The Non-Linear Matter Power Spectrum in Warm Dark Matter Cosmologies

Abstract

We investigate the non-linear evolution of the matter power spectrum by using
a large set of high-resolution N-body/hydrodynamic simulations. The linear
matter power in the initial conditions is consistently modified to accommodate
warm dark matter particles which induce a small scale cut-off in the power as
compared to standard cold dark matter scenarios. The impact of such thermal
relics is addressed at small scales with k > 1 h/Mpc and at z < 5, which are
particularly important for the next generation of Lyman-alpha forest, weak
lensing and galaxy clustering surveys. We quantify the mass and redshift
dependence of the warm dark matter non-linear matter power and we provide a
fitting formula which is accurate at the ~2% level below z=3 and for masses
m_wdm > 0.5 keV. The role of baryonic physics (cooling, star formation and
feedback recipes) on the warm dark matter induced suppression is also
quantified. Furthermore, we compare our findings with the halo model and show
their impact on the cosmic shear power spectra.

Full text

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arXiv:1107.4094v1 [astro-ph.CO] 20 Jul 2011
Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 22 July 2011 (MN LATEX style file v2.2)
The Non-linear Matter Power Spectrum in Warm Dark
Matter Cosmologies
M. Viel1,2, K. Markoviˇ c3,4,5, M. Baldi3,4, J. Weller3,4,5
1INAF - Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, I-34131 Trieste, Italy (viel@oats.inaf.it)
2INFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, Italy
3University Observatory Munich, Ludwig-Maximilian University, Scheinerstr. 1, 81679, Munich, Germany
4Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garching, Germany
5Max-Planck-Institut for Extraterrestrial Physics, Giessenbachstr., 85748 Garching, Germany
22 July 2011
ABSTRACT
We investigate the non-linear evolution of the matter power spectrum by using a large
set of high-resolution N-body/hydrodynamic simulations. The linear matter power
in the initial conditions is consistently modified to accommodate warm dark matter
particles which induce a small scale cut-off in the power as compared to standard
cold dark matter scenarios. The impact of such thermal relics is addressed at small
scales with k > 1hMpc−1and at z < 5, which are particularly important for the next
generation of Lyman-α forest, weak lensing and galaxy clustering surveys. We quantify
the mass and redshift dependence of the warm dark matter non-linear matter power
and we provide a fitting formula which is accurate at the ∼ 2% level below z = 3
and for masses mWDM? 0.5 keV. The role of baryonic physics (cooling, star formation
and feedback recipes) on the warm dark matter induced suppression is also quantified.
Furthermore, we compare our findings with the halo model and show their impact on
the cosmic shear power spectra.
Key words: Cosmology: theory – large-scale structure of the Universe – dark matter,
methods: numerical – gravitational lensing: weak
1 INTRODUCTION
The increasing amount of observational data available and
the numerical tools developed for their interpretation have
allowed to enter the so-called era of precision cosmology. At
the present time, the concordance cosmological model based
on a mixture of cold dark matter and a cosmological con-
stant must thereby be tested in new regimes (both in space
and in time) and using as many as possible observations and
techniques in order to either confirm or disprove it.
Among the many different observables the non-linear
matter power spectrum is a crucial ingredient since it al-
lows to describe the clustering properties of matter at small
scales and low-redshift, where linear theory is not reliable.
However, non-linear physical processes that could affect this
observable should be accurately modelled if quantitative re-
sults on the nature of dark matter are desidered.
Warm Dark Matter (WDM) is an intriguing possibil-
ity for a dark matter candidate whose velocity dispersion
is intermediate between those of cold dark matter and hot
dark matter (e.g. light neutrinos). In this scenario, at scales
smaller than the free-streaming cosmological perturbations
are erased and gravitational clustering is significantly sup-
pressed. If such particles are initially in thermal equilib-
rium they have a smaller temperature and affect smaller
scales than those affected by neutrinos, in addition WDM
produces a distinctive suppression feature at such scales as
compared to that induced by neutrinos. For example, ther-
mal relics of masses at around 1 keV which constitute all
of the dark matter have a free-streaming scale that is com-
parable to that of galaxies, well into the non-linear regime.
Among the different warm dark matter candidates a special
role is played by the sterile neutrino with mass in the keV
scale (Boyarsky et al. (2009a)). Warm dark matter has been
advocated originally to solve some putative problems that
are present in cold dark matter scenarios at small scales
(see Col´ ın et al. (2000); Bode et al. (2001)), however it is
at present controversial whether these tensions with cold
dark matter predictions can be solved by modifying the na-
ture of dark matter particles or by baryonic process (e.g.
Trujillo-Gomez et al. (2010)).
In the present paper we wish to quantify the impact
of a warm dark matter relic on the non-linear power spec-
trum by using a set of N-body/hydrodynamic simulations
of cosmological volumes at high resolution. Investigating
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M. Viel et al.
WDM scenarios in a cosmological setting has been done by
means of N-body codes in order to carefully quantify the
impact of such a candidate in terms of halo mass function,
structure formation, halo density properties (Bode et al.
(2001); Col´ ın et al. (2008); Colombi et al. (2009)) and par-
ticular care needs to be placed in addressing properly nu-
merical/convergence issues Wang & White (2007). In gen-
eral, while the WDM induced suppression transfer func-
tion can be reliably estimated in the linear regime (e.g.
Viel et al. (2005); Lesgourgues & Tram (2011)), the non-
linear suppression has not been investigated. A recent at-
tempt to obtain the non-linear matter power at small scales
by modifying the halo model has however been done in
Smith & Markovic (2011).
The analysisof matter
scales has been performed in recent years by differ-
ent groups but focussing mainly on baryon physics (e.g.
Rudd et al. (2008); Guillet et al. (2010); Casarini et al.
(2011); van Daalen et al. (2011)) such as feedback and cool-
ing) without considering how these properties are modified
in WDM scenarios.
Different contraints can be obtained by using several
astrophysical probes. For example by using SDSS Lyman-
α observables such as the transmitted Lyman-α flux power
very competitive constraints in the form of lower limits have
been obtained (Viel et al. 2005, 2006; Seljak et al. 2006).
