Warmth Elevating the Depths Shallower Voids With Warm Dark Matter

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Mon. Not. R. Astron. Soc. 000, ???? (2014) Printed 20 November 2014 (MN LATEX style file v2.2)

Warmth Elevating the Depths:

Shallower Voids with Warm Dark Matter

Lin F. Yang1, Mark C. Neyrinck1, Miguel A. Aragon-Calvo2 and Joseph Silk 1,3,41Department of Physics & Astronomy, The Johns Hopkins University, 3400 N Charles Street, Baltimore, MD 21218, USA2Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA3Institut dAstrophysique de Paris- 98 bis boulevard Arago-75014 Paris, France4Beecroft Institute of Particle Astrophysics and Cosmology, Department of Physics, University of Oxford,

Denys Wilkinson Building, 1 Keble Road, Oxford OX1 3RH, UK

in original form

ABSTRACT

Warm dark matter (WDM) has been proposed as an alternative to cold dark matter(CDM), to resolve issues such as the apparent lack of satellites around the Milky Way.Even if WDM is not the answer to observational issues, it is essential to constrain thenature of the dark matter. The effect of WDM on haloes has been extensively studied,but the small-scale initial smoothing in WDM also affects the present-day cosmic weband voids. It suppresses the cosmic sub-web inside voids, and the formation of bothvoid haloes and subvoids. InN-body simulations run with different assumed WDMmasses, we identify voids with the zobov algorithm, and cosmic-web componentswith the origami algorithm. As dark-matter warmth increases, the initial-conditionssmoothing increases, and the number of voids and subvoids is suppressed. Also, voiddensity profiles change, their shapes become flatter inside the void radius, while edgesof the voids remain unchanged. Also, filaments and walls become cleaner, as the sub-structures in between have been smoothed out; this leads to a clear, mid-range peakin the density PDF. These distinct features of voids make them possible observational

indicators of the warmth of dark matter.Key words: galaxies: halos methods: N-body simulations cosmology: theory cosmology: dark matter

1 INTRODUCTION

There is overwhelming evidence (e.g. Frenk & White 2012)for the existence of dark matter (DM), whose nature is stillunknown. Although many direct or indirect detection ex-periments (e.g. Akerib et al. (2014); Agnese et al. (2013);Aprile (2013); Ackermann et al. (2014)) have occurred andare ongoing, no conclusive result has been reported.

It has been long known (Peebles 1980; Bertone et al.2005) that the matter of the universe is dominated by DM.For structure formation, the velocity distribution of DMparticles plays an important role. In the standard CDMmodel, the DM is assumed to be entirely cold from thestandpoint of structure formation. The velocity dispersionis negligible at the era of matter-radiation equality (teq),and structure formation proceeds in a bottom-up fashion.Smaller structures form at first, then larger ones. This modelhas only a few parameters, that have been determined withhigh precision. However, several problems remain unsolved

E-mail: [email protected] E-mail: [email protected]

on sub-galactic scales. First, the missing satellite problem:simple arguments applied to CDM-only simulations implythat thousands or hundreds of dwarf galaxies are expectedin the local group and halo of the Milky Way, however onlyof order 10 of them were found ( Moore et al. 1999a; Mateo1998). Second, CDM predicts concentrated density profilesin the central region, e.g. r1 in the NFW (Navarro et al.

1997) profile, whereas many studies of galaxy rotation curveshave concluded that the density approaches a constant inthe core (Moore et al. 1999b; Ghigna et al. 2000). Third,the number of dwarf galaxies expected in local voids may beless than a CDM model would predict (Peebles 2001).

