11institutetext: IRAP, Université de Toulouse, CNRS, CNES, UPS, Toulouse,
France 22institutetext: Université St Joseph; Faculty of Sciences, Beirut,
Lebanon 33institutetext: Université PSL, Observatoire de Paris, Sorbonne
Université, CNRS, LERMA, F-75014, Paris, France
33email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Cluster counts III. $\Lambda$CDM extensions and the cluster tension

Ziad Sakr 1122 Stéphane Ilić 3311 Alain Blanchard 11

(Received ; accepted )

Despite the success of the Lambda cold dark matter ($\Lambda$CDM) cosmological
model, current estimations of the amplitude of matter fluctuations
($\sigma_{8}$) show an appreciable difference between its value inferred from
the cosmic microwave background (CMB) angular power spectrum ($C_{\ell}$) and
those obtained from cluster counts. Neutrinos or a modification of the growth
of structures had been previously investigated as the possible origin of this
discrepancy. In this work we examine whether further extensions to the
$\Lambda$CDM model could alleviate the tension. To this end, we derived
constraints on the parameters subject to the discrepancy, using CMB $C_{\ell}$
combined with cluster counts from the Sunyaev–Zel’dovich (SZ) sample with a
free dark energy equation of state parameter, while allowing the cluster mass
calibration parameter $(1-b)$ to vary. This latter is degenerate with
$\sigma_{8}$, which translates the discrepancy within the $\Lambda$CDM
framework into one between $(1-b)\sim 0.6$, corresponding to constraints on
$\sigma_{8}$ obtained from CMB, and $(1-b)\sim 0.8$, the value adopted for the
SZ sample calibration. We find that a constant $w$, when left free to vary
along with large priors on the matter density ([0.1,1.0]) and the Hubble
parameters ([30,200]), can reduce the discrepancy to less than 2$\sigma$ for
values far below its fiducial $w$ = -1. However, such low values of $w$ are
not allowed when we add other probes like the baryonic acoustic oscillation
(BAO) feature angular diameter distance measured in galaxy clustering surveys.
We also found, when we allow to vary in addition to $w$ a modification of the
growth rate through the growth index $\gamma$, that the tension is alleviated,
with the $(1-b)$ likelihood now centred around the Planck calibration value of
$\sim$ 0.8. However, here again, combining CMB and cluster counts with
geometrical distance probes restores the discrepancy, with the $(1-b)$
preferred value reverting back to the $\Lambda$CDM value of $\sim$ 0.6. The
same situation is observed when introducing, along with $w$ and $\gamma$,
further extensions to $\Lambda$CDM (e.g. massive neutrinos), although these
extensions reduce the tension to 2$\sigma$, even when combined with BAO
datasets. We also explore other common extensions by comparing two cases:
allowing a dynamical $w$ following a CPL parametrisation in addition to a
constant growth index, and when the growth index is expanded through a second
parameter $\gamma_{1}$ along with a constant $w$. In the former we reach the
same conclusions as with the case of a constant $w$ and $\gamma$, where the
discrepancy was alleviated only if we do not constrain $w$ by BAO, while in
the latter case, we observe that introducing $\gamma_{1}$ drives $(1-b)$
towards lower values that would instead increase the discrepancy on
$\sigma_{8}$. We conclude that none of these common extensions to $\Lambda$CDM
is able to fix the discrepancy and a misdetermination of the calibration
factor is the most preferred explanation. Finally, we investigate the effect
on our posteriors from limiting the Hubble constant priors to the usual common
adopted range of [30,100].

###### Key Words.:

Galaxies: clusters: general – large-scale structure of Universe – cosmological
parameters – dark energy – cosmic background radiation

##  1 Introduction

Cluster counts are considered a powerful tool for constraining cosmological
parameters (Oukbir & Blanchard, 1992; Mohr, 2005; Vikhlinin et al., 2009;
Allen et al., 2011). Their abundance is primarily sensitive to two parameters,
the matter density $\Omega_{\rm m}$ and the current amplitude of matter
fluctuations, $\sigma_{\rm 8}$ (Bahcall et al., 1997; Blanchard & Bartlett,
1998). The value of the latter, constrained from the Planck mission (Planck
Collaboration et al., 2016b) assuming the Lambda cold dark matter
($\Lambda$CDM) model, was found to be in tension with the value determined
from Sunyaev–Zel’dovich (SZ) cluster counts at lower redshifts (Planck
Collaboration et al., 2014). Similar discrepancies, although with different
amplitudes, were also found when using other large-scale structure local
probes, such as weak gravitational lensing detected clusters from the Canadian
Cluster Comparison Project (CCCP) (Hoekstra et al., 2015), Sunyaev Zel’dovich
identified galaxy clusters from the South Pole Telescope (SPT) survey (de Haan
et al., 2016), and gravitational lensing cluster mass measurements from the
Cluster Lensing And Supernova and Hubble (CLASH) survey (Penna-Lima et al.,
2017), while only few, such as the Weighing the Giants (WtG) weak lensing
cluster survey (Applegate et al., 2014), claim a more appropriate selection
function and yield values of $\sigma_{8}$ in agreement with those from the
Planck mission. Even though the early release from the large-scale Dark Energy
Survey (DES) (Abbott et al., 2018) or more recent ones (Amon et al., 2022)
with constraints from cosmic shear measurements and the latest Planck cosmic
microwave background (CMB) observations (Planck Collaboration et al., 2020)
have shown a lesser tension, the discrepancy is still present, especially when
the CMB $\sigma_{8}$ derived constraints are compared to those from the latest
galaxy clusters surveys (Kirby et al., 2019; Bocquet et al., 2019; Asgari et
al., 2020; Abdullah et al., 2020).
The origin of this tension is still under investigation. On the CMB side, the
high accuracy achieved by the Planck mission in measuring the CMB temperature
angular power spectrum in the main release (Planck Collaboration et al.,
2016b), or the final polarisation spectra in Planck Collaboration et al.
(2020), as well as other CMB experiments like the latest release of the
Atacama Cosmology Telescope (ACT) combined with the Wilkinson Microwave
Anisotropy Probe (WMAP) Aiola et al. (2020), leaves small room for any large
systematics that could be responsible for a misdetermination of the
cosmological parameters by this mission. Moreover, an analysis by Motloch & Hu
(2020) of the final release of the Planck satellite data on the anomaly
observed between the lensing gravitational potential and the lensing obtained
from the smoothing of the acoustic peaks in the temperature and polarisation
power spectra found that marginalising over this component cannot
substantially resolve the $\sigma_{8}$ tension with low-redshift measurements.
This is not the case for the cluster counts, where substantial systematics
remain when trying to relate the observable abundance of clusters to the
theoretical mass function. This is suggested from the scatter found between
the values of the calibrations of the different scaling relations connecting
cluster mass to observable quantities, such as X-ray emission from cluster hot
gases, cluster richness, other optical properties, or the SZ effect. This
uncertainty in the determination of the mass calibration parameter could have
a strong impact on the discrepancy since it was shown early on by Planck
Collaboration et al. (2014) that adopting a different value for the mass
observable parameter could alleviate the tension and that adopting a different
value for the scaling calibration may serve as a solution to fix the
discrepancy. Therefore, several studies investigated the impact of
modifications to the way the mass observable relations are determined on the
cluster abundance observable with relative success when trying to account for
it as solution to the $\sigma_{8}$ discrepancy. One line targeted the cooling
and heating processes closely related to cluster mass determination, such as
the work by McDonald et al. (2017), Henson et al. (2017), and Mummery et al.
(2017), who showed, using hydrodynamic simulations, that using a different
mass weighted cooling scheme or taking into account non-thermal pressure
coming from bulk and turbulent motions of gas in the intracluster medium (Shi
et al., 2016; Ettori & Eckert, 2022) could reduce the hydrostatic bias,
although still below the amount needed to reconcile the observed cluster
number count with CMB. Another study was performed by the second release of
the Planck collaboration (Planck Collaboration et al., 2016a) on the impact of
the signal-to-noise ratio (S/N) on the number of detections, but found that it
would be necessary for Planck to have missed nearly 40% of the clusters with
predicted SZ S/N ¿ 7 in order for the SZ and CMB cluster number count to match
with a level of scatter with cluster structure inconsistent with the
predictions of current hydrodynamical simulations. More recently, Lovisari et
al. (2020) investigated the differences in the mass estimates obtained through
independent X-ray, weak-lensing, and dynamical studies in the Planck-ESZ
cluster sample and found an average value insufficient to reconcile the Planck
cluster abundance and CMB results. Using simulations, Castro et al. (2020) and
Debackere et al. (2021) studied the effect of baryonic physics on cluster
statistics and found it subdominant compared to other systematics for current
available observations, similarly to Bocquet et al. (2016). Salvati et al.
(2021) showed that we need to drop the self-similarity hypothesis and assume a
change of $\sim$ 4 $\sigma$ in the slope of the scaling relation to reconcile
CMB and cluster counts similar to Andreon (2014), while Ghirardini et al.
(2021) and Wicker et al. (2022) investigated a possible redshift evolution of
the mass observable bias and found a value for the amplitude of the bias that
is consistent with that expected from hydrodynamical simulations and local
measurement, similarly to Salvati et al. (2019).
Another source of misdetermination of the cluster mass comes from possible
overlooked biases from the weak lensing proxy. Rasia et al. (2012) showed that
the scatter in the mass lensing relation could be reduced by discarding
clusters having an irregular morphology or clusters influenced by
substructure-rich environment. Another bias according to Valageas (2014) could
arise from galaxies that are not distributed randomly in space, but are
correlated with the underlying density field, resulting in a source-lens
clustering. Bahe et al. (2012) and Hamana et al. (2012) found that the
standard method for identifying clusters through a search of the peak heights
in the lensing convergence map is not well suited to determining the cluster
mass as the peaks are poorly correlated with their virial mass. More recently,
Meneghetti et al. (2020) reported that the observed cluster substructures are
more efficient lenses than predicted by the CDM simulation. This could be the
reason why (see the study by Lovisari et al. (2020)) the mean mass ratios
inferred from the Weak Lensing (WL) masses of different projects vary by a
large amount, from the APEX-SZ survey Klein et al. (2019) showing a very small
bias, to the Local Cluster Sub-structure Survey (LoCuSS) (Zhang et al., 2008)
and the combined Multi Epoch Nearby Cluster Survey (MENeaCS) with the Canadian
Cluster Comparison Project (CCCP) Herbonnet et al. (2020) showing a more
significant value ($1-b\sim$ 0.77), to the aforementioned WtG (Applegate et
al., 2014) pointing to the largest deviation ($1-b\sim$ 0.6), a value that
would alleviate the tension between the Planck results. The Planck SZ
collaboration also elaborated a novel method for measuring cluster masses
based on matching lensing of the CMB temperature anisotropies behind clusters
(Melin & Bartlett, 2015) and obtained $1-b\sim 1.01\pm 0.2$, a value that
increases the discrepancies although with greater statistical uncertainty.
More recently, mixed results have been found by Ingoglia et al. (2022) and
Lesci et al. (2022) when using lensing to calibrate the mass of a sample of
clusters from the third KiDS release; the first is in agreement with the
Planck’s clusters calibration, while the second is consistent with CMB
inferred counts. Murray et al. (2022) presented weak lensing mass estimates
for a sample of 458 galaxy clusters from the redMaPPer Sloan Digital Sky
Survey and found a tension between the mass–richness relation and that from
DES collaboration cluster sample. Finally, Schrabback et al. (2021) performed
weak lensing mass calibration on distant SPT galaxy clusters from expanded
follow-up deep observations and found it 30% lower than needed to reconcile a
Planck cosmology with the observed SPT-SZ cluster counts.
Other halo properties than the halo mass that could impact the abundance of
clusters should also not be forgotten, such as formation time, concentration,
and the dependence of the spatial distribution of dark matter halos, all
commonly called halo assembly bias (see Sunayama et al. (2016), and reference
therein), but also the cluster triaxiality or shape and orientation impact on
the number of detections (see Zhang et al. (2022) for a recent assessment on
the DES detection of clusters) or other environmental effects, such as the
density of the surrounding background (Manolopoulou et al., 2020), and the
impact on X-ray clusters.
To summarise, almost all cluster-based probes show a discrepancy with the CMB
probe on the $\sigma_{8}$ constraints when using their mass-observable
calibration obtained by different methods (see Penna-Lima et al. (2017) and
Salvati et al. (2018), for a compilation) and only few (some of those
mentioned above) suggest that there are improper selection functions, due to a
neglected Eddington bias (Battaglia et al., 2016) or a Malmquist bias (Ascaso
et al., 2017) for example, that should be taking into consideration and could
result in a substantial reduction of the discrepancy. Consequently, this has
motivated us to consider other alternatives as possible origins of this
tension.

