.0063+0.014}$ | $0.8146_{-0.0085-0.016}^{+0.0078+0.017}$ | $0.8138_{-0.0084-0.015}^{+0.0077+0.016}$ | $0.8145_{-0.0074-0.013}^{+0.0065+0.014}$ | $0.817_{-0.010-0.018}^{+0.009+0.019}$
$H_{0}\,[{\rm km/s/Mpc}]$ | $67.81_{-0.89-1.5}^{+0.68+1.7}$ | $67.75_{-0.70-1.2}^{+0.59+1.3}$ | $67.93_{-0.81-1.4}^{+0.62+1.5}$ | $68.04_{-0.60-1.1}^{+0.53+1.1}$ | $68.03_{-0.52-0.99}^{+0.46+0.98}$ | $69.9_{-1.1-1.9}^{+0.9+2.0}$
$S_{8}$ | $0.828_{-0.017-0.034}^{+0.017+0.033}$ | $0.829_{-0.013-0.025}^{+0.013+0.025}$ | $0.825_{-0.016-0.031}^{+0.016+0.030}$ | $0.823_{-0.013-0.025}^{+0.013+0.025}$ | $0.823_{-0.010-0.020}^{+0.010+0.020}$ | $0.796_{-0.016-0.031}^{+0.016+0.032}$
$r_{\rm{drag}}\,[{\rm Mpc}]$ | $147.02_{-0.31-0.60}^{+0.30+0.59}$ | $147.01_{-0.27-0.54}^{+0.27+0.53}$ | $147.04_{-0.29-0.56}^{+0.29+0.58}$ | $147.09_{-0.27-0.54}^{+0.27+0.51}$ | $147.10_{-0.24-0.52}^{+0.27+0.48}$ | $147.17_{-0.31-0.62}^{+0.32+0.59}$

Table 2: 68% and 95% CL on the free cosmological parameters of the
$\Lambda$ddm scenario (above the line) and the derived ones (below the line).
Figure 4: We show the 1-dimensional posterior distributions of some important
parameters and the 2-dimensional joint contours at 68% and 95% CL between some
of the model parameters for the $\Lambda$ddm scenario using various
cosmological probes and their combinations. $H_{0}$ and $r_{\rm drag}$ are
given in [km/s/Mpc] and [Mpc], respectively.