The constraints are modified if the WDM is assumed to ac-
count only partially to dark matter (Boyarsky et al. 2009a)
or if the initial linear suppression for a sterile neutrino
is considered (Boyarsky et al. 2009b). Alternatively, con-
straints on WDM models can be placed by using the evo-
lution and size of small scale structure in the local vol-
ume high resolution simulations (Tikhonov et al. 2009) the
simulated Milky Way haloes to probe properties of satel-
lite galaxies (Polisensky & Ricotti 2011; Lovell et al. 2011);
large scale structure data (Abazajian 2006); the formation
of the first stars and galaxies in high resolution simula-
tions (Gao & Theuns 2007); weak lensing power spectra and
cross-spectra (Markovic et al. 2011; Semboloni et al. 2011);
the dynamics of the satellites (Knebe et al. 2008); the abun-
dance of sub-structures (Col´ ın et al. 2000); the inner prop-
erties of dwarf galaxies (Strigari et al. 2006); the mass func-
tion in the local group as determined from radio observa-
tions in HI (Zavala et al. 2009); the clustering properties of
galaxies at small scales (Coil et al. 2008) and the proper-
ties of satellites as inferred from semi-analytical models of
galaxy formation (Macci` o & Fontanot 2010).
We believe that most of the astrophysical probes used
so far in order to constrain the small scale properties of dark
matter could benefit from a comprehensive numerical model-
ing of the non-linear matter power. The present work aims at
providing such a quantity by using N-body/hydrodynamic
simulations and the findings could also be useful for fu-
ture surveys such as PanSTARRS (Kaiser et al. 2002),
HETDEX (Hill et al. 2008), DES (Abbott et al. 2005),
LSST (Ivezic et al. 2008), EUCLID (Refregier et al. 2010)
or WFIRST1.
The plan of the paper is as follows. In Section 2 we
present our set of simulations and the code we use in or-
power spectraatsmall
1http://wfirst.gsfc.nasa.gov/
linear size (Mpc/h)mWDM(keV) soft. (kpc/h)
12.5
12.5
25a
25
50
50
100
100
25
25
25a,b,c
25
25
12.5
6.25
–
1
–
1
–
1
–
1
0.62
0.62
1.25
1.25
2.5
2.5
5
5
1.25
1.25
1.25
1.25
1.25
0.625
0.33
0.25
0.5
1
2
4
1
1
Table 1. Summary of the simulations performed. Linear box-size,
mass of warm dark matter particle and gravitational softening
are reported in comoving units (left, center and right columns,
respectively). The particle-mesh (PM) grid is chosen to be equal
to N1/3
hydrodynamic processes (a simplified star formation recipe and
radiative processes for the gas) and with full hydrodynamics with
the standard multiphase modelling of the interstellar medium and
strong kinetic feedback in the form of galactic winds. Simulations
(a) have been also run at lower resolution NDM= 3843and for
different values of σ8, Ωm and H0. Simulation (b) has been run
by switching the initial velocities of warm dark matter particles
off and by increasing the linear size of the PM grid by a factor
3. Simulation (c) has been run with NDM = 6403dark matter
particles with a softening of 1 kpc/h to z = 0.5.
DMwith NDM= 5123. Simulations (a) have been run with
der to investigate the non-linear suppression on the total
matter power. Section 3 contains the main results of the
present work and the description of the checks made in or-
der to present a reliable estimate of the WDM non-linear
suppression: we focus on numerical convergence, box-size,
baryonic physics, particles’ velocities and the effect induced
by cosmological parameters on the WDM power. As an ap-
plication of the findings of Section 3 we present in Section
4 the weak lensing power and cross-spectra for a realistic
future weak lensing survey and compare these results with
those that could be obtained by using either linear-theory
or halo models (Section 5). We conclude with a summary in
Section 6.
2 THE SIMULATIONS
Our set of simulations has been run with the parallel hydro-
dynamic (TreeSPH: Tree-Smoothed Particle Hydrodynam-
ics) code GADGET-2 based on the conservative ‘entropy-
formulation’ of SPH (Springel 2005). Most of the runs use
the TreePM (Tree-Particle Mesh) N-body set-up and con-
sist only of dark matter particles, however for few runs, in
order to test the impact of baryonic physics, we switched
hydrodynamic processes on.
The cosmological reference model corresponds to a
‘fiducial’ ΛCDM Universe with parameters, at z = 0,
Ωm = 0.2711, ΩΛ = 0.7289, Ωb = 0.0451, ns = 0.966, and
H0 = 70.3 km s−1Mpc−1and σ8 = 0.809. This model is in
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Non-linear Matter Power in WDM Cosmologies
3
agreement with the recent constraints obtained by WMAP-
7 year data (Komatsu et al. 2011) and by other large scale
structure probes. The initial (linear) power spectrum is gen-
erated at z = 99 with the public available software CAMB
2(Lewis et al. 2000) and then modified to simulate warm
dark matter (see below).
We consider different box sizes in order to both ad-
dress the large scale power and (more importantly) the ef-
fect of resolution. The gravitational softening is set to be
1/40-th of the mean linear inter-particle separation and is
kept fixed in comoving units. The dimension of the PM grid,
which is used for the long-range force computation, is cho-
sen to be equal to the number of particles unless for a single
case in which a finer grid is used. The simulations follow a
cosmological periodic volume filled with 5123dark matter
particles (an equal number of gas particles is used for the
hydrodynamic simulations), unless for two cases in which
a smaller and larger number of particles is chosen in or-
der to check for numerical convergence of matter power.
We mainly focus on warm dark matter masses around 1
keV. For such a mass, the characteristic cut-off in the power
spectrum appears at scales of about k ∼ 1.5 hMpc−1and
reaches 50% at k = 6 hMpc−1: these scales are non-linear
and thereby require high-resolution N-body techniques to be
modeled. However, in order to be conservative we present
results for the following mWDMvalues: 0.25, 0.5, 1, 2 and 4
keV. These limits could be easily converted to masses for
a sterile neutrino particle produced in the so-called stan-
dard Dodelson-Widrow scenario (Dodelson & Widrow 1994)
and corresponds to ms = 0.7,1.66,4.4,11.1,28.1 keV. Note
that physically motivated scenarios for example based on
non-resonant production mechanisms have been proposed,
however the simulations carried in the present work cannot
be strictly applied to those since they require a non-trivial
modification of the linear transfer function, as discussed by
Boyarsky et al. (2009b).
The initial conditions for warm dark matter particles
are generated using the procedure described in Viel et al.
(2005) and that we briefly summarize here. The linear
ΛCDM power is multiplied by the following function:
T2
lin(k) ≡ PWDM(k)/PΛCDM(k) = (1 + (αk)2ν)−5/ν,
?1keV
mWDM
where ν=1.12 and α has units of h−1Mpc (e.g.
Hansen et al. 2002). We stress that the above equation is
an approximation which is strictly valid only at k < 5 − 10
hMpc−1. Below this scale the warm dark matter power spec-
trum could be described by a more complicated function and
acoustic oscillations are present (see for example the recent
work in Lesgourgues & Tram 2011).