Among these alternatives, one was an extension to the $\Lambda$CDM model by
adding massive neutrinos. The latter escape matter potentials on scales below
their free streaming distance, slowing down the growth of matter perturbations
(Brandbyge et al., 2010) and lowering the value of $\sigma_{8}$. Therefore,
several studies, with more or less favourable conclusions, tried to test
whether a non-minimal neutrino mass helps in reducing the gap between the two
constrained values of $\sigma_{8}$ (Dvorkin et al., 2014; Giusarma et al.,
2014; Dvorkin et al., 2014; Di Valentino et al., 2016). In the companion
papers Sakr et al. (2018, hereafter SK18,) and Ilić et al. (2019, hereafter
IL19), we also examined with Markov chain Monte Carlo (MCMC) techniques the
role of neutrinos in fixing this discrepancy using an X-ray cluster sample or
SZ detected cluster sample, in combination with CMB datasets and additional
secondary probes like BAO and Ly-$\alpha$, and found that massive neutrinos
are not able to alleviate the tension because the introduction of massive
neutrinos modifies the confidence contours on $\sigma_{8}$, from cluster
counts and those from the CMB data, in the same direction of the
$\sigma_{8}-\Omega_{m}$ degeneracy, thus does not reduce the gap in the value
of $\sigma_{8}$ obtained from the two probes. We explored another possible
origin for this discrepancy through a modification of general relativity. This
could result in a change in the growth rate of structures, which is
parametrised in function of the so-called growth index $\gamma$ (Peebles,
1980). Different values of $\gamma$, corresponding to different modified
gravity models, change the expected number of clusters, yielding different
constraints on $\sigma_{8}$. Therefore, in (SK18) and (IL19), we investigated
the change in $\sigma_{8}$ constraints when allowing a free $\gamma$ in the
presence of massive neutrinos. However, unlike other works, we left the mass
observable scaling factor free to vary when combining the two aforementioned
probes. This data-driven approach was motivated by Blanchard & Douspis (2005),
where $\sigma_{8}$ constraints were found correlated with those of the mass
calibration parameter. Therefore, any new parameter showing correlation with
the mass observable one would be able, within the allowed constraints, to fix
the discrepancy. In SK18, we found that $\gamma$ satisfies this condition,
independently from the value of the neutrino mass, thus indicating its
ability, unlike neutrinos, to fix the discrepancy. However, this is only
possible for values far from the fiducial $\Lambda$CDM values that are not in
agreement with constraints from other growth probes. Moreover, we also found
in (IL19), when using instead the deeper SZ cluster detected sample that
although $\gamma$ still correlates with the calibration parameter, it cannot
reach the value necessary to fix the discrepancy, due to the constraining
power of the SZ sample probing a deeper and wider redshift range.
Drawing from the same idea, we propose here to explore the impact on the
aforementioned discrepancy of another extension to $\Lambda$CDM through a
change in the value of the dark energy equation of state (EoS) parameter $w$.
One way of doing this is by allowing a different constant value for $w$ than
the $\Lambda$CDM equivalent $-1$. Another would be through the
phenomenological relation $w=w_{0}+(1-a)\,w_{a}$ (Chevallier & Polarski, 2001;
Linder, 2003) mimicking a redshift dependent scalar field or other equivalent
model capable of inducing similar changes in dark energy density. There were
studies investigating the effect of a variation of EoS on the constraints of
the cosmological parameters, especially $\sigma_{8}$, using cluster counts or
probes at the local universe sensitive to the growth of structures. However,
unlike the method we followed in the companion papers of this work, they were
all mostly done after fixing the cluster observable mass calibration to that
determined from the hydrodynamical simulations. Among these works, Zhao et al.
(2017) used CMB $C_{\ell}$ combined with SZ cluster counts to put constraints
on the dark energy EoS parameter $w$ and neutrinos mass when the two are left
free to vary. They found that a $w$ different from its fiducial value, is
favoured by this combination and impacts the weighing of neutrinos, thus
indirectly suggesting that a $w$, alone or in combination with neutrinos,
could be a solution to the discrepancy. In addition, Joudaki et al. (2017)
compared the $\sigma_{8}$ constraints from local weak gravitational lensing
and that inferred from the CMB $C_{\ell}$ and found that a value for the dark
energy EoS parameter far from the equivalent fiducial $\Lambda$CDM value
$w=-1$ could alleviate the tension. Recently, Bolliet et al. (2018); Salvati
et al. (2018) and Bocquet et al. (2019), using another galaxy cluster-based
probe, the galaxy cluster power spectrum in combination with CMB $C_{\ell}$,
showed that a $w\neq-1$ is favoured over $\Lambda$CDM. Here we also
investigate whether the variation of the dark energy parameter, alone or
combined with neutrinos, could significantly reduce the cluster counts
discrepancy and alleviate the tension. Furthermore, we test how the
$\sigma_{8}$ \- $\Omega_{\rm m}$ degeneracy behaves when we allow the growth
index to vary along with the dark energy EoS parameter. We perform this study,
using a SZ cluster sample in combination with CMB Planck $C_{\ell}$ as our
main datasets in a MCMC parameter inference analysis. In line with our
previous investigations, we leave the cosmological parameters of the standard
model, the nuisance parameters, and the calibration parameter free to vary in
a model independent approach, letting the data fix the last parameter. To
alleviate possible degeneracies, we also consider an additional constraining
probe, the BAO feature angular distance, which is more sensitive to certain
expansion parameters (e.g. the Hubble constant) or to certain redshift ranges
than the others.
This study is organized in the following way. In Sect. 2 we review briefly the
theoretical determination of the cluster counts observable within the
extensions we consider: the dark energy density EoS parameter, the growth
index, and the neutrino mass. In Sect. 3 we describe the datasets we used and
how we implemented them in the MCMC analysis. We then present the results
obtained in Sect. 4 along with further evidence tests in Sect. 5 and conclude
in Sect. 6.

