# Constraining decaying dark matter with BOSS data
and the effective field theory of large-scale structures

Théo Simon  Electronic address<EMAIL_ADDRESS>Laboratoire Univers
& Particules de Montpellier (LUPM), CNRS & Université de Montpellier
(UMR-5299),Place Eugène Bataillon, F-34095 Montpellier Cedex 05, France
Guillermo Franco Abellán  Laboratoire Univers & Particules de Montpellier
(LUPM), CNRS & Université de Montpellier (UMR-5299),Place Eugène Bataillon,
F-34095 Montpellier Cedex 05, France  Peizhi Du  C.N. Yang Institute for
Theoretical Physics, Stony Brook University, Stony Brook, NY, 11794, USA
Vivian Poulin  Laboratoire Univers & Particules de Montpellier (LUPM), CNRS &
Université de Montpellier (UMR-5299),Place Eugène Bataillon, F-34095
Montpellier Cedex 05, France  Yuhsin Tsai  Department of Physics, University
of Notre Dame, IN 46556, USA

###### Abstract

We update cosmological constraints on two decaying dark matter models in light
of BOSS-DR12 data analyzed under the Effective Field Theory of Large-Scale
Structures (EFTofLSS) formalism, together with Planck, Pantheon and other BOSS
measurements of the baryonic acoustic oscillation (BAO). In the first model, a
fraction $f_{\rm dcdm}$ of cold dark matter (CDM) decays into dark radiation
(DR) with a lifetime $\tau$. In the second model (recently suggested as a
potential resolution to the $S_{8}$ tension), all the CDM decays with a
lifetime $\tau$ into DR and a massive warm dark matter (WDM) particle, with a
fraction $\varepsilon$ of the CDM rest mass energy transferred to the DR.
Using numerical codes from the recent literature, we perform the first
calculation of the mildly non-linear (matter and galaxy) power spectra with
the EFTofLSS for these two models. In the case of DR products, we obtain the
constraints $f_{\rm dcdm}\lesssim 0.022$ (95% C.L.) for lifetimes shorter than
the age of the universe, and $\tau/f_{\rm dcdm}\gtrsim 250$ Gyr in the long-
lived regime assuming $f_{\rm dcdm}\to 1$. We show that Planck data
contributes the most to these constraints, with EFTofBOSS providing a marginal
improvement over conventional BAO and redshift space distortions
($f\sigma_{8}$) data. In the case of DR and WDM decay products, we find that
EFTofBOSS data significantly improves the constraints at 68% C.L. on the CDM
lifetime with a $S_{8}$ prior from KiDS-1000. We show that, in order to fit
EFTofBOSS data while lowering $S_{8}$ to match KiDS-1000, the best-fit model
has a longer lifetime $\tau=120$ Gyr, with a larger kick velocity $v_{\rm
kick}/c\simeq\varepsilon\simeq 1.2\%$, than that without EFTofBOSS ($\tau=43$
Gyr, $\varepsilon=0.6\%$). We anticipate that future surveys will provide
exquisite constraints on such models.

††preprint: LUPM:22-005, YITP-SB-2022-04

##  I Introduction

The $\Lambda$ cold dark matter ($\Lambda$CDM) model provides outstanding
explanation for a wide variety of early universe data, such as Cosmic
Microwave Background (CMB) and Big Bang Nucleosynthesis (BBN), as well as late
universe observations of Large Scale Structure (LSS) including the Baryon
Acoustic Oscillation (BAO), and uncalibrated luminosity distance to SuperNovae
of type Ia (SNIa). Despite this remarkable success, the $\Lambda$CDM model
does not teach us about the intrinsic nature of its dark sector, made up of
both cold dark matter (CDM) and dark energy (DE). In addition, as the accuracy
of cosmological observations has improved, the concordance cosmological model
starts showing several experimental discrepancies. The most famous and
important cosmological puzzle, the so-called Hubble tension Verde  _et al._
(2019), corresponds to a large discrepancy ($\sim 4-5\sigma$) between the
local determination of $H_{0}$ from a variety of methods – and in particular
the cosmic distance ladder based on cepheid-calibrated SNIa by the SH0ES team
Riess  _et al._ (2021)–, and its determination using CMB data under the
assumption that the universe is described by the $\Lambda$CDM model Aghanim
_et al._ (2020a). Another intriguing cosmological conundrum, the one at the
heart of this study, is a less significant but older tension ($\sim$ 2-3
$\sigma$) between the weak-lensing111More precisely, there even exists a
‘lensing is low’ anomaly when comparing galaxy clustering and weak lensing
data within the $\Lambda$CDM cosmology Leauthaud  _et al._ (2017); Lange  _et
al._ (2021); Amon  _et al._ (2022). Asgari  _et al._ (2021); Abbott  _et al._
(2022); Hikage  _et al._ (2019); Amon  _et al._ (2022) and CMB Aghanim  _et
al._ (2020a); Aiola  _et al._ (2020) determinations of the amplitude of the
local matter fluctuations, parameterized as
$S_{8}=\sigma_{8}\sqrt{\Omega_{m}/0.3}$, where $\Omega_{m}$ is the current
total matter abundance, and $\sigma_{8}$ corresponds to the root mean square
of matter fluctuations on a $8h^{-1}$Mpc scale, with
$h=H_{0}/(100\,\textrm{km/s/Mpc})$, and is defined as follows

$\sigma_{8}=\int\frac{k^{3}}{2\pi}P_{m}(k)W_{8}^{2}(k)d\ln{k}.$ (1)

Here $P_{m}(k)$ is the linear matter power spectrum, and $W_{8}(k)$ is a
window function describing a sphere (in Fourier space) with a (historically
chosen) radius of $8h^{-1}$Mpc.

Barring unknown systematic errors (see e.g. Freedman (2021); Riess  _et al._
(2021); Amon  _et al._ (2022) for discussion), these discrepancies might be
the first clue about the intrinsic nature of the $\Lambda$CDM dark sector. On
the one hand, the resolution of the Hubble tension most likely involves new
physics in the pre-recombination era, through a decrease of the sound horizon
before recombination Bernal  _et al._ (2016); Aylor  _et al._ (2019); Knox and
Millea (2019); Camarena, David and Marra, Valerio (2021); Efstathiou (2021);
Schöneberg  _et al._ (2021), such as model involving dark radiation and/or new
neutrino properties Kreisch  _et al._ (2020); Berbig  _et al._ (2020); Ghosh
_et al._ (2020); Forastieri  _et al._ (2019); Escudero and Witte (2020, 2021);
Blinov and Marques-Tavares (2020); Ghosh  _et al._ (2021); Archidiacono and
Gariazzo (2022); Aloni  _et al._ (2021), early dark energy Karwal and
Kamionkowski (2016); Poulin  _et al._ (2019); Smith  _et al._ (2020);
Niedermann and Sloth (2021, 2020); Ye and Piao (2020), modified gravity Renk
_et al._ (2017); Umiltà  _et al._ (2015); Ballardini  _et al._ (2016); Rossi
_et al._ (2019); Braglia  _et al._ (2020a); Zumalacarregui (2020); Abadi and
Kovetz (2021); Ballardini  _et al._ (2020); Braglia  _et al._ (2020b); Di
Valentino  _et al._ (2016); Bahamonde  _et al._ (2021); Raveri (2020); Yan
_et al._ (2020); Frusciante  _et al._ (2020); Solà Peracaula  _et al._ (2019,
2020); Ballesteros  _et al._ (2020); Braglia  _et al._ (2021); Desmond  _et
al._ (2019); Lin  _et al._ (2019) or exotic recombination Chiang and Slosar
(2018); Hart and Chluba (2020); Sekiguchi and Takahashi (2021); Jedamzik and
Pogosian (2020); Cyr-Racine  _et al._ (2021) (for review, see Refs. Di
Valentino  _et al._ (2021a); Schöneberg  _et al._ (2021)). On the other hand,
the resolution of the $S_{8}$ tension requires a suppression in the matter
power spectrum for $k\sim 0.1-1\ h/$Mpc in order to reduce the value of the
$\sigma_{8}$ parameter (see Eq. 1), which can be achieved through a number of
models that take into account new hypothetical properties of dark matter (DM)
and/or DE Lesgourgues  _et al._ (2016); Buen-Abad  _et al._ (2015); Chacko
_et al._ (2016); Buen-Abad  _et al._ (2018); Brinckmann and Lesgourgues
(2019); Heimersheim  _et al._ (2020); Cyr-Racine  _et al._ (2021); Gómez-
Valent  _et al._ (2020); Di Valentino  _et al._ (2020); Lucca (2021); Abellan
_et al._ (2020); Di Valentino  _et al._ (2021b); Bansal  _et al._ (2021).

Decaying cold dark matter (DCDM) models, in which dark matter is unstable on
cosmological time-scale and decays into invisible products, have been proposed
as potential resolutions to cosmic tensions Enqvist  _et al._ (2015a);
Berezhiani  _et al._ (2015); Blinov  _et al._ (2020); Vattis  _et al._ (2019);
Abellan  _et al._ (2020); Abellán  _et al._ (2021). In the past it was found
that DM models with purely radiation decay products can neither resolve the
Hubble tension nor the $S_{8}$ tension Chudaykin  _et al._ (2016, 2018);
Poulin  _et al._ (2016); Clark  _et al._ (2020); Haridasu and Viel (2020);
Nygaard  _et al._ (2020), while DM models with massive decay products can
resolve the $S_{8}$ tension, as the massive particle produced during the decay
act as a WDM component, reducing power on scale below the free-streaming
length at late-times Abellan  _et al._ (2020); Abellán  _et al._ (2021).
Beyond recent observational tensions, the study of these models is important
from the particle physics point of view, as it addresses the question of the
stability of DM on long cosmological time-scales. In the literature, there are
many models involving the existence of DM decays at late-times, such as models
with R-parity violation Berezinsky  _et al._ (1991); Kim and Kim (2002), super
Weakly Interacting Massive particles (super WIMPs) Covi  _et al._ (1999); Feng
_et al._ (2003, 2003); Allahverdi  _et al._ (2015), sterile neutrinos
Abazajian  _et al._ (2012); Drewes  _et al._ (2017) or models with an
additional U(1) gauge symmetry Chen  _et al._ (2009); Choi  _et al._ (2020a,
b); Choi and Yanagida (2021). Besides cosmic tensions, some DCDM models were
proposed as a way to explain the excess of events in the electronic recoils
reported by the Xenon1T collaboration Choi  _et al._ (2020c); Xu and Zheng
(2020); Dutta  _et al._ (2021); Abellan  _et al._ (2020); Abellán  _et al._
(2021). In addition, DCDM models with massive daughters have also been
suggested as a potential solution to the small (subgalactic) scales structure
problem of CDM (e.g. Lin  _et al._ (2001); Sánchez-Salcedo (2003); Cembranos
_et al._ (2005); Kaplinghat (2005); Strigari  _et al._ (2007); Borzumati  _et
al._ (2008); Peter  _et al._ (2010); Peter and Benson (2010); Choi  _et al._
(2020b)).

In this article, we deal with DCDM with two type of decay products: (i) the
DCDM $\to$ DR model, where the decay products is only composed of a (massless)
dark radiation (DR) component, and (ii) the DCDM $\to$ WDM+DR model, where the
decay products are one massive WDM component and one DR component. Previous
works have limited themselves to the impact of DCDM decay at the background
and linear perturbations level, deriving constraints (and hints) on these
models from a combination of Planck CMB, BAO and uncalibrated luminosity
distance to SN1a data. Here, we go beyond previous works by making use of the
Effective Field Theory of Large Scale Structures (EFTofLSS) to describe the
mildly non-linear regime of the galaxy clustering power spectrum and derive
improved constraints thanks to the EFTofLSS applied to BOSS data. The main
objectives of this paper are: i) perform the first-ever computation of the
mildly non-linear regime in DCDM models with massive and massless decay
products through the EFTofLSS; ii) test whether current BOSS data can lead to
stronger constraints on these models; and iii) check whether these constraints
can put pressure on DCDM models that resolve the $S_{8}$ tension.

Our paper is structured as follows: in Sec. II, we briefly review the EFTofLSS
formalism, the observable at hand and the public codes available to perform
our analyses; in Sec. III, we introduce the models and present the non-linear
power spectrum computed with the EFTofLSS; in Sec. IV, we present the results
of comprehensive monte-carlo markov chain analyses of the DCDM model and
discuss the implications of these constraints for the $S_{8}$ tension; we
eventually conclude in Sec. V. App. A is dedicated to comparing results of the
EFTofLSS with N-body simulations in the DCDM$\to$DR model, while App. B
details the scope of our computation in the DCDM$\to$WDM+DR model. Finally
App. C, App. D and App. E present additional results of the MCMC analyses for
completeness.

##  II The galaxy power spectrum from the EFTofLSS formalism

Although an exhaustive review of the EFTofLSS is beyond the scope of this
paper222The first formulation of the EFTofLSS was carried out in Eulerian
space in Refs. Carrasco  _et al._ (2012); Baumann  _et al._ (2012) and in
Lagrangian space in Porto  _et al._ (2014). Once this theoretical framework
was established, many efforts were made to improve this theory and make it
predictive, such as the understanding of renormalization Pajer and Zaldarriaga
(2013); Abolhasani  _et al._ (2016), the IR-resummation of the long
displacement fields Senatore and Zaldarriaga (2014); Baldauf  _et al._ (2015);
Senatore and Zaldarriaga (2015); Senatore and Trevisan (2018); Lewandowski and
Senatore (2020); Blas  _et al._ (2016), and the computation of the two-loop
power spectrum Carrasco  _et al._ (2014a, b). Then, this theory was developed
in the framework of biased tracers (such as galaxies and halos) in Refs.
Senatore (2015); Mirbabayi  _et al._ (2015); Angulo  _et al._ (2015); Fujita
_et al._ (2020); Perko  _et al._ (2016); Nadler  _et al._ (2018)., in this
section, we briefly discuss the software tools that are available in the
literature making use of the EFTofLSS to analyze the full-shape of the galaxy
clustering power spectrum as measured by BOSS. The relevant observables are
the multipoles of the galaxy power spectrum, which are obtained through
Legendre polynomials ($\mathcal{L}_{l}$) decomposition:

$P_{l}(z,k)=\frac{2l+1}{2}\int^{1}_{-1}d\mu\mathcal{L}_{l}(\mu)P_{\text{gg}}(z,k,\mu),$
(2)

where $z$ is the redshift, $\mu=\hat{z}\cdot\hat{k}$ is the the angle between
the line-of-sight $\bm{z}$ and the wavevector of the Fourier mode $\bm{k}$,
and $P_{\text{gg}}(z,k,\mu)$ is the redshift-space (non-linear) galaxy power
spectrum at one-loop order (see the appendix of Ref. D’Amico  _et al._ (2021)
for the formal expression). This expression includes the “Alcock-Paczynski
transformation” which takes into account the fact that the observation uses
artificial cosmological parameters to convert redshifts as well as celestial
coordinates into Cartesian coordinates. The two main contributions to
$P_{\text{gg}}(z,k,\mu)$ are the monopole ($l=0$) and the quadrupole ($l=2$).
Currently, there are two codes in the literature that model non-linear effects
on the power spectrum at one-loop (including a proper infrared resummation
(IR) Senatore and Zaldarriaga (2014); Baldauf  _et al._ (2015); Senatore and
Zaldarriaga (2015); Senatore and Trevisan (2018); Lewandowski and Senatore
(2020); Blas  _et al._ (2016) and a number of observational systematics
corrections beyond the Alcock-Paczynski effect Alcock and Paczynski (1979),
such as window functions Beutler  _et al._ (2019) and fiber collisions Hahn
_et al._ (2017)) through the EFTofLSS method and which allows us to determine
the monopole and the quadrupole of the galaxy power spectrum: i) the
PyBird333https://github.com/pierrexyz/pybird code D’Amico  _et al._ (2021)– a
python module that determines the non-linear matter power spectrum from the
linear one returned by a Boltzmann code such as
CLASS444https://lesgourg.github.io/class_public/class.html Blas  _et al._
(2011) or CAMB555https://camb.info/ Lewis  _et al._ (2000), and ii) the CLASS-
PT666https://github.com/Michalychforever/CLASS-PT code Chudaykin  _et al._
(2020)– which is a stand-alone extension of the CLASS code. Both codes take
into account the same effects with respect to a standard linear Boltzmann
code, and in particular make use of the “FFTLog method” Hamilton (2000);
Simonovic  _et al._ (2018) to compute the one-loop power spectrum and the IR
resummation. Given that our DCDM $\to$ WDM+DR study makes use of an
independent extension to the CLASS code, we will rely on the PyBird code. We
provide a comparison between the two codes in the context of the DCDM$\to$DR
model (already implemented in CLASS-PT) in App. A. One might wonder whether
the EFTofLSS formalism must be extended to properly described the models under
consideration. In App. A and App. B, we argue that the current formalism (and
the codes in their standard form) is sufficient to describe the DCDM models
given present constraints and precision of the data. Yet, we anticipate that
the formalism will need to be developed further for future surveys such as
Euclid Amendola  _et al._ (2018) and the LSST/Vera Rubin Observatory (VRO)
Alonso  _et al._ (2018), which will reach sub-percent precision.

The data we use, in order to confront the non-linear galaxy power spectrum
forecasts with the observations, are made of three different sky-cut from BOSS
DR12 Alam  _et al._ (2017); Reid  _et al._ (2015); Kitaura  _et al._ (2016):
LOWZ NGC, CMASS NGC and CMASS SGC. LOWZ corresponds to the the BOSS DR12 data
including the BAO post-reconstruction for $0.2<z<0.43$ and has an effective
redshift $z_{eff,\text{LOWZ}}=0.32$, while CMASS corresponds to the the BOSS
DR12 data also including the BAO post-reconstruction for $0.43<z<0.7$ and has
an effective redshift $z_{eff,\text{CMASS}}=0.57$ (see Ref. Zhang  _et al._
(2021)). This data set will be called, in the following, ‘EFTofBOSS data’.
Finally, it is worth noting that the EFTofLSS method has been tested against
various simulations (D’Amico  _et al._ (2021); d’ Amico  _et al._ (2020);
Colas  _et al._ (2020); Nishimichi  _et al._ (2020)), and it has been
highlighted that the BOSS full shape can only be evaluated up to
$k_{\text{max}}\sim 0.2h\text{Mpc}^{-1}$, where the BOSS full shape
corresponds to the combination of the monopoles and quadrupoles of the power
spectra of LOWZ NGC, CMASS NGC and CMASS SGC. To be more precise, we consider
that $k_{\text{max,LOWZ}}=0.2h\text{Mpc}^{-1}$ and
$k_{\text{max,CMASS}}=0.23h\text{Mpc}^{-1}$. Finally, we mention that the
PyBird code makes use of ten additional nuisance parameters per sky-cut to
describe various aspects of the EFTofLSS, although some of them are
analytically marginalized over777When discussing best-fits however, we follow
the approach described in the App. C of Ref. D’Amico  _et al._ (2021). or
neglected, and only two extra parameters per sky-cut are required in the
analysis. Their role and definition are detailed for instance in Ref. Perko
_et al._ (2016).