Although better modelling of hydrodynamics and feed-back processes may solve these problems (e.g. Hoeft et al.2006), changing the DM itself could also help resolve some ofthe issues. Warm dark matter (WDM) has been an attrac-tive alternative since the 1980s (e.g. Schaeffer & Silk 1988).But it was disfavored since it requires an additional param-eter, the particle mass. Recently, the WDM model has re-ceived some interest since it can reproduce all the successfulCDM results on large scales, but also solve some small-scale

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issues. The key feature separating WDM from CDM modelsis the lack of initial small-scale fluctuations. The WDM hasslightly larger velocity dispersion atteq, giving a smoothingof initial fluctuations at a free-streaming length determinedby the WDM particle mass. From particle physics, the orig-inally favored WDM candidate was a gravitino (e.g. Mo-roi et al. 1993); more recently, a sterile neutrino (Boyarskyet al. 2009) has seen attention. Both theoretical and numer-ical studies (e.g. Bode et al. (2001)) have explored WDMmodels, and observational constraints have been put on themass of WDM particles.Viel et al. (2008) give a lower limitofmX = 1.2 keV from Lyman-forest data, which increasesto 4 keV if Sloan Digital Sky Survey (SDSS) and HIRESdata are included. Other independent studies (Miranda &Maccio 2007)also give consistent limits.

While previous studies on WDM mainly focused on theformation of halos or other dense structures, there has beensome work investigating the cosmic web itself. Schneideret al. (2012) studied voids in a WDM scenario, but focuson the halo population within them, finding that voids areemptier (of haloes and substructure) in WDM. Below, westudy the dark-matter density itself, which follows the op-posite trend, growing in density in WDM.Reed et al.(2014)also studied the large-scale-structure traced out by galaxiesin a WDM scenario, including a study of galaxy environ-ment. They found that WDM makes very little difference inthe usual observables of the galaxy population, when usingsubhalo abundance matching (SHAM) to identify galaxies.That is, they found that the subhaloes in the dark matter,above a mass threshold where halo formation is not sub-stantially disrupted from the loss of power in WDM, arearranged in nearly the same way in WDM and CDM. Hy-drodynamic simulations that include star formation indicatethat stars may form in filaments instead of haloes if the darkmatter is quite warm (Gao & Theuns 2007; Gao et al. 2014),an issue related to the low complexity of dark-matter halostructure in WDM(Neyrinck 2014b).

Voids are large underdense regions, occupying the ma-jority of the volume of the Universe, and are valuable cosmo-logical probes. For example, via the Alcock-Paczynski test,voids serve as a powerful tool to detect the expansion his-tory of the universe (e.g.Ryden(1995); Lavaux & Wandelt(2012)).Clampitt et al. (2013);Li et al.(2012)have also pro-posed voids as a probe of modified gravity. The abundance ofvoids may be sensitive to initial conditions (Kamionkowskiet al. 2009), hence voids may serve as probes of the early uni-verse. Previous work has also looked at voids in the contextof WDM. Tikhonov et al. (2009) measured the abundanceof mini-voids, which become scarcer in a WDM model. Re-cently,Clampitt & Jain (2014) detected void lensing at a sig-nificance of 13, raising hopes for void density-profile mea-surements using lensing. The few-parameter universal formfor void density profiles that Hamaus et al. (2014) foundwill likely help in extracting cosmological information fromvoids.

In this paper, we study how properties of voids in thecosmic web change in a WDM scenario, with different ini-tial power-spectrum attenuations corresponding to differentWDM masses. We analyze these simulations with the zobovvoid-finder (Neyrinck 2008) and the origami (Falck et al.2012), filament, wall and halo classifier. The paper is laidout as follows. In section 2, we introduce our warm dark

matter N-body simulations. In section 3, we analyze the fullcosmic web of DM in a WDM scenario. In section 4, weintroduce our void detection methods and shows the voidstatistical properties. In section 5, we show the distinct fea-tures of void density profiles for different DM settings. Wegive our conclusions and discussion in section 6.

2 SIMULATIONS

We simulate the evolution of CDM and WDM universes bya similar method to that ofAngulo et al. (2013). We startby computing the initial power spectrum of density fluctu-ations with the camb code (Lewis et al. 2000) using vanilla(C)DM cosmological parameters (h = 0.7, M = 0.3, = 0.7, b = 0.045, 8 = 0.83, ns = 0.96). For WDMmodels, we attenuate the power on small scales using thefitting formula below. We use the Zeldovich approximation(Zeldovich 1970) to impart initial displacements and ve-

locities at initial redshiftz= 127 to particles on the initiallattice of 5123 particles in a periodic box of size 100 h1 Mpc.To incorporate the effect of a thermally produced relic

WDM particle, we apply the following fitting formula to theWDM transfer functions (Bode et al. 2001),

TWDM = TCDM(k)[1 + (k)2]5.0, (1)

where the cutoff scale,

= 0.05

m0.4

0.15 h

0.65

1.3 mdm1keV

1.15

1.5

gX

0.29.

