# Mixed dark matter: matter power spectrum and halo mass function

G. Parimbelli  G. Scelfo  S. K. Giri  A. Schneider  M. Archidiacono  S. Camera
M. Viel

###### Abstract

We investigate and quantify the impact of mixed (cold and warm) dark matter
models on large-scale structure observables. In this scenario, dark matter
comes in two phases, a cold one (CDM) and a warm one (WDM): the presence of
the latter causes a suppression in the matter power spectrum which is allowed
by current constraints and may be detected in present-day and upcoming
surveys. We run a large set of $N$-body simulations in order to build an
efficient and accurate emulator to predict the aforementioned suppression with
percent precision over a wide range of values for the WDM mass,
$M_{\mathrm{wdm}}$, and its fraction with respect to the totality of dark
matter, $f_{\mathrm{wdm}}$. The suppression in the matter power spectrum is
found to be independent of changes in the cosmological parameters at the 2%
level for $k\lesssim 10\ h/$Mpc and $z\leq 3.5$. In the same ranges, by
applying a baryonification procedure on both $\Lambda$CDM and CWDM simulations
to account for the effect of feedback, we find a similar level of agreement
between the two scenarios. We examine the impact that such suppression has on
weak lensing and angular galaxy clustering power spectra. Finally, we discuss
the impact of mixed dark matter on the shape of the halo mass function and
which analytical prescription yields the best agreement with simulations. We
provide the reader with an application to galaxy cluster number counts.

##  1 Introduction

In the last two decades, the $\Lambda$ cold dark matter ($\Lambda$CDM) model
has settled as the standard cosmological theory. With a small number of
parameters, it provides an accurate description of the Universe at large
scales. It is well known, however, that there are considerable uncertainties
and potential inconsistencies at galactic and sub-galactic scales, which could
be related to the cold and collisionless nature of dark matter (DM). In
particular, the $\Lambda$CDM model has been claimed to overestimate the number
of satellite galaxies with respect to actual observations (“missing satellites
problem”, e.g. [1, 2, 3, 4, 5, 6, 7]). Moreover, halo profiles are found to be
much more diverse for galaxies with similar rotation curves (“diversity
problem” [8, 9]) and with constant density cores (“cusp/core problem”, e.g.
[10, 11, 12, 13]) with respect to what $\Lambda$CDM predicts. Finally, it has
been argued that in order to match the observed dwarf galaxy population around
the Milky Way, the most massive subhalos in $\Lambda$CDM simulations need to
remain without stars, which seems at odds with the standard picture of galaxy
formation (“too-big-to-fail problem”, e.g. [14, 15, 4, 16, 17]).

Although properly accounting for baryons can partially solve these problems
(e.g. [18, 19, 20]), the persistence of some issues [21], together with the
null result of laboratory searches of dark matter particles [22], motivated
the study of non-cold dark matter scenarios. Several models have been proposed
in this direction (e.g. interacting DM [9, 23], axions [24]). Among them one
of the long standing solutions of the cold dark matter (CDM) crisis is
provided by dark matter that decouples from the primordial plasma when still
relativistic (like standard neutrinos), but soon afterwards becomes non-
relativistic. In this picture, dark matter is warm (WDM) and it is able to
free-stream until late times, damping all the density perturbations below a
characteristic “free-streaming scale”. This particular scale is in turn
associated to a free-streaming mass $M_{\mathrm{fs}}$, a threshold below which
structure formation is heavily suppressed [25]. We can write

$M_{\mathrm{fs}}(\Omega_{\mathrm{wdm}},M_{\mathrm{wdm}})=10^{10}\left(\frac{\Omega_{\mathrm{wdm}}}{0.3}\right)^{1.45}\left(\frac{h}{0.65}\right)^{3.9}\left(\frac{1\
\mathrm{keV}}{M_{\mathrm{wdm}}}\right)^{3.45}\ M_{\odot}/h\,,$ (1.1)

with $M_{\mathrm{wdm}}$ is the WDM particle mass, $\Omega_{\mathrm{wdm}}$ its
density parameter and $h$ the dimensionless Hubble parameter. Free-streaming
becomes astrophysically relevant for masses of $\mathcal{O}$(keV) and below:
typical candidates are gravitinos [26, 27, 28] and sterile neutrinos [29, 30].
Currently, the tighest constraints come from Lyman-$\alpha$ data [31], in
particular from the combination of the XQ-100 flux power spectrum and
HIRES/MIKE, which yields $M_{\mathrm{wdm}}>5.3$ keV at 95% confidence level
(C.L.). The latter limit does however depend on a given set of astrophysical
priors and on assumptions on the reionization models and on the thermal
history of the intergalactic medium. In Ref. [32], the authors used strong
lensing to probe the free-streaming scale and the subhalo mass function: their
result is $M_{\mathrm{wdm}}>5.2$ keV at 95% C.L., while a recent study of
Milky Way satellites [33] found $M_{\mathrm{wdm}}>2.02$ keV at 95% C.L., a
limit that it is tightened to $M_{\mathrm{wdm}}>3.99$ keV when modelling the
effect of reionization. More recently, in Ref. [34] strong gravitational
lensing with extended sources, the Lyman-$\alpha$ forest, and the number of
luminous satellites in the Milky Way were analyzed jointly in a consistent
framework in order to find lower limits on thermal WDM and sterile neutrinos,
finding $M_{\mathrm{wdm}}>6.048$ keV at 95% C.L.. With these numbers in mind,
upcoming large-scale structure surveys, such as Euclid,111https://www.euclid-
ec.org/ DESI,222https://www.desi.lbl.gov/ and LSST at the Vera C. Rubin
Observatory,333https://www.lsst.org/ are likely not to improve much these
constraints using galaxy clustering and weak lensing only, due to the fact
these observables will not probe the scales where the suppression in the
density perturbations and in turn on the power spectrum occurs.

However, a scenario where DM comes in two phases, a cold one and a warm one,
called mixed dark matter or cold and warm dark matter (CWDM), constitutes an
intriguing possibility and a very simple extension to the $\Lambda$CDM model
for which weak lensing and galaxy clustering could provide some interesting
constraint. In this picture we have, besides the WDM mass, an additional
parameter, which is the WDM fraction with respect to the total DM amount:

$f_{\mathrm{wdm}}=\frac{\Omega_{\mathrm{wdm}}}{\Omega_{\mathrm{cdm}}+\Omega_{\mathrm{wdm}}},$
(1.2)

where $\Omega_{\mathrm{cdm}}$ and $\Omega_{\mathrm{wdm}}$ are the density
parameters for CDM and WDM, respectively. These scenarios typically display a
suppression in the power spectrum, which is shallower than WDM alone, with the
exception that it can occur already at relatively large scales. For instance,
the combination of a low $M_{\mathrm{wdm}}$ with a low $f_{\mathrm{wdm}}$ is
in agreement with observations and can in principle be detected by large-scale
structure surveys. Previous works on CWDM have focused on the physics at the
halo/galactic scale, highlighting how models sharing similar free-streaming
lengths but with different combinations of $f_{\mathrm{wdm}}$ and
$M_{\mathrm{wdm}}$ can produce halos with different properties below masses of
$10^{11}\ M_{\odot}/h$ due to the different behaviour of the power spectra at
small scales [35, 36]. Another work [37] used Lyman-$\alpha$ data in
combination with WMAP5 [38] to constrain CWDM models, finding that
$f_{\mathrm{wdm}}<0.2$ is allowed independently of the WDM mass. More
recently, Ref. [39] carried out an extensive study of mixed dark matter
models, fixing the WDM temperature to the Standard Model neutrino one in order
to obtain the correct $f_{\mathrm{wdm}}$ and also considering the fermionic or
bosonic nature of these particles. They found, combining data from Planck,
BOSS DR11 and Milky Way satellites $f_{\mathrm{wdm}}<0.29\ (0.23)$ for
fermions (bosons) in the mass range $1-10$ keV and $f_{\mathrm{wdm}}<0.43\
(0.45)$ in the mass range $10-100$ keV.