Initial velocities for warm dark matter particles are
drawn from a Fermi-Dirac distribution and added to the
proper velocity assigned by linear theory: the r.m.s. veloc-
ity dispersion associated to their thermal motion is 27.9,
11.5, 4.4. 1.7, 0.7 km/s for mWDM=0.25,0.5,1,2,4 keV, re-
spectively. The typical r.m.s. velocity dispersion for the dark
matter particles of the ΛCDM runs is ∼ 27 km/s, so at least
for masses above 1 keV the thermal WDM motion is a small
α(mWDM) = 0.049
?1.11 ?ΩWDM
0.25
?0.11?h
0.7
?1.22
(1)
2http://camb.info/
fraction of the physical velocity dispersion assigned by the
Zel’dovich approximation.
When baryonic physics is included we consider the fol-
lowing processes: i) radiative cooling and heating, ii) star
formation processes, iii) feedback by galactic winds. The
rationale is to see at which level these processes impact on
the non-linear matter power at small scales in terms of warm
dark matter suppression. Thus, we are not aiming at explor-
ing in a comprehensive way the impact of these processes on
the non-linear power at small scales. (e.g. van Daalen et al.
2011; Casarini et al. 2011): the baryonic simulations are
used only to quantify the impact of such processes on the
suppression induced by warm dark matter w.r.t. cold dark
matter scenarios.
Radiative cooling and heating processes are followed for
a primordial mix of hydrogen and helium by assuming a
mean Ultraviolet Background similar to that produced by
quasars and galaxies and implemented in Katz et al. (1996).
This background gives naturally a hydrogen ionization rate
Γ12 ∼ 1 at high redshift and an evolution of the physical
state of the intergalactic medium (IGM) which is in agree-
ment with observations (e.g. Bolton et al. 2005). The star
formation criterion for the default runs is a very simple one
that converts in collisionless stars all the gas particles whose
temperature falls below 105K and whose density contrast
is larger than 1000 (more details can be found in Viel et al.
2004). This prescription is usually called “QLYA” (quick
Lyman-α ) since it is very efficient in quantitatively describ-
ing the Lyman-α forest and the low density IGM. We also
run a simulation with the full multi-phase description of
the interstellar medium (ISM) and with kinetic feedback in
the form of strong galactic winds as in Springel & Hernquist
(2003). The chosen speed of the wind is 483 km/s and both
the ISM modelling and this feedback mechanisms are ex-
pected to impact on the distribution of baryons and thus on
the total matter power spectrum. We note that simulations
that include baryons are significantly slower than the default
(dark matter only runs) and therefore our constraints will
be mainly derived by the former simulations.
In the following, the different simulations will be indi-
cated by two numbers, (N1,N2): N1 is the size of the box in
comoving Mpc/h and N2 is the cubic root of the total num-
ber of gas particles in the simulation. The mass per dark
matter particle is 8.7 × 106M⊙/h for the default (25,512)
simulations. This mass resolution allows to adequately sam-
ple the free-streaming mass for the models considered here.
In Figure 1 we show the projected dark matter density
as extracted from the default (25,512) runs in the ΛCDM
case (left) and WDM case (right) for mWDM=1 keV. This
WDM particle mass is already ruled out at a significant level
by Lyman-α forest observations (e.g. Seljak 2005; Viel et al.
2006). The different rows refer to z = 0,2,5 from top to
bottom, respectively. In this Figure it is instrumental to see
how the clustering proceeds differently in the two scenarios
and while there are large differences below the Mpc scale at
z = 5 between the two cosmic webs, these differences are
largely erased by non-linear evolution at z = 0,2.
The main features of the simulations are summarized in
Table 1.
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M. Viel et al.
0
5
10
15
20
25
y [ h-1 Mpc ]
ΛCDMWDM
-1.0 -0.5 0.0
log (1+δDM)
0.51.01.52.0
Figure 1. “Visual” inspection of the redshift evolution of cosmic structures in the ΛCDM and WDM (mWDM=1 keV) scenarios (left
and right columns, respectively) for the defaults (25,512) runs. From the top to the bottom rows we show a 2.5 h−1Mpcthick slice of
the projected dark matter density at z = 0, 2 and 5 respectively. At z = 0 the clustering properties of the dark matter at scales k < 10
hMpc−1are indistinguishable in the two scenarios, while at z = 2,5 the WDM model has a suppression in power of about 5% and 25%
at k = 10 hMpc−1.
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Non-linear Matter Power in WDM Cosmologies
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1 10100
k (h/Mpc)
-50
-40
-30
-20
-10
0
10
100(PWDM-PLCDM)/PLCDM
z=1
1 10 100
k (h/Mpc)
-50
-40
-30
-20
-10
0
10
z=0
1 10100
-50
-40
-30
-20
-10
0
10
100(PWDM-PLCDM)/PLCDM
z=5
(100,512)
(50,512)
(25,512)
(25,384)
1 10 100
-50
-40
-30
-20
-10
0
10
z=2
Figure 2. Percentage difference between warm dark matter non-linear power spectrum and cold dark matter for the different runs. The
mass of the warm dark matter particle is kept fixed to mWDM= 1 keV. Blue, black, green curves refer to 100, 50, 25 h−1Mpcrespectively
and with a fixed number of particles NDM= 5123. The orange curves refer to 25 h−1Mpcand has a fixed number of particles NDM= 3843
The continuous lines represent the large scale estimate of the power, while the dashed ones describe the small scale power obtained with
the folding method (see text). The four panels represent different redshifts at z = 0,1,2,5 (bottom right, bottom left, top right and top
left, respectively). The dotted line plotted at z = 0 and z = 5 is the redshift independent linear suppression between the two models.
3 RESULTS
In this Section we describe the main results obtained from
our sample of simulations. The power spectrum is computed
from the distribution of the different sets of particles (dark
matter, gas and stars) separately and for the total matter
component by performing a CIC (Cloud-In-Cell) assignment
to a grid of the same size of the PM grid. The CIC ker-
nel is also deconvolved when getting the density at the grid
points (e.g. Viel et al. 2010)). We also show a small scale
estimate (k > 10 hMpc−1) of the power spectrum obtained
with the folding method described in (Jenkins et al. 1998;
Colombi et al. 2009), although this power spectrum will not
be used quantitatively.
We will plot the suppression in power spectrum as a per-
centage difference between WDM and ΛCDM matter power
spectra, normalized by the default ΛCDM power spectrum.