##  III Non-Linear power spectrum in DCDM cosmologies

In this section, we review the models of decaying dark matter considered in
this work, and present the first computation of the non-linear power spectra
in these cosmologies. We consider two different DCDM models (both are limited
to decay into the dark sector): one in which a fraction of dark matter decays
into massless particles, and the second one in which all of the dark matter
experiences two-body decay into massive and massless particles.

###  III.1 Dark Radiation decay products (DCDM $\to$ DR model)

####  III.1.1 Presentation of the model

In the first model we consider, the cold DM sector is partially composed of an
unstable particle (denoted as DCDM) that decays into a non-interacting
relativistic particle (denoted as DR). The rest of the DM is considered stable
and we refer to it as the standard CDM. In addition to the standard six
$\Lambda$CDM parameters, there are two free parameters describing the lifetime
of DCDM $\tau$ (or equivalently the decay width $\Gamma=\tau^{-1}$), as well
as the fraction of DCDM to total dark matter at the initial time $a_{\rm
ini}\to 0$:

$f_{\rm dcdm}\equiv\frac{\omega_{\rm dcdm}(a_{\rm
ini})}{\omega_{\text{tot,dm}}(a_{\rm ini})},$ (3)

with $\omega_{\text{tot,
dm}}\\!\equiv\\!\omega_{\text{dcdm}}+\omega_{\text{cdm}}$. With these
definitions, in the limit of large $\tau$ and/or small $f_{\rm dcdm}$, one
recovers the $\Lambda$CDM model.

The evolution of the homogeneous energy densities of the decaying dark matter
and dark radiation is given by (see e.g. Refs. Audren  _et al._ (2014);
Enqvist  _et al._ (2015b); Poulin  _et al._ (2016)):

$\displaystyle\dot{\bar{\rho}}_{\text{dcdm}}+3\mathcal{H}\bar{\rho}_{\text{dcdm}}$
$\displaystyle=-a\Gamma\bar{\rho}_{\text{dcdm}},$ (4)
$\displaystyle\dot{\bar{\rho}}_{\text{dr}}+4\mathcal{H}\bar{\rho}_{\text{dr}}$
$\displaystyle=a\Gamma\bar{\rho}_{\text{dcdm}},$ (5)

where $\mathcal{H}$ is the conformal Hubble parameter,

$\mathcal{H}^{2}(a)=\frac{8\pi Ga^{2}}{3}\sum_{i}\bar{\rho}_{i}(a),$ (6)

with

$\displaystyle\sum_{i}\bar{\rho}_{i}(a)$
$\displaystyle=\bar{\rho}_{\text{cdm}}(a)+\bar{\rho}_{\text{dcdm}}(a)+\bar{\rho}_{\text{dr}}(a)$
$\displaystyle+\bar{\rho}_{\gamma}(a)+\bar{\rho}_{\nu}(a)+\bar{\rho}_{b}(a)+\bar{\rho}_{\Lambda}.$
(7)

To describe the evolution of the linearly perturbed universe, we consider the
usual synchronous gauge, where the scalar part of the perturbed metric is
written as Ma and Bertschinger (1995)

$ds^{2}=a^{2}(\tau)\left[-d\tau^{2}+(\delta_{ij}+h_{ij}(\bm{x},t))dx^{i}dx^{j}\right]\,.$
(8)

Here $\tau$ is the conformal time, and $h_{ij}(\bm{x},\tau)$ is defined as

$\displaystyle h_{ij}(\bm{x},\tau)$ $\displaystyle=\int\
d^{3}ke^{i\bm{k}.\bm{x}}[\hat{k}_{i}\hat{k}_{j}h(\bm{k},\tau)$
$\displaystyle+\left(\hat{k}_{i}\hat{k}_{j}-\frac{1}{3}\delta_{ij}\right)6\eta(\bm{k},\tau)]\,.$
(9)

$h$ denotes the trace of $h_{ij}$, while $\eta$ corresponds to the other
traceless scalar degree of freedom of the metric perturbation in Fourier
space. Additionally, we consider the frame co-moving with the DCDM (and CDM)
fluid, such that $\theta_{\text{dcdm}}=\partial_{i}v^{i}_{\text{dcdm}}=0$,
where $\theta_{\text{dcdm}}$ is the divergence of the DCDM velocity
$v^{i}_{\text{dcdm}}$. As a result, the energy density perturbation of the
DCDM component,
$\delta_{\text{dcdm}}\equiv\rho_{\text{dcdm}}/\bar{\rho}_{\text{dcdm}}-1$,
follows the same evolution as standard CDM:

$\dot{\delta}_{\text{dcdm}}=-\frac{\dot{h}}{2}.$ (10)

The evolution of the linear perturbations of the DR integrated phase-space
distribution multipoles is governed by the following hierarchy of equations
Audren  _et al._ (2014); Enqvist  _et al._ (2015b); Poulin  _et al._ (2016):

$\displaystyle\dot{F}_{{\rm{dr}},0}$
$\displaystyle=-kF_{{\rm{dr}},1}-\frac{2}{3}r_{\rm dr}\dot{h}+\dot{r}_{\rm
dr}\delta_{\rm{dcdm}},$ (11) $\displaystyle\dot{F}_{{\rm{dr}},1}$
$\displaystyle=\frac{k}{3}F_{{\rm{dr}},0}-\frac{2k}{3}F_{{\rm{dr}},2},$ (12)
$\displaystyle\dot{F}_{{\rm{dr}},2}$
$\displaystyle=\frac{2k}{5}F_{{\rm{dr}},1}-\frac{3k}{5}F_{{\rm{dr}},3}+\frac{4}{15}r_{\rm
dr}(\dot{h}+6\dot{\eta}),$ (13) $\displaystyle\dot{F}_{{\rm{dr}},\ell}$
$\displaystyle=\frac{k}{(2\ell+1)}\left[\ell
F_{{\rm{dr}},\ell-1}-(\ell+1)F_{{\rm{dr}},\ell+1}\right]\ \ \ \ \ \ (\ell\geq
3).$ (14)

In the previous equations we have introduced $r_{\rm{dr}}\equiv
a^{4}\bar{\rho}_{\rm{dr}}(a)/\rho_{c,0}$ following Ref. Poulin  _et al._
(2016), where $\rho_{c,0}$ is the critical density today. In the scenario
under study, we have:

$\dot{r}_{\rm{dr}}=a\Gamma(\bar{\rho}_{\rm{dcdm}}/\bar{\rho}_{\rm{dr}})r_{\rm{dr}}.$
(15)

We also note that the first three multipoles are simply related to elements of
the perturbed stress-energy tensor as
$F_{{\rm{dr}},0}=r_{\rm{dr}}\delta_{\rm{dr}}$,
$F_{{\rm{dr}},1}=(4r_{\rm{dr}}/3k)\theta_{\rm{dr}}$, and
$F_{{\rm{dr}},2}=2\sigma_{\rm{dr}}r_{\rm{dr}}$. In order to truncate the
hierarchy of Eqs. (11)-(14), we adopt the scheme proposed in Ref. Ma and
Bertschinger (1995) for massless neutrinos.

These equations have been implemented in the Boltzmann code CLASS, and the
impact of DCDM $\to$ DR decay on the (linear) CMB and matter power spectrum
has been studied in details in the literature Audren  _et al._ (2014); Enqvist
_et al._ (2015b); Poulin  _et al._ (2016). In App. A we present a comparison
of the EFTofLSS calculation with N-body simulations performed in Ref. Hubert
_et al._ (2021). The results obtained from these two methods agree upto sub-
percent difference for $k\lesssim 0.2h/$Mpc and $z=0$, justifying that one can
safely analyze the (mildly) non-linear galaxy power spectrum with the
EFTofLSS.

####  III.1.2 The non-linear power spectrum

Thanks to the PyBird code, we plot in Fig. 2 the residuals of the non-linear
matter power spectra of the DCDM $\to$ DR model with respect to that of the
$\Lambda$CDM model at $z=0$. We also represent the associated linear matter
power spectra obtained from the CLASS code. In addition, we plot in Fig. 2 the
residuals of the monopole and quadrupole of the galaxy power spectra of this
model. In these figures, we set the $\Lambda$CDM parameters888For
completeness, note that the shape of the residuals of the galaxy and matter
power spectra depend on the values of the EFT nuisance parameters, especially
at large $k$. According to the notation of Ref. D’Amico  _et al._ (2021), for
the numerical evaluation we set $c_{s}=1\cdot({\rm Mpc}/h)^{2}$ for the matter
power spectra, and $b_{1}=2$, $b_{2}=1$, $b_{3}=0.5$, $b_{4}=0$,
$c_{ct}=0.5\cdot({\rm Mpc}/h)^{2}$, $c_{r,1}=2\cdot({\rm Mpc}/h)^{2}$ and
$c_{r,2}=c_{\epsilon,0}=c_{\epsilon,1}=c_{\epsilon,2}=0$ for the galaxy power
spectra. In practice, these parameters are optimized when quoting best-fits,
to ensure that they take realistic values. to their best-fit values from the
analysis of ‘Planck \+ Pantheon + EFTofBOSS + Ext-BAO’ (as described in sec.
IV). Finally, we simply vary the two parameters $f_{\rm dcdm}$ and $\tau$ to
isolate their cosmological effects : in the left panels, we fix $f_{\rm
dcdm}=1$ and vary $\tau\in[0.1,1000]$ Gyr, while in the right panel we fix
$\tau=1$ Gyr and vary $f_{\rm dcdm}\in[0.1,1]$.

Figure 1: Left \- Residuals of the linear (dashed lines) and non-linear matter
power spectrum (solid lines) for $f_{\rm dcdm}$ set to 1 and $\tau=$ 0.1, 1,
10, 100 Gyr. Residuals are taken with respect to the $\Lambda$CDM model at
$z=0$. Right \- The same, but this time $\tau$ is set to 1 Gyr and $f_{\rm
dcdm}=$ 0.1, 0.4, 0.7, 1.

Figure 2: Left \- Residuals of the monopole and the quadrupole of the galaxy
power spectrum for $f_{\rm dcdm}$ set to 1 and $\tau=$ 0.1, 1, 10, 100 Gyr.
Residuals are taken with respect to the $\Lambda$CDM model at $z=0$. Right \-
The same, but this time $\tau$ is set to 1 Gyr and $f_{\rm dcdm}=$ 0.1, 0.4,
0.7, 1.

From Figs. 2 and 2, one can see that the monopole of the galaxy power spectrum
shows a behavior very similar to that of the linear matter spectrum. For a
realistic choice of EFT parameters, it shows an almost scale-independent power
suppression due to two main reasons Audren  _et al._ (2014); Poulin  _et al._
(2016). First, the decay of DCDM decreases the duration of the matter
dominated era (and at fix $h$, a smaller $\Omega_{m}$/larger
$\Omega_{\Lambda}$), implying a shift of the power spectrum towards large
scales, i.e. towards small wavenumbers. Second, DCDM models involve a larger
ratio of $\omega_{\rm b}/\omega_{\rm cdm}$ compared to the $\Lambda$CDM model
due to the decay. Both effects manifests as a strong suppression of the small-
scale power spectrum, and the latter effect leads to an additional modulation
of the BAO amplitude visible as wiggles in Figs. 2 and 2. Moreover, we note
that the non-linear matter power spectrum shows a stronger scale-dependent
suppression compared to the linear power spectrum at $k\gtrsim
0.1h/\text{Mpc}$. On the other hand, the power suppression gets less strong
with larger $k$ in the monopole of the galaxy power spectrum, an effect
indicating a potential degeneracy with the EFT parameters. Finally, and as
expected, deviations with respect to $\Lambda$CDM increases as $\tau$
decreases and/or $f_{\rm dcdm}$ increases for the monopole as well as for the
quadrupole.

####  III.1.3 Preliminary study

To gauge the impact of using the EFTofBOSS data in our analyses of the DCDM
$\to$ DR model, we first perform a preliminary study in which we consider a
set of DCDM parameters laying at the 95% C.L.999From here on, we quote one-
sided bounds at 2$\sigma$ (95 % C.L.) and two-sided bounds at 1$\sigma$ (68%
C.L.). derived from Planck data, and compute the $\chi^{2}$ of the EFTofBOSS
data after optimising the EFT nuisance parameters. The goal is to check the
extent to which EFT nuisance parameters can lead to effects degenerate with
those of the DCDM with a quick analysis. We set all $\Lambda$CDM parameters to
their best-fit values from the analysis of ‘Planck \+ Pantheon + EFTofBOSS +
Ext-BAO’ (see Sec. IV). We perform two analyses: (i) we set $\tau=0.1$ Gyr and
take the upper bound on $f_{\rm dcdm}$ from our ‘Planck \+ Pantheon + Ext-BAO
(no Ly-$\alpha$)’, i.e. $f_{\rm dcdm}=0.0203$ (see Tab. 5), and (ii) we set
$f_{\rm dcdm}=1$ (i.e. all the dark matter decays), while we take the lower
bound of $\tau$ from our ‘Planck \+ Pantheon + Ext-BAO (no Ly-$\alpha$)’
analysis, i.e. $\tau=248.4$ Gyr (see Tab. 5). We show in Tab. 1 the $\chi^{2}$
associated to the EFTofBOSS data, and we plot in Fig 3, using the PyBird code,
the residuals (with respect to $\Lambda$CDM from the ‘Planck \+ Pantheon +
EFTofBOSS + Ext-BAO’ analysis) of these studies. To gauge the impact of EFT
nuisance parameters, in this latter figure, we show residuals with and without
the optimization procedure (in the latter case, we simply set the EFT nuisance
parameters to those of $\Lambda$CDM). This preliminary study allows us to
highlight two important points. Firstly, the optimization procedure has washed
out the suppression due to decay, which implies that the effect of the EFT
nuisance parameters are (at least partly) degenerate with that of the decay.
Secondly (and consequently), for these two analyses where we have chosen DCDM
parameters that are excluded at 95% C.L., we obtain a $\chi^{2}$ very close to
that of the $\Lambda$CDM best-fit model of the full analysis, suggesting that
EFTofBOSS data may not provide strong additional constraints to this model.
Naturally, it does not prevent the model to potentially yield an improved fit
over $\Lambda$CDM once all (cosmological and nuisance) parameters are
optimized against the data, and we will check our naive results against a full
analysis in Sec. IV.

Parameter |  $f_{\rm dcdm}=0.0203$ & $\tau=0.1$ Gyr |  $f_{\rm dcdm}=1$ & $\tau=248.4$ Gyr
---|---|---
$\chi^{2}_{\text{CMASS NGC}}$ | 41.3 | 40.7
$\chi^{2}_{\text{CMASS SGC}}$ | 43.9 | 44.0
$\chi^{2}_{\text{LOWZ NGC}}$ | 33.4 | 33.6
$\chi^{2}_{\text{EFTofBOSS}}$ | 118.6 | 118.3
$\footnotesize{\chi^{2}_{\rm min}(\text{DCDM})-\chi^{2}_{\rm min}(\Lambda\text{CDM})}$ | +0.8 | +0.5
Table 1: $\chi^{2}$ of each sky-cut of the EFTofBOSS data set for our DCDM
$\to$ DR preliminary study. We also indicated the $\Delta\chi^{2}$ with
respect to the analogous $\Lambda$CDM best-fit model (EFTofBOSS analysis in
Tab. 7). Figure 3: Residuals of the monopole and the quadrupole of our DCDM
$\to$ DR preliminary study with respect to $\Lambda$CDM model (EFTofBOSS
analysis in Tab. 3) for the three sky-cuts of the EFTofBOSS data. For the
solid lines we optimized the EFT nuisance parameters, while for the dotted
lines we set the EFT nuisance parameters to those of the $\Lambda$CDM
(EFTofBOSS analysis in Tab. 3).

###  III.2 Warm Dark Matter decay products (DCDM $\to$ WDM+DR model)

####  III.2.1 Presentation of the model

We now turn to a DCDM model where the entirety of the DM sector is considered
unstable (i.e. $f_{\rm dcdm}=1$ in the language of the first model), decaying
into dark radiation and a massive particle, which will act as WDM. As before,
we assume the decay products do not interact with the standard model
particles. The DCDM sector is now described by the DCDM lifetime $\tau$, and
the fraction $\varepsilon$ of rest-mass energy carried away by the massless
particle given by Blackadder and Koushiappas (2014)

$\varepsilon=\frac{1}{2}\left(1-\frac{m^{2}_{\text{wdm}}}{m^{2}_{\text{dcdm}}}\right),$
(16)

where $m_{\text{dcdm}}$ and $m_{\text{wdm}}$ are the mother and daughter
particle masses respectively. The accurate computation of the cosmological
impact of the DCDM sector requires to follow the evolution of the phase space
distribution of the warm particle produced during the decay. The full set of
equations is described in Ref. Aoyama  _et al._ (2014); Abellán  _et al._
(2021). We summarize here the sets of equations describing the evolution of
the background energy densities of the dark components, as well as the linear
perturbations in a fluid approximation, valid well within the horizon.

