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07Draft version February 13, 2013Preprint typeset using LATEX style emulateapj v. 05/04/06

ON THE STRUCTURE OF DARK MATTER HALOS AT THE DAMPING SCALE OF THE POWERSPECTRUM WITH AND WITHOUT RELICT VELOCITIES

Pedro ColınCentro de Radiostronomıa y Astrofısica, Universidad Nacional Autonoma de Mexico, Apartado Postal 72-3 (Xangari), 58089 Morelia,

Michoacan, Mexico

Octavio Valenzuela and Vladimir Avila-ReeseInstituto de Astronomıa, Universidad Nacional Autonoma de Mexico, C.P. 04510, Mexico, D.F., Mexico

Draft version February 13, 2013

ABSTRACT

We report a series of high–resolution cosmological N-body simulations designed to explore theformation and properties of dark matter halos with masses close to the damping scale of the primordialpower spectrum of density fluctuations. We further investigate the effect that the addition of a randomcomponent, vrms, into the particle velocity field has on the structure of halos. We adopted as a fiducialmodel the Λ warm dark matter (ΛWDM) cosmology with a non–thermal sterile neutrino mass of 0.5keV. The filtering mass corresponds then to Mf = 2.6 × 1012h−1M⊙. Halos of masses close to Mf

were simulated with several million of particles. The results show that, on one hand, the inner densityslope of these halos (at radii <∼ 0.02 the virial radius Rv) is systematically steeper than the onecorresponding to the Navarro-Frenk-White (NFW) fit or to the cold dark matter counterpart. On theother hand, the overall density profile (radii larger than 0.02 Rv) is less curved and less concentratedthan the NFW fit, with an outer slope shallower than -3. For simulations with vrms, the inner halodensity profiles flatten significantly at radii smaller than 2–3 h−1kpc (<∼ 0.010Rv − 0.015Rv). Aconstant density core is not detected in our simulations, with the exception of one halo for which theflat core radius is ≈ 1h−1kpc. Nevertheless, if “cored” density profiles are used to fit the halo profiles,the inferred core radii are ≈ (0.1− 0.8)h−1kpc, in rough agreement with theoretical predictions basedon phase–space constrains, and on dynamical models of warm gravitational collapse. A reduction ofvrms by a factor of 3 produces a modest decrease in core radii, less than a factor of 1.5. We discuss theextension of our results into several contexts, for example, to the structure of the cold DM micro–halosat the damping scale of this model.

Subject headings: cosmology:dark matter — galaxies:halos — methods:N–body simulations

1. INTRODUCTION

The nature of dark matter (DM) is one of the mostintriguing and fundamental problems in cosmology andparticle physics. The standard hypothesis assumes thatdark matter is made of non–baryonic collisionless ele-mental particles that become non–relativistic very earlyin the history of the Universe (cold). This minimal sce-nario, named Cold Dark Matter (CDM), has successfullyexplained the observed structure of the universe at largescales, like the two–point correlation function of galax-ies and the Cosmic Microwave Background Radiation(CMBR) anisotropies (see for recent results Springel etal. 2006; Spergel et al. 2006). Confrontation of modelpredictions with obervations turns out to be more com-plicated at galactic scales, because non–linear dynamicsand baryonic processes may distort considerably the un-derlaying DM distribution. Thus, the predictions of theCDM scenario and its variants at the scale of galaxies,are an active subject of study.