##  2 Cluster abundance and extensions to $\Lambda$CDM

Cluster abundance formalism and theoretical calculations in the $\Lambda$CDM
model or extensions to it, along with sample construction and selection
functions for the SZ detected cluster samples, is described in Planck
Collaboration et al. (2016b), SK18, and IL19. Here we briefly summarise most
of the theory and methods used when massive neutrinos or modified gravity are
introduced, along with a dark energy EoS parameter different from the fiducial
$\Lambda$CDM.

###  2.1 Mass function and cluster counts

Under the general hypothesis of self-similarity, the halo mass function (HMF)
can be written in a simple form (Blanchard et al., 1992) :

$n(m)=-\frac{\bar{\rho}}{m}\frac{\mathrm{d}{\rm\nu}}{\mathrm{d}m}\mathcal{F}({\rm\nu}).$
(1)

Here $\rho_{0}$ is the mean matter density today and $\mathcal{F}({\rm\nu})$
is a functional form of ${\rm\nu}=\delta_{c}/\sigma(m)$, the normalised
amplitude of fluctuations, where $\delta_{c}$ is the linear density threshold
at the present time for non-linear collapse and $\sigma(M)$ is the rms of the
linear density perturbations within a sphere of radius $R$ that contains mass
$M=4\pi/3\,\rho_{0}R^{3}$. The original form of the function $\mathcal{F}$
derived by Press & Schechter (1974) is

$\mathcal{F}_{\rm
PS}({\rm\nu})=\sqrt{\frac{2}{\pi}}\exp\left[-\frac{{\rm\nu}^{2}}{2}\right].$
(2)

A more refined evaluation of $\mathcal{F}$ has been the subject of numerous
investigations and continuous improvements of simulations that has allowed a
more accurate evaluation of the mass function such as Despali et al. (2016)
(DP16) HMF we use in this work, which updated the original evalutation of
Sheth & Tormen (1999),

$\mathcal{F}_{\rm
D}(\nu)=A\sqrt{\frac{2a}{\pi}}\left[1+\left(\frac{1}{a\nu^{2}}\right)^{p}\right]\
\ \exp\left[-\frac{a\nu^{2}}{2}\right],$ (3)

with new values for the fitted coefficients $A$, $a$, and $p$. Then, in order
to compare the theoretical HMF to the actual measured abundance of clusters of
galaxies, we need a relation between the cluster mass entering the mass
function and its observable quantities. This is done in the present study
using a sample of SZ detected clusters, where the distribution of clusters
function of redshift and S/N is written as

$\frac{dN}{dzdq}=\int d\Omega_{\rm mask}\int
d{M_{500}}\,\frac{dN}{dzd{M_{500}}d\Omega}\,P[q|{\bar{q}_{\rm
m}}({M_{500}},z,l,b)],$ (4)

with

$\frac{dN}{dzd{M_{500}}d\Omega}=\frac{dN}{dVd{M_{500}}}\frac{dV}{dzd\Omega},$
(5)

and the quantity $P[q|{\bar{q}_{\rm m}}({M_{500}},z,l,b)]$ being the
distribution of $q$ given the mean S/N value, ${\bar{q}_{\rm
m}}({M_{500}},z,l,b)$, predicted by the model for a cluster of mass
${M_{500}}$ (i.e. the overdensity defined at 500 with respect to the critical
density of the universe) and redshift $z$ located at Galactic coordinates
$(l,b)$.

We relate this quantity to the measured integrated Compton $y$-profile
$\bar{Y}_{500}$ using the scaling relations

$E^{-\beta}(z)\left[\frac{{D_{\rm
A}}^{2}(z){\bar{Y}_{500}}}{\mathrm{10^{-4}\,Mpc^{2}}}\right]=Y_{\ast}\left[{\frac{h}{0.7}}\right]^{-2+\alpha}\left[\frac{(1-b)\,{M_{500}}}{6\times
10^{14}\,M_{\odot}}\right]^{\alpha},$ (6)

where ${D_{\rm A}}$ is the angular diameter distance and $E(z)=H(z)/H_{0}$,
while $\alpha$, $\beta$, and $Y_{\ast}$ are additional parameters in the SZ
scaling law, along with $(1-b)$, which serves to link $M_{500}$ to $M_{\rm
X}$, the cluster mass determined from X-ray observations, playing the role of
the calibration parameter obtained from comparison with hydrostatic
simulations. There is an additional scaling parameter $\sigma_{\rm{ln}\,Y}$
that corresponds to the scatter of the log-normal distribution that the
measured $Y_{500}$ is assumed to follow with $\bar{Y}_{500}$ the mean, which
we also take into account in our analysis.

###  2.2 Extensions to the $\Lambda$CDM model and their implementation

####  2.2.1 Dark energy

A possible origin behind the discrepancy on $\sigma_{8}$ is that between the
time of recombination and the present time, a deviation from the $\Lambda$CDM
equivalent dark energy fluid EoS value changes the growth of structure in such
a way that the $\sigma_{8}$ is modified to the desired quantity.
The most common way to describe this deviation is by choosing a dark energy
EoS parameter $w$ constant $\neq-1$ as a first simple extension of
$\Lambda$CDM, the $w$CDM. We could also consider further expansions of the
dark energy EoS through the CPL phenomenological parametrisation mentioned in
Sect. 1:

$w=w_{0}+(1-a)w_{a}.$ (7)

This will modify the dark energy density $\rho_{\Lambda}$ entering
$\Omega_{\Lambda}$ in the Friedmann-Lemaître equations to

$\rho_{\Lambda\left(mod\right)}=\rho_{\Lambda}\left(1+z\right)^{3\left(1+\omega_{0}+\omega_{a}\right)}\exp\left(-3\omega_{a}\dfrac{z}{z+1}\right),$
(8)

with influence on the equation governing the growth of structure,

$\delta^{\prime\prime}(a)+\left[\frac{3}{a}+\frac{H^{\prime}(a)}{H(a)}\right]\delta^{\prime}(a)-\frac{3}{2}\Omega_{\rm
m}(a)\frac{H_{0}^{2}}{H^{2}(a)}\frac{\delta(a)}{a^{5}}=0\,,$ (9)

where a $w$ constant or redshift dependent enters the expression of $H(a)$ and
$\Omega_{\rm m}(a)$.
In addition to accounting for $w$ in the density budget and the growth rate in
the halo mass function, we leave the value of $\delta_{c}$ equal to its
fiducial value $\sim 1.686$, since Horellou & Berge (2005) and Pace et al.
(2010) have shown that $\delta_{c}$ remains almost unchanged when considering
dark energy with a $w\neq-1$ that is constant or evolving with redshift.
For the HMF multiplicity, several studies (see Baldi, 2012, and reference
therein) showed that the standard fitting performed in the $\Lambda$CDM
framework still provides a good fit to the results from the simulations with
constant $w\neq$-1 and parametrised EoS dark energy cosmologies, providing we
correct the growth rate of linear density perturbations to the value induced
by the background scaling of the dark energy density, as we do here.
Therefore, we can still assume the same functional HMF and its redshift
evolution provided by DP16 :

$\displaystyle a=$ $\displaystyle\,0.4332x^{2}+0.2263x+0.7665\,,$
$\displaystyle p=$ $\displaystyle\,-0.1151x^{2}+0.2554x+0.2488\,,$
$\displaystyle A=$ $\displaystyle\,-0.1362x+0.3292\,.$

Here $x=\log{(\Delta_{500}(z)/\Delta_{\rm{Vir}}(z))}$ is used to convert from
the virial mass definition to the critical cluster mass definition the SZ
sample selection function is built upon.
Finally, we rely on Aghanim et al. (2009), who showed that considering dark
energy cosmologies does not significantly alter the standard galaxy cluster
scaling relations, to keep the same functional form for Eqs. 5 and 6, which
are therefore kept when varying the EoS parameter $w$.