(2)

h1Mpc,

Here, mdm

is the mass of the WDM particle (or the effec-tive sterile neutrino); gX is the number of degrees of free-dom that the WDM particle contributes to the number den-sity (in our case, 3/2). In our set of simulations, we applied = 0, 0.05, 0.1 and 0.2 h1 Mpc, corresponding first toCDM, and then to WDM particle masses 1.4, 0.8 and 0.4keV. Of course, even with = 0, the interparticle spacingof 0.2 h1 Mpc imposes a cutoff in the initial power spec-trum; the true CDM cosmic web would have much morestructure. Also notice that only in the most extreme caseof 0.2 h1 Mpc is comparable to the interparticle sepa-ration. The reason that differences show up even when isbelow this scale is the broad shape of the attenuation de-scribed in Eq. (2).Bode et al. (2001) define a perhaps moremeaningful half-mode scale radius Rs, via T(/Rs) = 1/2;this quantity is 6. We hold 8 fixed in the linear powerspectrum. This changes the large-scale amplitude, but veryslightly since the smoothing kernel acts on scales well below8 h1 Mpc.

We have not included the thermal velocity kicks to theindividual particles to our simulation, for two reasons. First,particles in the simulation are averages over a statistical en-semble of particles, so it is unclear how to implement thethermal velocity in the initial conditions. Also, this ther-mal velocity would be negligible for our results. For the values we use, the RMS velocity distribution is of order 1km/s (Angulo et al. 2013), while typical velocities of par-ticles inside a void are of order 50 km/s at radius around0.5rv (Hamaus et al. 2014).

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Shallower Voids with Warm Dark Matter 3

0

2 5

5 0

7 5

1 0 0

0 2 5 5 0 7 5

x [ M p c / h ]

0

2 5

5 0

7 5

1 0 0

0 2 5 5 0 7 5 1 0 0

x [ M p c / h ]

- 4 . 0 0

0 . 1 7

4 . 3 3

8 . 5 0

Figure 1.LTFE density field slices, showing ln(1 + ). From top left to bottom right: CDM, WDM with = 0.05 h1 Mpc, WDM with= 0.1 h1 Mpcand WDM with = 0.2 h1 Mpc.

3 THE COSMIC WEB IN A WDM SCENARIO

Fig. 1 shows a slice of an LTFE (Lagrangian TessellationField Estimator; Abel et al. 2012) density field of the sim-ulation. This estimator makes use of the the fact that, un-der only gravity, the 3D manifold of DM particles evolvein phase space without tearing, conserve phase space vol-ume and preserve connectivity of nearby points. Hence thesheets (or streams) formed by the initial grids are assumedto remain at constant mass in the final snapshot. Using theinitial grid position (or the Lagrangian coordinates) of eachparticle, the density of each stream could be calculated. Weimplemented an OpenGL code of LTFE to estimate the den-sity field in a time efficient way. The differences of densityfields are clear: small-scale structures are smoothed out inthe WDM simulations. Fig.2shows mass-weighted 1-point

PDFs (probability density functions) at z= 0 from the sim-

ulations, for each. We measured the density at each parti-cle using the Voronoi Tessellation Field Estimator (Schaap& van de Weygaert 2000;van de Weygaert & Schaap 2009,VTFE). In the VTFE, each particle occupies a Voronoi cell,a locus of space closer to that particle than to any other par-ticle. The density VTFE+ 1 = V /Vat a particle is set bythe volume Vof its cell. This density measure is in a senseLagrangian, but only strictly so without multi-streaming.