This is the first of a series of two papers in which we investigate the impact
of CWDM models on the main cosmological large-scale structure observables that
will be probed in upcoming surveys: in particular, we will focus on the
angular power spectra of galaxy clustering and cosmic shear, on the halo mass
function as well as on the reconstructed (theoretical) matter power spectrum.
While for both galaxy clustering and weak lensing the link to the observable
is clear, the halo mass function cannot be directly probed and further
assumptions (on e.g. stellar content or luminosity of galaxies) residing in
dark matter halos should be made. However, using the few existing theoretical
prescriptions for reproducing the mass function, we decided to provide just a
qualitative investigation on observed quantities, in particular the cluster
number count.

To do so, we run a large set of high-resolution cosmological $N$-body
simulations over a wide range of values in the plane
$M_{\mathrm{wdm}}-f_{\mathrm{wdm}}$ in order to build an emulator able to
predict the suppression in the non-linear matter power spectrum with respect
to $\Lambda$CDM with percent precision. This would allow us to improve upon
currently existing fitting functions for CWDM, already provided in Ref. [40].
However, the focus of that paper was on strong gravitational lensing and
therefore the scales involved were much smaller than this work. Those fitting
functions were obtained by using only 6 different models in the
$M_{\mathrm{wdm}}-f_{\mathrm{wdm}}$ plane in rather small boxes (10 Mpc/$h$)
and this does not allow us to connect properly to the linear regime. We also
compare our results with the fitting formula by Ref. [41] for WDM only. In a
companion paper [42] we will present a detailed Markov Chain Monte Carlo
(MCMC) forecast on CWDM in future surveys.

This paper is organized as follows. In Section 2 we present the set of
simulations we run and use for our theoretical predictions. In Section 3 we
focus on the matter power spectrum: we describe how the suppression due to
CWDM looks like, we show how we build the emulator, we investigate possible
dependencies on the cosmological parameters and check whether the
baryonification effect [43, 44] is independent from the DM model assumed
($\Lambda$CDM or CWDM). In the following Sections we investigate and discuss
the impact of CWDM models on some fundamental cosmological observables such as
cosmic shear spectra (Section 4) and halo mass functions (Section 5). Finally,
in Section 6 we draw our conclusions.

##  2 Simulations and dataset

The creation of an emulator requires a thorough sampling of the whole
parameter space: this implies that a large number of different simulations has
to be run for a wide range of values for $f_{\mathrm{wdm}}$ and
$M_{\mathrm{wdm}}$. On the other hand, we also want to use cosmological
observables for which a reliable theoretical prescription already exists, like
the halo mass function [45, 46]. We therefore split our simulation set into
two sub-sets. The “main” set samples the $(f_{\mathrm{wdm}},M_{\mathrm{wdm}})$
parameter space on an almost regular grid (20 points total). We rely on boxes
of 80 $h^{-1}$ comoving Mpc of linear size: this allows us to reconnect with
the linear regime at the largest scales, while not being subjected to
numerical fragmentation or resolution effects (see below) at the smallest
scales of interest for future surveys ($k\lesssim 10\ h/$Mpc). Then, we choose
the values $f_{\mathrm{wdm}}=0.25,0.5,0.75,1$ and
$M_{\mathrm{wdm}}=0.1,0.3,0.5,1.5$ keV as our parameters. The range of masses
has been chosen as follows. For masses larger than 1.5 keV the differences
with $\Lambda$CDM at the power spectrum level are below percent level at
scales $\lesssim 10\ h/$Mpc (we also run a set with $M_{\mathrm{wdm}}=3.0$ keV
as a further check); for masses smaller than 0.1 keV and $f_{\mathrm{wdm}}=1$
the suppression in the matter power spectrum starts to occur at wavenumbers
comparable to the box size itself, also altering the value of $\sigma_{8}$
which we want to keep fixed for all of our runs. On the other hand, the
“extra” set randomly samples the parameter space (54 points in total) and its
only purpose is to populate the training set for our matter power spectrum
emulator. Figure 1 shows all the simulations that we run. In particular, red
dots correspond to the “main” set, while the blue ones refer to the “extra”
set.

Figure 1: Sampling of the $(f_{\mathrm{wdm}},M_{\mathrm{wdm}})$ parameter
space used in this work. Red dots refer to the “main” set of simulations and
are performed on an almost regular grid in both axes (dots relative to the
$M_{\mathrm{wdm}}=3.0$ keV set are not shown here). This set is mainly used to
test the validity of theoretical predictions for the observables we consider
here. Blue dots mark the “extra” set, created with the purpose of populating
randomly the parameter space to train the emulator for the suppression in the
matter power spectrum with respect to the $\Lambda$CDM model.

Our simulations have been run with the tree-particle mesh (TreePM) code
Gadget-III [47]. The simulations follow the gravitational evolution of
$512^{3}$ particles from an initial redshift of $z_{\mathrm{in}}=99$ to $z=0$.
All the particles were initialized assuming they were CDM and neglecting
thermal velocities: we explicitly checked that for $M_{\mathrm{wdm}}\geq 0.3$
keV the inaccuracy introduced by this assumption does not exceed 1.5% in the
matter power spectrum overall suppression for all redshifts and scales
$k\lesssim 5\ h/$Mpc. Thus, the differences among the different models are to
be found in the initial power spectrum, computed with CLASS [48], and
therefore in the initial displacement field generated with a modified version
of the N-GenIC
software,444https://github.com/franciscovillaescusa/N-GenIC_growth using
second-order Lagrangian perturbation theory.

The fiducial cosmological parameters are set to
$\Omega_{\mathrm{m}}=\Omega_{\mathrm{cdm}}+\Omega_{\mathrm{wdm}}+\Omega_{\mathrm{b}}=0.315$,
$\Omega_{\mathrm{b}}=0.049$, $h=0.674$, $n_{\mathrm{s}}=0.965$,
$\sigma_{8}=0.811$. With this choice of parameters, a single CDM particle has
a mass of $\sim 3.3\times 10^{8}\ \ M_{\odot}/h$. For each simulation we run
four different realizations: two standard ones and two where the phases of the
initial density field have been flipped. We do so to reduce effects due to
small-scale cross-correlations: as such, when computing matter power spectra
and halo mass functions, we will always take the average of the 4
realizations. We take 8 different snapshots, equally spaced from $z=3.5$ to
$z=0$.

We compute the matter power spectra using the Pylians3
code555https://github.com/franciscovillaescusa/Pylians3. We assign particles
to a grid of size 1024 using the Cloud-In-Cell mass-assignment scheme. This
ensures that the Nyquist frequency $k_{\mathrm{Nyq}}\approx 40\ h/$Mpc is much
larger than the maximum $k\sim 10\ h/$Mpc we are interested in in our
analysis. To further make sure that our results are converged at the smallest
scales, we run an additional simulation set with 10243 particles while keeping
the same box size. Figure 2 summarizes this test. We plot the percent
difference in the suppression of the matter power spectrum when considering
the simulation with 512 and 1024 particles per side. As it can be seen, for
all the redshifts considered and for all the scales of interest the difference
falls within $\sim 0.5\%$.

Figure 2: Percentage difference in the matter power spectrum suppression for
two simulations with same $f_{\mathrm{wdm}}=0.75$, $M_{\mathrm{wdm}}=0.5$ keV
and same boxsize $L=80$ Mpc/$h$ but with different number of particles (512
and 1024), to investigate the effects of resolution. Different colors label
different redshifts, the dark and light grey shaded areas represent the 1% and
2% regions.