####  2.2.2 Growth index

The growth rate $f=d\ln\rm D/d\ln a$, where $D$ is the growth of perturbations
$\delta(z)=\delta_{0}D(z)$, could be approximated by
$f=\Omega_{m}^{\gamma}(z)$ with $\Omega_{m}(z)$ the matter density at $z$ and
$\gamma$, called the growth index. The growth rate in $\Lambda$CDM is well
approximated when the growth index is set to $\sim 0.545$ (Lightman &
Schechter, 1990; Lahav et al., 1991) and takes different values in other
modified gravity models (Linder, 2005). Since late large-scale structures
growth could influence the cluster number counts, and thus impact the
$\sigma_{8}$ values, a solution to the discrepancy from allowing a free growth
index was investigated in SK18.
However, since the effect of $\gamma$ on the growth rate could also be
degenerate with those of $w$, we investigate in this work how the constraints
on the cosmological parameters, in particular $\Omega_{\rm m}$ and
$\sigma_{8}$, behave when we allow both $\gamma$ and $w$ the dark energy
parameter to vary, following

$\sigma_{\gamma,wCDM}(z)=\frac{D_{\Lambda
CDM}(z_{*})}{D_{\gamma,CDM}(z_{*})}\frac{D_{\gamma,CDM}(z)}{D_{\Lambda
CDM}(z)}\,\sigma_{wCDM}(z).$ (10)

In addition, we consider a redshift dependent growth index following a recent
higher order parametrisation proposed by Steigerwald et al. (2014):

$\gamma=\gamma_{0}+\gamma_{1}\ln\left(\Omega_{m}\left(z\right)\right).$ (11)

####  2.2.3 Massive neutrinos

Neutrinos, when massive, contribute to the matter density as part of the
cosmological energy density budget with implications on the CMB spectrum.
However, although they are part of the matter density at late time, they do
not contribute to the growth of the linear matter power spectrum, an
ingredient of the cluster counts probe, but rather reduce it by a factor
$(1-f_{\rm\nu})^{2}$, where $f_{\rm\nu}\equiv\Omega_{\rm\nu}/\Omega_{\rm m}$
(Lesgourgues & Pastor, 2006).
There is an additional non-linear effect from massive neutrinos that needs to
be taken into account when evaluating clusters counts. Simulations by Costanzi
et al. (2013) and Castorina et al. (2014) have shown that we get a better
match between a simulated and calculated mass function if we use what is known
as the CDM prescription, where the matter density in the calculation is
replaced with the cold dark matter density and the power spectrum is replaced
by the cold dark matter power spectrum.
Similar to the companion papers (SK18), in order to investigate if the above
effects from neutrinos could mix with those of the aforementioned $\Lambda$CDM
extensions and influence their ability to fix the discrepancy, we further
consider here the case with a free neutrino mass in addition to a free dark
energy EoS parameter and a free growth index $\gamma$.
Finally, massive neutrinos are considered in the standard scenario of particle
physics where the effective number of relativistic particles $N_{\rm eff}$ is
equal to $3.046$. However, beyond the standard model, the physics may alter
this value due to the presence of extra relativistic contributions. Therefore,
in Appendix B, the effect of varying $N_{\rm eff}$ with $w\neq-1$ is also
discussed.

##  3 Datasets and methods

As one of our primary probes, we use the CMB $C_{\ell}$ of the temperature,
polarisation, and their cross-correlations from the publicly available
datasets of the Planck mission (Aghanim et al., 2020) and their publicly
provided likelihood. We combine them with the SZ detected cluster sample PSZ2
containing a total of 439 clusters (Planck Collaboration et al., 2016a)
spanning the redshift range from z$\sim$0.0 to z$\sim$1.3. We further
constrain the dark energy EoS using the scale of the baryonic acoustic
oscillation (BAO) of the galaxy distribution measurements (Beutler et al.,
2011; Ross et al., 2015; Alam et al., 2017). The inference of the constraints
on the parameters is done using standard MCMC techniques where observations
are confronted with theoretical angular power spectra calculated with the
Boltzmann solver CLASS (Lesgourgues, 2011). CLASS allows the effects of
different neutrino masses and species to be added, as well as different dark
energy evolution scenarios. The total neutrino mass $\sum m_{\rm\nu}$ varies
in the most general configuration with three degenerate massive active
neutrinos. Later the $N_{\rm eff}$ parameter is also left free to vary. Then
using the scaling relation of Eq. 6 with the calibration parameter $(1-b)$ for
the SZ clusters, we relate the mass to the SZ signal to construct the redshift
bins cluster counts. All is implemented in our SZ cluster counts module in the
framework of the parameter inference Monte Python code (Audren et al., 2013).
When running MCMC chains, we mainly let the normalisation parameter $(1-b)$
for SZ free to vary along with the six CMB cosmological parameters. For the
other cluster scaling parameters, we fix $\beta$ and $Y_{\ast}$ to their best
values obtained from simulations $\beta\sim 0.66$ and log$Y_{\ast}=-0.186$,
while leaving $\alpha$ free. We also leave $\sigma_{\rm{ln}\,Y}$ free with
Gaussian priors derived from its best fit $(=0.075\pm 0.01)$.
We note that we do not take into account cross-correlations between different
experiments as they are all, except the CMB measurements, almost independent
and do not overlap in time or space with each others. Later in this work, we
need to compute the Bayesian evidence (with respect to the model with the
adopted Planck collaboration calibration) of the alternative extensions we
take into consideration for addressing the $\sigma_{8}$ tension. Considering a
dataset $\mathbf{x}$ and two different models ${\cal M}_{i}$ and ${\cal
M}_{j}$, described by the parameters $\boldsymbol{\theta}_{i}$ and
$\boldsymbol{\theta}_{j}$. The Bayes factor of model ${\cal M}_{i}$ with
respect to model ${\cal M}_{j}$, denoted $B_{ij}$, is given by

$\displaystyle B_{ij}\equiv\frac{\int
d\boldsymbol{\theta}_{i}\,\pi(\boldsymbol{\theta}_{i}|{\cal M}_{i}){\cal
L}(\mathbf{x}|\boldsymbol{\theta}_{i},{\cal M}_{i})\,,}{\int
d\boldsymbol{\theta}_{j}\,\pi(\boldsymbol{\theta}_{j}|{\cal M}_{j}){\cal
L}(\mathbf{x}|\boldsymbol{\theta}_{j},{\cal M}_{j})\,,}\,,$ (12)

where $\pi(\boldsymbol{\theta}_{i}|{\cal M}_{i})$ is the prior for the
parameter $\boldsymbol{\theta}_{i}$, and ${\cal
L}(\mathbf{x}|\boldsymbol{\theta}_{i},{\cal M}_{i})$ is the likelihood of the
data given the model parameter $\boldsymbol{\theta}_{i}$. A Bayes factor
$B_{ij}>1$ (or equivalently $\ln B_{ij}>0$) indicates that model ${\cal
M}_{i}$ is more strongly supported by data than model ${\cal M}_{j}$.
Finally, for the purpose of quantifying the degree of correlation between the
parameters inferred from MCMC, we later calculate the Pearson coefficient
$r_{xy}$ for a set of paired values of the parameters we want to investigate,
following the formula

$r_{xy}=\frac{n\sum x_{i}y_{i}-\sum x_{i}\sum y_{i}}{\sqrt{n\sum
x_{i}^{2}-(\sum x_{i})^{2}}\leavevmode\nobreak\ \sqrt{n\sum y_{i}^{2}-(\sum
y_{i})^{2}}},$ (13)

where $n$ is the sample size and $x_{i}$, $y_{i}$ are the individual sample
points indexed with $i$. The coefficient has a value between +1 and -1, where
1 is total positive linear correlation, 0 is no linear correlation, and -1 is
total negative linear correlation.

##  4 Results

In Planck Collaboration et al. (2014) and Ilić et al. (2015), it was shown
that the discrepancy in galaxy cluster counts, could translate into that of
the mass observable calibration parameter. In particular, in a MCMC analysis
combining CMB and cluster counts with a free calibration parameter, the
preferred value for the free parameter was found to be $(1-b)\sim 0.6$,
corresponding to the $\sigma_{8}$ obtained from Planck CMB datasets, while the
$(1-b)$ calibration value adopted for the Planck SZ cluster sample is equal to
$0.8$ more than 4 $\sigma$ away from the above best value. Massive neutrinos
were advocated as a possible solution to alleviate the tension. However, it
was found in SK18 and IL19 that the combination of CMB $C_{\ell}$ and cluster
counts still yield the same constraints on the calibration parameter,
independently from neutrino mass, indicating that the latter is not able to
fix the discrepancy. Another extension to $\Lambda$CDM by means of a growth
index $\gamma$ different from the fiducial $\Lambda$CDM value of $\sim 0.55$
was also investigated as a way to alleviate the tension. Following the same
method of leaving the calibration parameter free to vary when combining CMB
$C_{\ell}$ and cluster counts, a correlation between the calibration parameter
and the growth index was observed, suggesting that the latter could alleviate
the tension. However, this was only when using an X-ray cluster sample at very
low redshift, while the constraints from the SZ deeper Planck cluster sample
(IL19) are forbidding $\gamma$ from reaching values that would allow
$(1-b)\sim 0.8$. Here we follow the same approach as in SK18 and IL19,
specifically we combine the CMB angular power spectrum and cluster counts
probes in a MCMC analysis to constrain $\sigma_{8}$ and the other cosmological
parameters, leaving the mass calibration parameter free in order to be fixed
by data and allowing, instead of $\gamma$, a free dark energy EoS parameter
$w$ as our extension to $\Lambda$CDM. We also explore other cases where we
additionally allow $\gamma$ or neutrinos to vary alongside $w$ to investigate
the impact of these combinations on the discrepancy. We limit our study to the
SZ cluster sample only, since the X-ray sample is much less constraining, and
it was shown in IL19 that the combination of the two samples does not have any
significant influence on the constraints obtained from the SZ alone.