At = 0, this mass-weighted PDF shows two clearpeaks, noted byNeyrinck(2008). It was already clear thatthe higher-density peak, a roughly lognormal peak at e67 1000, comes from halo particles. Falck et al. (2012)firmly established this halo origin by classifying particleswith the origamialgorithm, into void (single-stream), wall,

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0 . 0 0 0

0 . 0 0 5

0 . 0 1 0

= 0

V

W

F

H

T o t a l

V o i d m o d e l

= 0 . 0 5 M p c / h

50

51 0

0 . 0 0 0

0 . 0 0 5

0 . 0 1 0

= 0 . 1 M p c / h

P

D

F

l n ( 1 + )

50

51 0

= 0 . 2 M p c / h

Figure 2.Mass-weighted PDFs of particle densities for the four simulations. A distinctive peak arises from filament and wall particles atmiddling densities in a WDM scenario. The V, W, F, and H curves add up to the total: they separate out void, filament, wall, and haloparticles, with crossings along 0, 1, 2, and 3 orthogonal axes. = 0 corresponds to CDM; as increases, the WDM becomes warmer.In this mass-weighted PDF, each particle (Lagrangian element of initial spacing 0.2 h1 Mpc) enters once. The dashed magenta curveshows the expression in Eq. (3).

filament, and halo morphologies. This algorithm counts thenumber of orthogonal axes along which a particle has beencrossed by any other particle, comparing the initial and finalconditions.

For = 0, the lumpy shape of the total PDF at lowdensities already suggests that there may be more than2 components. As increases, however, an intermediatewall+filament peak becomes unmistakable: the visual im-pression from the density-field maps that a greater fractionof the matter is in walls and filaments is obvious in the to-tal PDF as well. Again, the origami classification confirmsthis picture. As the WDM mass increases, particles movefrom low to high densities and morphologies, through thedifferent peaks. Curiously, the fractions in walls and fila-ments remain about constant, most of the particles trans-ferred from the void to halo peaks. There is a change in themean log-densities of the wall and filament peaks, however;while a substantial fraction of wall particles have


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Figure 3. z = 0 Voronoi particle densities ln(1 + ) on a 2DLagrangian sheet, with one particle per 0.2 h1 Mpc comovingLagrangian pixel. Each panel shows a 2D, 2562 slice, a quadrantof a full 5122 slice.

is the variance in spheres of radius L/N/(4/3)1/3 (whereL= 100 h1 Mpc is the box size, andN= 512) as calculatedfrom a camblinear power spectrum atz= 0, truncating thepower spectrum to zero at k > /(L/N).

In Figs. 3and4, Lagrangian-space density maps showwhere these various density regimes appear in the cosmicweb. Here, each pixel represents a particle, arranged on its

Figure 4. Same as Fig. 3, with origami morphologies added:black, red, and white contours separate void, wall, filament, andhalo particle morphologies.

initial lattice. In Fig.4, wart-like blobs within black contoursare haloes; these contract substantially in the mapping tocomoving z = 0 Eulerian space. The regions within whitecontours are void regions, which expand in comoving coor-dinates and come to fill most of the space; seeFalck et al.(2012)for more detail and an alternative plotting method.

As in Falck & Neyrinck (2014), the origami void re-

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gion, delineated by stream-crossing, does not pinch off intoan idealized catalog of cellular convex voids (e.g. Icke &van de Weygaert 1991; Neyrinck 2014a) in any of the sim-ulations. This percolation of the void volume is entirelyobvious even in 2D in Fig.4 at high. As decreases from0.2 h1 Mpc (going from WDM to CDM), increased small-scale structure gives density peaks along walls, but densityminima also appear along walls, also reaching lower densitiesat low. This is evident in the leftward shift of the wall peakwith decreasing in Fig.2. This lower-density environmentstunts growth, as in an open Universe (e.g. Sheth & van deWeygaert 2004). Thus, holes allowing percolation throughwalls are still there at low . Falck & Neyrinck (2014) de-scribe an origami definition of voids, giving a watershed-like catalog of voids incorporating the single-stream defini-tion, even though the walls do not entirely enclose them.Since this method was not fully developed when the presentproject was begun, we use the zobov void finder.

4 VOID DETECTION AND PROPERTIES

zobovfirst uses a Voronoi tessellation to get the density ofeach particle. After that, it uses each local density minimumas a seed and groups other particles around it using the wa-tershed algorithm, forming a zone, regions with a densityminimum and a ridge. These zones are combined into largerparent voids using essentially another application of the wa-tershed algorithm, giving a hierarchy of subvoids.