The halo mass function is computed with the ROCKSTAR software [49]. We first
identify all halos with masses larger than $10^{10}\ M_{\odot}/h$ through a
Friends-of-Friends (FoF) algorithm, with linking length 0.28 times the mean
interparticle separation. The virial mass in this paper is defined as the mass
enclosed in a region where the density is 200 times larger than the critical
density of the Universe. We operate a cut in sphericity, i.e. we consider only
halos with axes ratios $c/a>0.24$ and $b/a>0.34$, where $a>b>c$ are the three
semi-axes. This algorithm has been shown to remove the presence of an
artificial population of very elongated proto-halos which is typical in WDM
scenarios but rather insignificant in $\Lambda$CDM [46, 50]. Last, we operate
a second cut in mass and keep only halos with a mass larger than 10% of the
free-streaming mass in order to avoid effects from numerical fragmentation.

In Figure 3 we show a small region ($10\times 10\times 10$ Mpc/$h$) of three
different simulations, each with the same seed. The left one is a pure
$\Lambda$CDM simulation, the one on the right is a pure WDM simulation with
$M_{\mathrm{wdm}}=0.5$ keV, while the central one is a CWDM simulation also
with $M_{\mathrm{wdm}}=0.5$ keV but $f_{\mathrm{wdm}}=0.5$. Four different
snapshots at $z=0,1,2,3$ are shown from top to bottom. The free-streaming
length at $z=0$ is roughly 0.4 Mpc/$h$ in pure WDM, and 0.5 Mpc/$h$ in CWDM:
below this scale structures are less clustered and give rise to a suppression
in the matter power spectrum and, correspondingly, on the halo mass function.

Figure 3: A sample 10 Mpc/$h$ thick slice of Universe in three different
models of dark matter: the left panels display the $\Lambda$CDM Universe; the
central panels refer to a mixed DM model with $f_{\mathrm{wdm}}=0.5$ and
$M_{\mathrm{wdm}}=0.5$ keV; the right panels are relative to a pure WDM model
also with $M_{\mathrm{wdm}}=0.5$ keV. From top to bottom, the four snapshots
are at $z=0,1,2$ and $3$. It is clearly visible that below the free-streaming
length (roughly 1/20 of the size of the panel) structures and filaments become
less prominent with increasing WDM fraction, especially at high redshift.

##  3 Matter power spectrum

We start by analyzing the matter power spectrum. In this Section, we first
discuss how the suppression in the matter power spectrum looks like as a
function of $f_{\mathrm{wdm}}$, $M_{\mathrm{wdm}}$, and redshift (Section
3.1). We compare the results of our simulations to the theoretical prediction
from an emulator whose construction and testing is presented in Section 3.2.
We also show how these results extend and improve previous works in predicting
the non-linear power spectrum [40, 41, 51]. In Section 3.3, we assess possible
dependencies of the suppression on the cosmological parameters. Finally, we
show how baryonic processes are independent from the DM model assumed (Section
3.4).

###  3.1 Suppression of power spectrum

We summarize our results for what concerns matter power spectra in Figure 4.
Each subplot shows the suppression of power with respect to $\Lambda$CDM for a
given redshift (different rows) and WDM fraction (different columns). Color-
coded are the different WDM masses: red for 0.1 keV, blue for 0.3 keV, green
for 0.5 keV, and yellow for 1.5 keV. In particular, the dots represent the
results from our simulations, while solid lines display the performance of our
emulator (see Section 3.2). As a reference, with the same colors each subplot
shows the various scales at which the power spectrum starts to differ from the
$\Lambda$CDM one, computed as in Ref. [37]. This is referred to as free-
streaming horizon, i.e. the largest scale which is affected by WDM in cosmic
history. Finally, the grey shaded area denotes $k>10\ h$/Mpc, a rough estimate
of the scales which will not be probed by future large-scale structure
surveys. As we already mentioned, free-streaming is responsible for this
suppression 666We recall that we do not include thermal velocities of WDM in
our simulations. However, we remark here that we explicitly checked that
neglecting them has a small effect (see Section 2), i.e. the free-streaming as
imprinted in the initial power spectrum is enough to capture the suppression
of the matter power spectrum.. The effect is more pronounced for smaller WDM
masses (because of the larger thermal WDM velocities), for larger WDM fraction
(as the amount of matter subject to free-streaming increases), and for
increasing redshift (as the free-streaming length scales as
$1/k_{\mathrm{fs}}\propto(1+z)^{1/2}$ during matter domination and as the
high-redshift regime is closer to linear behaviour, where primordial
differences in matter power are more pronounced than in the non-linear
regime).

Figure 4: Each subplot shows the suppression of the matter power spectrum in
CWDM models with respect to $\Lambda$CDM for a given redshift (rows) and a
given $f_{\mathrm{wdm}}$ (columns). Different colors label different
$M_{\mathrm{wdm}}$: 0.1 keV in red, 0.3 keV in blue, 0.5 keV in green, and 1.5
keV in yellow. Dots represent the measurements from the “main” suite of our
simulations, while solid lines display the results obtained with the emulator
described in Section 3.2. Vertical colored dotted lines represent the scales
at which the suppression starts to kick in, as function of $f_{\mathrm{wdm}}$,
$M_{\mathrm{wdm}}$, and $z$, as computed in Ref. [37]. The grey shaded area on
the right marks the region $k>10\ h/$Mpc, i.e. an estimate of those
wavenumbers which will not be probed by upcoming large-scale structure
surveys.

###  3.2 Emulator: building, testing and performance

In order to find a model that best describes observations, we use a Markov
chain Monte Carlo (MCMC) framework. This method samples the parameter space,
comparing theoretical predictions with data. The typical number of samples
drawn in a cosmological MCMC is $\mathcal{O}(10^{5})$. However, our
simulations are computationally expensive ($>$ 5000 CPU-hours per set of
parameter values). Therefore we cannot directly explore the entire CWDM
parameter space.

In this section, we describe an emulator that can replace our simulation
procedure. An emulator can be imagined as a regression model learnt from
examples of our simulations, which is known as the training set. To create the
training set, we run 74 models (both the “main” and “extra” sets) by randomly
sampling the $(f_{\mathrm{wdm}},M_{\mathrm{wdm}})$ parameter space. As
described in the previous Section, we extract snapshots at 8 redshifts. The
parameter space sampling is shown in Figure 1.

Instead of directly emulating $P_{\mathrm{CWDM}}$, we emulate the power
spectrum ratio $P_{\mathrm{CWDM}}/P_{\mathrm{\Lambda CDM}}$, so that the
contribution of cosmic variance at the largest scales of the simulations is
removed.

Our data set is very large as we have 592 power spectra, each evaluated at 886
wavenumbers. To make our emulator memory efficient, we pre-process the data
set by reducing the dimensionality of our power spectra suppression data set
using principal component analysis (PCA). PCA is performed using the module
provided in scikit-learn package [52]. We keep the first 20 principal
components (PCs). With these PCs, we can reconstruct our data set within an
error of 2% for $k\lesssim 10~{}h/\mathrm{Mpc}$.

We use the Gaussian Process Regression (GPR) [53] to build our emulator.
Previous works have shown that GPR is apt to emulate cosmological power
spectrum, such as for non-linear matter distribution [54], Lyman-$\alpha$
forest [55], and the 21cm signal [56]. We use the GPR module provided in GPy
package [57]. This module finds a model that relates the input vector $\bm{x}$
= ($f_{\mathrm{wdm}},M_{\mathrm{wdm}},z$) to the output vector $\bm{y}$. In
our work, we will emulate the coefficients of the 20 PCs.