Two of the most controversial CDM predictions are:(i) the large abundance of subhalos in galaxy–sized halos(Kauffmann et al. 1993), and (ii) the cuspy inner densityprofile of dark halos (e.g., Navarro et al. 2004; Diemandet al. 2004). Based on comparison with observations, ithas been suggested that both predictions may indicate aflaw of the CDM scenario (Klypin et al. 1999a; Moore

et al. 1999a; Moore 1994; de Blok et al. 2001, Gentile etal. 2005,2007). However, these comparisons might be bi-ased by astrophysical processes in action during galaxyassembly and evolution, as for example the inhibitionof star formation in small subhalos (Bullock, Kravtsov,& Weinberg 2000; Benson et al. 2003; Governato et al.2007) or the halo core expansion due to energy or angularmomentum transfer from dark/baryonic structures (Ma& Boylan-Kolchin 2004; El-Zant et al. 2004; Weinberg& Katz 2007), but see also (Colın et al. 2006; Ceverino& Klypin 2007; Sellwood 2007). It has also been showedthat the disagreements may be a consequence of system-atics in the observational inferences, (e.g., Simon & Geha2007; Rhee et al. 2004; Hayashi & Navarro 2006; Valen-zuela et al. 2007).

It is also possible that slight modifications to the CDMparticle properties could solve or ameloriate the men-tioned potential problems if they persist (e.g., Spergel& Steinhardt 2000; Colın et al. 2002). As the precisionof observations and the control of systematics improve,the confrontation with model predictions opens a valu-able window for constraining the dark matter properties.Among the “slight” modifications with the CDM sce-nario is the introduction of warm particles, instead ofcold ones (Warm Dark Matter, hereafter WDM). WDMimplies two more degrees of freedom in comparison toCDM: (i) a damping (filtering) of the power spectrum at

2

some intermediate scale, λf , due to the relativistic freestreaming, and (ii) some primordial random velocity inthe dark particles, vrms (for CDM, the mass correspond-ing to λf , Mf , is comparable to planet masses, e.g., Die-mand et al. 2005; Profumo et al. 2006, and vrms ≈ 0).

Previous numerical studies have shown that WDM canbe very effective in reducing the amount of (sub)structurebelow the filtering mass Mf (e.g., Colın, Avila-Reese, &Valenzuela 2000; Avila-Reese et al. 2001; Bode et al.2001; Knebe et al. 2002). In addition, both the peculiardynamical formation history of these halos and its ran-dom vrms can also impose an upper limit on the phasespace density, potentially producing an observable coreof constant density (Hogan & Dalcanton 2000; Avila-Reese et al. 2001). If vrms has a thermal origin its am-plitude is linked directly to the mass of the WDM par-ticle (Hogan & Dalcanton 2000); notice, however, thatthe amplitude of the random velocities may depend onother physical factors not directly related to the parti-cle mass. This is the case, for instance, of gravitinosproduced non–thermally by late decays of the Next toLighest Supersymmetric Particle (NLSP) (Feng, Rajara-man & Takayama 2003; Strigari et al. 2007, see for arecent review Steffen 2006). Thus, an exploration of theeffect of a random velocity component independently ofthe dark matter particle mass seems to be necessary.

Predictions of the core radius in WDM halos have beencomputed assuming a King profile (Hogan & Dalcanton2000) or a subclass of the Zhao (1996) profiles (Strigariet al. 2006, herafter S2006). It is still unknown whichestimates give the more accurate value, yet these pre-dictions are necessary for comparison with observations(see e.g., S2006; Gilmore et al. 2007). Unfortunately,the predicted core radii for masses of the most popularWDM particle, the sterile neutrino, allowed by observa-tional constraints (& 2 keV, see §5 for references) arebelow the resolved scales in current simulations.

The structure of WDM halos of scales below the filter-ing radius λf might be different from their CDM coun-terparts not only in the central parts but also in theiroverall mass distribution. This is somehow expected be-cause the assembly history of these halos is different fromthe hierarchical one. Besides, they form later and haveconcentrations lower than those ones derived for CDMhalos (Avila-Reese et al. 2001; Bode et al. 2001; Knebe etal. 2002). However, it is controversial whether the shapeof the density profile differs systematically from the cor-responding CDM one (see for different results e.g., Husset al. 1999; Moore et al. 1999b; Knebe et al. 2003). Onthe other hand, WDM halos of mass close to Mf (withvrms set to 0) can be thought of as scaled-up versions ofthe first microhalos in a CDM cosmology. The earliestcollapse of CDM microhalos is a subject of considerablecurrent interest (e.g., Diemand et al. 2005; Gao et al.2005).