###  4.1 Dark energy equation of state parameter and the $\sigma_{8}$
discrepancy

In Fig. 1 we show confidence contours for the parameters $\sigma_{8}$ and
$\Omega_{m}$ along with $(1-b)$, the mass observable calibration parameter,
obtained from a MCMC analysis using CMB Planck datasets in combination with
the cluster counts from the Planck SZ detected sample, while allowing $w$, the
dark energy equation parameter, to vary. We also perform the same analysis,
but by combining it with BAO. In the first case (green contours and lines) the
preferred calibration is still low: $1-b\sim 0.64$; the tension with $1-b\sim
0.8$ is largely reduced, being less than 2$\sigma$. However, with $w$ tightly
correlated with $(1-b)$, the agreement happens for values of $w\sim-3$ far
below $-1$, pushing cosmological parameters in a region: $H_{0}\sim 140$
km/s/Mpc and $\Omega_{m}\sim 0.08$ largely excluded by other cosmological
probes. With the addition of the BAO data (red contours and lines) the $w$
variation is now limited to a small region around $w\sim-1$, while the $(1-b)$
preferred value returns to that of $\Lambda$CDM $\sim 0.6$ with Planck SZ
cluster calibration at more than 5$\sigma$ away.

Figure 1: Two-dimensional confidence contours at 68% and 95% for the
parameters $\Omega_{\rm m}$, $H_{0}$, $\sigma_{8}$, $1-b$, and $w$ derived
from Planck 2018 CMB datasets combined with the Planck 2015 SZ detected galaxy
cluster sample (green). Adding the BAO probe datasets, still with a free
constant $w$, tightens the constraints considerably. The vertical black dashed
line gives the Planck calibration value $\left(1-b\sim 0.8\right)$.

###  4.2 Dark energy and growth index interplay

As seen above, the dark energy EoS parameter is not able to reduce the
discrepancy to less than 2$\sigma$. This is mainly limited by the need of $w$
to accommodate both volume change with cluster sample redshift bins and the
variation in the growth rate in the sample bins. This was also more notable in
IL19 when we introduced a growth index $\gamma$ to alleviate the tension,
without success despite observing a strong correlation between $\gamma$ and
the calibration factor $(1-b)$. To see if the combination with the SZ sample
is able to simultaneously constrain two degrees of freedom when we allow $w$
and $\gamma$ to vary simultaneously, we show in Fig. 2 the results of a MCMC
analysis combining CMB datasets and SZ cluster counts, letting all the
cosmological parameters free to vary along with the growth index $\gamma$,
$w$, and the mass-observable normalisation factor. We also perform the same
analysis adding BAO probes. We observe in the first case (red lines and
contours) that the combination of $w$ with $\gamma$ succeeds in alleviating
the tension with the $(1-b)$ preferred value, clearly shifting to become
centred around $0.8$. However, this is obtained with the other cosmological
parameters shifting far away from their standard values, so that when we add
the BAO probes (blue lines and contours), we again obtain strong constraints
on all parameters around the fiducial $\Lambda$CDM values, and the Planck SZ
calibration value (dashed black vertical line) becomes barely reachable at
$3\sigma$ level.

We also show the case when we allow the mass of neutrinos to vary in Fig. 3.
It was found in SK18 and IL19 that the neutrino mass alone has no effect on
fixing the discrepancy; however, when considered along with $\gamma$, it was
found that $\gamma$ is then allowed to explore values that reduced the
discrepancy, though remaining far from fixing it. Here when we follow the same
scenario and allow massive neutrinos, but also a free $w$ along with $\gamma$,
as expected we still see, as in the case of a free $w$ and $\gamma$ with
massless neutrinos, that the tension is clearly alleviated (brown lines).
However, here again we lose this gain when we add BAO probes (orange lines),
with the latter able to constrain both $w$ and the neutrino mass, leaving
$\gamma$ alone unable to fix the discrepancy. Nevertheless, we note an
improvement with respect to the case of a free $w$ and $\gamma$ with massless
neutrinos, in the sense that the tension is reduced to the 2$\sigma$ level for
a neutrino mass of $\sim 0.4$eV.

Figure 2: Confidence contours at 68% and 95% for the parameters $(1-b)$,
$\sigma_{8}$, $\Omega_{\rm m}$, $H_{0}$, $w$, and $\gamma$ derived from Planck
2018 CMB datasets combined with the Planck 2015 SZ detected galaxy cluster
sample, then adding BAO probe datasets, all with a free constant $w$ and
$\gamma$. The vertical black dashed line gives the Planck calibration value
$\left(1-b\sim 0.8\right)$. Figure 3: Confidence contours at 68% and 95% for
the parameters $(1-b)$ , $\sigma_{8}$, $\Omega_{\rm m}$, $H_{0}$, $w$,
$\gamma$, and $\Sigma m_{{\rm\nu}}$ derived from Planck 2018 CMB datasets
combined with the Planck 2015 SZ detected galaxy cluster sample, then adding
BAO probe datasets, all with a free constant $w$, $\gamma$, and neutrino mass.
The vertical black dash line gives the Planck calibration value $\left(1-b\sim
0.8\right)$.

###  4.3 Dynamical dark energy EoS vs a redshift-dependent growth index

Here we focus, for generality and to allow more freedom for $w$ and $\gamma$,
on investigating two additional cases. In the first the free dark energy EoS
parameter evolves with time with two parameters $w_{0}$ and $w_{a}$ following
the CPL parametrisation (Eq. 7) along with a constant $\gamma$, while in the
second we instead allowed a redshift dependent growth index $\gamma$ following
Eq. 11 along with a free $w$ constant in time. For the two schemes we use the
parametrisation following Eq. 10 and as priors, $w_{0}\in[-3.0,-0.33]$ and
$w_{a}\in[-4.0,3.0]$ while $\gamma_{0}\in[0.0,2.0]$ and
$\gamma_{1}\in[-3.0,3.0]$.

In Fig. 4 we see that promoting the constant EoS parameter to one evolving
with redshift with another degree of freedom $w_{a}$ has an impact on the
discrepancy that is similar to the case when we allowed massive neutrinos
along with a constant $w$ and $\gamma$, since the tension is largely
alleviated (blue lines), with $1-b\sim 0.8$ falling at the maximum of
likelihood. However, the addition of BAO tightens the confidence contours
(orange lines) ruling out the Planck calibration, again by almost 3$\sigma$.
Moreover, as observed before, even without adding BAO, solving the discrepancy
would still need the matter density parameter as well as the Hubble constant
to explore unrealistic values (i.e. excluded by almost all the cosmological
probes) with $\Omega_{\rm m}$ heading below $0.1$ and $H_{0}$ exploring values
higher than $100.0$ km/s/Mpc.

Figure 4: Two-dimensional confidence contours at 68% and 95% for the
parameters $\Omega_{\rm m}$, $H_{0}$, $w_{0}$, and $w_{a}$ of a CPL
parametrisation of dark energy EoS, $\gamma$, and $(1-b)$ derived from Planck
2018 CMB datasets combined with the Planck 2015 SZ detected galaxy cluster
sample (blue contours), then adding BAO probe datasets (orange contours). The
vertical black dashed line gives the Planck calibration value $\left(1-b\sim
0.8\right)$.

The situation does not change when we consider instead a dynamical $\gamma$ as
in Fig. 5. The additional $\gamma_{1}$ degree of freedom only results in
slightly larger confidence contours (magenta lines) with respect to the case
of a constant growth index with a free $w$ (grey lines), while the
$\Omega_{\rm m}$ and $H_{0}$ constraints are outside the common values.
Moreover, when we combine with BAO, although it is not able to further
constrain the growth index, we observe that an evolving growth index drives
$(1-b)$ towards lower values (pink lines) that would rather increase the
discrepancy on $\sigma_{8}$.

Figure 5: Two-dimensional confidence contours at 68% and 95% for the
parameters $\Omega_{\rm m}$, $H_{0}$, $\gamma_{0}$, $\gamma_{1}$, and $w_{0}$
of a dynamical parametrisation of the growth index and $w_{0}$ and $(1-b)$,
derived from Planck 2018 CMB datasets combined with Planck 2015 SZ detected
galaxy cluster sample, then combined with BAO probe datasets. The vertical
black dashed line gives the Planck calibration value $\left(1-b\sim
0.8\right)$.

##  5 Further tests of evidence

In this section we try to further consolidate the previous conclusions by
adopting a model comparison approach, i.e. we consider that we are comparing
two models, the one with the free calibration parameter $(1-b)$ with the three
main extensions to $\Lambda$CDM ($w$, $w$ \+ $\gamma$, and $w$ \+ $\gamma$ \+
$\sum m_{\nu}$) against the one with $(1-b)\sim 0.8$, the calibration adopted
by Planck, allowed to vary but with a very narrow Gaussian error $\pm 0.01$
following (Planck Collaboration et al., 2014), also considering the previous
extensions. We then compare the Bayesian MCMC outcomes for the two models by
the classical 1D and 2D likelihood confidence contour separation in $\sigma$
and by using another method through the Bayesian evidence index (see Sect. 3
for a description and the method of evaluation).