A Gaussian process assumes any finite number of points in a parameter space to
be jointly Gaussian distributed as

$\displaystyle f(\bm{x})\sim\mathcal{N}(0,\mathcal{K})\ ,$ (3.1)

where $\mathcal{K}$ is the kernel function. This function models the
similarity between the data points in the training set. There exist various
choices of kernel functions [see e.g. 53]. In our work, we use the Matern
kernel [58, 53], which contains two hyper-parameters. These hyper-parameters
can be determined from the training set. Once we have learnt the kernel
function for our parameter space, we can predict the PCs at any point in our
training set. We then can reproduce the power spectra suppression from the
predicted PCs.

We use 90% of our data set, which was selected randomly from the full set, to
train the emulator, while we keep the remaining 10% to test its performance.
This test set contains $\sim 60$ data points. In Figure 5, we show the
percentage difference between the emulated and simulated power spectra
suppression for 12 representative data points from our test set. Most of the
emulated power spectra suppression are within 0.5% magnitude difference at all
scales. There are quite a few power spectra suppression where the difference
is larger. But this difference is only at high wavenumbers ($k\gtrsim 4\
h$/Mpc) and does not exceed $\sim 1.5\%$. As the emulation process has not
seen the test set data points during training, a good prediction capability at
these points hints that the emulator has learnt a generic model. Therefore
this emulator can be used to interpolate within the parameter space where it
is trained.

Figure 5: Percentage difference between the suppression in the matter power
spectrum as computed by the emulator and the one measured from randomly
selected simulations of the test set. Most of the predicted suppression fall
within the 1% region; only few ($\sim$5%) of them have a slightly worse
accuracy that anyway never exceeds 1.5%.

To have an overall picture of the performances of our emulator, we turn our
attention back to the solid lines of Figure 4. For all masses down to 0.1 keV
we recover the correct suppression at $<2\%$ level for all $z\leq 3.5$, for
all WDM fractions and masses. Minor problems may arise at high redshift
($z\gtrsim 3$), for high fractions and small masses. In these situations the
down-turn occurs at the very same size of the box, so that some numerical
errors are expected: these issues are related to the normalization of the
suppression, so to the simulations themselves rather than to the emulator.
While the accuracy of the emulator with respect to the simulations remains at
percent level, we detect a small ($\sim 2\%$) enhancement of the power
spectrum due to this effects in the range of scales comparable to the box size
itself ($0.1-0.3\ h$/Mpc). This has a little impact on the angular power
spectra of cosmic shear and galaxy clustering (see Section 4), where we have a
non-physical $\sim 1\%$ enhancement at $\ell\sim 200-500$. We remark anyway
that this does not represent a problem for the purposes of this paper. This
box-size effect only interests $f_{\mathrm{wdm}}\approx 1$ and the smallest
masses, a region which is already by far excluded by current constraints.
Moreover, this imprecision only concerns $z\sim 3$, where the typical window
functions for weak lensing and angular galaxy clustering are very close to
zero for benchmark future large-scale structure experiments.

Previous works have already tried to give a description of the non-linear
matter power spectrum in WDM or CWDM scenario. In particular, Refs [41, 51]
provided fitting formulae for the non-linear WDM suppression along the lines
of its linear counterpart [59]. Their claim was a 2% agreement for $z\leq 2$
and $M_{\mathrm{wdm}}\geq 0.5$ keV. We confirm that the agreement with our
simulations with $f_{\mathrm{wdm}}=1$ is good, although with a slightly lower
accuracy (5%) at $z\leq 2$ and $k<10\ h/$Mpc for masses $M_{\mathrm{wdm}}\geq
0.3$ keV. This accuracy however improves to 2% if we limit ourselves to scales
$k<3\ h/$Mpc. We also compared our simulations to the fitting formulae
provided by Ref. [40]. While these are in principle valid for both WDM and
CWDM, they were obtained by comparison to $N$-body simulations with a much
smaller boxsize (10 Mpc/$h$). The fit performs similarly to the one in Ref.
[41] for WDM scenarios. For CWDM we find an agreement of $\sim 5\%$ for $z\leq
2$ and $M_{\mathrm{wdm}}\geq 0.3$ keV up to scales of 10 $h$/Mpc, even though
the largest deviations occur at low redshift. Both these previous works
however fail to reproduce the suppression for $M_{\mathrm{wdm}}<0.3$ keV
already in the mildly non-linear regime.

###  3.3 Dependence on cosmological parameters

We investigate possible dependencies of the CWDM suppression on cosmological
parameters. While we do not expect that the suppression depends on parameters
such as $\Omega_{\mathrm{b}}$, $h$, or $n_{\mathrm{s}}$, there might be a
potential difference when we vary the overall spectrum amplitude $\sigma_{8}$
and the total matter content $\Omega_{\mathrm{m}}$. These two parameters are
those that are better constrained by weak lensing and photometric galaxy
clustering: in particular, the best constraints are typically achieved by
combining the latter two into a single parameter
$S_{8}=\sigma_{8}\left(\Omega_{\mathrm{m}}/0.3\right)^{\alpha}$, where
$\alpha$ is often set to 0.5 [60, 61, 62]. Interestingly, results from the
KiDS survey highlighted some tension on this parameter when analyzing the
cosmic shear in Fourier space rather than in configuration space [63, 64, 65].
Moreover, both of these values are in tension with Planck [66] so that
extensions to the $\Lambda$CDM scenario $-$ modifications of gravity, massive
neutrinos, WDM, baryon feedback (see also Section 3.4) $-$ are typically
invoked to try to solve this issue. It is essential therefore to examine what
happens in the $\Omega_{\mathrm{m}}-\sigma_{8}$ plane to the suppression of
the matter power spectrum also in the CWDM scenario.

We run a further set of CWDM simulations with larger and smaller
$\Omega_{\mathrm{m}}$ and $\sigma_{8}$ values, together with the corresponding
$\Lambda$CDM ones. We choose the differences to be
$\Delta\Omega_{\mathrm{m}}=\pm 0.02$ and $\Delta\sigma_{8}=\pm 0.045$. The
$\Omega_{\mathrm{m}}$ value is $\sim 3$ times the error forecast in Ref. [67]
by combining weak lensing and angular galaxy clustering. For $\sigma_{8}$, it
corresponds to $\sim 12$ times this error, but we made this choice on purpose
to take conservatively into account other possible effects that can alter the
overall spectrum normalization, like e.g. massive neutrinos. Our 8 different
cosmologies therefore have $\Omega_{\mathrm{m}}^{+}=0.335$,
$\Omega_{\mathrm{m}}^{-}=0.295$, $\sigma_{8}^{+}=0.856$,
$\sigma_{8}^{-}=0.766$ and the combinations of the two.

Figure 6:  Dependence of the CWDM suppression in the matter power spectrum
upon cosmological parameters. We run 8 simulations with larger and smaller
$\Omega_{\mathrm{m}}$, $\sigma_{8}$ values and their combinations (we show
here $f_{\mathrm{wdm}}=0.75$, $M_{\mathrm{wdm}}=0.5$ keV for CWDM). We compare
the suppression in the CWDM power spectrum for all the different cosmologies
with the same suppression in the fiducial one. In particular, different colors
of the lines are relative to different $\Omega_{\mathrm{m}}$: red when
$\Omega_{\mathrm{m}}$ is increased to 0.335, blue when it is decreased to
0.295. Different line styles refer to different values of $\sigma_{8}$: dashed
for 0.866, dotted for 0.766. The dark (light) shaded area marks the 1% (2%)
region, the vertical dotted line marks the scale $k=10\ h/$Mpc.