Parameters | CMB | CMB+lensing | CMB+Pantheon | CMB+BAO | CMB+lensing+Pantheon+BAO | CMB+R20
---|---|---|---|---|---|---
$\Omega_{\rm ddm}h^{2}$ | $0.1191_{-0.0015-0.0032}^{+0.0016+0.0029}$ | $0.1186_{-0.0013-0.0028}^{+0.0015+0.0026}$ | $0.1195_{-0.0014-0.0032}^{+0.0017+0.0029}$ | $0.1191_{-0.0014-0.0029}^{+0.0014+0.0028}$ | $0.1192_{-0.0011-0.0022}^{+0.0011+0.0022}$ | $0.1194_{-0.0015-0.0030}^{+0.0015+0.0029}$
$\Omega_{b}h^{2}$ | $0.02234_{-0.00016-0.00030}^{+0.00016+0.00031}$ | $0.02236_{-0.00015-0.00031}^{+0.00016+0.00030}$ | $0.02231_{-0.00015-0.00030}^{+0.00015+0.00030}$ | $0.02233_{-0.00015-0.00030}^{+0.00015+0.00029}$ | $0.02235_{-0.00015-0.00029}^{+0.00015+0.00029}$ | $0.02232_{-0.00015-0.00030}^{+0.00015+0.00028}$
$100\theta_{MC}$ | $1.04096_{-0.00031-0.00062}^{+0.00030+0.00061}$ | $1.04098_{-0.00031-0.00062}^{+0.00031+0.00060}$ | $1.04090_{-0.00031-0.00060}^{+0.00031+0.00062}$ | $1.04093_{-0.00031-0.00059}^{+0.00031+0.00058}$ | $1.04094_{-0.00030-0.00057}^{+0.00030+0.00058}$ | $1.04091_{-0.00030-0.00061}^{+0.00031+0.00060}$
$\tau$ | $0.0542_{-0.0073-0.015}^{+0.0074+0.015}$ | $0.0533_{-0.0074-0.015}^{+0.0074+0.015}$ | $0.0546_{-0.0081-0.015}^{+0.0073+0.016}$ | $0.0547_{-0.0082-0.015}^{+0.0073+0.016}$ | $0.0554_{-0.0079-0.014}^{+0.0068+0.015}$ | $0.0546_{-0.0080-0.015}^{+0.0075+0.016}$
$n_{s}$ | $0.9659_{-0.0043-0.0086}^{+0.0043+0.0087}$ | $0.9667_{-0.0042-0.0083}^{+0.0042+0.0085}$ | $0.9651_{-0.0044-0.0084}^{+0.0043+0.0084}$ | $0.9659_{-0.0041-0.0082}^{+0.0041+0.0085}$ | $0.9657_{-0.0039-0.0075}^{+0.0038+0.0075}$ | $0.9652_{-0.0043-0.0085}^{+0.0042+0.0086}$
${\rm{ln}}(10^{10}A_{s})$ | $3.045_{-0.015-0.029}^{+0.015+0.031}$ | $3.042_{-0.015-0.029}^{+0.014+0.030}$ | $3.046_{-0.015-0.031}^{+0.016+0.032}$ | $3.046_{-0.017-0.031}^{+0.015+0.033}$ | $3.047_{-0.016-0.027}^{+0.014+0.030}$ | $3.046_{-0.016-0.032}^{+0.016+0.033}$
$\alpha$ | $<0.0014\,<0.0035$ | $<0.0015\,<0.0035$ | $<0.0013\,<0.0032$ | $<0.0014\,<0.0031$ | $<0.0012\,<0.0027$ | $<0.0012<0.0030$
$w_{0}$ | $-1.58_{-0.34-0.41}^{+0.15+0.51}$ | $-1.57_{-0.32-0.39}^{+0.15+0.49}$ | $-1.026_{-0.037-0.073}^{+0.038+0.075}$ | $-1.020_{-0.056-0.12}^{+0.065+0.011}$ | $-1.019_{-0.031-0.063}^{+0.032+0.062}$ | $-1.182_{-0.049-0.096}^{+0.050+0.096}$
$\Omega_{m}$ | $0.193_{-0.055-0.067}^{+0.018+0.097}$ | $0.191_{-0.053-0.065}^{+0.017+0.095}$ | $0.304_{-0.011-0.022}^{+0.011+0.023}$ | $0.303_{-0.012-0.024}^{+0.012+0.023}$ | $0.3036_{-0.0075-0.015}^{+0.0076+0.015}$ | $0.264_{-0.011-0.019}^{+0.009+0.020}$
$\sigma_{8}$ | $0.977_{-0.044-0.15}^{+0.094+0.12}$ | $0.973_{-0.041-0.14}^{+0.089+0.011}$ | $0.823_{-0.014-0.027}^{+0.014+0.027}$ | $0.820_{-0.021-0.037}^{+0.019+0.039}$ | $0.820_{-0.011-0.021}^{+0.011+0.022}$ | $0.866_{-0.016-0.031}^{+0.016+0.032}$
$H_{0}\,[{\rm km/s/Mpc}]$ | $87_{-5-18}^{+12+15}$ | $88_{-5-18}^{+12+14}$ | $68.5_{-1.1-2.2}^{+1.1+2.2}$ | $68.5_{-1.6-2.6}^{+1.3+3.0}$ | $68.45_{-0.81-1.6}^{+0.81+1.6}$ | $73.4_{-1.3-2.6}^{+1.3+2.5}$
$S_{8}$ | $0.772_{-0.036-0.053}^{+0.023+0.060}$ | $0.766_{-0.036-0.050}^{+0.020+0.060}$ | $0.827_{-0.016-0.032}^{+0.016+0.032}$ | $0.824_{-0.013-0.027}^{+0.013+0.026}$ | $0.824_{-0.010-0.020}^{+0.010+0.021}$ | $0.813_{-0.015-0.030}^{+0.015+0.030}$
$r_{\rm{drag}}\,[{\rm Mpc}]$ | $147.02_{-0.30-0.58}^{+0.30+0.59}$ | $147.12_{-0.28-0.54}^{+0.28+0.55}$ | $146.98_{-0.30-0.58}^{+0.30+0.60}$ | $147.05_{-0.29-0.56}^{+0.28+0.57}$ | $147.06_{-0.25-0.51}^{+0.26+0.49}$ | $147.00_{-0.29-0.57}^{+0.29+0.57}$