We thus show in Fig. 6 the confidence contours of the $(1-b)$-$w$ parameters
for the three $\Lambda$CDM extensions for the two cases: a free calibration
parameter and $1-b\sim 0.8$ with a very restrictive Gaussian prior, and that
using the combination of clusters and CMB datasets along with the same scheme
but adding the BAO data. We clearly see in the first panel that the Planck
calibration is not reached even with a free $w$, while imposing $1-b\sim 0.8$
restricts $w$ closer to -1. Adding BAO further consolidates the finding by
shrinking the green contours to the red ones in Fig. 6. Allowing in addition a
free $\gamma$ in the second panel improves the situation and reduces to 1
$\sigma$ the discrepancy when comparing the red and grey contours, but the
addition of BAO increases it again to $\sim$3 $\sigma$. Instead, further
allowing neutrinos at the expense of more degrees of freedom, moves more
towards alleviating the tension, even if we assume a very good accuracy on
$(1-b)$, with the grey contours overlapping the brown ones. However, still
combining with BAO restores it back to $\sim$2 $\sigma$.

Figure 6: Confidence contours (68 and 95%) and posterior distributions for the
dark energy EoS parameter $w$ and mass calibration parameter $(1-b)$ when
combining CMB (with or without BAO) and SZ cluster data for three different
cases of allowing extensions to $\Lambda$CDM (detailed in the legends), in
comparison with confidence contours from the effect of a strong prior on
$(1-b)$ centred around the Planck calibration value of 0.8.

The values of the the Bayes factor in Table 1 reflects the previous
observations. Thus, we see in the first (respectively second) column,
corresponding to the Bayes factor computed for the three extensions for the
cases without (respectively with) BAO, that $\ln B_{ij}$ is always ¿ 1, which
we consider strong evidence that the model with $1-b\sim 0.8$ is not
preferred. We see that $\ln B_{ij}$ decreases when considering further
extensions, which is a sign that the addition of these extensions pushes
towards the model with the Planck calibration, however without preferring it.
We also observe that the combination with BAO in $\Lambda$CDM increases $\ln
B_{ij}$ as expected and depicted in Fig. 6 with the exception of the case with
only free $w$. The reason for this might be that BAO tightly restricts $w$ to
the same range for the two models, which is reflected in the calculation of
the Bayes factor that includes global contribution from the parameters
involved. These parameters are not compensated for by the difference in the 1D
likelihood of $(1-b)$, unlike the other two extensions where the higher
overlapping of the 1D confidence contours of the $(1-b)$ parameters cover the
gain from restricting $w$.

Table 1: Bayes factor ($\ln B_{ij}$) values for the $\Lambda$CDM extensions and mass observable parameters.  Parameters \ Data | CMB + Clust | CMB + Clust + BAO
---|---|---
$w$ | $20.64$ | $19.79$
$w$ \+ $\gamma$ | $4.42$ | $5.89$
$w$ \+ $\gamma$ \+ $\sum m_{\nu}$ | $1.81$ | $3.77$

##  6 Discussion and conclusion

In the present paper we examined whether a dark energy EoS parameter $w$ is
able to fix the discrepancy found in the value of $\sigma_{8}$ determined from
local probes like cluster counts with respect to that obtained at deep
redshifts from CMB $C_{\ell}$. Our strategy was to examine the constraints
that CMB and cluster abundance data yield, without further additional
assumption on clusters (i.e. leaving the calibration parameter of the mass
observable relation free to vary). In most of the cases the addition of
massive neutrinos was also considered. We found first, independently from
allowing or not massive neutrinos, that a dark energy EoS $w$, constant or
evolving with time, when left free, can fix the discrepancy only for $w$ well
below $-1$. However, these values are excluded by additional constraints from
BAO data.
Next we introduced, along with a free $w$, the $\gamma$ parametrisation to
model the effect of a modification of gravity on the growth rate. Though the
discrepancy is further reduced with the $(1-b)$ maximum likelihood value
becoming centred around 0.8; however, here again, combining it with
geometrical distance probes like BAO reverts the $(1-b)$ preferred value back
to the $\Lambda$CDM value of 0.6. The same is observed after introducing,
along with $w$ and $\gamma$, further extensions to $\Lambda$CDM like massive
neutrinos, where we found that a greater ease to the tension is found for
higher neutrino masses only when they are not constrained by the combination
with BAO datasets.
Finally, we compared the effect of allowing an evolving $w$ in time through
CPL parametrisation in addition to constant growth index against the case
where the index is made dynamical through a second parameter $\gamma_{1}$ in
addition to a constant $w$. In the first case, we reached the same conclusions
as with a constant $w$ and $\gamma$, where the discrepancy was alleviated only
if we do not further constrain $w$ by BAO, while in the second case we find
that an evolving growth index does not bring any improvement to the situation
since it drives $(1-b)$ towards lower values that instead increase the
discrepancy on $\sigma_{8}$.
To further consolidate our findings, we compared the models with the
extensions and a free calibration parameter against those with the same
extensions but with a calibration considered to be constrained to the per cent
level to the Planck calibration, and that by means of the common method of
comparing the confidence contours from MCMC chains or through another method,
based on comparing the values of the Bayes factors for each case. We found
that the two methods are in agreement with our previous findings and that the
values of the Bayes factor strongly supports that the proposed extensions are
not preferred over a change in the value of the mass observable calibration as
a way to fix the discrepancy.

We conclude then that, none of the extensions to $\Lambda$CDM model we tested
as possible solutions to solve the discrepancy is preferred over the
hypothesis of an improper calibration of the mass observable scaling relation.
However, if the Planck SZ calibration, $(1-b)\sim 0.8$, is consolidated in the
future (e.g. to $\sim$ 1% in the case we showed), the failure of the advocated
extensions to $\Lambda$CDM to fix the discrepancy (i.e. massive neutrinos,
dark energy, and a modified gravity through $\gamma$) would call for new
physics beyond the standard cosmological model common modifications.

This is especially true after finding, within the probes used in this work,
that the constraints on the influential parameters need to drastically shift
if we want to ease the tension : the matter density or dark energy EoS
parameter should reach very low values, around $\sim 0.1$ for $\Omega_{m}$ and
$<-3$ for $w$, while the Hubble constant and the growth index prefer very high
values, $>100$ for $H_{0}$ and $\sim 1.0$ for $\gamma$. This puts the
discrepancy at a high level in terms of the difficulty to solve, without
spoiling the remarkable concordance of the $\Lambda$CDM model (see e.g.
Benevento et al. (2022) for a relative success in doing so with a modified
gravity model). Within the current available measurements, in addition to the
numerous studies reviewed in the introduction of this work (see Sect. 1) there
have been some recent attempts to investigate and improve the bounds on the
mass observable calibration, such as using machine learning methods to reduce
the SZ flux-mass scatter (Wadekar et al., 2022), but only future generation
surveys, such as Euclid Laureijs et al. (2011), eRosita Borm et al. (2014),
DESI Aghamousa et al. (2016), Wfirst-Roman Spergel et al. (2015), LSST-Rubin
LSST Science Collaboration: Abell et al. (2009), SKA Bacon et al. (2020), and
NIKA2 SZ Large Program (LPSZ) Kéruzoré et al. (2022), will allow us to reach a
high reduction of the error on the calibration of the mass observable that
will confirm or not this discrepancy to a very high degree of evidence (see
also Pratt et al. 2019 for a review on the latest techniques and future
prospects).