Results of this further test are plotted in Figure 6, where we show the ratio
between the suppression of the power spectrum
$P_{\mathrm{CWDM}}/P_{\mathrm{\Lambda CDM}}$ in the varied cosmologies
(“cosmo”) and the one in our fiducial cosmology (“fid”). Differences are shown
in percent. Red colors correspond to cosmologies where $\Omega_{\mathrm{m}}$
is enhanced while light blue lines label cosmologies with a lower
$\Omega_{\mathrm{m}}$; analogously, dashed lines refer to cases with an
enhanced $\sigma_{8}$ and dotted lines to cases where $\sigma_{8}$ is
diminished. As it can be noticed from the figure, the effect of varying
$\Omega_{\mathrm{m}}$ is well below 1% even at the highest redshift we
consider ($z=3.5$). On the other hand, when we vary $\sigma_{8}$ the
suppression becomes slightly more cosmology-dependent at high redshift, helped
by the large variation we introduce in the parameter. However, this difference
is well within the 2% level at the scales we are interested in ($k\lesssim 10\
h/$Mpc) and the redshifts where upcoming surveys will be sensitive ($z\lesssim
2.5$). We are therefore confident in claiming that our emulator can be used to
predict the CWDM suppression also in the neighborhood of our fiducial set of
cosmological parameters and, in general, in the range of interest of
cosmological parameter for future surveys.

###  3.4 The baryonification model in CWDM models

Baryon feedback has been shown to be one of the leading mechanisms capable of
modifying the distribution of matter within dark matter halos up to relatively
large cosmological scales (see e.g. [68, 69]). From a cosmological point of
view, it constitutes an important systematic to be taken into account [70, 71,
72], while completely ignoring its effect on the matter power spectrum can
lead to a $\sim 5\sigma$ bias in the estimate of $\Omega_{\mathrm{m}}$ and
$\sigma_{8}$ [73, 74]. Since the observational constraints are still poor,
these phenomena are typically investigated through computationally expensive
hydrodynamic simulations. Moreover, the uncertainty caused by different AGN
feedback models can reach 50% for scales $k\leq 1\ h/$Mpc [75].

A novel approach circumventing the computational cost of the problem has been
first proposed by Ref. [44] and subsequently improved by Ref. [43]. In this
approach, called baryonification, baryon feedback is added on top of DM-only
simulations. In particular, the modification of the halo profiles is taken
into account through the displacement of DM particles from their positions.
Such displacement depends on five parameters directly related to the physics
of the gas: two parameters controlling the slope of the gas profile and its
dependence on the host-halo mass ($\mu$, $\log M_{c}$); one parameter setting
the maximum radius of gas ejection ($\theta_{\mathrm{ej}}$); two parameters
describing the central-galactic and total stellar fractions within the halo
($\eta_{\mathrm{cga}}$, $\eta_{\mathrm{tot}}$).

In this Section, we investigate whether the effects from baryons are separable
from the suppression induced by the CWDM model. While this separability has
been verified for the case of cosmologies with varying neutrino masses [76],
it remains untested for more general CWDM scenarios. We apply the
baryonification method to both our CWDM and $\Lambda$CDM simulations and
compare the results in terms of the relative suppression effects from baryonic
feedback. The results of this analysis is illustrated in Figure 7. For the
benchmark simulation with $f_{\mathrm{wdm}}=0.75$ and $M_{\mathrm{wdm}}=0.5$
keV, we show the percent-difference of the baryonification model effect on a
CWDM simulation with respect to the equivalent performed on top of the
corresponding $\Lambda$CDM one. Different colors label the 8 different
snapshots we took from $z=3.5$ to $z=0$. Shaded stripes represent the 1% (dark
grey) and 2% regions (light grey), while the scale $k=10\ h/$Mpc is marked
with a dashed black vertical line. The parameters we used for displacing the
particles ($\log(M_{c}\
[M_{\odot}/h])=13.8,\mu=0.21,\theta_{\mathrm{ej}}=4.0,\eta_{\mathrm{tot}}=0.32,\eta_{\mathrm{cga}}=0.6$)
correspond to a model in broad agreement both X-ray observations and
hydrodynamic simulations (see Ref. [43] for a more detailed discussion).

Figure 7 shows that the difference of the baryonic suppression between CDM and
our benchmark CWDM scenario remains below the percent level for $k\leq 5\
h/$Mpc (growing to 2-3 % for $k\leq 10\ h$/Mpc). This is significantly smaller
than the expected total baryonic suppression effect (which is of the order of
$\sim 10-30\%$) and of similar size of the expected precision of $N$-body
codes in the same range of scales [77]. We can therefore assume that CWDM
suppression is independent of baryonification. As a consequence, we can treat
baryonic and CWDM power suppression effects independently, which considerably
simplifies the analysis regarding the cosmological inference pipeline.

Figure 7: We apply the baryonification model on top of a $\Lambda$CDM as well
as of a CWDM simulation and measure the suppression in the matter power
spectrum due to baryon feedback in the two cases. In this Figure we plot the
“ratio of ratios”, i.e. the ratio of the suppressions of $P(k)$ in the
$\Lambda$CDM case and in the CWDM case: in particular, $P_{\mathrm{\Lambda
CDM}}$ and $P_{\mathrm{CWDM}}$ are the power spectra of the standard
$\Lambda$CDM and CWDM simulations, respectively, while $P_{\mathrm{BF+\Lambda
CDM}}$ and $P_{\mathrm{BF+CWDM}}$ are the power spectra of the $\Lambda$CDM
and CWDM simulations to which we have applied the baryonification model.
Different colors refer to the different snapshots of our simulations, the grey
shaded areas represent 1% (dark) and 2% (light) regions, respectively. The
simulation shown here has $f_{\mathrm{wdm}}=0.75$ and $M_{\mathrm{wdm}}=0.5$
keV; the parameters used are $\log(M_{c}\
[M_{\odot}/h])=13.8,\mu=0.21,\theta_{\mathrm{ej}}=4.0,\eta_{\mathrm{tot}}=0.32,\eta_{\mathrm{cga}}=0.6$
(see main text or Ref. [43] for an insight of the baryonification parameters).

##  4 Cosmic shear and galaxy clustering

The next observables we focus on in the framework of CWDM models are the
angular power spectra for weak lensing and galaxy clustering. Here, we want
simply to show the quantitative behaviour (together with some quantitative
discussion) of the CWDM suppression on projected spectra.

In this Section we assume the Limber approximation, valid for large multipoles
($\ell\gtrsim 10$) (see e.g. [78]). In this picture, assuming a flat Universe
and a single redshift bin for simplicity, the angular power spectra can be
written as

$C_{XY}(\ell)=\int\,\mathrm{d}z\,\frac{c}{H(z)}\ \frac{W_{X}(z)\
W_{Y}(z)}{\chi^{2}(z)}\ P\left(k=\frac{\ell+1/2}{\chi(z)},z\right),$ (4.1)

where $\chi(z)$ is the comoving distance to redsfhit $z$ and
$\\{X,Y\\}=\\{\mathrm{L,G}\\}$ so that LL, GG and GL stand for cosmic shear,
galaxy clustering and galaxy-galaxy lensing respectively. The two window
functions $W_{\mathrm{L}}(z)$ and $W_{\mathrm{G}}(z)$ are a measure of the
lensing efficiency and the galaxy bias of the sample, respectively, and they
are tightly related to the galaxy distribution in redshift $n(z)$. The galaxy
clustering window function is given by

$W_{\mathrm{G}}(z)=b(z)\ n(z)\ \frac{H(z)}{c},$ (4.2)

where for the bias we assume the functional form by Ref. [79], which reads

$b(z)=A+\frac{B}{1+\exp{[-(z-D)/C]}},$ (4.3)

with $\\{A,B,C,D\\}=\\{1.0,2.5,2.8,1.6\\}$.