The initial conditions for CDM and WDM have the same
phases and cosmological/astrophysical parameters in order
to highlight the effect of the warm dark matter free stream-
ing.
3.1 Resolution and box-size
In Figure 2 we show the percentage difference between the
non-linear total power spectrum of WDM for mWDM=1 keV
and ΛCDM runs. We subtract the shot-noise from all the
power spectrum estimates made. For our largest box-size
simulations the shot-noise is comparable to the actual mea-
sured power spectrum at z = 0 at k ∼ 150 hMpc−1, while
for the default simulations (25,512) of mWDM=1 (0.25) keV
the matter power spectrum is always above the shot-noise
level for z < 10 and for k < 20(7) hMpc−1.
This figure focuses on the resolution and box-size effects
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M. Viel et al.
1 10 1001000
k (h/Mpc)
-60
-40
-20
0
20
100(PWDM-PLCDM)/PLCDM
z=1
1 101001000
k (h/Mpc)
-60
-40
-20
0
20
z=0
1 10100 1000
-60
-40
-20
0
20
100(PWDM-PLCDM)/PLCDM
z=5
0.25 keV
0.5 keV
1 keV
2 keV
4 keV
1 10 1001000
-60
-40
-20
0
20
z=2
Figure 3. Percentage difference between warm dark matter non-linear power and cold dark matter for the different runs. The resolution
is kept fixed in this plot and only 25 h−1Mpcboxes are considered. Orange, green, black, blue and red curves refer to mWDM =
0.25,0.5,1,2,4 keV, respectively. The continuous lines represent the large scale estimate of the power, while the dashed ones describe the
small scale power obtained with the folding method (see text). The four panels represent different redshifts at z = 0,1,2,5 (bottom right,
bottom left, top right and top left, respectively). The dotted coloured curves plotted at z = 0 and z = 5 are the redshift independent
linear suppression between the different models.
and presents the ratio at four different redshifts z = 0,1,3,5
(bottom right, bottom left, top right, top left panels, respec-
tively) and for three different box-sizes (100, 50, 25 h−1Mpc
shown as blue, black and green curves). The dotted line
represents the redshift independent linear cut-off of Eq.1,
while the lower resolution (25,384) run is also plotted as
orange curves. Here there are two estimates for the power
spectrum: one at large scales (solid curves), the second at
smaller scales (dashed curves). We are primarily interested
in the power at scales k < 10 hMpc−1and thereby only the
large scale estimate will be used, however, we decide also
to show the power at smaller scales since physical and nu-
merical effects play a larger role in this range. We note that
the linear theory suppression is a good approximation only
at k < 1 hMpc−1. From Figure 2 one can see that there
is convergence up to k = 50 hMpc−1between (25,512) and
(25,384) runs in all the redshift range considered. The res-
olution used is thus sufficient for mWDM=1 keV particles.
Note that van Daalen et al. (2011) have recently found that
(100,512) ΛCDM simulations have sufficiently converged at
scales k < 10 hMpc−1. At k = 3(10) h−1Mpc and z = 5
there is already a 5 (50)% difference between the linear
and non-linear power. At z = 0,1,3 the differences be-
tween WDM and ΛCDM power is 1,3 and 5% respectively at
k = 10 hMpc−1. The maximum suppression dip is strongly
influenced by resolution and moves to larger wavenumbers
when the resolution increases. At k > 100 hMpc−1we note
a steep (resolution dependent) turn-over in the suppression
which is likely to be due to effects that impact on the halo
structure and which has also been found in numerical simu-
lations that include a fraction of the matter content in the
form of active neutrinos (Brandbyge et al. 2008; Viel et al.
2010).
We have checked that increasing the particle-mesh grid
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Non-linear Matter Power in WDM Cosmologies
7
by a factor three (i.e. PM=1536) has negligible impact on
the total matter power at scales k < 100 hMpc−1. In order
to test the robustness of our results in terms of shot-noise
level we have also run a WDM simulation with mWDM=1
keV and NDM = 6403particles and compared the power
spectra with the (25,512) and (25,384) runs: we confirm
very good agreement between these simulations at k < 20
hMpc−1in the redshift range considered in the present work.
More precisely, the (25,512) and (25,640) WDM runs agree
below the one percent level at k < 100 hMpc−1.
3.2 The effect of the mass of a warm dark matter
particle
Here we
mWDM on the non-linear matter power spectrum. The re-
sults are shown in Figure 3 where we report five different
masses for the (25,512) default runs. The masses refer to
mWDM=0.25,0.5,1,2 and 4 keV (orange, green, black, blue
and red curves, respectively) at z = 0,1,2 and 5 (bottom
right, bottom left, top right and top left, respectively). The
linear suppressions are also shown with dotted lines of the
corresponding colors. At z = 5 we can see large differences
between the models that become smaller with the redshift
evolution. The 20% suppression at k = 10 hMpc−1for the
mWDM=1 keV model becomes 2% at z = 1 and disappears
at z = 0. Basically the clustering properties of the dark mat-
ter are the same at scales above k ∼ 10 hMpc−1at least for
mWDM> 1 keV. The mWDM=0.5 keV model still presents a
7% suppression by z = 0, while the suppression is four times
larger at z = 2. The linear suppression is a very poor approx-
imation in the range of wavenumbers considered here even
at high redshift. At z = 1, which is particularly interesting
for weak lensing data, a 2% measurement of the non-linear
power is likely to be able to exclude models below the 1 keV
value (bottom left panel). The dip of maximum suppression
and the turn-over both move to larger scales as the mass
decreases.
We have also investigated the importance of the warm
dark matter velocities in the initial conditions by running a
simulation without adding the Fermi-Dirac drawn thermal
velocity to the dark matter particles. We tested this for a
mWDM=1 keV model and found differences always below
1% in terms of total matter power spectrum at the scales of
interest here.
address the effect ofa different value of
3.3 Baryonic effects
In this section we explore the effects of baryonic physics
on the warm dark matter suppression. Baryons amount to
about 17% of the total matter content and we expect that
astrophysical processes affecting their properties can impact
on the total matter distribution at small scales. We identify
three effects that are able to modify the clustering properties
of baryons: radiative processes, star formation and galactic
feedback. These processes are usually modelled by hydrody-
namic simulations of galaxy formation. Here, the main goal
is not to explore fully the many parameters governing these
important physical aspects, but rather to address their im-
pact in warm dark matter models by adopting prescriptions
that are widely used in the literature. There could well be
other astrophysical processes (radiative transfer effects, feed-
back from active galactic nuclei, etc.) that can also affect the
distribution of baryons and their clustering properties (see
for example van Daalen et al. 2011).