In this paper we explore by means of numerical sim-ulations the two questions mentioned above: the overallstructure of dark halos with masses close or just belowthe cutoff mass in the power spectrum of fluctuations,and the inner density profile of these halos when a ran-dom velocity is added to the particles. For our numericalstudy we use the truncated power spectrum correspond-ing to a non–thermal sterile neutrino of mW = 0.5 keV(Mf = 2.6×1012 h−1M⊙). We initially neglect the parti-

cle random velocity (vrms=0), and later we will considertwo values of vrms that cover the range of velocities cor-responding to thermal and non–thermal mW = 0.5 keVWDM particles. It is important to remark that our goalis not to study a specific WDM model but to explorein general the effects on halos of the power spectrumtruncation and the addition of random velocities to theparticles.

The structure of the paper is as follows. In §2 we de-scribe the cosmological model that we use for our inves-tigation : a WDM model with a filtering radius at thescale of Milky Way size halos. Halos of these scales wereresimulated with higher resolutions, first without addingthe corresponding random velocity, and then with addingthis velocity component to the particles. Details of thenumerical simulations carried out in this paper are givenin §3. The results from our different simulations are pre-sented in §4. Section §5 is devoted to a discussion of theresults and their implications. Finally, in §6 we presentthe main conclusions of the paper.

2. THE COSMOLOGICAL MODEL

The general cosmological background that we use forour numerical simulations corresponds to the popular flatlow–density model with Ω0 = 0.3, ΩΛ = 0.7 and h = 0.7(the Hubble constant in units of 100 km sec−1 Mpc−1).

For the experiments designed to explore the struc-ture of dark matter halos with masses close to Mf , weadopt an initial power spectrum corresponding to a non–thermal sterile neutrino of 0.5 keV. Even if this WDMmodel is ruled out by observations (see §5 for references),it is however adequate for the purposes stated in the In-troduction. As is shown below, the filtering mass Mf

corresponding to this WDM particle is of the order ofMilky Way–size halos, namely the halos that we are ableto follow with high resolution in a cosmological simula-tion. The high resolution of the simulations avoids thatthe formation of halos with a mass scale near to Mf willbe dominated by discreteness effects (see §3.1). Besides,for the resolution that we attain, we expect to resolve theinner regions where a flattening in the halo is predicteddensity profile is predicted for the case when vrmsis in-troduced.

We use here the transfer function Ts for the non–thermal sterile neutrino derived in Abazajian (2006a).The WDM power spectrum is then given by

PWDM (k) = T 2s (k)PCDM (k), (1)

whereTs(k) = [1 + (αk)ν ]−µ , (2)

and PCDM is the CDM power spectrum given by Klypin& Holtzman (1997). This fit is in excellent agreement,at the scales of interest, with the power spectrum ob-tained with linger, which is contained inside the cosmicspackage1. The parameter α is related to the mass of thesterile neutrino, ΩDM and h through

α = a( ms

1keV

)b(

ΩDM

0.26

)c (

h

0.7

)d

h−1Mpc, (3)

where a = 0.189, b = −0.858, c = −0.136, d = 0.692, ν =2.25, and µ = 3.08. The power spectrum is normalized

1 http://web.mit.edu/edbert/cosmics-1.04.tar.gz

3

to σ8 = 0.8, a value close to that estimated from thethird-release of WMAP (Spergel et al. 2006). Here σ8

is the rms of mass fluctations estimated with the top-hatwindow of radius 8h−1Mpc.