On the other hand, the window function for cosmic shear takes into account two
contributions, the former coming from the cosmological signal ($\gamma$) and
the latter coming from spurious correlations of galaxy orientations coming
from pairs aligned by the tidal field (IA). Hence, we have

$\displaystyle W_{\mathrm{L}}(z)$ $\displaystyle=$ $\displaystyle
W_{\gamma}(z)+\mathcal{F}_{\mathrm{IA}}(z)\ W_{\mathrm{IA}}(z),$ (4.4)
$\displaystyle W_{\gamma}(z)$ $\displaystyle=$
$\displaystyle\frac{3}{2}\Omega_{\mathrm{m}}\frac{H_{0}^{2}}{c^{2}}\chi(z)(1+z)\int_{z}^{\infty}\,\mathrm{d}z^{\prime}\
n(z^{\prime})\ \frac{\chi(z^{\prime})-\chi(z)}{\chi(z^{\prime})},$ (4.5)
$\displaystyle W_{\mathrm{IA}}(z)$ $\displaystyle=$ $\displaystyle n(z)\
\frac{H(z)}{c},$ (4.6)

where

$\mathcal{F}_{\mathrm{IA}}(z)=-\frac{A_{\mathrm{IA}}\mathcal{C}_{1}\Omega_{\mathrm{m}}}{D_{1}(z)}(1+z)^{\eta_{\mathrm{IA}}}\left[\frac{\left<L\right>(z)}{L_{*}(z)}\right]^{\beta_{\mathrm{IA}}},$
(4.7)

$\mathcal{C}_{1}=0.0134$ is fixed as it is completely degenerate with the
intrinsic alignment amplitude parameter $A_{\mathrm{IA}}$, $D_{1}(z)$ is the
linear growth factor. The intrinsic alignment parameters have fiducial values
$\\{A_{\mathrm{IA}},\eta_{\mathrm{IA}},\beta_{\mathrm{IA}}\\}=\\{1.72,-0.41,2.17\\}$.
Finally, $\left<L\right>(z)/L_{*}(z)$ is the mean luminosity of the sample in
units of the typical luminosity at a given redshift. This is taken from Ref.
[67], where the authors took the luminosity functions of early and late type
galaxies separately and joined them assuming a given fraction of ellipticals.
The resulting function fairly reproduces fig. C.1 of Ref. [80] in a certain
$z$ range and is subsequently extrapolated to match our own redshift range
777Private communication with V.F. Cardone.. This model is an extension of the
so called non-linear alignment model [67] first introduced in Ref. [81].

Figure 8: Ratio of the angular power spectrum $C(\ell)$ in a CWDM scenario
with respect to a pure $\Lambda$CDM model for weak lensing (top row), galaxy-
galaxy lensing (middle row), and galaxy clustering (bottom row). Different
columns report the suppression for different WDM fractions, while the WDM mass
information is color-coded: red for 0.1 keV, blue for 0.3 keV, green for 0.5
keV, and yellow for 1.5 keV. For simplicity, for this plot a single redshift
bin has been used. The golden shaded area represents cosmic variance for a
survey with the same specifics of Euclid. The vertical lines represent the
multipole at which shot/shape noise equals the cosmological signal, depending
on the number of sample galaxies per square arcminute (the number written on
the side of the line itself). The vertical grey shaded areas remove the
multipole regions that are likely not be used in the cosmological exploitation
(see text for details): the light and dark area represent a pessimistic and an
optimistic setting, respectively. In particular, $\ell_{\mathrm{min}}=10$,
enough to ensure the validity of the Limber’s approximation, while
$\ell_{\mathrm{max}}=1500,750,750$ for LL, GL, GG in the optimistic case and
$\ell_{\mathrm{max}}=5000,3000,3000$ for LL, GL, GG in the optimistic case,
respectively.

In Figure 8, we show the ratio between the angular power spectra in the CWDM
scenario with respect to $\Lambda$CDM for the three different cases of weak
lensing (top row), galaxy-galaxy lensing (middle row), and galaxy clustering
(bottom row). Different columns label different WDM fractions and different
colors of the solid lines refer to different WDM masses: red for 0.1 keV, blue
for 0.3 keV, green for 0.5 keV, and yellow for 1.5 keV. The presence of the
two grey shaded areas represents multipoles not considered for cosmological
exploitation in the Euclid forecasts [67]: light grey areas are eliminated
when we use a pessimistic range of multipoles; dark shaded areas are excluded
even in the most optimistic scenario. In particular, the lower limit is set to
$\ell_{\mathrm{min}}=10$ due to the fact that at lower mulitpoles the Limber’s
approximation is not valid. For the maximum multipole we assume
$\ell_{\mathrm{max}}=1500,750,750$ for LL, GL, GG in the pessimistic case and
$\ell_{\mathrm{max}}=5000,3000,3000$ for the optimistic case. These numbers
are chosen following the lines of Ref. [67]. The values
$\ell_{\mathrm{max}}=5000$ and 3000 are rather optimistic for upcoming
surveys, as the signal-to-noise generally saturates quickly above $\ell\gtrsim
500-1000$. On the other hand, neglecting non-Gaussian contributions (like they
do) in the data covariance matrix results in an unjustified boost in the
signal-to-noise. For cosmic shear, for instance, the signal-to-noise with a
full covariance matrix (i.e. including non-Gaussian contributions) up to
$\ell_{\mathrm{max}}=5000$ is the same to the one with a Gaussian-only
covariance matrix cut at $\ell_{\mathrm{cut}}\sim 1500$. The golden shaded
area represents the cosmic variance limit for a survey with a sky coverage of
$f_{\mathrm{sky}}=0.363$. Finally, the vertical lines mark the point where
shot/shape noise equals the cosmological signal, depending on how many
galaxies are in the sample: we show it for 3, 10, and 30 galaxies per square
arcminute, which are reasonable numbers for upcoming surveys. We assume that
galaxies follow the distribution

$n(z)\propto z^{2}\exp\left[-\left(\frac{z}{z_{0}}\right)^{3/2}\right],$ (4.8)

with $z_{0}=0.636$. The non-linear matter power spectra for $\Lambda$CDM
models are computed with the HMcode2020 halofit version implemented in CAMB
[82], while to account for the presence of CWDM we use the emulator we built
in Section 3.2888This is the reason why for $M_{\mathrm{wdm}}=0.1$ keV and
$f_{\mathrm{wdm}}>0.5$ there is a small non-physical positive bump at
mutipoles $\ell\sim 300$, see Section 3.1 for a more detailed explanation..

Figure 8 is organized in such a way that, if the underlying cosmology is
$\Lambda$CDM, each model whose line falls outside the golden shaded area at
multipoles lower than the ones where shape/shot noise becomes dominant can in
principle be excluded. In general and as expected, it is easier to exclude
lower values of $M_{\mathrm{wdm}}$ and high values of $f_{\mathrm{wdm}}$, for
which the suppression of the matter power spectrum is more pronounced. In the
optimistic scenario and accounting for a low noise (30 arcmin-2), cosmic shear
alone could be able in principle to exclude $M_{\mathrm{wdm}}\lesssim 0.3$ keV
for $f_{\mathrm{wdm}}>0.75$. Galaxy clustering exhibits a less pronounced
suppression but galaxy bias enhances the signal enough to allow to go to
higher multipoles before being dominated by shot noise: all in all, for
$f_{\mathrm{wdm}}>0.75$ masses smaller than $\sim 0.5$ keV can already be
excluded when sampling 10 galaxies arcmin-2. The suppression in galaxy-galaxy
lensing, finally, has an intermediate behaviour between the previous two, but
it has the advantage of being noise-free: even for the lowest WDM fraction we
may be able to exclude $M_{\mathrm{wdm}}<0.3$ keV. One can also increase the
signal-to-noise by dividing galaxies into more redshift bins and combining the
three observables. Of course, this plot and this analysis have a few caveats.
First, we are completely ignoring other sources of uncertainty, like super-
sample covariance [83, 84] or non-Gaussian contributions that can suppress the
signal-to-noise especially at high multipoles. Moreover, here we are fixing
our cosmology: we expect a worsening of the posteriors when relaxing this
assumption and in particular when marginalizing over parameters like
$\Omega_{\mathrm{m}}$ and $\sigma_{8}$.