In Figure 4 we plot the warm dark matter suppression
for the default simulation of mWDM=1 keV for three differ-
ent cases: pure dark matter (green curves); a hydrodynamic
simulation that include cooling and heating by an ultra-
violet background and the simple star formation criterion
able to simulate the Lyman-α forest (“BARYONS+QLYA”
run in blue); a hydrodynamic simulation that does in-
clude the full (sophisticated compared to QLYA implemen-
tation described above) star formation model based on the
multi-phase description of the ISM (Springel & Hernquist
2003) and strong galactic feedback in the form of winds
(“BARYONS+SF+WINDS” in black). Unfortunately, due
to the fact that hydrodynamic simulations are slower than
dark matter only runs it was not possible to carry the this
last simulation down to z = 0 and it was stopped at z = 1.2.
All of these processes can significantly change the clus-
tering of baryons especially at intermediate scales where
baryon pressure is important (k ∼ 1 hMpc−1), and where
they are not expected to trace the dark matter, and at
smaller scales given the complex interplay between feedback
and star formation processes. Cooling and heating modify
the thermal properties of the gas and are important espe-
cially for the low density IGM; the star formation criterion
determines how much gas is turned into stars within the po-
tential wells of dark matter haloes; galactic winds displace
gas out of the galaxies into the low density IGM and usu-
ally in a hot phase that prevents subsequent cooling. Since
the cosmic structure is different between CDM and WDM
models in general we do not expect the warm dark matter
suppression to be exactly the same between two simulations
that share the same astrophysical prescriptions. From Fig-
ure 4, one can see that the dark matter only simulations
and the one with radiative cooling and QLYA star forma-
tion are in good agreement at the percent level up to k = 10
hMpc−1, while at smaller scales there are significant differ-
ences and it is clear that the presence of baryons and star
formation greatly affects the maximum suppression and the
turn-over. Note that differences much larger than 10% be-
tween simulations implementing different radiative processes
(e.g. metal cooling) or feedback recipes are expected at
k > 20 hMpc−1in ΛCDM models (see e.g. Rudd et al. 2008;
Guillet et al. 2010; van Daalen et al. 2011). In the z = 0
panel we also show the difference in the power spectra of
ΛCDM and WDM models by normalizing to the correspond-
ing dark matter only model: in such a way we highlight the
effect of cooling produced by the baryons and not the WDM
signature. The two percentage differences are shown as cyan
(WDM) and red (ΛCDM) curves: the WDM universe when
filled with baryons that can cool has more power than a cor-
responding ΛCDM universe filled with the same baryon frac-
tion. The quantity Pnl,WDM,cooling/Pnl,WDM,dmonly is about
5% larger than Pnl,ΛCDM,cooling/Pnl,ΛCDM,dmonly at k = 10
hMpc−1and z = 5, becomes only 2 % larger at z = 1, and by
z = 0 there are no differences at the same wavenumber be-
tween the two quantities. The cooling of baryons inside the
potential wells of dark matter haloes produces further col-
lapse of structures and in general increases the (total) matter
power spectrum. It is thus likely that in the WDM model
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M. Viel et al.
1 10100
k (h/Mpc)
-50
-40
-30
-20
-10
0
100(PWDM-PΛCDM)/PΛCDM
z=1.2
1 10100
-50
-40
-30
-20
-10
0
100(PWDM-PΛCDM)/PΛCDM
z=5
DM only
BARYONS+QLYA
BARYONS+SF+WINDS
1 10 100
-50
-40
-30
-20
-10
0
z=3
1 10 100
k (h/Mpc)
-40
-20
0
20
z=0
Figure 4. Percentage difference between warm dark matter non-linear power and cold dark matter for different runs that incorporate
baryonic physical processes. The simulations refer to a 25 h−1Mpcbox and mWDM=1 keV. The green curves refer to the pure dark
matter simulations; blue curves refer to simulations that include baryons, cooling and a simplified recipe for star formation that turns
into collisionless stars all the gas particle below T=105K and denser than 1000 times the mean density (QLYA); black curves are
instead obtained by using the default criterion of multi-phase star formation of Springel (2005) and feedback in the form of strong kinetic
driven winds (this simulation was stopped at z = 1.2). The continuous lines represent the large scale estimate of the power, while the
dashed ones describe the small scale power obtained with the folding method (see text). The four panels represent different redshifts at
z = 0,1.2,3,5 (bottom right, bottom left, top right and top left, respectively). In the z = 0 panel (note the different scale for the y−axis)
we also show as the red and cyan curves the percentage of the matter power spectra that include and do not include cooling for ΛCDM
(red) and WDM (cyan) models, respectively (i.e. 100 × (Pbaryons+QLYA
mat
− PDMONLY
mat
)/PDMONLY
mat
).
the baryons cool slightly more efficiently than in the corre-
sponding ΛCDM since at high redshift the collapse of haloes
around the WDM cutoff is rapid and small scale modes af-
fected by cooling grow more rapidly than in CDM: this is
also the trend found by Gao & Theuns (2007) from the anal-
ysis of cooling at very high resolution and high redshift in
hydrodynamic simulations.
The warm dark matter suppression is thereby highly
influenced by astrophysics effects at k = 100 hMpc−1and
in general we expect a a suppression of about 2-3% at
k = 10 hMpc−1and at z > 1.5 for the mWDM=1 keV case
once baryons are included, while this discrepancy becomes
smaller at lower redshifts.
To sum up, any attempt to recover the total non lin-
ear matter power at z < 5 and at scales k = 1 − 10
hMpc−1should heavily rely on a proper modelling of astro-
physical aspects such as radiative processes, feedback and
star formation recipes.
3.4 Other cosmological parameters
To test the robustness of our results we extended the set of
simulations by exploring also other cosmological parameters,
namely: Ωm, H0 and σ8. In order to do that we modified the
input linear ΛCDM parameter calculated by CAMB and vary
one parameter at a time. Some parameters, such as σ8 (or
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Non-linear Matter Power in WDM Cosmologies
9
0.11.0 10.0 100.01000.0
k (h/Mpc)
0.90
0.95
1.00
1.05
1.10
(PWDM/PLCDM)/((PWDM/PLCDM)REF
z=1
0.11.0 10.0 100.0 1000.0
k (h/Mpc)
0.90
0.95
1.00
1.05
1.10
z=0
0.1 1.010.0100.01000.0
0.90
0.95
1.00
1.05
1.10
(PWDM/PLCDM)/((PWDM/PLCDM)REF
z=5
σ8=0.87
σ8=0.809
σ8=0.75
0.11.0 10.0100.0 1000.0
0.90
0.95
1.00
1.05
1.10
z=2
Figure 5. Impact of a different σ8 value in terms of WDM-induced suppression. The four panels represent different redshifts at z =
0,1,2,5 (bottom right, bottom left, top right and top left, respectively) for the (25,512) with mWDM=1 keV. Green represents the
(σ8= 0.809) reference case, while the two other curves indicate the suppression for σ8= 0.87 (black) and σ8= 0.75 (blue).