## References

  * Abbott et al. (2018) Abbott, T. M. C., Abdalla, F. B., Alarcon, A., et al. 2018, Phys. Rev. D, 98, 043526
  * Abdullah et al. (2020) Abdullah, M. H., Klypin, A., & Wilson, G. 2020, ApJ, 901, 90
  * Aghamousa et al. (2016) Aghamousa, A. et al. 2016 [arXiv:1611.00036]
  * Aghanim et al. (2009) Aghanim, N., da Silva, A. C., & Nunes, N. J. 2009, A&A, 496, 637
  * Aghanim et al. (2020) Aghanim, N. et al. 2020, Astron. Astrophys., 641, A5
  * Aiola et al. (2020) Aiola, S. et al. 2020, JCAP, 12, 047
  * Alam et al. (2017) Alam, S., Ata, M., Bailey, S., et al. 2017, MNRAS, 470, 2617
  * Allen et al. (2011) Allen, S. W., Evrard, A. E., & Mantz, A. B. 2011, ARA&A, 49, 409
  * Amon et al. (2022) Amon, A. et al. 2022, Phys. Rev. D, 105, 023514
  * Andreon (2014) Andreon, S. 2014, Astron. Astrophys., 570, L10
  * Applegate et al. (2014) Applegate, D. E., von der Linden, A., Kelly, P. L., et al. 2014, MNRAS, 439, 48
  * Ascaso et al. (2017) Ascaso, B., Mei, S., Bartlett, J. G., & Benítez, N. 2017, MNRAS, 464, 2270
  * Asgari et al. (2020) Asgari, M., Tröster, T., Heymans, C., et al. 2020, A&A, 634, A127
  * Audren et al. (2013) Audren, B., Lesgourgues, J., Benabed, K., & Prunet, S. 2013, J. Cosmology Astropart. Phys., 2, 001
  * Bacon et al. (2020) Bacon, D. J. et al. 2020, Publ. Astron. Soc. Austral., 37, e007
  * Bahcall et al. (1997) Bahcall, N. A., Fan, X., & Cen, R. 1997, ApJ, 485, L53
  * Bahe et al. (2012) Bahe, Y. M., McCarthy, I. G., & King, L. J. 2012, Mon. Not. Roy. Astron. Soc., 421, 1073
  * Baldi (2012) Baldi, M. 2012, Physics of the Dark Universe, 1, 162
  * Battaglia et al. (2016) Battaglia, N., Leauthaud, A., Miyatake, H., et al. 2016, J. Cosmology Astropart. Phys., 8, 013
  * Benevento et al. (2022) Benevento, G., Kable, J. A., Addison, G. E., & Bennett, C. L. 2022 [arXiv:2202.09356]
  * Beutler et al. (2011) Beutler, F., Blake, C., Colless, M., et al. 2011, MNRAS, 416, 3017
  * Blanchard & Bartlett (1998) Blanchard, A. & Bartlett, J. G. 1998, A&A, 332, L49
  * Blanchard & Douspis (2005) Blanchard, A. & Douspis, M. 2005, A&A, 436, 411
  * Blanchard et al. (1992) Blanchard, A., Valls-Gabaud, D., & Mamon, G. A. 1992, A&A, 264, 365
  * Bocquet et al. (2019) Bocquet, S., Dietrich, J. P., Schrabback, T., et al. 2019, ApJ, 878, 55
  * Bocquet et al. (2016) Bocquet, S., Saro, A., Dolag, K., & Mohr, J. J. 2016, Mon. Not. Roy. Astron. Soc., 456, 2361
  * Bolliet et al. (2018) Bolliet, B., Comis, B., Komatsu, E., & Macías-Pérez, J. F. 2018, MNRAS, 477, 4957
  * Borm et al. (2014) Borm, K., Reiprich, T. H., Mohammed, I., & Lovisari, L. 2014, Astron. Astrophys., 567, A65
  * Brandbyge et al. (2010) Brandbyge, J., Hannestad, S., Haugbølle, T., & Wong, Y. Y. Y. 2010, J. Cosmology Astropart. Phys., 9, 014
  * Castorina et al. (2014) Castorina, E., Sefusatti, E., Sheth, R. K., Villaescusa-Navarro, F., & Viel, M. 2014, J. Cosmology Astropart. Phys., 2, 049
  * Castro et al. (2020) Castro, T., Borgani, S., Dolag, K., et al. 2020, Mon. Not. Roy. Astron. Soc., 500, 2316
  * Chevallier & Polarski (2001) Chevallier, M. & Polarski, D. 2001, International Journal of Modern Physics D, 10, 213
  * Costanzi et al. (2013) Costanzi, M., Villaescusa-Navarro, F., Viel, M., et al. 2013, J. Cosmology Astropart. Phys., 12, 012
  * de Haan et al. (2016) de Haan, T., Benson, B. A., Bleem, L. E., et al. 2016, ApJ, 832, 95
  * Debackere et al. (2021) Debackere, S. N. B., Schaye, J., & Hoekstra, H. 2021, Mon. Not. Roy. Astron. Soc., 505, 593
  * Despali et al. (2016) Despali, G., Giocoli, C., Angulo, R. E., et al. 2016, MNRAS, 456, 2486
  * Di Valentino et al. (2016) Di Valentino, E., Giusarma, E., Lattanzi, M., et al. 2016, Physics Letters B, 752, 182
  * Dvorkin et al. (2014) Dvorkin, C., Wyman, M., Rudd, D. H., & Hu, W. 2014, Phys. Rev. D, 90, 083503
  * Ettori & Eckert (2022) Ettori, S. & Eckert, D. 2022, Astron. Astrophys., 657, L1
  * Ghirardini et al. (2021) Ghirardini, V. et al. 2021, Astrophys. J., 910, 14
  * Giusarma et al. (2014) Giusarma, E., Di Valentino, E., Lattanzi, M., Melchiorri, A., & Mena, O. 2014, Phys. Rev. D, 90, 043507
  * Hamana et al. (2012) Hamana, T., Oguri, M., Shirasaki, M., & Sato, M. 2012, Mon. Not. Roy. Astron. Soc., 425, 2287
  * Henson et al. (2017) Henson, M. A., Barnes, D. J., Kay, S. T., McCarthy, I. G., & Schaye, J. 2017, Mon. Not. Roy. Astron. Soc., 465, 3361
  * Herbonnet et al. (2020) Herbonnet, R., Sifon, C., Hoekstra, H., et al. 2020, Mon. Not. Roy. Astron. Soc., 497, 4684
  * Hoekstra et al. (2015) Hoekstra, H., Herbonnet, R., Muzzin, A., et al. 2015, MNRAS, 449, 685
  * Horellou & Berge (2005) Horellou, C. & Berge, J. 2005, MNRAS, 360, 1393
  * Ilić et al. (2015) Ilić, S., Blanchard, A., & Douspis, M. 2015, A&A, 582, A79
  * Ilić et al. (2019) Ilić, S., Sakr, Z., & Blanchard, A. 2019, A&A, 631, A96
  * Ingoglia et al. (2022) Ingoglia, L. et al. 2022, Mon. Not. Roy. Astron. Soc., 511, 1484
  * Joudaki et al. (2017) Joudaki, S., Mead, A., Blake, C., et al. 2017, MNRAS, 471, 1259
  * Kéruzoré et al. (2022) Kéruzoré, F., Artis, E., Macías-Pérez, J.-F., et al. 2022, EPJ Web Conf., 257, 00025
  * Kirby et al. (2019) Kirby, M., Rozo, E., Morris, R. G., et al. 2019, arXiv e-prints, arXiv:1910.13548
  * Klein et al. (2019) Klein, M., Israel, H., Nagarajan, A., et al. 2019, Mon. Not. Roy. Astron. Soc., 488, 1704
  * Lahav et al. (1991) Lahav, O., Lilje, P. B., Primack, J. R., & Rees, M. J. 1991, MNRAS, 251, 128
  * Laureijs et al. (2011) Laureijs, R. et al. 2011 [arXiv:1110.3193]
  * Lesci et al. (2022) Lesci, G. F. et al. 2022, Astron. Astrophys., 659, A88
  * Lesgourgues (2011) Lesgourgues, J. 2011, ArXiv e-prints [arXiv:1104.2932]
  * Lesgourgues & Pastor (2006) Lesgourgues, J. & Pastor, S. 2006, Phys. Rep, 429, 307
  * Lightman & Schechter (1990) Lightman, A. P. & Schechter, P. L. 1990, ApJS, 74, 831
  * Linder (2003) Linder, E. V. 2003, Physical Review Letters, 90, 091301
  * Linder (2005) Linder, E. V. 2005, Phys. Rev. D, 72, 043529
  * Lovisari et al. (2020) Lovisari, L., Ettori, S., Sereno, M., et al. 2020, Astron. Astrophys., 644, A78
  * LSST Science Collaboration: Abell et al. (2009) LSST Science Collaboration: Abell, P. A., Allison, J., Anderson, S. F., et al. 2009, arXiv e-prints, arXiv:0912.0201
  * Manolopoulou et al. (2020) Manolopoulou, M., Hoyle, B., Mann, R. G., Sahlen, M., & Nadathur, S. 2020, Mon. Not. Roy. Astron. Soc., 500, 1953
  * McDonald et al. (2017) McDonald, M. et al. 2017, Astrophys. J., 843, 28
  * Melin & Bartlett (2015) Melin, J.-B. & Bartlett, J. G. 2015, Astron. Astrophys., 578, A21
  * Meneghetti et al. (2020) Meneghetti, M. et al. 2020, Science, 369, 1347
  * Mohr (2005) Mohr, J. J. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 339, Observing Dark Energy, ed. S. C. Wolff & T. R. Lauer, 140
  * Motloch & Hu (2020) Motloch, P. & Hu, W. 2020, Phys. Rev. D, 101, 083515
  * Mummery et al. (2017) Mummery, B. O., McCarthy, I. G., Bird, S., & Schaye, J. 2017, Mon. Not. Roy. Astron. Soc., 471, 227
  * Murray et al. (2022) Murray, C., Bartlett, J. G., Artis, E., & Melin, J.-B. 2022, Mon. Not. Roy. Astron. Soc., 512, 4785
  * Oukbir & Blanchard (1992) Oukbir, J. & Blanchard, A. 1992, A&A, 262, L21
  * Pace et al. (2010) Pace, F., Waizmann, J.-C., & Bartelmann, M. 2010, MNRAS, 406, 1865
  * Pearson (1895) Pearson, K. 1895, Proceedings of the Royal Society of London Series I, 58, 240
  * Peebles (1980) Peebles, P. J. E. 1980, The large-scale structure of the universe
  * Penna-Lima et al. (2017) Penna-Lima, M., Bartlett, J. G., Rozo, E., et al. 2017, A&A, 604, A89
  * Planck Collaboration et al. (2014) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2014, A&A, 571, A29
  * Planck Collaboration et al. (2016a) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016a, A&A, 594, A27
  * Planck Collaboration et al. (2016b) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016b, A&A, 594, A13
  * Planck Collaboration et al. (2020) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6
  * Pratt et al. (2019) Pratt, G. W., Arnaud, M., Biviano, A., et al. 2019, Space Sci. Rev., 215, 25
  * Press & Schechter (1974) Press, W. H. & Schechter, P. 1974, ApJ, 187, 425
  * Rasia et al. (2012) Rasia, E. et al. 2012, New J. Phys., 14, 055018
  * Ross et al. (2015) Ross, A. J., Samushia, L., Howlett, C., et al. 2015, MNRAS, 449, 835
  * Sakr et al. (2018) Sakr, Z., Ilić, S., Blanchard, A., Bittar, J., & Farah, W. 2018, A&A, 620, A78
  * Salvati et al. (2018) Salvati, L., Douspis, M., & Aghanim, N. 2018, A&A, 614, A13
  * Salvati et al. (2019) Salvati, L., Douspis, M., Ritz, A., Aghanim, N., & Babul, A. 2019, Astron. Astrophys., 626, A27
  * Salvati et al. (2021) Salvati, L. et al. 2021 [arXiv:2112.03606]
  * Schrabback et al. (2021) Schrabback, T. et al. 2021, Mon. Not. Roy. Astron. Soc., 505, 3923
  * Sheth & Tormen (1999) Sheth, R. K. & Tormen, G. 1999, MNRAS, 308, 119
  * Shi et al. (2016) Shi, X., Komatsu, E., Nagai, D., & Lau, E. T. 2016, Mon. Not. Roy. Astron. Soc., 455, 2936
  * Spergel et al. (2015) Spergel, D. et al. 2015 [arXiv:1503.03757]
  * Steigerwald et al. (2014) Steigerwald, H., Bel, J., & Marinoni, C. 2014, J. Cosmology Astropart. Phys., 5, 042
  * Sunayama et al. (2016) Sunayama, T., Hearin, A. P., Padmanabhan, N., & Leauthaud, A. 2016, Mon. Not. Roy. Astron. Soc., 458, 1510
  * Valageas (2014) Valageas, P. 2014, Astron. Astrophys., 561, A53
  * Vikhlinin et al. (2009) Vikhlinin, A., Kravtsov, A. V., Burenin, R. A., et al. 2009, ApJ, 692, 1060
  * Wadekar et al. (2022) Wadekar, D., Thiele, L., Villaescusa-Navarro, F., et al. 2022 [arXiv:2201.01305]
  * Wicker et al. (2022) Wicker, R., Douspis, M., Salvati, L., & Aghanim, N. 2022 [arXiv:2204.12823]
  * Zhang et al. (2008) Zhang, Y. Y., Finoguenov, A., Boehringer, H., et al. 2008, Astron. Astrophys., 482, 451
  * Zhang et al. (2022) Zhang, Z. et al. 2022 [arXiv:2202.08211]
  * Zhao et al. (2017) Zhao, M.-M., Li, Y.-H., Zhang, J.-F., & Zhang, X. 2017, MNRAS, 469, 1713