The background energy densities evolve as follows Aoyama  _et al._ (2014):

$\displaystyle\dot{\bar{\rho}}_{\text{dcdm}}+3\mathcal{H}\bar{\rho}_{\text{dcdm}}$
$\displaystyle=-a\Gamma\bar{\rho}_{\text{dcdm}},$ (17)
$\displaystyle\dot{\bar{\rho}}_{\text{wdm}}+3(1+w)\mathcal{H}\bar{\rho}_{\text{wdm}}$
$\displaystyle=(1-\varepsilon)a\Gamma\bar{\rho}_{\text{dcdm}},$ (18)
$\displaystyle\dot{\bar{\rho}}_{\text{dr}}+4\mathcal{H}\bar{\rho}_{\text{dr}}$
$\displaystyle=\varepsilon\Gamma a\bar{\rho}_{\text{dcdm}}\,$ (19)

where $w=\bar{P}_{\text{wdm}}/\bar{\rho}_{\text{wdm}}$ is the equation of
state of the massive daughter particle. In the limit of large $\tau$ or small
$\varepsilon$, one recovers the $\Lambda$CDM model, while setting
$\varepsilon=1/2$ leads to a decay solely into massless particles.

In the synchronous gauge comoving with the DCDM fluid, the linear perturbation
equations for the parent particle and DR daughter is still given by Eq. (10)
and Eqs. 11-14, respectively. However, the quantity $r_{\rm dr}$ now satisfies

$\dot{r}_{\rm{dr}}=a\varepsilon\Gamma(\bar{\rho}_{\rm{dcdm}}/\bar{\rho}_{\rm{dr}})r_{\rm{dr}},$
(20)

where the parameter $\varepsilon$ now affects the amount of energy transferred
to the DR. Regarding the WDM linear perturbations, it is unfortunately not
possible to integrate out the dependency on momenta as it is done for the DR
species. In general one has to follow the evolution of the full phase-space
distribution, which becomes very computationally demanding (see Ref. Abellán
_et al._ (2021) for the expression of the full Boltzmann hierarchy).
Nevertheless, it was shown in Ref. Abellán  _et al._ (2021) that, well within
the horizon, the dynamics of the WDM perturbations can be well approximated by
the following set of fluid equations:

$\displaystyle\dot{\delta}_{\text{wdm}}$
$\displaystyle=-3\mathcal{H}(c_{s}^{2}-\omega)\delta_{\text{wdm}}-(1+\omega)\left(\theta_{\text{wdm}}+\frac{\dot{h}}{2}\right)$
$\displaystyle+(1-\varepsilon)a\Gamma\frac{\bar{\rho}_{\text{dcdm}}}{\bar{\rho}_{\text{wdm}}}(\delta_{\text{dcdm}}-\delta_{\text{wdm}}),$
(21) $\displaystyle\dot{\theta}_{\text{wdm}}$
$\displaystyle=-\mathcal{H}(1-3c_{g}^{2})\theta_{\text{wdm}}+\frac{c_{s}^{2}}{1+\omega}k^{2}\delta_{\text{wdm}}-k^{2}\sigma_{\text{wdm}}$
$\displaystyle-(1-\varepsilon)a\Gamma\frac{1+c_{g}^{2}}{1+\omega}\frac{\bar{\rho}_{\text{dcdm}}}{\bar{\rho}_{\text{wdm}}}\theta_{\text{wdm}},$
(22)

where $c_{s}$ is the WDM sound speed in the synchronous gauge, i.e.
$c_{s}^{2}=\delta P_{\text{wdm}}/\delta\rho_{\text{wdm}}$, and $c_{g}$ is the
WDM adiabatic sound speed, i.e.
$c_{g}^{2}=\dot{\bar{P}}_{\text{wdm}}/\dot{\bar{\rho}}_{\text{wdm}}$, which
one can write in the following form:

$\displaystyle c_{g}^{2}$
$\displaystyle=w\left(5-\frac{\mathfrak{p}_{\text{wdm}}}{\bar{P}_{\text{wdm}}}-\frac{\bar{\rho}_{\text{dcdm}}}{\bar{\rho}_{\text{wdm}}}\frac{a\Gamma}{3w\mathcal{H}}\frac{\varepsilon^{2}}{1-\varepsilon}\right)$
$\displaystyle\times\left[3(1+w)-\frac{\bar{\rho}_{\text{dcdm}}}{\bar{\rho}_{\text{wdm}}}\frac{a\Gamma}{\mathcal{H}}(1-\varepsilon)\right]^{-1}.$
(23)

In this latter equation, $\mathfrak{p}_{\text{wdm}}$ is the pseudo-pressure
(introduced in the context of the fluid equations for massive neutrinos
Lesgourgues and Tram (2011)), which corresponds to a higher momenta integral
of the WDM homogeneous phase space distribution, reducing to the standard
pressure in the relativistic limit. Solving the fluid equations requires
specifying the sound speed $c_{s}$, which was found to be well described by
the following formula:

$c_{s}^{2}(k,\tau)=c_{g}^{2}[1+0.2\times(1-2\varepsilon)\sqrt{k/k_{\rm
fs}}]\,$ (24)

where the free-streaming scale $k_{\rm fs}$ of the WDM is computed as:

$k_{\text{fs}}(\tau)=\sqrt{\frac{3}{2}}\frac{\mathcal{H}(\tau)}{c_{g}(\tau)}\,.$
(25)

The free-streaming scale corresponds to the scale at which pressure (coming
from the ‘velocity kick’ received during the decay process) suppresses
perturbations of the WDM compared to those of the DCDM. In other word, on
scales $k<k_{\rm fs}$, one has $\delta_{\text{wdm}}=\delta_{\text{dcdm}}$,
while on scale $k>k_{\rm fs}$ the WDM perturbations are suppressed and exhibit
oscillations over time.

To obtain the linear CMB and matter power spectrum, we make use of an
extension of the CLASS code101010https://github.com/PoulinV/class_decays
described in Ref. Abellán  _et al._ (2021), and we determine the non-linear
galaxy power spectrum using the PyBird code. We have argued in previous
section and in App. A, through direct comparison with N-body simulations, that
PyBird can safely be used to describe DM decays with massless decay products.
Unfortunately, we do not have access to such N-body simulations in the case of
massive decay products. A priori, the problem is not the decay per se (as we
have seen for the massless decay products). Rather, contrarily to the case of
massless daugthers, the massive daughter may develop perturbations whose
contribution to the total matter power spectrum can be highly non-trivial. In
App. B, following Refs Senatore and Zaldarriaga (2017); de Belsunce and
Senatore (2019), which treated the similar case of massive neutrinos, we argue
that the corrections to the EFTofLSS necessary to fully capture the model-
specific effects can be neglected for most of the parameter space of interest,
as the fractional contribution of the WDM to the DM density is small (in
particular for the best-fit model that we derive), or the free-streaming scale
exceeds the scale-cut considered in the analysis.

####  III.2.2 The non-linear power spectrum

We plot in Fig. 5 the residuals of the non-linear matter power spectra of the
DCDM $\to$ WDM+DR model with respect to that of the $\Lambda$CDM model at
$z=0$. We also represent the associated linear matter power spectra obtained
from the CLASS code, exactly as in Fig. 4 of Ref. Abellán  _et al._ (2021). In
addition, we plot in Fig. 5 the residuals of the monopole and quadrupole of
the galaxy power spectra of this model. In these figures, the cosmological
parameters are taken from the DCDM $\to$ WDM+DR best-fit model of Ref. Abellan
_et al._ (2020), while the nuisance parameters are set as in Figs. 2 and 2. In
the left panels, we fix $\varepsilon=0.1$ and vary $\tau\in[10,300]$ Gyr,
while in the right panel we fix $\tau=30$ Gyr and vary
$\varepsilon\in[0.001,0.5]$.

Figure 4: Left \- Residuals of the linear (dashed lines) and non-linear matter
power spectrum (solid lines) for $\varepsilon$ set to 0.1 and $\tau=$ 10, 30,
100, 300 Gyr. Residuals are taken with respect to the $\Lambda$CDM model at
$z=0$. Right \- The same, but this time $\tau$ is set to 30 Gyr and
$\varepsilon=$ 0.001, 0.01, 0.1, 0.5.

Figure 5: Left \- Residuals of the monopole and the quadrupole of the galaxy
power spectrum for $\varepsilon$ set to 0.1 and $\tau=$ 10, 30, 100, 300 Gyr.
Residuals are taken with respect to the $\Lambda$CDM model at $z=0$. Right \-
The same, but this time $\tau$ is set to 30 Gyr and $\varepsilon=$ 0.001,
0.01, 0.1, 0.5.

As for the case of the DCDM $\to$ DR model, we obtain a very similar behaviour
between the linear matter power spectrum and the monopole of the galaxy power
spectrum, except for a mild monotonic reduction of the power suppression at
larger $k$’s in the monopole of the galaxy power spectrum (due to the choice
of EFT parameters, this reduction of the suppression may change for different
values). The presence of a warm dark matter component which does not cluster
on small scales suppresses the matter power spectrum as well as the galaxy
power spectrum, and $\tau$ – which sets the abundance of the WDM species today
– controls the amplitude of the power suppression, while $\varepsilon$
controls the cutoff scale. One can see in Fig. 5 that the suppression of the
galaxy spectrum increases as $\tau$ decreases (left panel), while the
suppression starts to occur on larger scales as $\varepsilon$ increases (right
panel). Once $\varepsilon=0.5$, the free-streaming scale $k_{\text{fs}}$
becomes equivalent to the Hubble horizon, and the effects become identical to
that of the DCDM $\to$ DR model presented before. Note that because of the
effect of the WDM, the $\varepsilon=0.1$ case has a stronger suppression than
the $\varepsilon=0.5$ (pure dark radiation) case. Moreover, we find (see Fig.
5) that the non-linear correction slightly modulates the slope of the power
suppression compared to the linear matter power spectrum. It always leads to a
stronger suppression than the linear one at large enough $k$ (for
$\varepsilon\gtrsim 0.1$, the modulation occurs at $k\gtrsim
0.1h/\text{Mpc}$). However, for smaller $\varepsilon$ (see the
$\varepsilon=0.01$ case for example), the modulation can appear as a milder
power suppression compared to the linear one in the range of validity of the
EFT at one-loop order.

####  III.2.3 Preliminary study

Parameter | Best-fit
---|---
$\chi^{2}_{\text{CMASS NGC}}$ | 41.2
$\chi^{2}_{\text{CMASS SGC}}$ | 44.5
$\chi^{2}_{\text{LOWZ NGC}}$ | 34.4
$\chi^{2}_{\text{EFTofBOSS}}$ | 120.1
$\footnotesize{\chi^{2}_{\rm min}(\text{DCDM})-\chi^{2}_{\rm min}(\Lambda\text{CDM})}$ | +3.1
Table 2: $\chi^{2}$ of each sky-cut of the EFTofBOSS data set for our DCDM
$\to$ WDM+DR preliminary study. We also indicated the $\Delta\chi^{2}$ with
respect to the analogous $\Lambda$CDM best-fit model (EFTofBOSS + $S_{8}$
analysis in Tab. 7). Figure 6: Residuals of the monopole and the quadrupole of
our DCDM $\to$ WDM+DR preliminary study with respect to $\Lambda$CDM model
(EFTofBOSS analysis in Tab. 3) for the three sky-cuts of the EFTofBOSS data.
For the solid lines we optimized the EFT nuisance parameters, while for the
dotted lines we set the EFT nuisance parameters to those of the $\Lambda$CDM
(EFTofBOSS analysis in Tab. 3).

Similarly to the case of the DCDM $\to$ DR model, we perform a preliminary
study to test whether the EFTofBOSS data can further constrain the DCDM $\to$
WDM+DR model that resolves the $S_{8}$ tension. We fix cosmological
parameters111111The analysis performed in Refs. Abellan  _et al._ (2020);
Abellán  _et al._ (2021) made use of a $S_{8}$ prior that includes information
from BOSS Heymans  _et al._ (2020). For consistency and to avoid double
counting information, we re-performed the analysis (see Sec. IV) with a prior
derived from KiDS-1000 data alone. to those obtained from the joint analysis
of Planck data, Pantheon SN1a data, a compilation of BAO data and the $S_{8}$
measurements by KiDS-1000 Asgari  _et al._ (2021). We optimize the EFT
nuisance parameters of the galaxy power spectrum to check the extent to which
they can lead to effects degenerate with those of the DCDM. We show in Tab. 2
the $\chi^{2}$ associated to the EFTofBOSS data, while in Fig. 6, using the
PyBird code, we plot the residuals with respect to the best-fit $\Lambda$CDM
model from the analysis of ‘Planck \+ Pantheon + EFTofBOSS + Ext-BAO’ (see
Sec. IV). In this figure, we represent residuals with and without the EFT
optimization procedure (in the latter case, we simply set the EFT nuisance
parameters to those of $\Lambda$CDM). As before, one can see that the effects
of the DCDM are strongly reduced once EFT nuisance parameters are optimized,
suggesting a strong degeneracy between the DCDM and the EFT parameters.
Nevertheless, for this preliminary study, the $\chi^{2}$ is degraded by $+3.1$
compared to the best-fit $\chi^{2}$ obtained in the $\Lambda$CDM model for the
full analysis. Contrary to the preliminary study of the DCDM $\to$ DR model
for which we obtained a $\chi^{2}$ close to that of the $\Lambda$CDM model, we
anticipate that the EFTofBOSS data can provide additional constraining power
to this model.

##  IV A comprehensive MCMC analysis of the DCDM models

###  IV.1 Data and method

We now perform a Monte Carlo Markov Chain (MCMC) analyses, confronting these
two DCDM models with recent cosmological observations. To do so, we make use
of the MontePython-v3 code Brinckmann and Lesgourgues (2019) interfaced with
our modified CLASS version. We perform various analyses from a combination of
the following data sets:

  * •

Planck: The low-$l$ CMB TT, EE, and the high-$l$ TT, TE, EE data, as well as
the gravitational lensing potential reconstruction from Planck 2018 Aghanim
_et al._ (2020a, b).

  * •

Pantheon: The Pantheon SNIa catalogue, spanning redshifts $0.01<z<2.3$ Scolnic
_et al._ (2018).

  * •

Ext-BAO: The BAO measurements from 6dFGS at $z=0.106$, SDSS DR7 at $z=0.15$
Beutler  _et al._ (2011); Ross  _et al._ (2015); Alam  _et al._ (2017), and
the joint constraints from eBOSS DR14 Ly-$\alpha$ absorption auto-correlation
at $z=2.34$ and cross-correlation with quasars at $z=2.35$ de Sainte Agathe
_et al._ (2019); Blomqvist  _et al._ (2019).

  * •

BOSS BAO/$\bm{f\sigma_{8}}$: The measurements of the BAO and the redshift
space distortion $f\sigma_{8}(z)$ from the CMASS and LOWZ galaxy samples of
BOSS DR12 at $z=0.38$, 0.51, and 0.61 Alam  _et al._ (2017).

  * •

$\bm{S_{8}}$: The KIDS-1000 cosmic shear measurement of
$S_{8}=0.759^{+0.024}_{-0.021}$, modeled as a a split-normal likelihood Asgari
_et al._ (2021).

  * •

EFTofBOSS: The CMASS and LOWZ data sets of the EFTofBOSS data (see Sec. II).

Our analyses always include Planck, Pantheon and Ext-BAO data. However, we
quantify the impact of EFTofBOSS data and the $S_{8}$ prior by performing
analyses with and without these data. When not including the EFTofBOSS data,
we make use of the conventional BOSS BAO/$f\sigma_{8}$ data. We use Planck
conventions for the treatment of neutrinos and include two massless and one
massive species with $m_{\nu}=0.06$ eV Aghanim  _et al._ (2020a). We impose a
large flat prior on the dimensionless baryon energy density $\omega_{b}$, the
Hubble parameter today $H_{0}$, the logarithm of the variance of curvature
perturbations centered around the pivot scale $k_{p}=0.05$ Mpc-1 (according to
the Planck convention), $\ln(10^{10}\mathcal{A}_{s})$, the scalar spectral
index $n_{s}$, and the re-ionization optical depth $\tau_{\rm reio}$. We
assume our MCMC chains to be converged when the Gelman-Rubin criterion
$R-1<0.05$ Gelman and Rubin (1992). Finally, we extract the best-fit
parameters from the procedure highlighted in appendix of Ref. Schöneberg  _et
al._ (2021).