As), only boost the amplitude of the power spectrum but do
not change the shape. However the boost of power can have
also effects on the non-linear level. We choose the following
parameters for the WDM and corresponding ΛCDM runs:
Ωm = 0.22,0.32, H0 = 62,78 km/s/Mpc and σ8 = 0.75,0.87.
When calculating the suppression we always normalize both
the simulations to the same σ8 value (σ8 = 0.75,0.809 and
0.87). Since the WDM cut-off, for the WDM models in-
vestigated here, appears at much smaller scales than those
probed by the σ8 normalization, this requirement, together
with the fact that the WDM suppression has a very distinc-
tive feature, will make the suppression nearly independent
from any other parameter probed. The range explored by the
H0 values produces a maximum ±2% difference in terms of
the WDM suppression compared to the reference H0 = 70.3
km/s/Mpc case at k = 1−10 hMpc−1and at z < 3, while at
z = 5 there is a 5% difference at k = 10 hMpc−1. The Ωm
parameter produces a maximum difference of 1% at z < 3 in
the same range of wavenumbers and about 5% at z = 5 and
k = 10 hMpc−1. A slightly larger impact is the one induced
by a different choice of σ8 that we show in Figure 5 where
the WDM induced suppression is divided by the reference
case: it is clear that the large (10 %) differences in place at
z = 5 are largely canceled by the non-linear growth and are
at the ±2% level at z = 1−2 and at the 3% level at k = 10
hMpc−1and z = 0.
Motivated by the present findings we regard our non-
linear cutoff and its redshift dependence as robust at least for
the range of cosmological parameters investigated at z < 3,
for mWDM? 0.5 keV and at k = 1 − 10 hMpc−1: in fact
the differences are at the ± 2% level and in the next section
we will provide a fitting formula with a comparable level of
accuracy. Larger masses for mWDMwill only result in smaller
differences in terms of WDM suppression.
We also notice that degenerate features with the non-
linear WDM suppression might arise in the context of non-
standard models of dark energy, as e.g. interacting dark en-
ergy scenarios (see e.g. Baldi 2011). The investigation of such
possible degeneracies goes beyond the scope of the present
paper.
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M. Viel et al.
3.5 An analytical fitting formula
Inspired by the corresponding formula for the linear sup-
pression, we have found the following fitting formula to be
a good approximation of the late time evolution of the non-
linear suppression with an accuracy at the 2% level at z < 3
and for masses larger than mWDM=0.5 keV:
T2
nl(k) ≡ PWDM(k)/PΛCDM(k) = (1 + (αk)νl)−s/ν,
?1keV
mWDM
with ν = 3, l = 0.6 and s = 0.4.
We have chosen as a pivot redshift z = 1 since this is the
redshift where accurate weak lensing data will be available.
The accuracy of this fitting procedure is discussed below and
shown in Figure 7.
α(mWDM,z) = 0.0476
?1.85 ?1 + z
2
?1.3
, (2)
4WEAK LENSING SHEAR POWER
SPECTRA
Following Markovic et al. (2011) and Smith & Markovic
(2011), we examine the effect of the fitting function in Equa-
tion 2 on the weak lensing power spectrum. Weak gravita-
tional lensing is the distortion found in images of distant
galaxies due to the deflection of light from these galaxies by
the gravitational potential wells of intervening matter. For
a review, see for example Bartelmann & Schneider (2001).
The advantage of gravitational lensing is that unlike other
Large Scale Structure data, it does not require a knowl-
edge of galaxy bias for the derivation of the properties of
the underlying dark matter density field and is, at least on
large scales, independent of baryonic physics. In other words,
the weak lensing power spectrum directly probes the mat-
ter power spectrum. However, weak lensing measures the
matter power spectrum at low redshifts. For this reason it
is necessary to have available robust models of non-linear
structure. For a survey able to probe angular multipoles
from l ∼ 20 up to l ∼ 2 × 104, in the redshift range of
z = 0.5−2.0, the corresponding range of wavenumbers must
be k ∼ 0.005 − 15 hMpc−1. Note that the matter power at
k > 10 hMpc−1only has a significant contribution to the
weak lensing power spectrum at lower redshifts, where how-
ever the lensing power is lower.
Future weak lensing surveys accompanied by extensive
photometric redshift surveys will be able to disentangle the
contribution to weak lensing by dark matter at different red-
shifts, by binning source galaxies into tomographic bins (Hu
1999). By cross and auto correlations of the lensing power in
these bins, the three dimensional dark matter distribution
can be reconstructed. An existing example of such a recon-
struction is the COSMOS field (Massey et al. 2007). Such
tomography probes the non-linear matter power spectrum
at different redshifts.
We use HALOFIT (Smith et al. 2003) to calculate non-
linear corrections to the approximate linear matter power
spectrum (Ma 1996). We then apply Equation 2 to approx-
imate the WDM effects and find the weak lensing power
spectrum (e.g. Takada & Jain 2004):
Cij(l) =
?χH
0
dχlWi(χl)Wj(χl)χ−2
l
Pnl
?
k =
l
χl,χl
?
, (3)
where χl(zl) is the comoving distance to the lens at redshift
zl and Wi is the lensing weight in the tomographic bin i:
?zmax
zl
Wi(zl) =
4πG
al(zl)c2ρm,0χl
ni(zs)χls(zs,zl)
χs(zs)
dzs
, (4)
where we assume a flat universe and al(zl) is the scale factor
at the redshift of the lens, ρm,0 is the matter energy density
today and ni(zs) is the normalised redshift distribution of
sources in the i-th tomographic bin. We bin the multipoles
into 20 bins.
In order to asses detectability of WDM by future weak
lensing surveys, we calculate predicted error bars on the
weak lensing power spectrum using the covariance matrix
formalism (Takada & Jain 2004) and assuming errors for a
future realistic Weak Lensing survey as in Markovic et al.