As in Avila-Reese et al. (2001), we have defined thefree–streaming (damping) wavenumber, kf , as the k forwhich the WDM transfer function T 2

s (k) decreased to0.5, and compute the corresponding filtering mass in thelinear power spectrum as

Mf =4π

3ρ

(

λf

2

)3

, (4)

where ρ is the present–day mean density of the universe.The filtering wavelength λf = 2π/kf of a non–thermalsterile neutrino of 0.5 keV is 3.9 h−1Mpc, which corre-sponds to a filtering mass Mf = 2.6× 1012 h−1M⊙. Therandom component was linearly added to the velocitiescalculated with the Zel’dovich approximation at the on-set of the simulation.

A number of Milky Way–size halos are simulated withthe WDM power spectrum given by eq. (1) and withvrms=0. In this way we isolate the effects of the powerspectrum filtering on the structure of the halos. Later,the same halos are resimulated but adding random veloc-ities to the particles. We approximate the shape of theparticle phase space distribution function (DF) with thecorresponding thermal one; i.e, we use a thermal equi-librium Fermi-Dirac DF. This is a good approximationfor a non–thermal sterile neutrino of 0.5 keV (Abazajian2006a). Further, as a first approximation, we assumethat the amplitude of the DF of our non–thermal ster-ile neutrinos is equal to the one corresponding to theirthermal partners with mW=0.5 keV (Hogan & Dalcanton2000). With this assumption for DF shape and ampli-tude, the neutrino random velocity at z = 40, the initialredshift in our simulations, is ≈ 4 km/s.

The particle velocity dispersion in the linear regimedecreases adiabatically with the expansion. The conven-tion is to define vrms in terms of the value extrapolatedto the present epoch, vrms(z) = vrms(0)(1+ z); therefore,for our case vrms(0) ≈ 0.1 km/s. We should note thatthe rms velocity amplitudes of sterile neutrinos and ther-mal particles of the same mass can not be the same, butlarger because the former become non–relativistic laterthan the latter. A rough calculation shows that for the0.5 keV sterile neutrino, vrms should be approximatelythree times higher than for a 0.5 keV thermal WDM par-ticle, i.e. vrms(0) ≈ 0.3 km/s. We resimulate halos withboth values of the random velocity, vrms(0) ≈ 0.1 and 0.3km/s, having in mind that our main goal is not to studya particular WDM model but to explore the structure ofthe dark halos with different initial conditions .

3. NUMERICAL SIMULATIONS

A series of high resolution simulations of Milky Waysize galactic halos are performed using the Adaptive Re-finement Tree (ART) N-body code (Kravtsov, Klypin, &Khokhlov 1997) in its multiple mass scheme (Klypin etal. 2001). The ART code achieves high spatial resolu-tion by refining the root grid in high-density regions withan automated refinement algorithm.

In all of our experiments we first run a low–mass resolu-tion (LMR) simulation of a box of Lbox = 10h−1Mpc on

a side and 643 or 1283 particles. We then select a spher-ical region centered on a Milky Way size halo of radius∼ 3 times the virial radius2 of the chosen halo. TheLagrangian region corresponding to the z = 0 sphericalvolume is identified at z = 40 and resampled with ad-ditional small–scale waves (Klypin et al. 2001). Halonames (first column in Table 1) are denoted by a capitalletter followed by the effective number of particles thatwere used in the high resolution zone and a subindexthat indicates the value of vrms(0) (in km/s; reported incolumn 2)3. The mass per particle, mp, in the high–resolution region is given in column (5) while in column(6) we give the virial mass of the halo, Mv. The latteralong with mp can be used to compute the number ofparticles inside a halo. The global expansion timestep∆a0, and the formal spatial resolution hfor –measuredby the size of a cell in the finest refinement grid– aregiven in columns (3) and (4). ART integrates the equa-tions using the expansion factor a as the time variablesuch that at z = 0, a = 1.