Table 3: 68% and 95% CL on the free cosmological parameters of the $w_{0}$ddm
model (above the horizontal line) and the derived ones (below the horizontal
line). Figure 5: We show the 1-dimensional posterior distributions of some
important parameters and the 2-dimensional joint contours at 68% and 95% CL
between some of the model parameters for the $w_{0}$ddm scenario using various
cosmological probes and their combinations. $H_{0}$ and $r_{\rm drag}$ are
given in [km/s/Mpc] and [Mpc], respectively.

###  IV.2 ddm and dynamical dark energy: The $\bm{w_{0}}$ddm and
$\bm{w_{0}w_{a}}$ddm models

Next, we consider an extended model in which the DE equation of state is a
constant, $w_{0}$, and hence it is expected that the effects of the dynamical
dark energy can be realized from the behaviour of this ddm scenario. In Table
3 we present the constraints at 68% CL and 95% CL on the cosmological
parameters of the $w_{0}$ddm scenario. The corresponding 1D posterior
distributions and 2D contour plots are shown in Fig. 5.

As in our previous analysis a robust upper limit on $\alpha$ is obtained for
all the dataset combinations, also when the R20 prior is included. Actually,
the upper limits on $\alpha$ in this scenario are almost similar to those
bounding the $\Lambda$ddm model. Moreover, since $\alpha$ is consistent with
zero, the best-fit values of the cosmological parameters are similar to those
constraining the $w_{0}$CDM scenario; see e.g., Table II of Ref. Anchordoqui
_et al._ (2021b).

Since this $w_{0}$ddm scenario solves both the $H_{0}$ and $S_{8}$ tensions,
we can safely combine Planck and the R20 prior to understand if the resolution
of the Hubble tension will happen at the price of $\alpha\neq 0$ or a phantom
dark energy $w_{0}<-1$. We conclude that the solution is due to the latter
because $\alpha$ remains consistent with zero. We can see that the combination
CMB+R20 fixes $H_{0}$ in agreement with SH0ES at $1\sigma$ level, but the
replacement of $w_{0}$ by $\Lambda$ exacerbates the $S_{8}$ tension. Moreover,
as we can see in Fig. 5, the introduction of $w_{0}$ breaks the correlation
between $\alpha$ and the parameters of interest, such as $H_{0}$, $r_{\rm
drag}$, $S_{8}$, and $\Omega_{m}$.

Finally, we considered a model in which the dark energy equation of state has
a dynamical nature in terms of the CPL parametrization. In Table 4 we show the
constraints at 68% CL and 95% CL on the cosmological parameters of this
scenario. The corresponding 2D contour plots are shown in Fig. 6. This further
extension of the dynamical DE sector does not change the conclusions of the
previous analyses: $\alpha$ does not correlate with the other parameters of
the model, and we have a robust upper limit on $\alpha$ for all the dataset
combinations. The constraints on the cosmological parameters are similar to
those obtained in the $w_{0}w_{a}$CDM model, because $\alpha$ is always
consistent with zero.