In the default parameter-free algorithm output, thewhole field is a single large super-void with many levels ofsub-voids, which is difficult to use directly. We therefore re-quired the zones added to a void to have core density lessthan , the mean density of the simulation. zobovmeasuresthe statistical significance of voids, compared to a Poissonprocess. The probability a void is real depends on the den-sity contrast, defined as the ratio of the minimum densityon the ridge separating the void from another void to thevoids minimum density. To focus on voids with low discrete-ness effects, we analyze voids with significance larger than3 according to this density contrast criterion and measuretheir properties. We found about 7600, 6700, 3900, and 1400voids for the 4 simulations. Consistently with the resultsof Tikhonov et al. (2009), the number of voids decreasesas the cut-off scale increases. Compared to Hamaus et al.(2014), our much higher mass resolution (compared to theirsparse-sampling to better approximate a galaxy sample) al-lows much smaller voids to be detected in the matter field.We acknowledge that these smallest structures would likelynot show up in a galaxy survey unless it was extremely deep.Even then, small voids may be smeared out by redshift-spacedistortions. However, some of these small (sub)voids wouldbe in low-density regions, with smaller redshift-space distor-tions smearing them out. And also, we emphasize that manyof these small void cores, at low sampling, would also likelybe centers of larger, parent voids, that are not necessarilyincluded in our pruned catalog.

Comparing to Aragon-Calvo & Szalay (2013);Aragon-Calvo et al.(2010), which considered the hierarchical struc-ture of voids in different levels of smoothing, we do not ex-plicitly take out the sub-voids in each level of the hierarchybut use their effective sizes to characterize them. Apparently,

1 0

- 1

1 0

0

1 0

1

r

e f f

[ M p c / h ]

0

- 4

0

- 3

0

- 2

0

- 1

C D M

= 0 . 0 5

= 0 . 1

= 0 . 2

Figure 5. The void radius reff distribution for different darkmatter models.

larger voids show up in lower levels of the hierarchy tree. Fig.5shows distributions of void effective radius, reff, defined viaVr = 4/3r

3eff. The size of the voids in our simulations peaks

roughly at 2 h1 Mpc, and shifts slightly outward at high (moving to WDM). The abundances of voids around theinterparticle spacing, 0.2 h1 Mpc, start to become noisy. Inthe right tails (reff 3 Mpc) of Fig. 5, there are more largevoids in a WDM scenario, but not dramatically so. This may

depend more on the void definition than the distribution ofcore densities, however, since the reported radius of a voidobviously depends crucially on where its boundary is drawn.While at radius 1Mpc, the abundance depression on theleft and increasing at the right of the figure clearly showsthe effect that sub-structures in larger voids have been sup-pressed. Fig. 6 shows the minimum core Voronoi densitydistribution for voids and sub-voids (in units of the meandensity ). It shows that the centers of voids become shal-lower in WDM. This is a simple physical effect: in WDM, theinitial density PDF on the scale of the interparticle spacingis narrower than in CDM, because of the small-scale attenu-ation. This results in a narrower particle density distributionatz= 0, as shown in Sec. 3. In particular, density minima,

in the low-density tail, increase in density.

5 DENSITY PROFILES

We take special interest in void density profiles, as theyhave been measured by several different authors recently(Hamaus et al. 2014; Clampitt & Jain 2014; Pisani et al.2014; Nadathur et al. 2014; Ricciardelli et al. 2014). Weshow density profiles in Fig. 8. As found by Hamaus et al.(2014), smaller voids are, on average, deeper.

Our voids are mostly in the radius range 1-10 h1 Mpc.We divided the voids into two bins of effective radius, 1-5and 5-10 h1 Mpc. We measured the density profiles start-ing from void centers using linear radial bins. For radial

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1 0

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V o i d C o r e D e n s i t y /

0

- 3

0

- 2

0

- 1

0

0

0

1

0

2

C D M

= 0

.0 5

= 0

.1

= 0

.2

Figure 6. The void core density distribution for different darkmatter models.