$\Lambda$CDM
---
Parameter | w/ EFTofBOSS | w/ EFTofBOSS + $S_{8}$
$100\ \omega_{\rm b}$ |  $2.242(2.245)_{-0.015}^{+0.014}$ |  $2.247(2.248)\pm 0.014$
$\omega_{\text{cdm}}$ | $0.1191(0.1191)\pm 0.00095$ | $0.1184(0.1184)\pm+0.00089$
$H_{0}/[{\rm km/s/Mpc}]$ | $67.76(67.80)_{-0.44}^{+0.42}$ | $68.05(68.07)\pm 0.41$
$\text{ln}(10^{10}A_{s})$ | $3.048(3.049)_{-0.016}^{+0.015}$ | $3.043(3.043)_{-0.016}^{+0.015}$
$n_{s}$ | $0.9666(0.9676)\pm 0.0039$ | $0.9680(0.9687)\pm 0.0039$
$\tau_{\rm reio}$ | $0.0571(0.0574)_{-0.0085}^{+0.0075}$ | $0.0555(0.0549)_{-0.0078}^{+0.0077}$
$\Omega_{\rm m}$ | $0.3098(0.3093)_{-0.0058}^{+0.0057}$ | $0.3057(0.3055)\pm 0.0053$
$\sigma_{8}$ | $0.8097(0.8102)_{-0.0065}^{+0.0063}$ | $0.8056(0.8055)\pm 0.0062$
$S_{8}$ | $0.82(0.82)\pm 0.01$ | $0.813(0.813)_{-0.0096}^{+0.0094}$
$\chi^{2}_{\rm min}$ | 3927.0 | 3933.0
$Q_{\rm DMAP}\equiv\sqrt{\chi^{2}_{\rm min}({\rm w/}~{}S_{8})-\chi^{2}_{\rm min}({\rm w/o}~{}S_{8})}$ | 2.4$\sigma$
Table 3: The mean (best-fit) $\pm 1\sigma$ errors of the cosmological
parameters from our ‘Planck \+ Pantheon + EFTofBOSS + Ext-BAO’ and ‘Planck \+
Pantheon + EFTofBOSS + Ext-BAO + $S_{8}$’ analyses for the $\Lambda$CDM model.
For each data set we also report its best-fit $\chi^{2}$. DCDM$\to$DR
---
Parameter | w/ EFTofBOSS | w/ EFTofBOSS + $S_{8}$
$\Gamma/[{\rm Gyr}^{-1}]$ |  unconstrained (4.8) |  unconstrained (5.8)
$f_{\rm dcdm}$ | $<0.0216(1.62\cdot 10^{-4})$ | $<0.0242(1.67\cdot 10^{-4})$
$100\ \omega_{\rm b}$ | $2.236(2.244)\pm 0.015$ | $2.241(2.248)_{-0.015}^{+0.016}$
$\omega_{\text{cdm}}$ | $0.1187(0.1191)\pm 0.0010$ | $0.1180(0.1184)_{-0.00093}^{+0.001}$
$H_{0}/[{\rm km/s/Mpc}]$ | $67.98(67.77)_{-0.48}^{+0.46}$ | $68.30(68.10)_{-0.47}^{+0.44}$
$\text{ln}(10^{10}A_{s})$ | $3.051(3.049)_{-0.016}^{+0.015}$ | $3.047(3.045)_{-0.016}^{+0.015}$
$n_{s}$ | $0.9650(0.9671)_{-0.004}^{+0.0042}$ | $0.9660(0.9687)_{-0.0043}^{+0.0044}$
$\tau_{\rm reio}$ | $0.0577(0.0572)_{-0.0079}^{+0.0073}$ | $0.0562(0.0557)_{-0.0077}^{+0.0074}$
$\Omega_{\rm m}$ | $0.3069(0.3097)\pm 0.0061$ | $0.3026(0.3050)_{-0.0057}^{+0.0059}$
$\sigma_{8}$ | $0.8110(0.8101)_{-0.0066}^{+0.0063}$ | $0.8071(0.8061)_{-0.0063}^{+0.0062}$
$S_{8}$ | $0.82(0.82)\pm 0.01$ | $0.811(0.813)_{-0.0095}^{+0.0097}$
$\chi^{2}_{\rm min}$ | 3927.0 | 3933.0
$\footnotesize{\chi^{2}_{\rm min}(\text{DCDM})-\chi^{2}_{\rm min}(\Lambda\text{CDM})}$ | 0.0 | 0.0
$Q_{\rm DMAP}\equiv\sqrt{\chi^{2}_{\rm min}({\rm w/}~{}S_{8})-\chi^{2}_{\rm min}({\rm w/o}~{}S_{8})}$ | 2.4$\sigma$
Table 4: The mean (best-fit) $\pm 1\sigma$ errors of the cosmological
parameters from our ‘Planck \+ Pantheon + EFTofBOSS + Ext-BAO’ and ‘Planck \+
Pantheon + EFTofBOSS + Ext-BAO + $S_{8}$’ analyses for the DCDM $\to$ DR
model. For each data set we also report its best-fit $\chi^{2}$, and the
$\Delta\chi^{2}$ with respect to the analogous $\Lambda$CDM best-fit model.
DCDM $\to$ DR
---
Data sets | $f_{\rm dcdm}$ |  $\tau$ (for $f_{\rm dcdm}=1$)
Planck | $<0.0205$ |  $>$246.3 Gyr
Planck \+ Pantheon + Ext-BAO (no Ly-$\alpha$) | $<0.0203$ |  $>$248.4 Gyr
Planck \+ Pantheon + BOSS BAO/$f\sigma_{8}$ \+ Ext-BAO (no Ly-$\alpha$) | $<0.0190$ |  $>$260.4 Gyr
Planck \+ Pantheon + BOSS BAO/$f\sigma_{8}$ \+ Ext-BAO |  $<$0.0219 |  $>$250.0 Gyr
Planck \+ Pantheon + EFTofBOSS + Ext-BAO |  $<$0.0216 |  $>$249.6 Gyr
Table 5: The 95% C.L. limit on $f_{\rm dcdm}$ for the standard DCDM $\to$ DR
analysis, and the 95% C.L. limit on $\tau$ for the DCDM $\to$ DR analysis
where $f_{\rm dcdm}$ is fixed to the unit. Let us recall that ‘Ext-BAO’ refers
to the BAO measurements from 6dFGS, SDSS DR7, and the joint constraints from
eBOSS DR14 Ly-$\alpha$ auto-correlation and cross-correlation. For some data
sets we removed the Ly-$\alpha$ constraints (’no Ly-$\alpha$’) to explicitly
show its impact.

###  IV.2 Dark Radiation decay products

Let us recall that in the case of the DCDM $\to$ DR model we have two
additional parameters: $\Gamma=\tau^{-1}$, the decay rate of DCDM, and $f_{\rm
dcdm}$, the fraction of DCDM with respect to the total DM. In the MCMC
analyses, we impose flat priors on $\Gamma$ and $f$:

$\displaystyle 0\leq$ $\displaystyle~{}\Gamma/{\rm Gyr}^{-1}\leq 10,$
$\displaystyle 0\leq$ $\displaystyle~{}f_{\rm dcdm}\leq 1.$

Our results for the analyses with and without $S_{8}$ prior are presented in
Tab. 4, while the results of the analyses of $\Lambda$CDM against the same
data sets are given in Tab. 3. The $\chi^{2}$ of the EFTofBOSS data are
reported in Tab. 7. In Fig. 7, we display the 1D and 2D posteriors of
$\left\\{\Gamma/{\rm Gyr}^{-1},f_{\rm dcdm},H_{0},S_{8},\Omega_{m}\right\\}$
for the DCDM $\to$ DR model with and without the EFTofBOSS data set. In App.
C, we represent the same figure, but this time with and without the $S_{8}$
prior (and with the EFTofBOSS data set for both). Without the $S_{8}$ prior,
the $\Delta\chi^{2}$ with respect to $\Lambda$CDM is compatible with
zero121212The improvement is below the precision of ${\cal O}(0.1)$ that we
estimated on the minimization, and we therefore simply quote
$\Delta\chi^{2}=0.0$. Hereinafter, we follow the same approach when reporting
other $\Delta\chi^{2}$. (see Tab. 4), implying that the data does not favor
the DCDM $\to$ DR model. From Fig. 7, one can see that the inclusion of the
EFTofBOSS data does not improve the constraint on this model, which is
consistent with the ‘naive’ analysis presented in sec. III.1.3. Moreover, we
show that when adding the $S_{8}$ prior, the $\Delta\chi^{2}$ with respect to
$\Lambda$CDM is still compatible with zero (and the model does not provide a
good fit to the $S_{8}$ prior) while the constraints on $\Gamma$ and $f_{\rm
dcdm}$ are largely unaffected. We conclude (as in past studies) that this
model does not resolve the $S_{8}$ tension.

To summarize our results, and present the most up-to-date constraints on DCDM
with massless decay products, in Tab. 5 we compare the 95% C.L. limits
obtained for $f_{\rm dcdm}$ and $\tau$ when successively adding data sets. To
obtain the bounds on $f_{\rm dcdm}$ (in the ‘short-lived’ regime), we
marginalize over the parameter $\Gamma$ in the range described above. On the
other hand, to obtain the $\tau$ limits (in the ‘very long-lived’ regime), we
fix $f_{\rm dcdm}=1$ in our MCMC analyses, i.e. we assume that all DM decays.
Note that, for $f_{\rm dcdm}\to 1$, one can interpret our constraints as a
limit on the ratio $\tau/f_{\rm dcdm}$, as discussed in Ref. Poulin  _et al._
(2016). From Tab. 5, one can deduce:

  1. •

The strongest constraints are obtained when considering ‘Planck \+ Pantheon +
BOSS BAO/$f\sigma_{8}$ \+ Ext-BAO (no Ly-$\alpha$)’. In that case, we find
$f_{\rm dcdm}<0.0190$ (in the short-lived regime), and $\tau/f_{\rm
dcdm}>260.4$ Gyr (for $f_{\rm dcdm}\to 1$).

  2. •

On the other hand the inclusion of Ly-$\alpha$ BAO data slightly reduce the
constraints. This is consistent with the fact that these data are compatible
with $\Lambda$CDM only at the $1.7\sigma$ level de Sainte Agathe  _et al._
(2019); Blomqvist  _et al._ (2019), favoring lower energy density at high-$z$
Poulin  _et al._ (2018). Additionally, we find that constraints with the
EFTofBOSS data are the same as those with the standard redshift space
distortion $f\sigma_{8}$ information. Our fiducial constraints, including all
data, is therefore $f_{\rm dcdm}<0.0216$, and $\tau/f>249.6$ Gyr.

  3. •

Our constraints are somewhat different than those derived in Ref. Nygaard  _et
al._ (2020), which considering Planck 2018 + BAO data (see Tab. 2 of this
reference) found $f_{\rm dcdm}<0.0262$ at 95 % C.L. and $\tau/f_{\rm
dcdm}>268.8$ Gyr. Our constraints are stronger on $f_{\rm dcdm}$, compatible
with the fact that we include more data, but weaker on $\tau$, which may be
explained by the fact that their posteriors never quite reach $f_{\rm
dcdm}\sim 1$, as necessary to derive constraints in the ‘very long-lived’
regime.

Figure 7: 2D posterior distributions of the DCDM $\to$ DR model with and
without the EFTofBOSS data set. The gray shaded bands refer to the joint
$S_{8}$ measurement from KiDS-1000 + BOSS + 2dFLens Heymans  _et al._ (2020).
Figure 8: 2D posterior distributions of the DCDM $\to$ WDM+DR model with and
without the EFTofBOSS data set. We took into account the $S_{8}$ prior from
KIDS-1000 for these two MCMC analyses. The gray shaded bands refer to the
joint $S_{8}$ measurement from KiDS-1000 + BOSS + 2dFLens.

###  IV.3 Warm Dark Matter decay products

DCDM$\to$WDM+DR
---
Parameter | w/ EFTofBOSS | w/ EFTofBOSS + $S_{8}$
$\text{log}_{10}(\Gamma/[{\rm Gyr}^{-1}])$ |  unconstrained (-2.98) |  $2.21(-2.08)_{-0.6}^{+1.5}$
$\text{log}_{10}(\varepsilon)$ |  unconstrained (-3.84) | $-2.30(-1.92)_{-1.10}^{+0.84}$
$100\ \omega_{\rm b}$ | $2.242(2.245)_{-0.014}^{+0.014}$ | $2.245(2.242)_{-0.015}^{+0.014}$
$\omega_{\rm dcdm}^{\rm ini}$ | $0.1192(0.1190)_{-0.0009}^{+0.00089}$ | $0.1188(0.1192)_{-0.00099}^{+0.00084}$
$H_{0}/[{\rm km/s/Mpc}]$ | $67.78(67.82)_{-0.42}^{+0.41}$ | $67.97(67.73)_{-0.42}^{+0.44}$
$\text{ln}(10^{10}A_{s})$ | $3.049(3.051)_{-0.016}^{+0.015}$ | $3.046(3.052)_{-0.016}^{+0.015}$
$n_{s}$ | $0.9668(0.9679)\pm 0.0039$ | $0.9676(0.9670)\pm 0.0039$
$\tau_{\rm reio}$ | $0.0571(0.0584)_{-0.0080}^{+0.0071}$ | $0.0564(0.0584)_{-0.0077}^{+0.0074}$
$\Omega_{\rm m}$ | $0.3090(0.3089)_{-0.0057}^{+0.0055}$ | $0.3064(0.3094)_{-0.0058}^{+0.0055}$
$\sigma_{8}$ | $0.806(0.811)_{-0.014}^{+0.012}$ | $0.790(0.763)_{-0.010}^{+0.027}$
$S_{8}$ | $0.818(0.823)_{-0.012}^{+0.016}$ | $0.798(0.775)_{-0.012}^{+0.025}$
$\chi^{2}_{\rm min}$ | 3927.0 | 3929.3
$\footnotesize{\chi^{2}_{\rm min}(\text{DCDM})-\chi^{2}_{\rm min}(\Lambda\text{CDM})}$ | 0.0 | $-3.8$
$Q_{\rm DMAP}\equiv\sqrt{\chi^{2}_{\rm min}({\rm w/}~{}S_{8})-\chi^{2}_{\rm min}({\rm w/o}~{}S_{8})}$ | 1.5$\sigma$
Table 6: The mean (best-fit) $\pm 1\sigma$ errors of the cosmological parameters from our ‘Planck \+ Pantheon + EFTofBOSS + Ext-BAO’ and ‘Planck \+ Pantheon + EFTofBOSS + Ext-BAO + $S_{8}$’ analyses for the DCDM $\to$ WDM+DR model. For each data set we also report its best-fit $\chi^{2}$, and the $\Delta\chi^{2}$ with respect to the analogous $\Lambda$CDM best-fit model. |  $\Lambda$CDM | DCDM $\to$ DR | DCDM $\to$ WDM+DR
---|---|---|---
| w/ EFTofBOSS | w/ EFTofBOSS + $S_{8}$ | w/ EFTofBOSS | w/ EFTofBOSS + $S_{8}$ | w/ EFTofBOSS | w/ EFTofBOSS + $S_{8}$
$\chi^{2}_{\text{CMASS NGC}}$ | 40.3 | 39.2 | 40.4 | 39.2 | 40.2 | 40.8
$\chi^{2}_{\text{CMASS SGC}}$ | 44.0 | 44.3 | 44.0 | 44.3 | 44.1 | 43.8
$\chi^{2}_{\text{LOWZ NGC}}$ | 33.5 | 33.5 | 33.5 | 33.5 | 33.5 | 33.7
$\chi^{2}_{\text{EFTofBOSS}}$ | 117.8 | 117.0 | 117.9 | 117.0 | 117.8 | 118.3
Table 7: $\chi^{2}$ of each sky-cut of the EFTofBOSS data set for our ‘Planck
\+ Pantheon + EFTofBOSS + Ext-BAO’ and ‘Planck \+ Pantheon + EFTofBOSS + Ext-
BAO + $S_{8}$’ analyses for $\Lambda$CDM, DCDM $\to$ DR and DCDM $\to$ WDM+DR
models.

We now turn to the case of the DCDM $\to$ WDM+DR model, described by the
parameters $\Gamma=\tau^{-1}$, the decay rate of DCDM, and $\varepsilon$, the
fraction of DCDM rest mass energy converted into DR. Note that in this
section, we trade the density of DM today, $\omega_{\rm cdm}$, for the initial
density of DM (before decays occur) at $a\to 0$, $\omega_{\text{dcdm}}^{\rm
ini}$. For a stable particle, we simply have $\omega_{\text{dcdm}}^{\rm
ini}\equiv\omega_{\rm cdm}$ as defined previously. In the MCMC analyses, we
imposed logarithmic priors131313For discussions about the impact of prior
choices, see the appendix of Ref. Abellán  _et al._ (2021) on $\varepsilon$
and $\Gamma$, and a flat prior on $\omega_{\text{dcdm}}^{\rm ini}$:

$\displaystyle-4\leq$ $\displaystyle~{}\text{log}_{10}(\Gamma/[{\rm
Gyr}^{-1}])\leq 1,$ $\displaystyle-4\leq$
$\displaystyle~{}\text{log}_{10}(\varepsilon)\leq\text{log}_{10}(0.5),$
$\displaystyle 0\leq$ $\displaystyle~{}\omega_{\text{dcdm}}^{\rm ini}\leq 1.$

We present our results for the analyses with and without $S_{8}$ prior in Tab.
6, while the $\chi^{2}$ of the EFTofBOSS data of these analysis are reported
in Tab. 7. All relevant $\chi^{2}$ per experiment are given in App. D. In Fig.
8, we display the 1D and 2D posteriors of
$\left\\{\text{log}_{10}(\Gamma/[{\rm
Gyr}^{-1}]),\text{log}_{10}(\varepsilon),H_{0},S_{8},\Omega_{m}\right\\}$ for
the DCDM $\to$ WDM+DR model with and without the EFTofBOSS data set, always
including the $S_{8}$ prior. Posteriors without the $S_{8}$ prior are shown in
App. C.

####  IV.3.1 Estimating the tension with the $S_{8}$ measurement

Without the $S_{8}$ prior, the total $\chi^{2}$ does not show any improvement
(see Tab. 6) and the data do not favor the DCDM $\to$ WDM+DR model. In fact,
in the absence of the $S_{8}$ prior, it seems that one could derive apparently
strong constraints on these models141414In Ref. Abellán  _et al._ (2021), it
was shown through a mock data analysis that Planck data alone could not detect
the best-fit model required to explain the $S_{8}$ tension, artificially
leading to strong constraints on the DCDM model.. Yet, once the $S_{8}$
likelihood is included, we find $\Delta\chi^{2}=-3.8$ (for 2 extra degrees of
freedom) at virtually no cost in $\chi^{2}$ for other likelihoods (see App.
D): the inclusion of the $S_{8}$ prior help in opening up the degeneracy with
the DCDM parameters, without degrading the fit to the host of cosmological
data, as stressed in Refs. Abellan  _et al._ (2020); Abellán  _et al._ (2021).

Nevertheless, the DCDM model is not statistically favored over $\Lambda$CDM,
as the preference over $\Lambda$CDM is currently solely driven by the low
$S_{8}$ prior, for which we have used a value only in mild $\sim 2.4\sigma$
tension with the $\Lambda$CDM prediction151515Different $S_{8}$ priors would
lead to different preferences. The preference could also be made stronger at
fixed $\varepsilon$ (see Ref. Abellán  _et al._ (2021)).. We can estimate the
residual tension between data sets within the various models by computing the
‘difference in maximum a posterior’ ($Q_{\rm DMAP}$ statistics Raveri and Hu
(2019)) between the $\chi^{2}$ obtained with and without the $S_{8}$ prior.
The tension estimator gives $Q_{\rm DMAP}\equiv\sqrt{\chi^{2}_{\rm min}({\rm
w/}~{}S_{8})-\chi^{2}_{\rm min}({\rm w/o}~{}S_{8})}=1.5\sigma$ in the
DCDM$\to$WDM+DR model, as compared to $2.4\sigma$ in the $\Lambda$CDM and
DCDM$\to$DR models.