We started the simulations at z = 40 because the powerat the Nyquist frequency at this redshift is in the linearregime. Note that spurious noise can influence the evo-lution of simulations that include vrms if they start tooearly. The noise might be particularly important whenthe generated vrms have amplitudes comparable to thoseof the Zel’dovich peculiar velocities. In order to showthis, we started a simulation at z = 100 and evolved itup to z = 40. At z = 100 the vrms velocities are on av-erage 2.5 times greater than at z = 40. Figure 1 showsthe measured initial power spectrum of the simulationstarted at z = 40 (solid line) and the power spectrumfor the simulation started at z = 100 and measured atz = 40 (squares). The latter simulation developed spuri-ous power at frequencies higher than about 0.6 h/Mpc.We run another simulation started at z = 100, but withnoaddition of a relic velocity component. In this casewe did not detect the evolution of spurious power byz = 40. In order to make sure that this spurious noisedoes not appear when the simulation is started at z = 40we repeated the experiment but now the onset of thesimulation is set at z = 40 and the power spectra aremeasured at z = 20. Unlike the previous case, no differ-ences between the power spectra were detected. In otherwords, as far as the initial power spectrum is concernedit does not matter if the simulation is started at z = 40o z = 20.

Concern may arise about the structure of halos simu-lated in a relatively small Lbox = 10h−1Mpc box, spe-cially in a WDM cosmology, where there is a scale belowwhich the power spectrum exponentially drops to zero.Avila-Reese et al. (2001) discussed this potential issueand concluded that in order to be confident about thesimulated halo structure, a box size greater than the fil-tering length λf should be used. For our simulationsLbox is 2.5 larger than λf . In any case, we also exper-imented with other box sizes, namely, 15h−1Mpc and20h−1Mpc (not shown in the Tables), for the vrms = 0

2 This radius is defined as the radius at which the mean overden-sity is equal to the predicted by a top-hat spherical collapse whichis 337 for our selected cosmological model.

3 The random velocity component was linearly added to thevelocities correspondingt to the Zeldovich approximation at theonset of the simulation.

4

TABLE 1Model and Halo Parametersa

name tag vrms ∆a0 hfor mp Mv Vmax Rv c1/5 cNFW Lgρ1% rc rS+c rS+

c,30

km/s 10−3 kpc/h 106M⊙/h 1012M⊙/h km/s kpc/h [h2M⊙/pc3] kpc/h kpc/h kpc/h(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

A2560.0 0.0 1.0 0.305 4.96 2.84 224.1 286 5.6 8.3 -1.44 – – –A2560.1 0.1 1.0 0.305 4.96 2.37 218.2 228 5.0 5.9 -1.44 0.082 1.151 0.572B2560.0 0.0 0.5 0.305 4.96 4.36 253.2 330 5.1 7.7 -1.46 – – –B2560.1 0.1 0.5 0.305 4.96 4.35 250.8 330 5.0 6.7 -1.54 0.098 1.380 0.565C5120.0 0.0 0.5 0.152 0.62 1.29 160.7 222 4.2 6.4 -1.53 – – –C5120.1 0.1 0.5 0.152 0.62 1.26 158.9 220 4.3 4.7b -1.79 0.093 1.265 0.689C5120.3 0.3 0.5 0.152 0.62 1.28 158.5 221 4.3 3.6b -1.99 0.121 1.527 0.827D5120.0 0.0 0.5 0.152 0.62 1.28 160.1 221 4.3 6.1 -1.54 – – –D5120.1 0.1 0.5 0.152 0.62 1.24 160.9 219 4.6 4.2b -1.89 0.106 1.389 0.816D5120.1b 0.1 0.5 0.152 0.62 1.25 159.6 219 4.3 4.8b -1.77 0.091 1.246 0.598D5120.3b 0.3 0.5 0.152 0.62 1.26 160.4 220 4.3 3.7 -1.96 0.118 1.498 0.792E5120.0 0.0 1.0 0.152 0.62 2.13 197.3 262 4.9 6.4 -1.57 – – –E5120.1 0.1 1.0 0.152 0.62 2.06 192.3 259 4.7 5.7b -1.66 0.092 1.287 0.657E5120.3 0.3 1.0 0.152 0.62 2.14 197.0 262 4.9 5.3b -1.71 0.101 1.380 0.519

aAll the halos presented in this Table were resimulated from a Lbox 10h−1Mpc box simulation.bNote that the NFW function does not provide a good fit to the density profiles of halos with vrms > 0. However, for completeness,

we report here the value of cNFW obtained from the fit.

