# 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.

**0**Bookmarks

**·**

**91**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**We review the status of sterile neutrino dark matter and discuss astrophysical and cosmological bounds on its properties as well as future prospects for its experimental searches. We argue that if sterile neutrinos are the dominant fraction of dark matter, detecting an astrophysical signal from their decay (the so-called 'indirect detection') may be the only way to identify these particles experimentally. However, it may be possible to check the dark matter origin of the observed signal unambiguously using its characteristic properties and/or using synergy with accelerator experiments, searching for other sterile neutrinos, responsible for neutrino flavor oscillations. We argue that to fully explore this possibility a dedicated cosmic mission - an X-ray spectrometer - is needed.Physics of the Dark Universe. 06/2013; 1(s 1–2). - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We present updated constraints on the free-streaming of warm dark matter (WDM) particles derived from an analysis of the Lya flux power spectrum measured from high-resolution spectra of 25 z > 4 quasars obtained with the Keck High Resolution Echelle Spectrometer (HIRES) and the Magellan Inamori Kyocera Echelle (MIKE) spectrograph. We utilize a new suite of high-resolution hydrodynamical simulations that explore WDM masses of 1, 2 and 4 keV (assuming the WDM consists of thermal relics), along with different physically motivated thermal histories. We carefully address different sources of systematic error that may affect our final results and perform an analysis of the Lya flux power with conservative error estimates. By using a method that samples the multi-dimensional astrophysical and cosmological parameter space, we obtain a lower limit mwdm > 3.3 keV (2sigma) for warm dark matter particles in the form of early decoupled thermal relics. Adding the Sloan Digital Sky Survey (SDSS) Lya flux power spectrum does not improve this limit. Thermal relics of masses 1 keV, 2 keV and 2.5 keV are disfavoured by the data at about the 9sigma, 4sigma and 3sigma C.L., respectively. Our analysis disfavours WDM models where there is a suppression in the linear matter power spectrum at (non-linear) scales corresponding to k=10h/Mpc which deviates more than 10% from a LCDM model. Given this limit, the corresponding "free-streaming mass" below which the mass function may be suppressed is 2x10^8 Msun/h. There is thus very little room for a contribution of the free-streaming of WDM to the solution of what has been termed the small scale crisis of cold dark matter.Physical review D: Particles and fields 06/2013; 88(4). - SourceAvailable from: Emiliano Sefusatti[Show abstract] [Hide abstract]

**ABSTRACT:**We re-evaluate the extragalactic gamma-ray flux prediction from dark matter annihilation in the approach of integrating over the nonlinear matter power spectrum, extrapolated to the free-streaming scale. We provide an estimate of the uncertainty based entirely on available N-body simulation results and minimal theoretical assumptions. We illustrate how an improvement in the simulation resolution, exemplified by the comparison between the Millennium and Millennium II simulations, affects our estimate of the flux uncertainty and we provide a "best guess" value for the flux multiplier, based on the assumption of stable clustering for the dark matter perturbations described as a collision-less fluid. We achieve results comparable to traditional Halo Model calculations, but with a much simpler procedure and a more general approach, as it relies only on one, directly measurable quantity. In addition we discuss the extension of our calculation to include baryonic effects as modeled in hydrodynamical cosmological simulations and other possible sources of uncertainty that would in turn affect indirect dark matter signals. Upper limit on the integrated power spectrum from supernovae lensing magnification are also derived and compared with theoretical expectations.01/2014; 441(3).

Page 1

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

c ? 0000 RAS

Page 2

2

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

c ? 0000 RAS, MNRAS 000, 000–000

Page 3

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.

c ? 0000 RAS, MNRAS 000, 000–000

Page 4

4

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.

c ? 0000 RAS, MNRAS 000, 000–000

Page 5

Non-linear Matter Power in WDM Cosmologies

5

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

c ? 0000 RAS, MNRAS 000, 000–000

Page 6

6

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

c ? 0000 RAS, MNRAS 000, 000–000

Page 7

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

c ? 0000 RAS, MNRAS 000, 000–000

Page 8

8

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

c ? 0000 RAS, MNRAS 000, 000–000

Page 9

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.

c ? 0000 RAS, MNRAS 000, 000–000

Page 10

10

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

c ? 0000 RAS, MNRAS 000, 000–000

Page 11

Non-linear Matter Power in WDM Cosmologies

11

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-

c ? 0000 RAS, MNRAS 000, 000–000

Page 12

12

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-

c ? 0000 RAS, MNRAS 000, 000–000

Page 13

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.