##  Appendix A Hubble constant prior range and its impact on the posteriors

So far, when using only CMB datasets and cluster counts, we allowed large
priors for the Hubble constant enabling it to potentially reach $H_{0}\sim$
200.0, well above the current constraints from other probes like BAO for
example. The motivation was to permit $w$ to explore values well below $-1$
that may reconcile CMB and cluster counts constraints on $\sigma_{8}$. We
observed in Fig. 1 that the discrepancy on $\sigma_{8}$, which directly
translates into that on $(1-b)$, was alleviated for $w\ll$ -1 where the
$(1-b)$ likelihood reached $\sim$ 0.8, the Planck SZ sample calibration, to
less than 2$\sigma$ and for values of $H_{0}\sim$ 160.0. The same was also
observed in other cases, in particular in Fig. 3 where we also allowed the
growth index $\gamma$ to vary. Here we repeated, in Fig. 7 and Fig. 8, the
previous analysis, but limiting the prior range of the Hubble constant to
$\in[30,100]$, in line with the current constraints on $H_{0}$. We observe
then three main differences between the two posteriors for the two considered
$H_{0}$ priors:

  * •

The discrepancy is not alleviated in the free $w$ only case with (1-b) maximum
likelihood centred far from $1-b\sim 0.8$ with the latter falling at a level
of more than 3$\sigma$. The same is seen in the free $w$ and $\gamma$ cases
where the Planck calibration still falls at the 3$\sigma$ level. This is
especially seen in the $w$ plus $\gamma$ case where the _cut_ in the $H_{0}$
prior does not trivially translate in the posterior contours with also a
change in the likelihood maximum value.

  * •

The preferred values of $w$ in the $H_{0}$ tight prior case are now close to
-1, while they were far below that value for the $H_{0}$ large prior case with
the $w$ corresponding to $\Lambda$CDM falling at the $3\sigma$ level.

  * •

The correlation between $w$ and $(1-b)$, found in the $H_{0}$ large prior case
as a sign of its ability to fix the discrepancy, has vanished in the tight
prior case, while that of the modified gravity related parameter $\gamma$ with
$(1-b)$ is still strong, indicating its ability, unlike $w$, to solve the
discrepancy. This can also be seen in Table 2 where the correlation
coefficient described in Sect. 3 is calculated for different cases with
$h\in[30,100]$. We then see that the index for $(1-b)$ \- $\gamma$ reaches
high values, above 0.5, contrary to $(1-b)$ \- $w_{0}$, which remains at very
low values, well below the median for the correlation index.

Figure 7: Two-dimensional confidence contours at 68% and 95% for the
parameters $\Omega_{\rm m}$, $\sigma_{8}$, $(1-b)$, and $w$ derived from
Planck 2018 CMB datasets combined with the Planck 2015 SZ detected galaxy
cluster sample for $H_{0}\in[30,100]$ (blue lines) in comparison with
$H_{0}\in[30,200]$ (red lines). Figure 8: Confidence contours at 68% and 95%
for the parameters $(1-b)$ , $\sigma_{8}$, $\Omega_{\rm m}$, $w$, and $\gamma$
derived from Planck 2018 CMB datasets combined with the Planck 2015 SZ
detected galaxy cluster sample for $H_{0}\in[30,100]$ (red lines) in
comparison with $H_{0}\in[30,200]$ (blue lines). Table 2: Evaluations of the
Pearson correlation coefficient for the pair $(1-b)$ \- $\gamma$ vs $(1-b)$ \-
$\omega_{0}$, the inferred parameters from MCMC using different combinations
of the CMB datasets and the SZ detected cluster sample.

|  $(1-b)$ \- $\gamma$ |  $(1-b)$ \- $\omega_{0}$
---|---|---
|  |
SZ+CMB | $0.711$ | $0.295$
SZ+CMB+$M_{\rm\nu}$ | $0.736$ | $-0.017$
SZ+CMB+BAO | $0.750$ | $-0.010$
SZ+CMB+$w_{0}-w_{a}$ | $0.749$ | $-0.024$
SZ+CMB+$\gamma_{0}-\gamma_{1}$ | $0.702$ | $0.241$

##  Appendix B Effective relativistic number of neutrinos in combination with
$w$

Figure 9: Two-dimensional confidence contours at 68% and 95% for the
parameters $\Omega_{\rm m}$, $\sigma_{8}$, $w_{0}$, free constant dark energy,
and the effective number of neurinos $N_{\rm eff}$ in color distribution
representation derived from MCMC analysis combining CMB datasets and a local
X-ray cluster sample.

In the main text we showed that allowing massive neutrinos has no substantial
effect in helping $w$ to fix the discrepancy. This was while keeping the
number of neutrino species $N_{\rm eff}\sim 3.046$, the value favoured by
earth experiments. Here we leave $N_{\rm eff}$ free in addition to $w$ while
fixing the mass calibration parameter to the Planck calibration. We see in
Fig. 9 that the $N_{\rm eff}$ confidence contours are in agreement with the
fiducial value $3.046$ as well as $w$ with -1, mainly because the variation in
$N_{\rm eff}$ induce changes to the correlation $w$ \- $\Omega_{\rm m}$ in
opposite directions to that between $w$ and $\sigma_{8}$, as indicated from
the colour scheme we used to represent the variation in $N_{\rm eff}$. This
results in keeping the $\sigma_{8}-\Omega_{\rm m}$ contours unchanged and in
agreement with fiducial $N_{\rm eff}$, clearly shows that varying the $N_{\rm
eff}$ parameter does not help $w$ in reconciling cluster counts and CMB
$C_{\ell}$ constraints on $\sigma_{8}$.

###### Acknowledgements.

We thank the anonymous referees for their valuable and constructive comments
and suggestions on our manuscript. Ziad Sakr was supported by a grant of
excellence from the Agence des Universités Francophones (AUF). Ziad Sakr
acknowledges support from the IRAP Toulouse and IN2P3 Lyon computing centers.
Stéphane Ilic thanks the Centre national d’études spatiales which supports his
postdoctoral research contract.