The absence of correlation between $\alpha$ and the dark energy equation of
state can be deduced by comparing the upper and bottom panels of Fig. 1. This
is because even if the effects of $\alpha$ and $w_{0}$ are similar in the
low-$\ell$ multipole range, contrarily to $w_{0}$, $\alpha$ affects strongly
the position of the peaks and the smoothing of the damping tail. Since this
high multipole region is extremely well constrained by the peak structure of
the CMB data, the addition of external datasets or extensions of the minimal
$\Lambda$ddm model do not affect the upper bound on $\alpha$.

Parameters | CMB | CMB+lensing | CMB+Pantheon | CMB+BAO | CMB+lensing+Pantheon+BAO | CMB+R20
---|---|---|---|---|---|---
$\Omega_{\rm ddm}h^{2}$ | $0.1189_{-0.0015-0.0032}^{+0.0016+0.0030}$ | $0.1185_{-0.0013-0.0028}^{+0.0015+0.0027}$ | $0.1195_{-0.0014-0.0032}^{+0.0017+0.0030}$ | $0.1196_{-0.0014-0.0027}^{+0.0014+0.0027}$ | $0.1195_{-0.0011-0.0023}^{+0.0012+0.0022}$ | $0.1191_{-0.0015-0.0032}^{+0.0016+0.0029}$
$\Omega_{b}h^{2}$ | $0.02235_{-0.00016-0.00031}^{+0.00016+0.00031}$ | $0.02238_{-0.00016-0.00031}^{+0.00016+0.00031}$ | $0.02232_{-0.00016-0.00030}^{+0.00015+0.00030}$ | $0.02232_{-0.00015-0.00029}^{+0.00015+0.00030}$ | $0.02232_{-0.00015-0.00029}^{+0.00014+0.00028}$ | $0.02234_{-0.00015-0.00031}^{+0.00015+0.00031}$
$100\theta_{MC}$ | $1.04096_{-0.00032-0.00063}^{+0.00031+0.00063}$ | $1.04100_{-0.00030-0.00058}^{+0.00030+0.00061}$ | $1.04089_{-0.00032-0.00065}^{+0.00033+0.00064}$ | $1.04089_{-0.00030-0.00063}^{+0.00033+0.00060}$ | $1.04091_{-0.00030-0.00059}^{+0.00030+0.00061}$ | $1.04093_{-0.00031-0.00059}^{+0.00031+0.00061}$
$\tau$ | $0.0544_{-0.0078-0.016}^{+0.0079+0.016}$ | $0.0530_{-0.0074-0.015}^{+0.0073+0.016}$ | $0.0542_{-0.0074-0.015}^{+0.0075+0.016}$ | $0.0539_{-0.0078-0.015}^{+0.0076+0.016}$ | $0.0544_{-0.0080-0.015}^{+0.0074+0.016}$ | $0.0545_{-0.0082-0.015}^{+0.0074+0.016}$
$n_{s}$ | $0.9662_{-0.0044-0.0088}^{+0.0043+0.0084}$ | $0.9672_{-0.0040-0.0081}^{+0.0040+0.0081}$ | $0.9654_{-0.0047-0.0085}^{+0.0044+0.0088}$ | $0.9649_{-0.0041-0.0084}^{+0.0045+0.0081}$ | $0.9652_{-0.0040-0.0079}^{+0.0040+0.0077}$ | $0.9658_{-0.0043-0.0084}^{+0.0043+0.0085}$