####  IV.3.2 Impact of EFTofBOSS data

Comparing to results without the EFTofBOSS data, for which we get161616This
number is different to that quoted in Refs. Abellan  _et al._ (2020); Abellán
_et al._ (2021) because we recall that we make use of a different $S_{8}$
prior from KiDS-1000 alone, that does not include information from BOSS data
and therefore has larger error bars. $\Delta\chi^{2}=-4.4$, we find that the
$\Delta\chi^{2}$ is only mildly degraded by the inclusion of EFTofBOSS data.
More precisely, the $\chi^{2}$ of the total EFTofBOSS data for the DCDM $\to$
WDM+DR model, given in Tab. 7, is only slightly larger than that for
$\Lambda$CDM ($\Delta\chi^{2}=1.3$) despite a much lower $S_{8}\simeq 0.775$
(at the best-fit) which yields a very good fit of the KiDS-1000 prior.
Comparing to the analysis with the BAO/$f\sigma_{8}$ measurement from BOSS-
DR12 (also presented in App. D), we note that these ‘compressed’ data already
showed a minor degradation of $\chi^{2}$ compared to $\Lambda$CDM
($\Delta\chi^{2}=1.1$). We conclude that BOSS-DR12 data are in good agreement
with the DCDM $\to$ WDM+DR model, but have a non-negligible impact, as the
‘naive’ analysis presented in sec. III.2.3 suggested.

More precisely, one can see in Fig. 8 that the main impact of EFTofBOSS data
is to cut in the $\log_{10}(\Gamma/{\rm Gyr}^{-1})-\log_{10}(\varepsilon)$
degeneracy, excluding too large values of $\log_{10}(\Gamma/{\rm Gyr}^{-1})$.
In App. E we show that including the EFTofBOSS data does not shift the
$\Lambda$CDM parameters. Therefore, the EFTofLSS significantly improves the
constraints on the $\tau=\Gamma^{-1}$ parameter at $1\sigma$:

$\displaystyle 1.61<\rm{log}_{10}(\tau/{\rm Gyr})<3.71~{}~{}~{}~{}(w/{\rm
EFTofBOSS})\,,$

to be compared with

$\displaystyle 1.31<\rm{log}_{10}(\tau/{\rm Gyr})<3.82~{}~{}~{}(w/o~{}{\rm
EFTofBOSS})\,.$

Additionally, we observe a notable evolution of the DCDM parameters of the
best-fit model compared to the analysis without EFTofBOSS (and with the
BAO/$f\sigma_{8}$ measurement from BOSS-DR12 instead): the best-fit model,
with the inclusion of the $S_{8}$ likelihood, now has $\Gamma=0.0083\
\rm{Gyr}^{-1}$ ($\tau=120$ Gyr) and $\varepsilon=0.012$, while previously
$\Gamma=0.023\ \rm{Gyr}^{-1}$ ($\tau=43$ Gyr) and $\varepsilon=0.006$. This
means that EFTofBOSS data favors longer lived DM models and therefore a
smaller fraction of WDM today $f_{\rm wdm}\equiv\bar{\rho}_{\rm
wdm}/(\bar{\rho}_{\rm dcdm}+\bar{\rho}_{\rm wdm})\simeq 10\%$ compared to
$f_{\rm wdm}\simeq 27\%$ previously, but a significantly larger kick velocity
$v_{\rm kick}/c\simeq\varepsilon$ (and therefore a larger free-streaming
scale).

It is instructive to compare these numbers with recent constraints derived
from observations of Milky Way satellites by the DES collaboration Mau  _et
al._ (2022). These constraints exclude ${\rm log}_{10}(\Gamma/{\rm
Gyr}^{-1})\gtrsim-1.5$ for ${\rm log}_{10}(v_{\rm kick}/c)\simeq{\rm
log}_{10}(\varepsilon)\gtrsim-4$. The best-fit model of our EFTofBOSS
analysis, and a large fraction of the 68% C.L., lie well within the allowed
region, but these observations certainly provide a crucial test of the DCDM
cosmology, as a deficit of satellites compared to $\Lambda$CDM is expected in
this model.

Figure 9: Residuals of the monopole and the quadrupole of the DCDM $\to$
WDM+DR model for EFTofBOSS data and EFTofBOSS data + $S_{8}$ prior. We
normalized these residuals as well as the data with the $\Lambda$CDM best-fit
(EFTofBOSS data). Figure 10: Residuals of linear (dashed lines) and non-linear
(solid line) matter power spectrum of the DCDM $\to$ WDM+DR model (EFTofBOSS
data + $S_{8}$ prior) for $z=$ 0, 0.2 (Euclid minimal redshift), 0.32
(effective redshift of the LOWZ sky-cut) and 0.57 (effective redshift of the
HIGHZ sky-cut). We normalized these residuals with the $\Lambda$CDM best-fit
(EFTofBOSS data).

####  IV.3.3 Towards high-accuracy measurements of the galaxy power spectrum

To gauge the importance of future surveys in constraining the DCDM $\to$
WDM+DR model, we show in Fig. 9 the residuals of the monopoles and quadrupoles
of the galaxy power spectrum between the DCDM $\to$ WDM+DR and $\Lambda$CDM
models. One can see that there are sub-percent differences between the models
that gives us hope to probe the DCDM model further. Indeed, future galaxy
clustering power spectrum data with higher precision and measurements at
additional redshift bins such as Euclid Amendola  _et al._ (2018), VRO Alonso
_et al._ (2018) and DESI Aghamousa, Amir and others (2016) have an expected
sensitivity that should allow us to detect these mild differences. In order to
estimate the impact of future observations on the preference of the DCDM $\to$
WDM+DR model with respect to the $\Lambda$CDM model, we plot in Fig. 10 the
residuals of the non-linear matter power spectrum171717We set here
$c_{s}=1\cdot({\rm Mpc}/h)^{2}$, which is an effective parameter of the one-
loop correction that can be interpreted as the effective sound speed of the
dark matter. between the best-fit of the DCDM $\to$ WDM+DR model (for the
‘EFTofBOSS + $S_{8}$’ analysis) and $\Lambda$CDM model (for the ‘EFTofBOSS’
analysis). We represent it for different redshifts, starting at the minimal
redshift probed by an experiment like Euclid Amendola  _et al._ (2018). Note
that at the level of the non-linear matter power spectrum, the suppression
with respect to the $\Lambda$CDM model at $z=0.32$ and $z=0.57$ corresponding
to current observations are more than one order of magnitude stronger than
what is seen in the residual of the monopole and quadrupole of the galaxy
power spectrum (see Fig. 9) due to the impact of the EFT parameters. This
shows that current theoretical uncertainties limit the ability to use galaxy
surveys to probe the DCDM model, and represent a major challenge for future
surveys. Additionally, we observe that as $z$ decreases, the deviation from
$\Lambda$CDM increases significantly because of the production of WDM through
the decay. We keep for future work to check through dedicated forecasts
whether accumulation of low redshift data, as well as the reduction of error
bars, will allow us to firmly detect or exclude the DCDM $\to$ WDM+DR model
that resolves the $S_{8}$ tension.

##  V Conclusions

In this paper, we have confronted two models of DCDM with BOSS DR12 galaxy
power spectrum data Alam  _et al._ (2017) as described by the EFTofLSS from
Refs. Senatore (2015); Mirbabayi  _et al._ (2015); Angulo  _et al._ (2015);
Fujita  _et al._ (2020); Perko  _et al._ (2016); Nadler  _et al._ (2018);
D’Amico  _et al._ (2021). We focused first on a model where a fraction of dark
matter decays into dark radiation, the DCDM $\to$ DR model, and second on a
model where all the dark matter decays into warm massive particles and dark
radiation particles, the DCDM $\to$ WDM+DR model. The latter model was
recently suggested as a possible resolution to the $S_{8}$ tension, the
mismatch between the determination of the $S_{8}$ parameter from Planck CMB
power spectrum Aghanim  _et al._ (2020a) and from weak lensing surveys by KiDS
Heymans  _et al._ (2020); Asgari  _et al._ (2021), CFHTLenS Heymans  _et al._
(2013) and DES Abbott  _et al._ (2022). We presented in Sec. III the first
calculation of the non-linear (matter and galaxy) power spectra in DCDM models
making use of recent progresses in the EFTofLSS. We then confronted in Sec. IV
these two models to a compilation of Planck TTTEEE and lensing power spectra,
BAO data from BOSS and eBOSS (including Ly-$\alpha$ data), uncalibrated
luminosity distance to SN1a from the Pantheon catalogue Scolnic  _et al._
(2018), as well as measurements of the monopole and quadrupole of the galaxy
power spectrum for three different sky-cut of BOSS-DR12 (see Ref. Zhang  _et
al._ (2021)), namely LOWZ NGC, CMASS NGC and CMASS SGC Alam  _et al._ (2017).
We compared the use of either the BAO/$f\sigma_{8}$ from that same release, or
the full shape of the galaxy power spectrum. Additionally, we tested the
ability of these models to resolve the $S_{8}$ tension by performing analyses
with and without prior on $S_{8}$ as measured by KiDS Asgari  _et al._ (2021).
Our results can be summarized as follows:

  1. •

We have derived the most up-to-date bound on the fraction of decaying dark
matter $f_{\rm dcdm}$, which is now $f_{\rm dcdm}<0.0216$ for short-lived
DCDM. We have also updated constraints on the lifetime of dark matter for the
case where $f_{\rm dcdm}\to 1$, namely $\tau/f_{\rm dcdm}>249.6$ Gyr. However,
we have found that the EFTofLSS does not provide significantly better
constraints to the cosmological parameters for the DCDM $\to$ DR model,
compared to the use of the standard BAO/$f\sigma_{8}$ data. In agreement with
past studies, we have found that these models do not help neither for the
$S_{8}$ nor for the $H_{0}$ tension, and the inclusion of EFTofBOSS data
doesn’t alter that conclusion.

  2. •

The DCDM $\to$ WDM+DR model can explain the low $S_{8}$ value measured by
KiDS-1000 while preserving the goodness of fit to other data set, including
EFTofBOSS data. The residual tension is $1.5\sigma$ compared to $2.4\sigma$ in
the $\Lambda$CDM model. Nevertheless, the model is not statistically favored
over $\Lambda$CDM ($\Delta\chi^{2}=-3.8$ for 2 degrees of freedom, roughly
corresponding to $1.5\sigma$). The inclusion of EFTofBOSS data only marginally
affects the preference.

  3. •

EFTofBOSS data however do significantly improve the 1-$\sigma$ constraint on
the DCDM lifetime for the DCDM $\to$ WDM+DR model, and when combined with the
$S_{8}$ prior, we now obtain $\rm{log}_{10}(\tau/\rm{Gyr})=2.21_{-0.6}^{+1.5}$
compared to $\rm{log}_{10}(\tau/\rm{Gyr})=1.92_{-0.61}^{+1.9}$ without the
EFTofBOSS. The constraints on $\rm{log}_{10}(\varepsilon)$ are however
slightly weaker than with BAO/$f\sigma_{8}$ measurements.

  4. •

The EFTofBOSS data also affects the best-fit model which, with the inclusion
of the $S_{8}$ likelihood, corresponds to a longer lived DM with $\tau=120$
Gyr (compared to $\tau=43$ Gyr previously) and a larger kick velocity $v_{\rm
kick}/c\simeq\varepsilon=1.2\%$ (compared to $v_{\rm kick}/c\simeq 0.6\%$
previously).

Looking forward, we expect future galaxy clustering power spectrum data, with
higher precision and measurements at additional redshift bins such as Euclid
Amendola  _et al._ (2018), VRO Alonso  _et al._ (2018) and DESI Aghamousa,
Amir and others (2016), to provide us with exquisite sensitivity to DM decays
into an invisible sector whether massive or massless. Moreover, as the error
bars decrease, it will likely be necessary to identify and account for the
corrections to be made to the EFTofLSS in order to capture all the specific
effects of the DCDM $\to$ WDM+DR model. We keep for future work to test
whether these surveys will be able to firmly detect or exclude the DCDM $\to$
WDM+DR model that resolves the $S_{8}$ tension.

###### Acknowledgements.

The authors are thankful to Pierre Zhang for his precious help with the PyBird
code. His explanations and advices were essential to the success of this
project. We also warmly thank Zakaria Chacko and Abhish Dev for their
contribution in the early stages of this project. This work has been partly
supported by the CNRS-IN2P3 grant Dark21. The authors acknowledge the use of
computational resources from the Excellence Initiative of Aix-Marseille
University (A*MIDEX) of the “Investissements d’Avenir” programme. This project
has received support from the European Union’s Horizon 2020 research and
innovation program under the Marie Skodowska-Curie grant agreement No
860881-HIDDeN. P.D. is supported in part by Simons Investigator in Physics
Award 623940 and NSF award PHY-1915093. Y.T. is supported by the NSF grant
PHY-2014165 and PHY-2112540.

##  Appendix A Comparison between the EFTofLSS and N-body methods for the DCDM
$\to$ DR model

Figure 11: Comparison between the residuals of the non-linear matter power
spectra predicted by the N-body simulation and the residuals of the linear
matter power spectra predicted by the CLASS code on the one hand (left panel),
and the residuals of the non-linear matter power spectra predicted by the
CLASS-PT and PyBird codes on the other hand (right panel). We compute these
power spectra for $z=0$ and for $\tau=32$ Gyr, while we varied $f_{\rm dcdm}$
from 0 to 1 with a step of 0.2.

Figure 12: Ratio between residuals of the non-linear matter power spectra
obtained from the N-body simulation and those obtained from the CLASS-PT and
the PyBird codes for $z=0$, $\tau=32$ Gyr and $f_{\rm dcdm}$ varying from 0 to
1 with a step of 0.2.

In this appendix, we compare the non-linear matter power spectrum obtained
through the EFTofLSS method with the results of dedicated N-body simulations
performed in Ref. Hubert  _et al._ (2021). Authors of Ref. Hubert  _et al._
(2021) have determined a fitting formula which describe the correction to the
non-linear matter power spectrum due to the DM decay compared to the
$\Lambda$CDM model, as a function of $\tau$, $f_{\rm dcdm}$, and the redshift
$z$. In Fig. 12, we compare this fitting formula, where we set $\tau=32$ Gyr
and $z=0$ and vary $f_{\rm dcdm}\in[0,1]$, with the linear matter power
spectrum of the CLASS code (left panel), and with the non-linear matter power
spectra from both the CLASS-PT and PyBird codes (right panel). Here, we set
the $\Lambda$CDM parameters to the values used in Ref. Hubert  _et al._
(2021). The left panel of this figure is intended as a reproduction of Fig. 1
of this reference for direct comparison, while the right panel presents the
comparison of interest. Indeed, from the right panel of Fig. 12, one can
clearly see that (i) the CLASS-PT and PyBird codes give very similar power
spectra for the DCDM $\to$ DR model181818We set $c_{s}=1\cdot({\rm
Mpc}/h)^{2}$, which is an effective parameter of the one-loop correction that
can be interpreted as the effective sound speed of the dark matter., (ii) the
deviation from $\Lambda$CDM predicted in these two EFTofLSS codes is very
close to that obtained through N-body simulation. In order to determine more
precisely the deviations between the EFTofLSS and the N-body methods, we plot,
in Fig. 12, the ratio between the residuals obtained with the N-Body
simulation and those obtained with the CLASS-PT and PyBird codes. One can see
that the difference is below the $\sim 1\%$ level until $k\sim
0.2h/\text{Mpc}$ (the maximum $k$ at which the EFTofLSS is valid at one-loop
order for a small $z$). Let us note that the difference between the N-body
simulation and the EFTofLSS power spectrum for $k\lesssim 0.02h/\text{Mpc}$ is
not relevant; it is merely due to the fact that the N-body fitting formula
does not encode this behavior for low $k$ (see Hubert  _et al._ (2021)), but
this $k$-range is well within the linear regime and does not necessitate a
correction. Let us also remark that the lower $f_{\rm dcdm}$ (or the longer
$\tau$), the smaller the difference between the residuals from the N-body and
those from the EFTofLSS method. Since current constraints only allow small
values of $f_{\rm dcdm}\lesssim 2.5\%$ or large lifetime $\tau\gtrsim 240$
Gyr, it is safe to use the PyBird (or CLASS-PT) code in their current form to
describe the DCDM$\to$DR model.

##  Appendix B Assessing the validity of the EFTofLSS in the DCDM $\to$ WDM+DR
model

Figure 13: Values of the WDM fraction at $z=0.32$ (left pannel) and the
redshift at which the WDM fraction becomes 1% (right pannel), in a region of
the $\rm{log}_{10}(\varepsilon)$-$\rm{log}_{10}(\Gamma/\rm{Gyr}^{-1})$ plane.
The $\Lambda$CDM parameters are fixed to the best-fit from the ‘Planck \+
Pantheon + EFTofBOSS + Ext-BAO + $S_{8}$’ analysis. The black lines indicate
the $1\sigma$ limits of this analysis, while the point highlighted in purple
indicates the best-fit. All the points below the yellow and red lines
correspond to models having a WDM free-streaming wavenumber $k_{\rm fs}$
larger than $k_{\rm max}=0.2\,h/\textrm{Mpc}$ and $k_{\rm
NL}=0.7\,h/\textrm{Mpc}$, respectively.

In this appendix, we discuss the validity of the EFTofLSS in the DCDM $\to$
WDM+DR model. In Ref. Senatore and Zaldarriaga (2017), the EFTofLSS was
extended to describe massive neutrinos, an extension to $\Lambda$CDM with
properties similar to that of the DCDM$\to$WDM+DR model. Indeed, the main
issue with employing the EFTofLSS to describe the DCDM$\to$WDM+DR model does
not lie in the effect of the decay itself (the effect of the decay on the
perturbed equations of the DCDM is identical to that of the DCDM $\to$ DR
model, which is captured by our formalism as discussed in App. A), but rather
in the production of a warm massive species which may contribute in a non-
trivial way to the power spectrum of galaxies. At the linear level, it was
found the massive decay products behave similarly to CDM at wavenumbers
smaller than the free-streaming scale $k_{\rm fs}$ (with $k_{\rm fs}$
approximately given by Eq. 25), but is strongly suppressed due to pressure
terms at larger wavenumbers similarly to WDM and hot DM such as neutrinos. In
Ref. Senatore and Zaldarriaga (2017), the contribution of neutrinos to the
total one loop power spectrum was computed, and it was found that the dominant
effect is a correction to the dark matter power spectrum that scales like
$16f_{\nu}$, where
$f_{\nu}\equiv\bar{\rho}_{\nu}/(\bar{\rho}_{\nu}+\bar{\rho}_{\rm cdm})\sim
1\%$, at $k>k_{\rm fs}$ and roughly half of that at $k<k_{\rm fs}$. The naive
${\cal O}(f_{\nu})$ contribution is enhanced by twice the logarithm of the
redshift of matter-radiation equality, as neutrinos are present from early
times. The log-enhanced contribution represents about $70\%$ of the
contribution to the total one loop power spectrum. Additionally, at leading
order, counter-terms can be captured by simply re-scaling the effective DM
sound speed $c_{s}^{2}$ and do not necessitate adding new parameters to the
dark-matter-only calculation. In the case of the DCDM model, the WDM is
produced at much later times. We plot in Fig. 13 (right panel) the redshift
$z_{1\%}$ at which the WDM contribution $f_{\rm wdm}\equiv\bar{\rho}_{\rm
wdm}/(\bar{\rho}_{\rm wdm}+\bar{\rho}_{\rm dcdm})$ reaches $\sim 1\%$. We also
represent the limit at 68% C.L. derived in our work. For the best-fit model
(shown in purple in the figure), $z_{1\%}\sim 5$. The log-enhancement from the
ratio of scale factor between $z\sim 5$ and $z_{eff,{\rm LOWZ}}=0.32$ is of
$\log[(1+z_{1\%})/(1+z_{0})]\approx 1-2$ compared to the $\log[(1+z_{\rm
eq})/(1+z_{0})]\approx 8$ in the neutrino study in Senatore and Zaldarriaga
(2017) that gives the $16f_{\nu}$ factor. We therefore expect the WDM
correction to only be comparable to the massive neutrino case even if the
energy density (today) ratio is $\approx 10$ times larger than neutrinos.