In Ref. [70], the authors focused on the possible degeneracies between baryon
feedback and massive neutrinos in cosmic shear spectra. In particular, they
found interesting degeneracy patterns between neutrino mass and both the
baryon feedback parameter $\log M_{c}$ and the intrinsic alignment parameter.
Since massive neutrinos could be considered WDM, we expect a similar behaviour
for the case of CWDM: we leave this study to a companion paper [42] where we
run MCMC forecasts for CWDM models in a Euclid-like survey, with a proper
marginalization over astrophysical, nuisance and cosmological parameters.

##  5 Halo mass function

The last physical quantity that we investigate is the halo mass function. We
focus on the mass function since many observable quantities are directly
linked to it, for example, the galaxy mass function, the (conditional) mass
function on the number of Milky Way satellites, the number of high-redshift
galaxies capable of driving reionization processes and even the strong lensing
signal. Even if the accurate modelling of the observables would require
astrophysical assumptions, the underlying dark matter mass function will
always be the fundamental ingredient of any theoretical effort.

The most rigorous way of deriving it is through the excursion set of peaks
[85, 86, 87] which extends the Press & Schechter formalism [88]. In this
framework, the number of halos per unit mass per unit volume can be written as

$\frac{\mathrm{d}n}{\mathrm{d}\ln M}=-\frac{\bar{\rho}}{M}\ f(\nu)\
\frac{\mathrm{d}\ln\sigma}{\mathrm{d}\ln M}.$ (5.1)

In the equation above, $f(\nu)$ is a universal, cosmology-independent function
of the peak height $\nu=\delta_{\mathrm{c}}/\sigma(M)$, where
$\delta_{\mathrm{c}}\approx 1.686$ is the linearly-extrapolated spherical
overdensity for collapse and $\sigma(M)$ is the root mean square mass
fluctuation

$\sigma^{2}(M)=\int\frac{\,\mathrm{d}k\ k^{2}}{2\pi^{2}}P_{\mathrm{lin}}(k)\
|W(kR)|^{2}.$ (5.2)

The mass $M$ is related to the radius $R$ depending on the kind of window
function chosen. In the $\Lambda$CDM framework, the window function $W(kR)$
that smooths the density field is typically chosen to be a top-hat in
configuration space, which in Fourier space translates to

$W(x)=\frac{3}{x^{3}}\left[\sin x-x\cos x\right].$ (5.3)

Moreover, the universal $f(\nu)$ function is often chosen to be the Sheth-
Tormen one [89, 90]:

$f(\nu)=A\ \sqrt{\frac{2q\nu^{2}}{\pi}}\left[1+(q\nu^{2})^{-p}\right]\
e^{-q\nu^{2}/2},$ (5.4)

with $A=\left[1+2^{-p}\ \Gamma(1/2-p)/\sqrt{\pi}\right]^{-1},\ p=0.3,\
q=0.707$. However, when dealing with free-streaming species or, in general,
with models where the power spectrum has a small-scale cut-off, the top-hat
filter is not the best suitable choice as it predicts an excess of low-mass
halos [45, 46, 91]. A sharp-$k$ filter was invoked by Ref. [45] to solve this
problem: our findings show that despite being able to predict fairly well the
low-mass suppression, this filter suffers from problems in modelling the
absolute mass function. More recently, Ref. [92] proposed the use of a
smooth-$k$ filter, namely

$W(kR)=\left[1+(kR)^{\beta}\right]^{-1},$ (5.5)

where $\beta$ is a free parameter that can be fitted against $N$-body
simulations. This two new filters have the advantage of being able to
alleviate the issues caused by the small-scale cut-off in the linear power
spectrum. The downside of using them is that they do not have a well-defined
mass associated to the filter scale, i.e. their integral over the volume
diverges. What is typically done to restore the scaling $M\propto R^{3}$ is to
introduce a second free-parameter $c$ to be fitted against simulations such
that

$M=\frac{4}{3}\pi(cR)^{3}.$ (5.6)

For the smooth-$k$ filter, Ref. [92] found that $\beta=4.8$ and $c=3.3$
provide a reasonable fit to $N$-body simulations, while Ref. [93] obtained
comparable results ($\beta=3.0$, $c=3.3$) in scenarios where dark matter
produces acoustic oscillations at small scales. We show our results in Figure
9. In each subplot, we show the suppression in the halo mass function with
respect to the $\Lambda$CDM case. The model used is analogous to the one
described above with the only difference that we use $q=1$ in Eq. 5.4 [45].
Different rows refer to different redshifts, different columns label different
WDM fractions while WDM masses are color-coded: 0.1 keV in red, 0.3 keV in
blue, 0.5 keV in green, and 1.5 keV in yellow. As already mentioned in Section
2, we define the halo mass as the mass enclosed in a radius where
$\rho>200\rho_{\mathrm{crit}}$. For reference, vertical dotted lines represent
the mass enclosed in a sphere of radius given by the free-streaming horizon
[37]. Solid lines represent the theoretical prediction using a smooth-$k$
filter and $(\beta,c)=(4.8,3.3)$. We find that this combination of parameters
performs slightly better than $(\beta,c)=(3.0,3.3)$, especially at low
redshift. We also show the results for a sharp-$k$ filter with $c=2.5$ (dashed
lines) [45]. For the case of CWDM, we conclude that both approaches work
similarly well, with the smooth-$k$ mass function providing slightly better
than the sharp-$k$ mass function at low and slightly worse at higher
redshifts. Finally, it may be noticed that at small $M_{\mathrm{wdm}}$ and
large $f_{\mathrm{wdm}}$ the halo mass function for CWDM models assumes larger
values than the $\Lambda$CDM one. This once again comes from the fact that we
chose to parametrize the power spectrum amplitude with $\sigma_{8}$ rather
than for $A_{\mathrm{s}}$ and it is connected to the non-physical “bump” we
were discussing in Sections 3.2 and 4.

Figure 9: Suppression in the halo mass function in a CWDM scenario with
respect to a $\Lambda$CDM cosmology, for a given redshift, WDM fraction and
mass. In particular, different columns label different WDM fractions,
different rows refer to different redshifts while the WDM mass is color-coded:
$M_{\mathrm{wdm}}$: 0.1 keV in red, 0.3 keV in blue, 0.5 keV in green, 1.5 keV
in yellow. Dots represent the measurements from our “main” suite of
simulations, while error bars are taken to be the variance of the 4
realizations. Halo mass is defined as the mass enclosed in a region where
$\rho>200\ \rho_{\mathrm{crit}}$. On the other hand, solid lines represent the
theoretical prediction from the excursion set theory when a smooth-$k$ window
function is used with parameters $\beta=4.8$, $c=3.3$ (see Ref. [92]); dashed
lines instead represent the same but for a sharp-$k$ filter (see Ref. [46]).
As a reference, we plot as vertical dotted lines the mass enclosed in a sphere
of radius given by the free-streaming horizon [37].