1 0

- 1

1 0

0

r

0 . 2

[ M p c / h ]

0

- 3

0

- 2

0

- 1

0

0

C D M

= 0 . 0 5

= 0 . 1

= 0 . 2

Figure 7.The distribution ofr0.2 for different Dark Matter mod-els.

bin [r, r +r), the density is simply 3Nr/4[(r+ r)3 r3]

whereNr is the number of particles detected in this bin. Weinvestigate two definitions of the center: a) the actual den-sity minimum of the void, as measured by the VTFE; andb) the volume centroid of the void, defined as

xiVi/

Vi,

where xi is the position of particle i belonging to the void,and Vi is the Voronoi volume of that particle. The volumecentroid would be easier to locate observationally than thedensity minimum. These profiles are further scaled by dif-ferent radii rs (i.e. reff or r0.2) using linear interpolation.Note that for the profiles starting from the density mini-mum, the Voronoi tessellation always guarantees a particlein the center, and hence a spike. We removed the central bin

to remove such an artificial spike. Error bars of each databin were measured using the standard deviation divided by

N, where N is the number of profiles stacked. The errorbars shown in this paper are all 2- errors.

One issue that remains curious to us is that since thevoids detected by zobovare highly non-regular, the effectiveradius may not be a good template for the critical size ofthe voids, especially for the purpose of stacking. We visuallychecked the voids and found that most of the irregularitiesare in the noisy edges of the voids. We therefore defineda radius r0.2 for voids such that at this point the averagedensity encompassed is 0.2, followingJennings et al.(2013).The distribution ofr0.2 is shown in Fig. 7. r0.2 was typicallyfrom 0.5 3 h1 Mpc. The distribution of voids of r0.2 0.5 h1 Mpc shows similar features as does reffdue to thesmall scale suppression effect of WDM. Again we used thecore particle as the center, scaled the void profiles measuredby shell bins by r0.2, and stacked them, shown in Fig. 9.The central densities are quite similar across different voids.Note thatr0.2 is typically 2-3 times smaller thanreff. This issurprisingly large, but may be the effect of including the full,generally irregularly shaped, density ridges around voids.As discussed inJennings et al. (2013), choosing a differentdefinition of void size simply moves voids among radius bins,but does not change their total abundance.

The shallowing of the density profile in central bins isentirely unsurprising if the profiles are measured from den-sity minima; this follows almost trivially from the increase indensity minima. Less obvious is the behaviour of void pro-files as measured from their volume centroids. The profilefrom the volume centroid is more observationally relevant,because there is some hope of inferring it from a dense galaxysample, while locating a 3D density minimum would be quitedifficult. In Fig.10, we show density profiles measured in thesame way as before, except from the volume centroids usingthe Voronoi volumes of all particles reported in the zobovvoid. These profiles are scaled by r0.2 and stacked in thesame radius bins as in the previous case. As previously, wetried to usereffto scale and stack, but result in highly noisyprofiles. The reason is the same: the shapes of the voids be-come irregular when their sizes are small, and there is somerandomness in whether part of a density ridge is included ornot, so the zobov effective radius can be noisy.

Either scaling the profile with r0.2 or reff, the densityprofiles show some universalities the central part of theprofile is relatively stable. For different DM settings, thecenter part of the profiles is clearly different. While the pro-files when scaling by r0.2 are less noisy than those scaled byreff, the profiles shapes are similar in both cases. It is reas-suring that the results hold whether the volume centroid orthe density minimum is used to measure the density profile.

In most previous profile studies, people use the effec-tive radius reff to scale the density profile and get univer-sal profiles a relatively flat central plateau, a sharp edgeand a compensated wall, tending to unity faraway. We ar-gue that using r0.2, the radius at which the mean encloseddensity reaches 0.2, as a scaling constant to find small voidprofiles is better. The void wall radius deviates from reffsig-nificantly with non-spherical shapes, whiler0.2 characterizesevery void in the context of a spherical-evolution model. Ifthe voids are self-similar, as stated inNadathur et al.(2014),r0.2 surely returns more consistent profiles. This is indeed

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1 0

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r / r

e f f

4

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2

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0

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e f f

= 1 . 0

5 . 0 M p c / h

= 0 . 0 0

= 0 . 0 5

= 0 . 1 0

= 0 . 2 0

1 0

- 1

1 0

0

r / r

e f f

. 0

. 5

. 0

. 5

. 0

r

e f f

= 5 . 0

1 0 . 0 M p c / h

= 0 . 0 0

= 0 . 0 5

= 0 . 1 0

= 0 . 2 0

Figure 8.The density profile measured and scaled with reff. The origin is the density minimum of each void. The left panel shows voidsin the 1-5 h1 Mpc radius bin; the right panel shows results from 5-10 h1 Mpc voids.