We plot in Fig. 13 left panel, the fractional contribution of WDM at $z=0.32$
(the effective redshift of the low-z surveys) as a function of $\Gamma$ and
$\varepsilon$. We also represent the limit at 68% C.L. derived in this work.
One can see that it is under $\sim 15\%$ as long as
$\textrm{log}_{10}(\Gamma/[{\rm Gyr}^{-1}])\lesssim-1.8$. Additionally, we
show the value of $\varepsilon-\Gamma$ for which the free-streaming scale
$k_{\rm fs}$ is equal to the maximum $k$ mode relevant for our analysis of
BOSS data ($k_{\rm max}=0.2\,h/$Mpc) and the maximum scale considered in the
EFT computation ($k_{\rm NL}=0.7\,h/$Mpc). In a large part of the parameter
space favored by our analysis for which $f_{\rm wdm}$ is not small, $k_{\rm
fs}$ exceeds $k_{\rm max}$ and therefore corrections should also be minor. An
improved EFT treatment including the effect of the massive decay product would
be necessary however to describe the power spectrum up to $k_{\rm NL}$. Given
current precision of the data and the large theoretical uncertainty already
present, the corrections to our calculation should be negligible, but more
work needs to be done to accurately describe the part of the parameter space
with large $\Gamma$ (and leading to large $f_{\rm wdm}$), in particular for
future surveys which can reach sub-percent precision at larger wave-numbers.

##  Appendix C The role of the $S_{8}$ prior

In this appendix, we present 2D posterior distributions obtained with and
without the $S_{8}$ prior (but with the EFTofBOSS data) in both DCDM
cosmologies. In the case of the DCDM $\to$ DR model, represented in Fig. 14,
the impact of the $S_{8}$ prior is minor. However, in the case of the DCDM
$\to$ WDM+DR model, represented in Fig. 15, it opens up a degeneracy with
$\\{\Gamma,\varepsilon\\}$ which can lead to low $S_{8}$ while preserving the
fit to other data sets. Without the $S_{8}$ prior, the DCDM model is not
favored by Planck data. As discussed in the main text, when the $S_{8}$ prior
is included, the fit to Planck data is not affected, while the DCDM model can
accommodate the lower $S_{8}$ value, contrarily to the $\Lambda$CDM model.
From the $Q_{\rm DMAP}$ statistics Raveri and Hu (2019), we can estimate the
residual tension as $Q_{\rm DMAP}\equiv\sqrt{\chi^{2}_{\rm min}({\rm
w/}~{}S_{8})-\chi^{2}_{\rm min}({\rm w/o}~{}S_{8})}=1.5\sigma$ in the
DCDM$\to$WDM+DR model, as compared to $2.4\sigma$ in the $\Lambda$CDM model
and DCDM$\to$DR model.

Figure 14: 2D posterior distributions of the DCDM $\to$ DR model reconstructed
from an analysis of Planck, Pantheon, Ext-BAO and EFTofBOSS data, with (blue)
and without (red) the $S_{8}$ prior from KiDS-1000. The gray shaded bands
refer to the joint $S_{8}$ measurement from KiDS-1000 + BOSS + 2dFLens.
Figure 15: 2D posterior distributions of the DCDM $\to$ WDM+DR model
reconstructed from an analysis of Planck, Pantheon, Ext-BAO and EFTofBOSS
data, with (blue) and without (red) the $S_{8}$ prior from KiDS-1000. The gray
shaded bands refer to the joint $S_{8}$ measurement from KiDS-1000 + BOSS +
2dFLens.

##  Appendix D Supplementary tables of $\chi^{2}_{\rm min}$ values per
experiment

In this appendix, we report the best-fit $\chi^{2}$ per experiment for both
$\Lambda$CDM (Tab. 8) and DCDM $\to$ WDM+DR (Tab. 9) models for our analyses
with the ‘BAO/$f\sigma_{8}$ \+ $S_{8}$’, ‘EFTofBOSS’ and ‘EFTofBOSS + $S_{8}$’
data.

$\Lambda$CDM
---
Data set | w/ BAO/$f\sigma_{8}$ \+ $S_{8}$ | w/ EFTofBOSS | w/ EFTofBOSS + $S_{8}$
Planck high-$l$ TTTEEE | 2349.0 | 2347.4 | 2349.3
Planck low-$l$ EE | 396.1 | 396.7 | 396.1
Planck low-$l$ TT | 22.8 | 23.0 | 22.7
Planck lensing | 9.6 | 8.9 | 9.6
Pantheon | 1027.0 | 1027.1 | 1027.0
Ext-BAO | 6.3 | 6.2 | 6.3
BOSS BAO/$f\sigma_{8}$ | 6.0 | $-$ | $-$
EFTofBOSS | $-$ | 117.8 | 117.0
$S_{8}$ | 5.3 | $-$ | 5.0
total $\chi^{2}_{\rm min}$ | 3821.9 | 3927.0 | 3933.0
Table 8: $\chi^{2}$ of each data set for our ‘Planck \+ Pantheon + BOSS
BAO/$f\sigma_{8}$ \+ Ext-BAO + $S_{8}$’, ‘Planck \+ Pantheon + EFTofBOSS +
Ext-BAO’ and ‘Planck \+ Pantheon + EFTofBOSS + Ext-BAO + $S_{8}$’ analyses for
the $\Lambda$CDM model. Since we rounded the $\chi^{2}$ of each experiment,
the total $\chi^{2}$ is only equal to the sum of each $\chi^{2}$ at ${\cal
O}(0.1)$ precision. DCDM$\to$WDM+DR
---
Data set | w/ BAO/$f\sigma_{8}$ \+ $S_{8}$ | w/ EFTofBOSS | w/ EFTofBOSS + $S_{8}$
Planck high-$l$ TTTEEE | 2347.7 | 2347.3 | 2348.0
Planck low-$l$ EE | 397.2 | 396.9 | 397.0
Planck low-$l$ TT | 23.1 | 23.0 | 23.2
Planck lensing | 8.9 | 8.9 | 9.1
Pantheon | 1027.2 | 1027.1 | 1027.2
Ext-BAO | 6.1 | 6.2 | 6.2
BOSS BAO/$f\sigma_{8}$ | 7.1 | $-$ | $-$
EFTofBOSS | $-$ | 117.8 | 118.3
$S_{8}$ | 0.2 | $-$ | 0.4
total $\chi^{2}_{\rm min}$ | 3817.5 | 3927.0 | 3929.2
Table 9: $\chi^{2}$ of each data set for our ‘Planck \+ Pantheon + BOSS
BAO/$f\sigma_{8}$ \+ Ext-BAO + $S_{8}$’, ‘Planck \+ Pantheon + EFTofBOSS +
Ext-BAO’ and ‘Planck \+ Pantheon + EFTofBOSS + Ext-BAO + $S_{8}$’ analyses for
the DCDM $\to$ WDM+DR model. Since we rounded the $\chi^{2}$ of each
experiment, the total $\chi^{2}$ is only equal to the sum of each $\chi^{2}$
at ${\cal O}(0.1)$ precision.

##  Appendix E $\Lambda$CDM parameters of the DCDM $\to$ WDM+DR model

In this appendix, we compare in Fig. 16 the $\Lambda$CDM parameters of the
DCDM $\to$ WDM+DR model obtained from the analyses with (blue) and without
(red) the EFTofBOSS data, while in Fig. 17 we represent the $\Lambda$CDM
parameters reconstructed from an analysis of Planck, Pantheon, Ext-BAO and
EFTofBOSS data, with (blue) and without (red) the $S_{8}$ prior. We also show
in this second figure the standard $\Lambda$CDM posteriors as a reference
(green). One can see that the $\Lambda$CDM parameters are left largely
unchanged in the DCDM $\to$ WDM+DR model. The decay into warm products only
affect the growth of structure at late-times with little impact on parameters
that could affect early time physics. This is essentially why cosmological
data other than those measuring $S_{8}$ (and potentially the growth of
structure at late-times) are unaffected by the DCDM, despite a lower $S_{8}$
value.

Figure 16: 2D posterior distributions of the DCDM $\to$ WDM+DR model with and
without the EFTofBOSS data set for the $\Lambda$CDM parameters. We took into
account the $S_{8}$ prior from KIDS-1000 for these two MCMC analyses. The gray
shaded bands refer to the joint $S_{8}$ measurement from KIDS-1000 + BOSS +
2dFLens.  Figure 17: 2D posterior distributions of the DCDM $\to$ WDM+DR model
with and without the $S_{8}$ prior from KiDS-1000. We also plot the 2D
posterior distribution for the $\Lambda$CDM model without this $S_{8}$ prior.
We took into account the EFTofBOSS data for these three MCMC analyses. The
gray shaded bands refer to the joint $S_{8}$ measurement from KiDS-1000 + BOSS
+ 2dFLens.