0 0.5 1 1.5 2

-10

-8

-6

-4

-2

0

Fig. 1.— Comparison of the power spectra measured at z = 40for the simulation started at z = 40 (solid line) and the one startedat z = 100 (squares). The longest plotted wavelength is Lbox whilethe highest frequency is (2π/Lbox) 256.

case, and found results similar to those reported here forthe 10h−1Mpc box.

The bound density maxima (BDM) group finding algo-rithm (Klypin et al. 1999b), or a variant of it (Kravtsovet al. 2004), is used to locate the halos in the simulationsand to generate their density profiles. The BDM findspositions of local maxima in the density field smoothedat the scale of interest and applies physically motivatedcriteria to test whether a group of particles is a gravita-tionally bound halo.

Aside from those halos shown in Table 1, for the haloD with vrms=0 we have also run a very high resolutionsimulation with about 31 million particles in the high

resolution zone. This halo was taken from the same 10h−1Mpc box run but with 1283 particles in the LMRmode. As far as we know this is the highest resolutionsimulation of a halo run in a WDM cosmology. The samehalo was also simulated with less resolution for a conver-gence test. The parameters of the sequence of halos Dare resumed in Table 2.

The simulations presented here differ in several aspectsfrom previous WDM simulations. First, it should be em-phasized that our aim rather than discussing a specificWDM model is to explore the influence of the truncationof the power spectrum and/or the addition of random ve-locities upon the structure of dark halos of masses closeto the truncation scale. For this aim we need to simu-late (a) halos with very high resolution, and (b) haloswith masses close to Mf . The halos simulated in Avila-Reese et al. (2001) had several times less particles thanthe best–resolved halos presented here and the aims inthat paper focused in exploring general halo propertiesfor a concrete WDM model. Other papers aimed to studythe properties of WDM halos (Bode et al. 2001; Knebeet al. 2002; Busha et al. 2006) focused more in the sta-tistical aspects than in details of the inner halo struc-ture; therefore, the halos in these papers had resolutionsmuch lower than those attained here. The properties ofthe WDM halos simulated here are in general agreementwith previous findings; for example, their concentrationsare systematically lower (Avila-Reese et al. 2001; Eke etal. 2001; Bode et al. 2001) and they form later (Knebeet al. 2002; Busha et al. 2006) than the correspondingΛCDM halos.

3.1. Discreteness effects

One of the motivations of this paper is to investigatethe structure of well resolved halos with masses close orbelow the damping (truncation) scale in the power spec-trum, Mf . The origin of these halos is controversial.Halos with masses close to Mf (truncation halos) couldbe formed by a quasi–monolithic collapse of filaments ofsize ∼ λf (e.g., Avila-Reese et al. 2001). They could also

5

TABLE 2High Resolution Halo D Parameters

Lbox name tag vrms timestep resolution mp Mvir

(h−1Mpc) (10−3) (h−1kpc) (h−1M⊙) (1012 h−1M⊙)(1) (2) (3) (4) (5) (6) (7)