${\rm{ln}}(10^{10}A_{s})$ | $3.045_{-0.016-0.033}^{+0.016+0.033}$ | $3.040_{-0.015-0.029}^{+0.014+0.031}$ | $3.045_{-0.015-0.032}^{+0.015+0.032}$ | $3.045_{-0.016-0.032}^{+0.016+0.032}$ | $3.045_{-0.015-0.030}^{+0.015+0.031}$ | $3.045_{-0.016-0.031}^{+0.016+0.031}$
$\alpha$ | $<0.0015\,<0.0035$ | $<0.0015\,<0.0035$ | $<0.0012\,<0.0031$ | $<0.0012\,<0.0028$ | $<0.0011\,<0.0026$ | $<0.0014<0.0033$
$w_{0}$ | $-1.17_{-0.58}^{+0.33}<-0.28$ | $-1.18_{-0.56}^{+0.37}<-0.34$ | $-0.88_{-0.14-0.25}^{+0.13+0.26}$ | $-0.59_{-0.27-0.53}^{+0.27+0.51}$ | $-0.962_{-0.078-0.15}^{+0.078+0.16}$ | $-0.81_{-0.16-0.51}^{+0.32+0.41}$
$w_{a}$ | $<-0.89\,<0.64$ | $<-0.81<0.73$ | $-0.71_{-0.58-1.3}^{+0.73+1.2}$ | $-1.19_{-0.75-1.5}^{+0.80+1.4}$ | $-0.24_{-0.27-0.60}^{+0.32+0.58}$ | $<-1.15\,<0.30$
$\Omega_{m}$ | $0.214_{-0.078-0.10}^{+0.022+0.15}$ | $0.211_{-0.073-0.09}^{+0.023+0.13}$ | $0.295_{-0.014-0.025}^{+0.013+0.028}$ | $0.339_{-0.026-0.049}^{+0.026+0.051}$ | $0.3044_{-0.0076-0.015}^{+0.0077+0.015}$ | $0.265_{-0.010-0.018}^{+0.009+0.020}$
$\sigma_{8}$ | $0.949_{-0.05-0.18}^{+0.12+0.15}$ | $0.946_{-0.51-0.17}^{+0.11+0.13}$ | $0.834_{-0.016-0.034}^{+0.018+0.031}$ | $0.794_{-0.026-0.045}^{+0.023+0.049}$ | $0.822_{-0.011-0.022}^{+0.011+0.022}$ | $0.868_{-0.016-0.032}^{+0.017+0.033}$
$H_{0}\,[{\rm km/s/Mpc}]$ | $84_{-7-22}^{+15+18}$ | $84_{-7-21}^{+14+17}$ | $69.5_{-1.4-2.8}^{+1.4+2.8}$ | $65.0_{-2.8-4.8}^{+2.2+5.0}$ | $68.43_{-0.81-1.6}^{+0.81+1.6}$ | $73.3_{-1.3-2.5}^{+1.3+2.6}$
$S_{8}$ | $0.783_{-0.046-0.063}^{+0.025+0.077}$ | $0.776_{-0.045-0.059}^{+0.024+0.071}$ | $0.827_{-0.016-0.033}^{+0.017+0.032}$ | $0.843_{-0.018-0.036}^{+0.018+0.034}$ | $0.828_{-0.011-0.022}^{+0.011+0.022}$ | $0.815_{-0.015-0.029}^{+0.015+0.030}$
$r_{\rm{drag}}\,[{\rm Mpc}]$ | $147.05_{-0.30-0.61}^{+0.30+0.60}$ | $147.15_{-0.27-0.56}^{+0.27+0.54}$ | $146.99_{-0.29-0.58}^{+0.29+0.62}$ | $146.98_{-0.29-0.57}^{+0.29+0.59}$ | $147.03_{-0.27-0.54}^{+0.27+0.53}$ | $147.03_{-0.29-0.57}^{+0.30+0.59}$