## References

  * Verde  _et al._ (2019) L. Verde, T. Treu, and A. G. Riess, “Tensions between the early and the late universe,” in  _Nature Astronomy 2019_ (2019) 1907.10625 .
  * Riess  _et al._ (2021) Adam G. Riess _et al._ , “A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team,” (2021), arXiv:2112.04510 [astro-ph.CO] .
  * Aghanim  _et al._ (2020a) N. Aghanim _et al._ (Planck), “Planck 2018 results,” Astron. Astrophys. 641, A6 (2020a), 1807.06209 .
  * Leauthaud  _et al._ (2017) Alexie Leauthaud _et al._ , “Lensing is Low: Cosmology, Galaxy Formation, or New Physics?” Mon. Not. Roy. Astron. Soc. 467, 3024–3047 (2017), arXiv:1611.08606 [astro-ph.CO] .
  * Lange  _et al._ (2021) Johannes U. Lange, Alexie Leauthaud, Sukhdeep Singh, Hong Guo, Rongpu Zhou, Tristan L. Smith, and Francis-Yan Cyr-Racine, “On the halo-mass and radial scale dependence of the lensing is low effect,” Mon. Not. Roy. Astron. Soc. 502, 2074–2086 (2021), arXiv:2011.02377 [astro-ph.CO] .
  * Amon  _et al._ (2022) A. Amon _et al._ , “Consistent lensing and clustering in a low-$S_{8}$ Universe with BOSS, DES Year 3, HSC Year 1 and KiDS-1000,” (2022), arXiv:2202.07440 [astro-ph.CO] .
  * Asgari  _et al._ (2021) Marika Asgari _et al._ (KiDS), “KiDS-1000 Cosmology: Cosmic shear constraints and comparison between two point statistics,” Astron. Astrophys. 645, A104 (2021), arXiv:2007.15633 [astro-ph.CO] .
  * Abbott  _et al._ (2022) T. M. C. Abbott _et al._ (DES), “Dark Energy Survey Year 3 results: Cosmological constraints from galaxy clustering and weak lensing,” Phys. Rev. D 105, 023520 (2022), arXiv:2105.13549 [astro-ph.CO] .
  * Hikage  _et al._ (2019) Chiaki Hikage _et al._ (HSC), “Cosmology from cosmic shear power spectra with Subaru Hyper Suprime-Cam first-year data,” Publ. Astron. Soc. Jap. 71, 43 (2019), arXiv:1809.09148 [astro-ph.CO] .
  * Aiola  _et al._ (2020) Simone Aiola _et al._ (ACT), “The Atacama Cosmology Telescope: DR4 Maps and Cosmological Parameters,” JCAP 12, 047 (2020), arXiv:2007.07288 [astro-ph.CO] .
  * Freedman (2021) Wendy L. Freedman, “Measurements of the Hubble Constant: Tensions in Perspective,” Astrophys. J. 919, 16 (2021), arXiv:2106.15656 [astro-ph.CO] .
  * Bernal  _et al._ (2016) Jose Luis Bernal, Licia Verde, and Adam G. Riess, “The trouble with H0,” JCAP 10, 019 (2016), 1607.05617 .
  * Aylor  _et al._ (2019) Kevin Aylor, MacKenzie Joy, Lloyd Knox, Marius Millea, Srinivasan Raghunathan, and W.L. Kimmy Wu, “Sounds Discordant: Classical Distance Ladder & $\Lambda$CDM -based Determinations of the Cosmological Sound Horizon,” Astrophys. J. 874, 4 (2019), 1811.00537 .
  * Knox and Millea (2019) Lloyd Knox and Marius Millea, “Hubble constant hunter’s guide,” (2019), 1908.03663 .
  * Camarena, David and Marra, Valerio (2021) Camarena, David and Marra, Valerio, “On the use of the local prior on the absolute magnitude of type ia supernovae in cosmological inference,”  (2021), 10.1093/mnras/stab1200, 2101.08641 .
  * Efstathiou (2021) George Efstathiou, “To H0 or not to H0?” Mon. Not. Roy. Astron. Soc. 505, 3866–3872 (2021), arXiv:2103.08723 [astro-ph.CO] .
  * Schöneberg  _et al._ (2021) Nils Schöneberg, Guillermo Franco Abellán, Andrea Pérez Sánchez, Samuel J. Witte, Vivian Poulin, and Julien Lesgourgues, “The $H_{0}$ Olympics: A fair ranking of proposed models,” (2021), arXiv:2107.10291 [astro-ph.CO] .
  * Kreisch  _et al._ (2020) Christina D. Kreisch, Francis-Yan Cyr-Racine, and Olivier Doré, “Neutrino puzzle: Anomalies, interactions, and cosmological tensions,” Phys. Rev. D 101, 123505 (2020), arXiv:1902.00534 [astro-ph.CO] .
  * Berbig  _et al._ (2020) Maximilian Berbig, Sudip Jana, and Andreas Trautner, “The Hubble tension and a renormalizable model of gauged neutrino self-interactions,” Phys. Rev. D 102, 115008 (2020), arXiv:2004.13039 [hep-ph] .
  * Ghosh  _et al._ (2020) Subhajit Ghosh, Rishi Khatri, and Tuhin S. Roy, “Can dark neutrino interactions phase out the Hubble tension?” Phys. Rev. D 102, 123544 (2020), arXiv:1908.09843 [hep-ph] .
  * Forastieri  _et al._ (2019) Francesco Forastieri, Massimiliano Lattanzi, and Paolo Natoli, “Cosmological constraints on neutrino self-interactions with a light mediator,” Phys. Rev. D 100, 103526 (2019), arXiv:1904.07810 [astro-ph.CO] .
  * Escudero and Witte (2020) Miguel Escudero and Samuel J. Witte, “A CMB search for the neutrino mass mechanism and its relation to the Hubble tension,” Eur. Phys. J. C 80, 294 (2020), arXiv:1909.04044 [astro-ph.CO] .
  * Escudero and Witte (2021) Miguel Escudero and Samuel J. Witte, “The hubble tension as a hint of leptogenesis and neutrino mass generation,” Eur. Phys. J. C 81, 515 (2021), arXiv:2103.03249 [hep-ph] .
  * Blinov and Marques-Tavares (2020) Nikita Blinov and Gustavo Marques-Tavares, “Interacting radiation after Planck and its implications for the Hubble Tension,” JCAP 09, 029 (2020), arXiv:2003.08387 [astro-ph.CO] .
  * Ghosh  _et al._ (2021) Subhajit Ghosh, Soubhik Kumar, and Yuhsin Tsai, “Free-streaming and Coupled Dark Radiation Isocurvature Perturbations: Constraints and Application to the Hubble Tension,” (2021), arXiv:2107.09076 [astro-ph.CO] .
  * Archidiacono and Gariazzo (2022) Maria Archidiacono and Stefano Gariazzo, “Two sides of the same coin: sterile neutrinos and dark radiation. Status and perspectives,” (2022), arXiv:2201.10319 [hep-ph] .
  * Aloni  _et al._ (2021) Daniel Aloni, Asher Berlin, Melissa Joseph, Martin Schmaltz, and Neal Weiner, “A Step in Understanding the Hubble Tension,” (2021), arXiv:2111.00014 [astro-ph.CO] .
  * Karwal and Kamionkowski (2016) Tanvi Karwal and Marc Kamionkowski, “Dark energy at early times, the Hubble parameter, and the string axiverse,” Phys. Rev. D 94, 103523 (2016), arXiv:1608.01309 [astro-ph.CO] .
  * Poulin  _et al._ (2019) Vivian Poulin, Tristan L. Smith, Tanvi Karwal, and Marc Kamionkowski, “Early dark energy can resolve the hubble tension,” Phys. Rev. Lett. 122, 221301 (2019), 1811.04083 .
  * Smith  _et al._ (2020) Tristan L. Smith, Vivian Poulin, and Mustafa A. Amin, “Oscillating scalar fields and the Hubble tension: a resolution with novel signatures,” Phys. Rev. D 101, 063523 (2020), arXiv:1908.06995 [astro-ph.CO] .
  * Niedermann and Sloth (2021) Florian Niedermann and Martin S. Sloth, “New early dark energy,” Phys. Rev. D 103, L041303 (2021), arXiv:1910.10739 [astro-ph.CO] .
  * Niedermann and Sloth (2020) Florian Niedermann and Martin S. Sloth, “Resolving the Hubble tension with new early dark energy,” Phys. Rev. D 102, 063527 (2020), arXiv:2006.06686 [astro-ph.CO] .
  * Ye and Piao (2020) Gen Ye and Yun-Song Piao, “Is the Hubble tension a hint of AdS phase around recombination?” Phys. Rev. D 101, 083507 (2020), arXiv:2001.02451 [astro-ph.CO] .
  * Renk  _et al._ (2017) Janina Renk, Miguel Zumalacárregui, Francesco Montanari, and Alexandre Barreira, “Galileon gravity in light of ISW, CMB, BAO and H0 data,” JCAP 10, 020 (2017), arXiv:1707.02263 [astro-ph.CO] .
  * Umiltà  _et al._ (2015) C. Umiltà, M. Ballardini, F. Finelli, and D. Paoletti, “CMB and BAO constraints for an induced gravity dark energy model with a quartic potential,” JCAP 08, 017 (2015), arXiv:1507.00718 [astro-ph.CO] .
  * Ballardini  _et al._ (2016) Mario Ballardini, Fabio Finelli, Caterina Umiltà, and Daniela Paoletti, “Cosmological constraints on induced gravity dark energy models,” JCAP 05, 067 (2016), arXiv:1601.03387 [astro-ph.CO] .
  * Rossi  _et al._ (2019) Massimo Rossi, Mario Ballardini, Matteo Braglia, Fabio Finelli, Daniela Paoletti, Alexei A. Starobinsky, and Caterina Umiltà, “Cosmological constraints on post-Newtonian parameters in effectively massless scalar-tensor theories of gravity,” Phys. Rev. D 100, 103524 (2019), arXiv:1906.10218 [astro-ph.CO] .
  * Braglia  _et al._ (2020a) Matteo Braglia, Mario Ballardini, William T. Emond, Fabio Finelli, A. Emir Gumrukcuoglu, Kazuya Koyama, and Daniela Paoletti, “Larger value for $H_{0}$ by an evolving gravitational constant,” Phys. Rev. D 102, 023529 (2020a), arXiv:2004.11161 [astro-ph.CO] .
  * Zumalacarregui (2020) Miguel Zumalacarregui, “Gravity in the Era of Equality: Towards solutions to the Hubble problem without fine-tuned initial conditions,” Phys. Rev. D 102, 023523 (2020), arXiv:2003.06396 [astro-ph.CO] .
  * Abadi and Kovetz (2021) Tal Abadi and Ely D. Kovetz, “Can conformally coupled modified gravity solve the Hubble tension?” Phys. Rev. D 103, 023530 (2021), arXiv:2011.13853 [astro-ph.CO] .
  * Ballardini  _et al._ (2020) Mario Ballardini, Matteo Braglia, Fabio Finelli, Daniela Paoletti, Alexei A. Starobinsky, and Caterina Umiltà, “Scalar-tensor theories of gravity, neutrino physics, and the $H_{0}$ tension,” JCAP 10, 044 (2020), arXiv:2004.14349 [astro-ph.CO] .
  * Braglia  _et al._ (2020b) Matteo Braglia, William T. Emond, Fabio Finelli, A. Emir Gumrukcuoglu, and Kazuya Koyama, “Unified framework for early dark energy from $\alpha$-attractors,” Phys. Rev. D 102, 083513 (2020b), arXiv:2005.14053 [astro-ph.CO] .
  * Di Valentino  _et al._ (2016) Eleonora Di Valentino, Alessandro Melchiorri, and Joseph Silk, “Cosmological hints of modified gravity?” Phys. Rev. D 93, 023513 (2016), arXiv:1509.07501 [astro-ph.CO] .
  * Bahamonde  _et al._ (2021) Sebastian Bahamonde, Konstantinos F. Dialektopoulos, Celia Escamilla-Rivera, Gabriel Farrugia, Viktor Gakis, Martin Hendry, Manuel Hohmann, Jackson Levi Said, Jurgen Mifsud, and Eleonora Di Valentino, “Teleparallel Gravity: From Theory to Cosmology,” (2021), arXiv:2106.13793 [gr-qc] .
  * Raveri (2020) Marco Raveri, “Reconstructing Gravity on Cosmological Scales,” Phys. Rev. D 101, 083524 (2020), arXiv:1902.01366 [astro-ph.CO] .
  * Yan  _et al._ (2020) Sheng-Feng Yan, Pierre Zhang, Jie-Wen Chen, Xin-Zhe Zhang, Yi-Fu Cai, and Emmanuel N. Saridakis, “Interpreting cosmological tensions from the effective field theory of torsional gravity,” Phys. Rev. D 101, 121301 (2020), arXiv:1909.06388 [astro-ph.CO] .
  * Frusciante  _et al._ (2020) Noemi Frusciante, Simone Peirone, Luis Atayde, and Antonio De Felice, “Phenomenology of the generalized cubic covariant Galileon model and cosmological bounds,” Phys. Rev. D 101, 064001 (2020), arXiv:1912.07586 [astro-ph.CO] .
  * Solà Peracaula  _et al._ (2019) Joan Solà Peracaula, Adria Gomez-Valent, Javier de Cruz Pérez, and Cristian Moreno-Pulido, “Brans–Dicke Gravity with a Cosmological Constant Smoothes Out $\Lambda$CDM Tensions,” Astrophys. J. Lett. 886, L6 (2019), arXiv:1909.02554 [astro-ph.CO] .
  * Solà Peracaula  _et al._ (2020) Joan Solà Peracaula, Adrià Gómez-Valent, Javier de Cruz Pérez, and Cristian Moreno-Pulido, “Brans–Dicke cosmology with a $\Lambda$-term: a possible solution to $\Lambda$CDM tensions,” Class. Quant. Grav. 37, 245003 (2020), arXiv:2006.04273 [astro-ph.CO] .
  * Ballesteros  _et al._ (2020) Guillermo Ballesteros, Alessio Notari, and Fabrizio Rompineve, “The $H_{0}$ tension: $\Delta G_{N}$ vs. $\Delta N_{\rm eff}$,” JCAP 11, 024 (2020), arXiv:2004.05049 [astro-ph.CO] .
  * Braglia  _et al._ (2021) Matteo Braglia, Mario Ballardini, Fabio Finelli, and Kazuya Koyama, “Early modified gravity in light of the $H_{0}$ tension and LSS data,” Phys. Rev. D 103, 043528 (2021), arXiv:2011.12934 [astro-ph.CO] .
  * Desmond  _et al._ (2019) Harry Desmond, Bhuvnesh Jain, and Jeremy Sakstein, “Local resolution of the Hubble tension: The impact of screened fifth forces on the cosmic distance ladder,” Phys. Rev. D 100, 043537 (2019), arXiv:1907.03778 [astro-ph.CO] .
  * Lin  _et al._ (2019) Meng-Xiang Lin, Marco Raveri, and Wayne Hu, “Phenomenology of Modified Gravity at Recombination,” Phys. Rev. D 99, 043514 (2019), arXiv:1810.02333 [astro-ph.CO] .
  * Chiang and Slosar (2018) Chi-Ting Chiang and Anže Slosar, “Inferences of $H_{0}$ in presence of a non-standard recombination,” (2018), arXiv:1811.03624 [astro-ph.CO] .
  * Hart and Chluba (2020) Luke Hart and Jens Chluba, “Updated fundamental constant constraints from Planck 2018 data and possible relations to the Hubble tension,” Mon. Not. Roy. Astron. Soc. 493, 3255–3263 (2020), arXiv:1912.03986 [astro-ph.CO] .
  * Sekiguchi and Takahashi (2021) Toyokazu Sekiguchi and Tomo Takahashi, “Early recombination as a solution to the $H_{0}$ tension,” Phys. Rev. D 103, 083507 (2021), arXiv:2007.03381 [astro-ph.CO] .
  * Jedamzik and Pogosian (2020) Karsten Jedamzik and Levon Pogosian, “Relieving the Hubble tension with primordial magnetic fields,” Phys. Rev. Lett. 125, 181302 (2020), arXiv:2004.09487 [astro-ph.CO] .
  * Cyr-Racine  _et al._ (2021) Francis-Yan Cyr-Racine, Fei Ge, and Lloyd Knox, “A Symmetry of Cosmological Observables, and a High Hubble Constant as an Indicator of a Mirror World Dark Sector,” (2021), arXiv:2107.13000 [astro-ph.CO] .
  * Di Valentino  _et al._ (2021a) Eleonora Di Valentino, Olga Mena, Supriya Pan, Luca Visinelli, Weiqiang Yang, Alessandro Melchiorri, David F. Mota, Adam G. Riess, and Joseph Silk, “In the realm of the Hubble tension—a review of solutions,” Class. Quant. Grav. 38, 153001 (2021a), arXiv:2103.01183 [astro-ph.CO] .
  * Lesgourgues  _et al._ (2016) Julien Lesgourgues, Gustavo Marques-Tavares, and Martin Schmaltz, “Evidence for dark matter interactions in cosmological precision data?” JCAP 02, 037 (2016), 1507.04351 .
  * Buen-Abad  _et al._ (2015) Manuel A. Buen-Abad, Gustavo Marques-Tavares, and Martin Schmaltz, “Non-Abelian dark matter and dark radiation,” Phys. Rev. D 92, 023531 (2015), arXiv:1505.03542 [hep-ph] .
  * Chacko  _et al._ (2016) Zackaria Chacko, Yanou Cui, Sungwoo Hong, Takemichi Okui, and Yuhsin Tsai, “Partially Acoustic Dark Matter, Interacting Dark Radiation, and Large Scale Structure,” JHEP 12, 108 (2016), arXiv:1609.03569 [astro-ph.CO] .
  * Buen-Abad  _et al._ (2018) Manuel A. Buen-Abad, Martin Schmaltz, Julien Lesgourgues, and Thejs Brinckmann, “Interacting dark sector and precision cosmology,” JCAP 01, 008 (2018), 1708.09406 .
  * Brinckmann and Lesgourgues (2019) Thejs Brinckmann and Julien Lesgourgues, “MontePython 3: boosted MCMC sampler and other features,” Phys. Dark Univ. 24, 100260 (2019), 1804.07261 .
  * Heimersheim  _et al._ (2020) Stefan Heimersheim, Nils Schöneberg, Deanna C. Hooper, and Julien Lesgourgues, “Cannibalism hinders growth: Cannibal Dark Matter and the $S_{8}$ tension,” (2020), 2008.08486 .
  * Gómez-Valent  _et al._ (2020) Adrià Gómez-Valent, Valeria Pettorino, and Luca Amendola, “Update on coupled dark energy and the $H_{0}$ tension,” Phys. Rev. D 101, 123513 (2020), arXiv:2004.00610 [astro-ph.CO] .
  * Di Valentino  _et al._ (2020) Eleonora Di Valentino, Alessandro Melchiorri, Olga Mena, and Sunny Vagnozzi, “Interacting dark energy in the early 2020s: A promising solution to the $H_{0}$ and cosmic shear tensions,” Phys. Dark Univ. 30, 100666 (2020), arXiv:1908.04281 [astro-ph.CO] .
  * Lucca (2021) Matteo Lucca, “Dark energy–dark matter interactions as a solution to the S8 tension,” Phys. Dark Univ. 34, 100899 (2021), arXiv:2105.09249 [astro-ph.CO] .
  * Abellan  _et al._ (2020) Guillermo F. Abellan, Riccardo Murgia, Vivian Poulin, and Julien Lavalle, “Hints for decaying dark matter from $S_{8}$ measurements,” (2020), 2008.09615 .
  * Di Valentino  _et al._ (2021b) Eleonora Di Valentino _et al._ , “Cosmology intertwined III: $f\sigma_{8}$ and $S_{8}$,” Astropart. Phys. 131, 102604 (2021b), arXiv:2008.11285 [astro-ph.CO] .
  * Bansal  _et al._ (2021) Saurabh Bansal, Jeong Han Kim, Christopher Kolda, Matthew Low, and Yuhsin Tsai, “Mirror Twin Higgs Cosmology: Constraints and a Possible Resolution to the $H_{0}$ and $S_{8}$ Tensions,” (2021), arXiv:2110.04317 [hep-ph] .
  * Enqvist  _et al._ (2015a) Kari Enqvist, Seshadri Nadathur, Toyokazu Sekiguchi, and Tomo Takahashi, “Decaying dark matter and the tension in $\sigma_{8}$,” JCAP 09, 067 (2015a), 1505.05511 .
  * Berezhiani  _et al._ (2015) Z. Berezhiani, A.D. Dolgov, and I.I. Tkachev, “Reconciling planck results with low redshift astronomical measurements,” Phys. Rev. D 92, 061303 (2015), 1505.03644 .
  * Blinov  _et al._ (2020) Nikita Blinov, Celeste Keith, and Dan Hooper, “Warm Decaying Dark Matter and the Hubble Tension,” JCAP 06, 005 (2020), arXiv:2004.06114 [astro-ph.CO] .
  * Vattis  _et al._ (2019) Kyriakos Vattis, Savvas M. Koushiappas, and Abraham Loeb, “Late universe decaying dark matter can relieve the $H_{0}$ tension,” Phys. Rev. D99, 121302 (2019), 1903.06220 .
  * Abellán  _et al._ (2021) Guillermo F. Abellán, Riccardo Murgia, and Vivian Poulin, “Linear cosmological constraints on 2-body decaying dark matter scenarios and robustness of the resolution to the $S_{8}$ tension,” (2021), arXiv:2102.12498 [astro-ph.CO] .
  * Chudaykin  _et al._ (2016) A. Chudaykin, D. Gorbunov, and I. Tkachev, “Dark matter component decaying after recombination: Lensing constraints with planck data,” Phys. Rev. D94, 023528 (2016), 1602.08121 .
  * Chudaykin  _et al._ (2018) A. Chudaykin, D. Gorbunov, and I. Tkachev, “Dark matter component decaying after recombination: Sensitivity to baryon acoustic oscillation and redshift space distortion probes,” Phys. Rev. D 97, 083508 (2018), arXiv:1711.06738 [astro-ph.CO] .
  * Poulin  _et al._ (2016) Vivian Poulin, Pasquale D. Serpico, and Julien Lesgourgues, “A fresh look at linear cosmological constraints on a decaying dark matter component,” Journal of Cosmology and Astroparticle Physics 2016, 036–036 (2016).
  * Clark  _et al._ (2020) Steven J. Clark, Kyriakos Vattis, and Savvas M. Koushiappas, “Cosmological constraints on late-Universe decaying dark matter as a solution to the H0 tension,” (2020), 2006.03678 .
  * Haridasu and Viel (2020) Balakrishna S. Haridasu and Matteo Viel, “Late-time decaying dark matter: constraints and implications for the H0-tension,” (2020), 2004.07709 .
  * Nygaard  _et al._ (2020) Andreas Nygaard, Thomas Tram, and Steen Hannestad, “Updated constraints on decaying cold dark matter,” (2020), arXiv:2011.01632 [astro-ph.CO] .