(2011) and Smith & Markovic (2011) with 8 redshift bins in
the range z = 0.5 − 2.0. We plot the resulting percentage
differences between WDM and CDM weak lensing power
spectra in figure 6. It is important to note that the error
bars in the figure do not fully characterise the sensitivity of
the power spectra, since there are additional correlations be-
tween the error bars of different bin combinations. Addition-
ally, there are correlations in the error bars on large l (small
scales) due to non-linearities. Further statistical tests using
the entire covariance matrix must be used in order to fully
account for the above correlations. For this plot we choose
only the 5−th and 8−th redshift bins, whose source galaxy
distributions have the mean at z ∼ 1.0 and 1.6 respectively.
These bins are chosen because they represent a range with
the maximal WDM effect as well as lensing signal. Note that
the upturn around l ∼ 103in the auto-correlation power
spectra of bins 5 and 8 is due to the dominance of shot
noise on those scales. This noise is due to intrinsic galaxy
ellipticities and can be eliminated by cross-correlating dif-
ferent redshift bins, as can also be seen in Figure 6 (see also
Takada & Jain 2004).
In the right panel of Figure 6 we plot the effects of
the 0.5 keV particle and since the black dashed line lies
far outside the error bars this is a strong indication that
such a particle can be ruled out (or detected) by a future
weak lensing survey. This is consistent with previous works
(Markovic et al. 2011; Smith & Markovic 2011). In the left
panel of Figure 6 we plot the effects of a 1 keV WDM parti-
cle: in this case it is more difficult to distinguish from CDM
(black dashed line), but the strongly affected cross power
spectra are still significantly different from their expected
values in ΛCDM.
5 COMPARISON WITH HALO MODEL
As described in Section 4, it is necessary to have a robust
model of non-linear structure in order to take full advan-
tage of future weak lensing data. For this reason we com-
pare the non-linear matter power spectra extracted from
the simulations with previously derived non-linear models.
The halo model of non-linear structure is based on the
assumption that large scale structure is made up of indi-
vidual objects occupying peaks in the matter overdensity
field (Press & Schechter 1974; Seljak 2000; Cooray & Sheth
2002). The most important elements of this model, the mass
function, the halo bias (Press & Schechter 1974) and the
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10
1
10
2
10
3
10
4
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
multipole l
100(Cκ,ij
CDM−Cκ,ij
WDM)/Cκ,ij
CDM


1 keV
1 keV 1 keV
z = (1.0,1.0)
z = (1.0,1.6)
z = (1.6,1.6)
10
1
10
2
10
3
10
4
multipole l
0.5 keV
0.5 keV 0.5 keV
Figure 6. The percentage WDM effect in auto- and cross-correlation power spectra of redshift bins at approximately z = 1 and z = 1.6,
respectively. All the lines are calculated from non-linear matter power spectra modified for WDM by the fitting function in Equation
2 for WDM particle masses of 1 keV (left panel) and 0.5 keV (right panel). In addition we plot predicted error bars for a future weak
lensing survey, dividing the multipoles into 20 redshift bins. Note that the error bars on auto and cross power spectra of different bins
are correlated and therefore in order to fully characterise the detectable differences between the WDM (solid lines) and CDM (dashed
black line at 0) models, one must know the entire covariance matrix for a survey. Note secondly that the auto power spectra of redshift
bins at z = 1 and z = 1.6 have an upturn around l ∼ 103. This is due to the dominance of shot noise on those scales. This upturn is not
present in the cross power spectrum, because through cross correlation this noise due to intrinsic galaxy ellipticities is eliminated.
halo density profile (Navarro et al. 1995) are based on the
assumptions that all dark matter in the universe is found in
haloes and that there is no observable suppression of small
scale overdensities from early-times free-streaming of dark
matter particles or late-times thermal velocities.
These are characteristic properties of CDM, but do not
apply to WDM. For this reason Smith & Markovic (2011)
modified the halo model by applying a specific prescription
to the non-linear contribution, in addition to suppressing the
initial density field, modelled by applying a transfer function
from Viel et al. (2005) to the linear matter power spectrum.
Such prescription consists in: i) treating the dark matter
density field as made up of two components: a smooth, linear
component and a non-linear component, both with power at
all scales; ii) introducing a cut-off mass scale, below which
no haloes are found; iii) suppressing the mass function also
above the cut-off scale and iv) suppressing the centres of
halo density profiles by convolving them with a Gaussian
function, whose width depended on the WDM relic thermal
velocity.
Here, we do not attempt to explore each of these ele-
ments with simulations individually, but rather compare the
final matter power spectra found from simulations and from
the WDM halo model of Smith & Markovic (2011).
Secondly, Smith et al. (2003) compared the standard
CDM halo model to CDM simulations of large scale struc-
ture formation and developed an analytical fit to the non-
linear corrections of the matter power spectrum, known as
HALOFIT. We apply these corrections to a linear matter
power suppressed by the Viel et al. (2005) WDM transfer
function (see equation 1).
We show the results of these comparisons in Figure
7. As before, we plot the percent differences between the
WDM and CDM matter power spectra obtained from our
simulations of WDM only. We show this for particle masses
of mWDM=1 keV (left panels) and mWDM=0.5 keV (right
panels) at redshifts z = 1 (top row) and z = 0.5 (bottom
row). We find that the WDM halo model is closest to sim-
ulations at redshift z = 1 for mWDM=1 keV, but that it
over-estimates the suppression effect at redshift z = 0.5 for
mWDM=0.5 keV WDM by about 5 percent on scales k > 1
hMpc−1. On scales k < 1 hMpc−1however, the HALOFIT
non-linear correction describes the simulations better than
the halo model, even though on smaller scales it severely
underestimates the suppression effect, which becomes worse
at lower redshifts. A further small modification of the WDM
halo model will improve its correspondence to the simula-
tions and allow to use it at small scales.
We additionally consider these models of non-linear
WDM structure to calculate the weak lensing power spec-
tra in order to explore the significance of using the correct
model. We again plot percentage differences between WDM
and CDM weak lensing power spectra in Figure 8. We show
only curves representing the cross correlation power spec-
trum of redshift bins at z = 1 and z = 1.6 for consistency
with Figure 6. We again examine WDM models with par-
ticle masses of mWDM=1 keV (left panel) and mWDM=0.5
keV (right panel). We also calculate the weak lensing power
spectra without non-linear corrections to the matter power
spectrum and note that this severely over-estimates the ef-
fect of WDM suppression. In the lensing calculation, the
HALOFIT non-linear corrections applied to the WDM sup-
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M. Viel et al.