Table 4: 68% and 95% CL on the free cosmological parameters of the
$w_{0}w_{a}$ddm model (above the horizontal line) and the derived ones (below
the horizontal line).  Figure 6: We show the 1-dimensional posterior
distributions of some important parameters and the 2-dimensional joint
contours at 68% and 95% CL between some of the model parameters for the
$w_{0}w_{a}$ddm scenario using various cosmological probes and their
combinations. $H_{0}$ and $r_{\rm drag}$ are given in [km/s/Mpc] and [Mpc],
respectively.

##  V Conclusions

We have investigated a specific cosmological set-up where the CDM sector is
disintegrated into dark radiation in the background of a homogeneous and
isotropic universe and the remaining matter components do not take part in
this dynamics. The dark energy sector has been assumed to be either non-
dynamical (the cosmological constant) or dynamical (characterised by some
equation of state). We considered a typical functional form of the
disintegration rate $\Gamma=\alpha{\cal H}$, where $\alpha>0$ describes the
strength of the disintegration. The scenario where the dark energy is the
cosmological constant has been proposed as a solution of the $H_{0}$ and
$S_{8}$ tensions Bringmann  _et al._ (2018) and recently confronted with 2015
Planck data Pandey  _et al._ (2020). We have shown in this paper that when the
latest 2018 Planck temperature and polarization data are used, the $H_{0}$
tension cannot be solved below $3\sigma$ when the cosmological constant is
considered. When Planck 2018 is combined with the other cosmological probes
only an upper limit for $\alpha$ is obtained, i.e. in agreement with no
disintegration. This is an interesting result which clearly indicates the
growing sensitivity in the cosmological probes. For instance, it is already
well known to the entire community that with the development of the
observational probes, the constraints on various cosmological parameters
(either model dependent or model independent) have been significantly
narrowed. This can be considered to be the development in the theoretical
model buildings in cosmology, and based on the results presented in this work
we recommend that the cosmological models tested with Planck 2015 should be
reexamined with Planck 2018.

To compare with the previous results in the literature we finally combined CMB
from Planck 2018 with the R20 prior, obtaining a reduction of the $H_{0}$ and
$S_{8}$ tensions at $1.4\sigma$ and $1\sigma$ level, respectively, at the
price of a $1\sigma$ indication for $\alpha\neq 0$. However this last CMB+R20
combination should be considered with caution because of the disagreement
within the datasets. We conclude that the addition of intermediate-redshift
data (from the Pantheon supernova type Ia dataset and baryon acoustic
oscillations) weakens the effectiveness of this model to address the $H_{0}$
and $S_{8}$ tensions; for details, see Table 2 and Fig. 4.

In a second phase of the investigation we studied two related models endowed
with dynamical dark energy $-$ one in which the dark energy equation of state
is constant with redshift, $w_{\rm de}(a)=w_{0}$, and another where $w_{\rm
de}(a)$ assumes the CPL parametrization $w_{\rm de}(a)=w_{0}+w_{a}(1-a)$. We
performed Monte Carlo Markov chain analyses for different combinations of data
sets and for different parameter sets. The results of the second-phase data-
analysis are encapsulated in Tables 3 (for $w_{0}$ddm) and 4 (for
$w_{0}w_{a}$ddm), and can be summarized as follows:

  * •

When the R20 prior is used in combination with a compilation of the latest CMB
measurements, the $H_{0}$ tension is resolved at the $1\sigma$ level but the
consideration of a dynamical dark energy component exacerbates the $S_{8}$
tension. In these cases, there is agreement between Planck and R20, making the
use of the latter safe. The solution of the $H_{0}$ tension is attributable to
the phantom character of the dark energy.

  * •

We do not find evidence for the disintegration of CDM into dark radiation in
any of the different combinations of observational datasets under study, i.e.
$\alpha$ is always consistent with zero.

The corresponding graphical variations of free parameters in $w_{0}$ddm and
$w_{0}w_{a}$ddm models are displayed in Figs. 5 and 6 respectively.

In summary, we have shown that when the Planck 2018 temperature and
polarization data are combined with other observations, such as the Hubble
diagram of type Ia supernovae and the large-scale structure of the universe as
traced by galaxies, there is no evidence for dark matter disintegrating at a
rate proportional to the Hubble expansion.

###### Acknowledgements.

LAA is supported by the U.S. National Science Foundation (NSF Grant
PHY-2112527). EDV acknowledges the support of the Addison-Wheeler Fellowship
awarded by the Institute of Advanced Study at Durham University. SP gratefully
acknowledges the financial support given by 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]. YW’s work was supported
by the National Natural Science Foundation of China under Grant No. 12075109,
11575075. WY’s work was supported by the National Natural Science Foundation
of China under Grants No. 11705079 and No. 11647153, and Liaoning
Revitalization Talents Program under Grant no. XLYC1907098.

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