We want to further address the discussion on the halo mass function by linking
it to actual observables, focusing on the cluster number counts. The cluster
abundance is particularly helpful in breaking the degeneracy between
$\Omega_{\mathrm{m}}$ and $\sigma_{8}$ thus providing tight constraints on
these two parameters (see e.g. [94, 95, 96]). We want now to qualitatively
investigate which CWDM models will be excluded in upcoming cluster surveys. In
the simple model we consider, the cumulative number of galaxy clusters of mass
larger than a threshold $M_{\mathrm{th}}$ is given by

$N(>M_{\mathrm{th}})=\int_{0}^{z_{\mathrm{max}}}\,\mathrm{d}z\
\frac{\,\mathrm{d}V}{\,\mathrm{d}z}\int_{M_{\mathrm{th}}}^{\infty}\,\mathrm{d}M\
\frac{\mathrm{d}n}{\mathrm{d}M}(z),$ (5.7)

where $\,\mathrm{d}V/\,\mathrm{d}z=4\pi f_{\mathrm{sky}}\ c/H(z)\
\chi^{2}(z)$, with $f_{\mathrm{sky}}=0.363$, and we fix $z_{\mathrm{max}}=2$
and $M_{\mathrm{th}}=10^{13.8}\ M_{\odot}/h$ [96]. We compute this quantity
for a large set of CWDM masses, generating with CLASS linear matter power
spectra up to very small WDM masses (even below 0.1 keV) and smoothing them
with a smooth-$k$ window function with parameters $(\beta,c)=(4.8,3.3)$. In
this part of the analysis, for all the models we decide to keep $A_{s}$ fixed,
rather than $\sigma_{8}$: in this way, we are able to push the WDM masses down
to values for which the suppression in the power spectrum occurs at scales
that influence significantly the value of $\sigma_{8}$ and in turn cluster
abundance.

Figure 10: The Figure shows the difference between the cluster number count in
CWDM models with different values of WDM fraction ($x$-axis) and WDM mass
($y$-axis) and $\Lambda$CDM models, normalized by its theoretical Poisson
uncertainty. The difference is color-coded; the contour lines mark the regions
where such difference is 1, 3, 5, 10, 100, 250, 500, 750 times the expected
error in counts assuming an underlying $\Lambda$CDM model.

In Figure 10 we show the difference between the cluster number count in CWDM
scenarios and the corresponding $\Lambda$CDM value, computed using eq. 5.7.
Such difference is then normalized by the theoretical uncertainty on the
number count itself, which is assumed to be Poissonian. In an ideal case,
models for which the difference exceeds 1-$\sigma$ will be excluded in future
surveys. However, a few caveats must be specified, as the cluster number count
is subject to a number of systematics. In particular, the complex cluster
physics must be taken into account through some effective scaling relations
that connect the theoretical mass function to a prediction of the distribution
of clusters in the observables of the survey. These scaling relations depend
in turn on some unknown nuisance parameters (see e.g. [96, 97]). Ref. [98]
showed how increasing the accuracy of the scaling relations leads to
remarkable improvements on the constraints of the cosmological parameters; on
the other hand, it causes the choice of the mass function, which for current
data represents a subdominant systematic source, to become relevant. In light
of this, we keep two conservative 3-$\sigma$ and 5-$\sigma$ discrepancies as
rule-of-thumb for the exclusion (or detection) of a given CWDM model. We can
crudely approximate the allowed regions as:

$\displaystyle M_{\mathrm{wdm}}$ $\displaystyle\gtrsim$
$\displaystyle\left[0.08+0.8\ f_{\mathrm{wdm}}^{0.82}\right]\
\mathrm{keV}\quad[3-\sigma]$ (5.8) $\displaystyle M_{\mathrm{wdm}}$
$\displaystyle\gtrsim$ $\displaystyle\left[0.06+0.7\
f_{\mathrm{wdm}}^{0.87}\right]\ \mathrm{keV}\quad[5-\sigma].$ (5.9)

##  6 Discussion and conclusions

The $\Lambda$CDM model has been shown to provide an extremely accurate
description of our Universe at large scales. There are, however, remaining
uncertainties at small cosmological scales, which have led to several claims
of tensions between theory and observations: these include e.g. the missing
satellites, the too-big-to-fail, the profile diversity and the cusp-core
problems. In order to solve or alleviate these tensions warm dark matter (WDM)
is often invoked. WDM introduces a suppression in the power spectrum on scales
smaller than the free-streaming length $\lambda_{\mathrm{fs}}$ or equivalently
at masses lower than the free-streaming mass (see Eq. 1.1). Current
constraints from Milky Way satellites show that $M_{\mathrm{wdm}}>2.02$ keV
[33], while Lyman-$\alpha$ studies set $M_{\mathrm{wdm}}>5.3$ keV [31] at 95%
C.L.. With these values, the suppression in the matter power spectrum occurs
at scales which are smaller than the ones probed by upcoming surveys like
Euclid. The current constraints do not forbid the intriguing possibility that
dark matter exists in two phases, a cold one and a warm one. This scenario,
called mixed dark matter or cold-warm dark matter (CWDM), is the object of
study of this paper.

In this work, we ran a large set of cosmological $N$-body simulations spanning
a wide range of parameter values in the plane
$M_{\mathrm{wdm}}-f_{\mathrm{wdm}}$, where $M_{\mathrm{wdm}}$ is the WDM
component mass and $f_{\mathrm{wdm}}$ is the WDM fraction with respect to
total DM. We used the outputs to compute the suppression of the matter power
spectrum with respect to the $\Lambda$CDM case and to build an emulator: this
is able to predict the suppression in power with an accuracy of $\sim 1.5\%$
over the range $0<f_{\mathrm{wdm}}<1,M_{\mathrm{wdm}}\gtrsim 0.1$ keV,
improving on previous existing fitting formulae [41, 40]. We also tested
whether the suppression depends on cosmological parameters, in particular
those which weak lensing and angular galaxy clustering are most sensitive to,
i.e. $\Omega_{\mathrm{m}}$ and $\sigma_{8}$. We showed that such dependence is
always below 2% at the scales and redshifts of interest, even for the most
extreme cases we consider, with $\sigma_{8}$ more than 10$\sigma$ away from
our fiducial value. We also demonstrated that the difference of the baryonic
suppression between CWDM and $\Lambda$CDM is much smaller than the expected
total suppression effect and of the same order of magnitude of the expected
precision of $N$-body codes for scales $k\lesssim 10\ h/$Mpc, thus proving
that baryonic effects can be treated independently from the DM model assumed.
We used the emulated suppression to qualitatively show the impact of CWDM on
weak lensing and angular galaxy clustering power spectra, focusing on which
combinations of WDM masses and fractions may in principle be detected in
upcoming surveys. Finally, we studied the halo mass function. First, we
confirmed that the smooth-$k$ filter prescription proposed by Ref. [92]
provides a good description both of the overall halo mass function and of its
suppression in the CWDM scenario. Then, using the same prescription, we linked
the halo mass function to an actual observable, the cluster number counts,
performing a semi-quantitative estimate on the CWDM models which could be
probed in upcoming surveys.

In a future paper [42] we plan to perform a full Monte Carlo Markov Chain
analysis on synthetic data to have more realistic forecasts on the
$M_{\mathrm{wdm}}$ and $f_{\mathrm{wdm}}$ parameters allowed by upcoming
surveys.

## Acknowledgements

GP, GS and MV are supported by the INFN PD51 INDARK grant. SG and AS are
supported by the Swiss National Science Foundation via the grant
PCEFP2_181157. GP acknowledges Alessandra Fumagalli and Vincenzo Cardone for
useful discussion and Nils Schöneberg for providing the code for generating
the initial power spectra. Simulations were performed on the Ulysses
SISSA/ICTP supercomputer.

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