−30
−25
−20
−15
−10
−5
0
100(PWDM(k)−PCDM(k))/PCDM(k)


1 keV
z = 1
Simulations
Fit to simulations
Modified halo model
Halofit
Linear
0.5 keV
z = 1
10
−1
10
0
10
1
10
2
−30
−25
−20
−15
−10
−5
0
1 keV
z = 0.5
k[h/Mpc]
100(PWDM(k)−PCDM(k))/PCDM(k)
10
−1
10
0
10
1
10
2
0.5 keV
z = 0.5
k[h/Mpc]
Figure 7. The comparison of different non-linear models at redshifts z = 1.0 (top panels) and z = 0.5 (bottom panels) for WDM
particles with masses 1 keV (left panels) and 0.5 keV (right panels). The blue diamonds represent the fractional differences calculated
from DM-only simulations from previous plots with the fiducial values for σ8. The blue solid lines are the corresponding analytical fits
from equation 2. The green solid lines are calculated using the modified halo model, whereas the green dashed line is the standard
HALOFIT. The dotted line is the effect as seen in the linear matter power spectrum.
pressed linear matter power spectrum seem to perform bet-
ter in describing the results of our WDM simulations than
than the WDM halo model. This is most likely due to the
fact that the range of wavenumbers that are better described
by the HALOFIT corrections, namely k < 1hMpc−1are sig-
nificantly more relevant to the weak lensing power spectrum
than the smaller scales where HALOFIT deviates from the
simulation results. Using the linear WDM suppression trans-
fer function as recently done in Semboloni et al. (2011) can
significantly overestimates the effect of WDM on the weak
lensing power at l > 1000 as compared to the CDM case.
6 CONCLUSIONS
By using a large set of N-body and hydrodynamic simula-
tions we have explored the non-linear evolution of the total
matter power. The focus of the present work is on small
scales and relatively low redshifts where non-linear effects
are important and need to be properly modelled with sim-
ulations. We checked for numerical convergence and box-
sizes/resolution effects in the range k = 1−10 hMpc−1. We
explored how different masses of a warm dark matter candi-
date affect the non-linear suppression as compared to a cor-
responding ΛCDM model that shares the same parameters
and astrophysical inputs. Our findings can be summarized
as follows:
- Cosmological volumes of linear size 25h−1comoving
Mpc are sufficient to sample the WDM suppression for
mWDM? 1 keV with great accuracy.
- The non-linear suppression induced by WDM is
strongly redshift dependent: however, by z = 0, up to k = 10
hMpc−1, there are virtually no differences between ΛCDM
and WDM models with mWDM? 1 keV.
- At higher redshifts differences are larger, being closer
to the linear suppression, and at z ∼ 1 there are differences
of the order of a few percent between the non-linear WDM
and ΛCDM power spectra.
- Baryonic physics and in particular radiative processes
for the gas component, the star formation criterion and
galactic feedback are likely to affect the matter power at
the 2-3 % level in the range k = 1 − 10 hMpc−1.
- We investigate how a change in Ωm, H0 and σ8 impact
on the non-linear power and in particular on the WDM sup-
pression when values different from our reference choice are
used and found small difference at the ± 2% level at the
scales considered here: thus the WDM cutoff has a distinc-
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Non-linear Matter Power in WDM Cosmologies
13
10
1
10
2
10
3
10
4
−10
−8
−6
−4
−2
0
100(Cκ,ij
CDM−Cκ,ij
WDM)/Cκ,ij
CDM
multipole l
z = (1.0,1.6)
1 keV
1 keV1 keV 1 keV
z = (1.0,1.6)z = (1.0,1.6) z = (1.0,1.6)


Fit to simulations
Modified halo model
Halofit
Linear
10
1
10
2
10
3
10
4
multipole l
z = (1.0,1.6)
0.5 keV
0.5 keV 0.5 keV 0.5 keV
z = (1.0,1.6)z = (1.0,1.6) z = (1.0,1.6)
Figure 8. The comparison of the impact of using different models of non-linear power spectra from figure 7 on the weak lensing power
spectrum. As above, the blue line is the fractional difference in percent between weak lensing power spectra calculated using the fitting
function found in this work (2). The green solid line is the weak lensing power spectrum calculated using the halo model modified for
WDM. The dashed green line is the same using standard HALOFIT. The dotted line is calculated by omitting all non-linear corrections.
It is evident that excluding such corrections causes a significant overestimation of the WDM effect. All the lines in this plot are calculated
from cross power spectra of the 5th and 8th tomographic bins (corresponding to z = 1.6 and z = 1, respectively) for WDM particle
masses of mWDM=1 keV (left panel) and mWDM=0.5 keV (right panel).
tive feature which is not degenerate with other cosmological
parameters also at a non-linear level.
- We provide a useful fit to the non-linear WDM induced
suppression in terms of a redshift-dependent transfer func-
tion; this fitting formula should agree to the actual measured
power at the 2% level at z < 3 and for masses above 0.5 keV.
- Reaching a higher accuracy (percent level) in terms of
WDM non-linear power would require a much more care-
ful analysis of astrophysical aspects related to the baryonic
component.
- We find that future weak lensing surveys will most
likely be powerful enough to rule out WDM masses smaller
than 1 keV, which is consistent with previous results of
Markovic et al. (2011) and Smith & Markovic (2011). Rul-
ing out models for masses larger than 1 keV would still be
possible by using the cross-correlation signal between differ-
ent redshift bins.
- Non-linear corrections to the matter power spectrum in
the WDM scenario obtained from HALOFIT correspond bet-
ter to the results of the WDM only simulation at scales k < 1
hMpc−1, but on smaller scales the WDM halo model of
Smith & Markovic (2011) performs better and suggests that
a further modification to the halo model might be needed,
especially for weak lensing power spectrum calculations.
We believe that future efforts aiming at measuring the cold-
ness of cold dark matter should investigate the non-linear
matter power in the range z = 0 − 5 either using weak
lensing observables or the small scale clustering of galax-
ies. These constraints can be particularly useful since they
are complementary to those that can be obtained from high
redshift Lyman-α forest data (e.g. BOSS/SDSS-III survey)
or galactic and sub-galactic observables in the local universe.
ACKNOWLEDGMENTS.
We are grateful to Robert Smith for providing tables to
calculate the halo model power spectrum. Numerical com-
putations were performed at the High Performance Com-
puter Cluster Darwin (HPCF) in Cambridge (UK) and at
the RZG computing center in Garching. We acknowledge
support from an ASI/AAE grant, ASI contracts Euclid-IC
(I/031/10/0), INFN PD51, PRIN MIUR, PRIN INAF 2009,
ERC-StG “cosmoIGM”. Post-processing od the simulations
has also been carried at CINECA (Italy) and COSMOS Su-
percomputer in Cambridge (UK). KM acknowledges support
from the International Max-Planck Research School.
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