1 0

- 1

1 0

0

r / r

0 . 2

4

3

2

1

0

r

0 . 2

= 0 . 5

1 . 0 M p c / h

= 0 . 0 0

= 0 . 0 5

= 0 . 1 0

= 0 . 2 0

1 0

- 1

1 0

0

r / r

0 . 2

. 0

. 5

. 0

. 5

. 0

. 5

. 0

r

0 . 2

= 1 . 0

3 . 0 M p c / h

= 0 . 0 0

= 0

.0 5

= 0 . 1 0

= 0 . 2 0

Figure 9. The density profile measured and scaled with r0.2. The origin is the density minimum of each void. The left panel shows thevoids in radius bin 0.5 1.0 h1 Mpc; the right panel shows the results of 1.0 3.0 h1 Mpc.

shown in Fig. 9 and Fig. 10. Since the slope of profiles isapparently the largest at r r0.2, a lensing measurementwould b e most sensitive if using this definition.

6 CONCLUSION AND DISCUSSION

We measured statistics and density profiles of voids in differ-ent DM settings, namely CDM, and WDM with character-istic scales = 0.05, 0.1 and 0.2 h1 Mpc. In summary, thevoids in WDM are shallower. They also tend to be larger,although this effect depends more on the void definition.The number of statistically significant voids is also smallerin WDM simulations. The main question this paper posesis whether we can use the void density profiles as a detec-tor for WDM, or some other process that attenuates initialsmall-scale power. Our answer is yes, in principle.

However, it is quite difficult to constrain matter densityprofiles observationally, even if the voids themselves can belocated using galaxies or other tracers, which becomes trickyat small radius because of issues such as redshift-space dis-tortions. Lensing could be used to constrain density profiles,as shown in (Clampitt & Jain 2014). However, lensing is onlysensitive to the gradient of the surface density, whose gradi-

ent becomes zero at the very centre of the void. Fortunately,our results do show a difference in the density gradient indifferent WDM scenarios. Another possible probe is the inte-grated Sachs-Wolfe effect (e.g. Granett et al. 2008), usefullysensitive to the potential, although that measure is difficultto detect for small voids because of the dominant primordialCMB. But there are other probes of the potential throughvoids, such as in fluctuations in the cosmic expansion rate asmeasured with supernovae. Another aspect of voids that wedid not measure, but would also be sensitive to matter den-

sity profiles, is the velocity field within voids; perhaps veryfaint tracers such as absorption lines can be used to con-strain these. Another possible direction to constrain WDMusing the properties is through the properties of filamentsand perhaps walls in WDM, which we quantitatively showedbecome more prominent; however, we leave a detailed studyof their properties such as density profiles to future work.

ACKNOWLEDGMENTS

We thank Alex Szalay for useful comments and for provid-ing access to high-performance computers for the numericalsimulations. LFY and MN are grateful for support from a

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= 0 . 0 0

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= 0

.1 0

= 0 . 2 0

1 0

- 1

1 0

0

r / r

0 . 2

. 4

. 2

. 0

. 8

. 6

. 4

. 2

. 0

. 2

r

0 . 2

= 1 . 0 3 . 0 M p c / h

= 0 . 0 0

= 0 . 0 5

= 0 . 1 0

= 0 . 2 0

Figure 10.The density profile measured and scaled with r0.2, measured from void volume centroids. The left panel is the results forvoids of radius 0.5-1 h1 Mpc; the right panel uses voids of radius 1-3 h1 Mpc.

grant in Data-Intensive Science from the Gordon and Betty

Moore and Alfred P. Sloan Foundations and from NSF grantOIA-1124403 and OCI-1040114. MN is thankful for supportfrom a New Frontiers in Astronomy and Cosmology grantfrom the Sir John Templeton Foundation. The research ofJS has also been supported at IAP by ERC project 267117(DARK) hosted by Universite Pierre et Marie Curie - Paris6.

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