  * Berezinsky  _et al._ (1991) V. Berezinsky, A. Masiero, and J.W.F. Valle, “Cosmological signatures of supersymmetry with spontaneously broken R parity,” Physics Letters B 266, 382–388 (1991).
  * Kim and Kim (2002) Hang Bae Kim and Jihn E. Kim, “Late decaying axino as CDM and its lifetime bound,” Physics Letters B 527, 18–22 (2002).
  * Covi  _et al._ (1999) Laura Covi, Jihn E. Kim, and Leszek Roszkowski, “Axinos as cold dark matter,” Phys. Rev. Lett. 82, 4180–4183 (1999), hep-ph/9905212 .
  * Feng  _et al._ (2003) J. L. Feng, A. Rajaraman, and F. Takayama, “Superweakly interacting massive particles,” Phys. Rev. Lett. 91, 011302 (2003), hep-ph/0302215 .
  * Feng  _et al._ (2003) Jonathan L. Feng, Arvind Rajaraman, and Fumihiro Takayama, “Superweakly interacting massive particle dark matter signals from the early universe,” Phys. Rev. D 68, 063504 (2003), hep-ph/0306024 .
  * Allahverdi  _et al._ (2015) Rouzbeh Allahverdi, Bhaskar Dutta, Farinaldo S. Queiroz, Louis E. Strigari, and Mei-Yu Wang, “Dark matter from late invisible decays to and of gravitinos,” Phys. Rev. D 91, 055033 (2015), 1412.4391 .
  * Abazajian  _et al._ (2012) K. N. Abazajian _et al._ , “Light Sterile Neutrinos: A White Paper,” (2012), arXiv:1204.5379 [hep-ph] .
  * Drewes  _et al._ (2017) M. Drewes _et al._ , “A white paper on kev sterile neutrino dark matter,” JCAP 01, 025 (2017), 1602.04816 .
  * Chen  _et al._ (2009) Chuan-Ren Chen, Fuminobu Takahashi, and T. T. Yanagida, “Gamma rays and positrons from a decaying hidden gauge boson,” Phys. Lett. B 671, 71–76 (2009), 0809.0792 .
  * Choi  _et al._ (2020a) Gongjun Choi, Motoo Suzuki, and Tsutomu T. Yanagida, “Degenerate Sub-keV fermion dark matter from a solution to the Hubble tension,” (2020a), 2002.00036 .
  * Choi  _et al._ (2020b) Gongjun Choi, Motoo Suzuki, and Tsutomu. T. Yanagida, “Degenerate fermion dark matter from a broken U(1)B-L gauge symmetry,” Phys. Rev. D 102, 035022 (2020b), 2004.07863 .
  * Choi and Yanagida (2021) Gongjun Choi and Tsutomu T. Yanagida, “Gravitino cosmology helped by a right handed (s)neutrino,” (2021), 2104.02958 .
  * Choi  _et al._ (2020c) Gongjun Choi, Motoo Suzuki, and Tsutomu T. Yanagida, “XENON1T anomaly and its implication for decaying warm dark matter,” Phys. Lett. B 811, 135976 (2020c), 2006.12348 .
  * Xu and Zheng (2020) Shuai Xu and Sibo Zheng, “Resolving xenon excess with decaying cold dark matter,” (2020), 2012.10827 .
  * Dutta  _et al._ (2021) Koushik Dutta, Avirup Ghosh, Arpan Kar, and Biswarup Mukhopadhyaya, “Decaying fermionic warm dark matter and XENON1T electronic recoil excess,” Phys. Dark Univ. 33, 100855 (2021), arXiv:2103.14664 [hep-ph] .
  * Lin  _et al._ (2001) W. B. Lin, D. H. Huang, X. Zhang, and R. Brandenberger, “Nonthermal Production of Weakly Interacting Massive Particles and the Subgalactic Structure of the Universe,” Phys. Rev. Lett. 86, 954–957 (2001), astro-ph/0009003 .
  * Sánchez-Salcedo (2003) F. J. Sánchez-Salcedo, “Unstable Cold Dark Matter and the Cuspy Halo Problem in Dwarf Galaxies,” ApJ Letters 591, L107–L110 (2003), astro-ph/0305496 .
  * Cembranos  _et al._ (2005) J. A. R. Cembranos, J. L. Feng, A. Rajaraman, and F. Takayama, “Superweakly interacting massive particle solutions to small scale structure problems,” Phys. Rev. Lett. 95, 181301 (2005), hep-ph/0507150 .
  * Kaplinghat (2005) Manoj Kaplinghat, “Dark matter from early decays,” Phys. Rev. D 72, 063510 (2005), astro-ph/0507300 .
  * Strigari  _et al._ (2007) Louis E. Strigari, Manoj Kaplinghat, and James S. Bullock, “Dark matter halos with cores from hierarchical structure formation,” Phys. Rev. D 75, 061303 (2007), astro-ph/0606281 .
  * Borzumati  _et al._ (2008) F. Borzumati, T. Bringmann, and P. Ullio, “Dark matter from late decays and the small-scale structure problems,” Phys. Rev. D 77, 063514 (2008), hep-ph/0701007 .
  * Peter  _et al._ (2010) Annika H. G. Peter, Christopher E. Moody, and Marc Kamionkowski, “Dark-matter decays and self-gravitating halos,” Phys. Rev. D 81, 103501 (2010), 1003.0419 .
  * Peter and Benson (2010) Annika H. G. Peter and Andrew J. Benson, “Dark-matter decays and milky way satellite galaxies,” Phys. Rev. D 82, 123521 (2010), 1009.1912 .
  * Carrasco  _et al._ (2012) John Joseph M. Carrasco, Mark P. Hertzberg, and Leonardo Senatore, “The effective field theory of cosmological large scale structures,” Journal of High Energy Physics 2012 (2012), 10.1007/jhep09(2012)082.
  * Baumann  _et al._ (2012) Daniel Baumann, Alberto Nicolis, Leonardo Senatore, and Matias Zaldarriaga, “Cosmological non-linearities as an effective fluid,” Journal of Cosmology and Astroparticle Physics 2012, 051–051 (2012).
  * Porto  _et al._ (2014) Rafael A. Porto, Leonardo Senatore, and Matias Zaldarriaga, “The Lagrangian-space Effective Field Theory of large scale structures,” Journal of Cosmology and Astroparticle Physics 2014, 022–022 (2014).
  * Pajer and Zaldarriaga (2013) Enrico Pajer and Matias Zaldarriaga, “On the renormalization of the effective field theory of large scale structures,” Journal of Cosmology and Astroparticle Physics 2013, 037–037 (2013).
  * Abolhasani  _et al._ (2016) Ali Akbar Abolhasani, Mehrdad Mirbabayi, and Enrico Pajer, “Systematic renormalization of the effective theory of Large Scale Structure,” Journal of Cosmology and Astroparticle Physics 2016, 063–063 (2016).
  * Senatore and Zaldarriaga (2014) Leonardo Senatore and Matias Zaldarriaga, “Redshift Space Distortions in the Effective Field Theory of Large Scale Structures,” (2014), arXiv:1409.1225 [astro-ph.CO] .
  * Baldauf  _et al._ (2015) Tobias Baldauf, Mehrdad Mirbabayi, Marko Simonovic, and Matias Zaldarriaga, “Equivalence principle and the baryon acoustic peak,” Physical Review D 92 (2015), 10.1103/physrevd.92.043514.
  * Senatore and Zaldarriaga (2015) Leonardo Senatore and Matias Zaldarriaga, “The IR-resummed Effective Field Theory of Large Scale Structures,” Journal of Cosmology and Astroparticle Physics 2015, 013–013 (2015).
  * Senatore and Trevisan (2018) Leonardo Senatore and Gabriele Trevisan, “On the IR-resummation in the EFTofLSS,” Journal of Cosmology and Astroparticle Physics 2018, 019–019 (2018).
  * Lewandowski and Senatore (2020) Matthew Lewandowski and Leonardo Senatore, “An analytic implementation of the IR-resummation for the BAO peak,” Journal of Cosmology and Astroparticle Physics 2020, 018–018 (2020).
  * Blas  _et al._ (2016) Diego Blas, Mathias Garny, Mikhail M. Ivanov, and Sergey Sibiryakov, “Time-sliced perturbation theory II: baryon acoustic oscillations and infrared resummation,” Journal of Cosmology and Astroparticle Physics 2016, 028–028 (2016).
  * Carrasco  _et al._ (2014a) John Joseph M. Carrasco, Simon Foreman, Daniel Green, and Leonardo Senatore, “The 2-loop matter power spectrum and the IR-safe integrand, volume=2014,” Journal of Cosmology and Astroparticle Physics , 056–056 (2014a).
  * Carrasco  _et al._ (2014b) John Joseph M. Carrasco, Simon Foreman, Daniel Green, and Leonardo Senatore, “The Effective Field Theory of Large Scale Structures at two loops,” Journal of Cosmology and Astroparticle Physics 2014, 057–057 (2014b).
  * Senatore (2015) Leonardo Senatore, “Bias in the effective field theory of large scale structures,” Journal of Cosmology and Astroparticle Physics 2015, 007–007 (2015).
  * Mirbabayi  _et al._ (2015) Mehrdad Mirbabayi, Fabian Schmidt, and Matias Zaldarriaga, “Biased tracers and time evolution,” Journal of Cosmology and Astroparticle Physics 2015, 030–030 (2015).
  * Angulo  _et al._ (2015) Raul Angulo, Matteo Fasiello, Leonardo Senatore, and Zvonimir Vlah, “On the statistics of biased tracers in the Effective Field Theory of Large Scale Structures,” Journal of Cosmology and Astroparticle Physics 2015, 029–029 (2015).
  * Fujita  _et al._ (2020) Tomohiro Fujita, Valentin Mauerhofer, Leonardo Senatore, Zvonimir Vlah, and Raul Angulo, “Very massive tracers and higher derivative biases,” Journal of Cosmology and Astroparticle Physics 2020, 009–009 (2020).
  * Perko  _et al._ (2016) Ashley Perko, Leonardo Senatore, Elise Jennings, and Risa H. Wechsler, “Biased Tracers in Redshift Space in the EFT of Large-Scale Structure,” (2016), arXiv:1610.09321 [astro-ph.CO] .
  * Nadler  _et al._ (2018) Ethan O. Nadler, Ashley Perko, and Leonardo Senatore, “On the bispectra of very massive tracers in the Effective Field Theory of Large-Scale Structure,” Journal of Cosmology and Astroparticle Physics 2018, 058–058 (2018).
  * D’Amico  _et al._ (2021) Guido D’Amico, Leonardo Senatore, and Pierre Zhang, “Limits on wCDM from the EFTofLSS with the PyBird code,” Journal of Cosmology and Astroparticle Physics 2021, 006–006 (2021).
  * Alcock and Paczynski (1979) C. Alcock and B. Paczynski, “An evolution free test for non-zero cosmological constant,” Nature 281, 358–359 (1979).
  * Beutler  _et al._ (2019) Florian Beutler, Emanuele Castorina, and Pierre Zhang, “Interpreting measurements of the anisotropic galaxy power spectrum,” Journal of Cosmology and Astroparticle Physics 2019, 040–040 (2019).
  * Hahn  _et al._ (2017) ChangHoon Hahn, Roman Scoccimarro, Michael R. Blanton, Jeremy L. Tinker, and Sergio Rodríguez-Torres, “The Effect of Fiber Collisions on the Galaxy Power Spectrum Multipoles,” Monthly Notices of the Royal Astronomical Society , stx185 (2017).
  * Blas  _et al._ (2011) Diego Blas, Julien Lesgourgues, and Thomas Tram, “The Cosmic Linear Anisotropy Solving System (CLASS). Part II: Approximation schemes,” Journal of Cosmology and Astroparticle Physics 2011, 034–034 (2011).
  * Lewis  _et al._ (2000) Antony Lewis, Anthony Challinor, and Anthony Lasenby, “Efficient computation of CMB anisotropies in closed FRW models,” Astrophys. J. 538, 473–476 (2000), arXiv:astro-ph/9911177 .
  * Chudaykin  _et al._ (2020) Anton Chudaykin, Mikhail M. Ivanov, Oliver H.E. Philcox, and Marko Simonovic, “Nonlinear perturbation theory extension of the Boltzmann code CLASS,” Physical Review D 102 (2020), 10.1103/physrevd.102.063533.
  * Hamilton (2000) A. J. S. Hamilton, “Uncorrelated modes of the nonlinear power spectrum,” Mon. Not. Roy. Astron. Soc. 312, 257–284 (2000), arXiv:astro-ph/9905191 .
  * Simonovic  _et al._ (2018) Marko Simonovic, Tobias Baldauf, Matias Zaldarriaga, John Joseph Carrasco, and Juna A. Kollmeier, “Cosmological perturbation theory using the FFTLog: formalism and connection to QFT loop integrals,” JCAP 04, 030 (2018), arXiv:1708.08130 [astro-ph.CO] .
  * Amendola  _et al._ (2018) Luca Amendola _et al._ , “Cosmology and fundamental physics with the Euclid satellite,” Living Rev. Rel. 21, 2 (2018), 1606.00180 .
  * Alonso  _et al._ (2018) David Alonso _et al._ (LSST Dark Energy Science), “The LSST Dark Energy Science Collaboration (DESC) Science Requirements Document,” (2018), 1809.01669 .
  * Alam  _et al._ (2017) Shadab Alam _et al._ (BOSS), “The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample,” Mon. Not. Roy. Astron. Soc. 470, 2617–2652 (2017), 1607.03155 .
  * Reid  _et al._ (2015) Beth Reid _et al._ , “SDSS-III Baryon Oscillation Spectroscopic Survey Data Release 12: galaxy target selection and large scale structure catalogues,” (2015), arXiv:1509.06529 [astro-ph.CO] .
  * Kitaura  _et al._ (2016) Francisco-Shu Kitaura, Sergio Rodríguez-Torres, Chia-Hsun Chuang, Cheng Zhao, Francisco Prada, Héctor Gil-Marín, Hong Guo, Gustavo Yepes, Anatoly Klypin, Claudia G. Scóccola, Jeremy Tinker, Cameron McBride, Beth Reid, Ariel G. Sánchez, Salvador Salazar-Albornoz, Jan Niklas Grieb, Mariana Vargas-Magana, Antonio J. Cuesta, Mark Neyrinck, Florian Beutler, Johan Comparat, Will J. Percival, and Ashley Ross, “The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: mock galaxy catalogues for the BOSS Final Data Release,” Monthly Notices of the Royal Astronomical Society 456, 4156–4173 (2016).
  * Zhang  _et al._ (2021) Pierre Zhang, Guido D’Amico, Leonardo Senatore, Cheng Zhao, and Yifu Cai, “BOSS Correlation Function Analysis from the Effective Field Theory of Large-Scale Structure,” (2021), arXiv:2110.07539 [astro-ph.CO] .
  * d’ Amico  _et al._ (2020) Guido d’ Amico, Jérôme Gleyzes, Nickolas Kokron, Katarina Markovic, Leonardo Senatore, Pierre Zhang, Florian Beutler, and Héctor Gil-Marín, “The cosmological analysis of the SDSS/BOSS data from the Effective Field Theory of Large-Scale Structure,” Journal of Cosmology and Astroparticle Physics 2020, 005–005 (2020).
  * Colas  _et al._ (2020) Thomas Colas, Guido d’ Amico, Leonardo Senatore, Pierre Zhang, and Florian Beutler, “Efficient cosmological analysis of the SDSS/BOSS data from the Effective Field Theory of Large-Scale Structure,” Journal of Cosmology and Astroparticle Physics 2020, 001–001 (2020).
  * Nishimichi  _et al._ (2020) Takahiro Nishimichi, Guido D’Amico, Mikhail M. Ivanov, Leonardo Senatore, Marko Simonovic, Masahiro Takada, Matias Zaldarriaga, and Pierre Zhang, “Blinded challenge for precision cosmology with large-scale structure: Results from effective field theory for the redshift-space galaxy power spectrum,” Physical Review D 102 (2020), 10.1103/physrevd.102.123541.
  * Audren  _et al._ (2014) Benjamin Audren, Julien Lesgourgues, Gianpiero Mangano, Pasquale Dario Serpico, and Thomas Tram, “Strongest model-independent bound on the lifetime of dark matter,” JCAP 12, 028 (2014), 1407.2418 .
  * Enqvist  _et al._ (2015b) Kari Enqvist, Seshadri Nadathur, Toyokazu Sekiguchi, and Tomo Takahashi, “Decaying dark matter and the tension in $\sigma$8,” Journal of Cosmology and Astroparticle Physics 2015, 067–067 (2015b).
  * Ma and Bertschinger (1995) Chung-Pei Ma and Edmund Bertschinger, “Cosmological perturbation theory in the synchronous and conformal newtonian gauges,” The Astrophysical Journal 455, 7 (1995).
  * Hubert  _et al._ (2021) Jonathan Hubert, Aurel Schneider, Doug Potter, Joachim Stadel, and Sambit K. Giri, “Decaying dark matter: Simulations and weak-lensing forecast,” (2021), arXiv:2104.07675 [astro-ph.CO] .
  * Blackadder and Koushiappas (2014) Gordon Blackadder and Savvas M. Koushiappas, “Dark matter with two- and many-body decays and supernovae type Ia,” Physical Review D 90 (2014), 10.1103/physrevd.90.103527.
  * Aoyama  _et al._ (2014) Shohei Aoyama, Toyokazu Sekiguchi, Kiyotomo Ichiki, and Naoshi Sugiyama, “Evolution of perturbations and cosmological constraints in decaying dark matter models with arbitrary decay mass products,” Journal of Cosmology and Astroparticle Physics 2014, 021–021 (2014).
  * Lesgourgues and Tram (2011) Julien Lesgourgues and Thomas Tram, “The Cosmic Linear Anisotropy Solving System (CLASS) IV: efficient implementation of non-cold relics,” JCAP 09, 032 (2011), 1104.2935 .
  * Senatore and Zaldarriaga (2017) Leonardo Senatore and Matias Zaldarriaga, “The Effective Field Theory of Large-Scale Structure in the presence of Massive Neutrinos,” (2017), arXiv:1707.04698 [astro-ph.CO] .
  * de Belsunce and Senatore (2019) Roger de Belsunce and Leonardo Senatore, “Tree-level bispectrum in the effective field theory of large-scale structure extended to massive neutrinos,” Journal of Cosmology and Astroparticle Physics 2019, 038–038 (2019).
  * Heymans  _et al._ (2020) Catherine Heymans _et al._ , “Kids-1000 cosmology: Multi-probe weak gravitational lensing and spectroscopic galaxy clustering constraints,” (2020) 2007.15632 .
  * Aghanim  _et al._ (2020b) N. Aghanim _et al._ (Planck), “Planck 2018 results. VIII. Gravitational lensing,” Astron. Astrophys. 641, A8 (2020b), arXiv:1807.06210 [astro-ph.CO] .
  * Scolnic  _et al._ (2018) D. M. Scolnic _et al._ , “The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample,” Astrophys. J. 859, 101 (2018), 1710.00845 .
  * Beutler  _et al._ (2011) Florian Beutler, Chris Blake, Matthew Colless, D.Heath Jones, Lister Staveley-Smith, Lachlan Campbell, Quentin Parker, Will Saunders, and Fred Watson, “The 6dF Galaxy Survey: baryon acoustic oscillations and the local Hubble constant,” Mon. Not. Roy. Astron. Soc. 416, 3017–3032 (2011), 1106.3366 .
  * Ross  _et al._ (2015) Ashley J. Ross, Lado Samushia, Cullan Howlett, Will J. Percival, Angela Burden, and Marc Manera, “The clustering of the SDSS DR7 main Galaxy sample I: A 4 per cent distance measure at z = 0.15,” Mon. Not. Roy. Astron. Soc. 449, 835–847 (2015), 1409.3242 .
  * de Sainte Agathe  _et al._ (2019) Victoria de Sainte Agathe _et al._ , “Baryon acoustic oscillations at z = 2.34 from the correlations of Ly$\alpha$ absorption in eBOSS DR14,” Astron. Astrophys. 629, A85 (2019), 1904.03400 .
  * Blomqvist  _et al._ (2019) Michael Blomqvist _et al._ , “Baryon acoustic oscillations from the cross-correlation of Ly$\alpha$ absorption and quasars in eBOSS DR14,” Astron. Astrophys. 629, A86 (2019), 1904.03430 .
  * Gelman and Rubin (1992) Andrew Gelman and Donald B. Rubin, “Inference from Iterative Simulation Using Multiple Sequences,” Statistical Science 7, 457 – 472 (1992).
  * Poulin  _et al._ (2018) Vivian Poulin, Kimberly K. Boddy, Simeon Bird, and Marc Kamionkowski, “Implications of an extended dark energy cosmology with massive neutrinos for cosmological tensions,” Phys. Rev. D97, 123504 (2018), 1803.02474 .
  * Raveri and Hu (2019) Marco Raveri and Wayne Hu, “Concordance and discordance in cosmology,” Phys. Rev. D 99, 043506 (2019), arXiv:1806.04649 [astro-ph.CO] .
  * Mau  _et al._ (2022) S. Mau _et al._ , “Milky Way Satellite Census. IV. Constraints on Decaying Dark Matter from Observations of Milky Way Satellite Galaxies,” (2022), arXiv:2201.11740 [astro-ph.CO] .
  * Aghamousa, Amir and others (2016) Aghamousa, Amir and others (DESI), “The DESI Experiment Part I: Science,Targeting, and Survey Design,” (2016), 1611.00036 .
  * Heymans  _et al._ (2013) Catherine Heymans _et al._ , “CFHTLenS tomographic weak lensing cosmological parameter constraints: Mitigating the impact of intrinsic galaxy alignments,” Mon. Not. Roy. Astron. Soc. 432, 2433 (2013), 1303.1808 .