REFERENCES

Abazajian K., 2006, Phys. Rev. D, 73, 063513

Abbott T., et al., 2005, arXiv:astro-ph/0510346

Baldi M., 2011, MNRAS, 414, 116

Bartelmann M., Schneider P., 2001, Physics Reports, 340,

291

Bode P., Ostriker J. P., Turok N., 2001, ApJ, 556, 93

Bolton J. S., Haehnelt M. G., Viel M., Springel V., 2005,

MNRAS, 357, 1178

Boyarsky A., Lesgourgues J., Ruchayskiy O., Viel M.,

2009a, JCAP, 5, 12

c ? 0000 RAS, MNRAS 000, 000–000

Page 14

14

M. Viel et al.

Boyarsky A., Lesgourgues J., Ruchayskiy O., Viel M.,

2009b, Physical Review Letters, 102, 201304

Brandbyge J., Hannestad S., Haugbølle T., Thomsen B.,

2008, JCAP, 8, 20

Casarini L., La Vacca G., Amendola L., Bonometto S. A.,

Macci` o A. V., 2011, JCAP, 3, 26

Coil A. L., Newman J. A., Croton D., Cooper M. C., Davis

M., Faber S. M., Gerke B. F., Koo D. C., Padmanabhan

N., Wechsler R. H., Weiner B. J., 2008, ApJ, 672, 153

Col´ ın P., Avila-Reese V., Valenzuela O., 2000, ApJ, 542,

622

Col´ ın P., Valenzuela O., Avila-Reese V., 2008, ApJ, 673,

203

Colombi S., Jaffe A., Novikov D., Pichon C., 2009, MN-

RAS, 393, 511

Cooray A., Sheth R., 2002, Physics Reports, 372, 1

Dodelson S., Widrow L. M., 1994, Phys. Rev. Lett., 72, 17

Gao L., Theuns T., 2007, Science, 317, 1527

Guillet T., Teyssier R., Colombi S., 2010, MNRAS, 405,

525

Hansen S. H., Lesgourgues J., Pastor S., Silk J., 2002, MN-

RAS, 333, 544

Hill G. J., et al., 2008, ASP Conf. Ser., 399, 115

Hu W., 1999, The Astrophysical Journal, 522, L21

Ivezic Z., et al., 2008, arXiv:0805.2366

Jenkins A., Frenk C. S., Pearce F. R., Thomas P. A., Col-

berg J. M., White S. D. M., Couchman H. M. P., Peacock

J. A., Efstathiou G., Nelson A. H., 1998, ApJ, 499, 20

Kaiser N., et al., 2002, Proc. SPIE Int. Soc. Opt. Eng.,

4836, 154

Katz N., Weinberg D. H., Hernquist L., 1996, ApJS, 105,

19

Knebe A., Arnold B., Power C., Gibson B. K., 2008, MN-

RAS, 386, 1029

Komatsu E., et al., 2011, ApJS, 192, 18

Lesgourgues J., Tram T., 2011, ArXiv e-prints

Lewis A., Challinor A., Lasenby A., 2000, Astrophys. J.,

538, 473

Lovell M., Eke V., Frenk C., Gao L., Jenkins A., Theuns

T., Wang J., Boyarsky A., Ruchayskiy O., 2011, ArXiv

e-prints

Ma C.-P., 1996, Astrophysical Journal v.471, 471, 13

Macci` o A. V., Fontanot F., 2010, MNRAS, 404, L16

Markovic K., Bridle S., Slosar A., Weller J., 2011, JCAP,

1, 22

Massey R., Rhodes J., Leauthaud A., Capak P., Ellis R.,

Koekemoer A., Refregier A., Scoville N., Taylor J. E., Al-

bert J., Berge J., Heymans C., Johnston D., Kneib J.-P.,

Mellier Y., Mobasher B., Semboloni E., Shopbell P., Tasca

L., Waerbeke L. V., 2007, arXiv, astro-ph

Navarro J. F., Frenk C. S., White S. D. M., 1995, arXiv,

astro-ph

Polisensky E., Ricotti M., 2011, Phys. Rev. D, 83, 043506

Press W. H., Schechter P., 1974, ApJ, 187, 425

Refregier A., et al., 2010, arXiv:1001.0061

Rudd D. H., Zentner A. R., Kravtsov A. V., 2008, ApJ,

672, 19

Seljak U., 2000, Monthly Notices of the Royal Astronomical

Society, 318, 203

Seljak U., Makarov A., McDonald P., Trac H., 2006, Phys-

ical Review Letters, 97, 191303

Seljak U. e. a., 2005, Phys. Rev. D, 71, 103515

Semboloni E., Hoekstra H., Schaye J., van Daalen M. P.,

McCarthy I. G., 2011, ArXiv e-prints

Smith R. E., Markovic K., 2011, ArXiv e-prints

Smith R. E., Peacock J. A., Jenkins A., White S. D. M.,

Frenk C. S., Pearce F. R., Thomas P. A., Efstathiou G.,

Couchman H. M. P., 2003, MNRAS, 341, 1311

Springel V., 2005, MNRAS, 364, 1105

Springel V., Hernquist L., 2003, MNRAS, 339, 289

Springel V., Hernquist L., 2003, Mon. Not. Roy. Astron.

Soc., 339, 312

Strigari L. E., Bullock J. S., Kaplinghat M., Kravtsov

A. V., Gnedin O. Y., Abazajian K., Klypin A. A., 2006,

ApJ, 652, 306

Takada M., Jain B., 2004, Monthly Notices of the Royal

Astronomical Society, 348, 897

Tikhonov A. V., Gottl¨ ober S., Yepes G., Hoffman Y., 2009,

MNRAS, 399, 1611

Trujillo-Gomez S., Klypin A., Primack J., Romanowsky

A. J., 2010, ArXiv e-prints

van Daalen M. P., Schaye J., Booth C. M., Dalla Vecchia

C., 2011, ArXiv e-prints

Viel M., Haehnelt M. G., Lewis A., 2006, MNRAS, 370,

L51

Viel M., Haehnelt M. G., Springel V., 2004, MNRAS, 354,

684

Viel M., Haehnelt M. G., Springel V., 2010, JCAP, 6, 15

Viel M., Lesgourgues J., Haehnelt M. G., Matarrese S.,

Riotto A., 2005, Phys. Rev. D, 71, 063534

Viel M., Lesgourgues J., Haehnelt M. G., Matarrese S.,

Riotto A., 2006, Physical Review Letters, 97, 071301

Wang J., White S. D. M., 2007, MNRAS, 380, 93

Zavala J., Jing Y. P., Faltenbacher A., Yepes G., Hoffman

Y., Gottl¨ ober S., Catinella B., 2009, ApJ, 700, 1779

c ? 0000 RAS, MNRAS 000, 000–000

#### View other sources

#### Hide other sources

- Available from Marco Baldi · Jun 2, 2014
- Available from ArXiv
- Available from arxiv.org