10 D10240.0 off 0.5 0.040 7.75 × 104 1.2810 D5120.0 off 0.5 0.152 6.20 × 105 1.2710 D2560.0 off 0.5 0.305 4.96 × 106 1.2710 D1280.0 off 0.5 0.610 3.97 × 107 1.25

be just the result of an incomplete collapse, highly de-viated from the spherical–symmetric case, of originallylarger structures assembled hierarchically (Busha et al.2006). On the other hand, it has been suggested that ha-los with masses considerably less than Mf form by frag-mentation of the shrinking filaments of size ∼ λf (e.g.,Valinia et al. 1997; Avila-Reese et al. 2001; Bode et al.2001; Gotz & Sommer-Larsen 2003; Knebe et al. 2003).However, it is also known that the filaments in Hot DarkMatter simulations that start from a cubic lattice breakup into regularly spaced clumps, which reflect the ini-tial grid pattern. Therefore, some of these halos seenin WDM simulations could be spurious, product of dis-creteness effects. Recently, Wang & White (2007) haveshown that this artifact is present even for a glass–likeinitial particle load (White 1996).

As Wang & White (2007) show, halos of masses smallerthan a given effective fraction of Mf , which depends onthe resolution of the simulation, will be spurious. We se-lected the WDM model (§2) and the number of particlesin the simulations in such a way that the halos studiedhere can neiher be fake nor affected by discreteness ef-fects. Following Wang & White (2007), in our modelwith 10243 effective number of particles, only structureswith masses lower than 1.3 ×109h−1M⊙ are candidate tobe spurious. For the models with 5123 and 1283 effectivenumber of particles, the masses of structures triggeredby the initial grid spacing are < 2.6 × 109h−1M⊙ and1.04 ×1010h−1M⊙, respectively. For our WDM model,the first halos to form have masses & 1012h−1M⊙, whichare far from the mentioned effective resolution limits.

4. RESULTS

We first present the results of our WDM simulationswith vrms = 0. In this way, we explore the inner structureof halos close to the truncation mass Mf , which could bescaled-up versions of the first CDM microhalos (of masses∼ 10−5M⊙ for a neutralino mass of 100 GeV). After-wards we present the results of the same simulations butintroducing random velocities to the particles with twodifferent amplitudes, vrms(0) = 0.1 and 0.3 km/s (see§2). The main goal of the last simulations is to explorethe predicted flattening in the halo inner density profileproduced by the addition of a random component to theparticle velocities(see section 1). Table 1 resumes themain properties of the resimulated halos, which were se-lected to have masses close to the truncation mass of theinitial power spectrum.

4.1. The Structure of Halos at the scale of Damping

Figures 2 and 3 show the spherically averaged den-sity profiles measured for the halos of masses around the

power–spectrum filtering mass, Mf , and with vrms=0.The first panel of Fig. 2 shows halos A2560.0 andB2560.0 (the latter was shifted by −1 in the log); the sec-ond and third panels show halos C5120.0 and E5120.0, re-spectively. In Fig. 3, the same halo D but simulated with4 different resolutions is shown. In both Figs., the thindashed lines are the best Navarro–White–Frenk (NFW,Navarro, Frenk, & White 1997) fits to the showed den-sity profiles. In the lower panels, the residuals of themeasured density profile and the NFW fit are plottedwith the same line coding as in the corresponding upperpanels.

Halos A2560.0 and B2560.0 in the left panel andC5120.0 and E5120.0 in the second and third panelsof Fig. 2, were simulated with formal spatial resolutionshfor = 0.305h−1kpc and 0.152h−1kpc, respectively (seeTable 1). Previous convergence studies for CDM halossimulated with the ART code have shown that the in-nermost halo density is reliable only for radii larger thanfour times hfor and containing more than 200 particles(Klypin et al. 2001). For all the density profiles shownin Figs. 2 and 3, the innermost plotted point correspondsto radii larger than h4 = 4×hfor by ∼ 30% and they con-tain more than 200 particles. The convergence analysisthat we have carried out for our WDM halos suggeststhat instead of h4, the innermost radius should be closeto 8 × hfor, h8.

Figure 3 compares the density profiles of halo D, whichwas re–simulated with four different resolutions sepa-rated each by a factor of eight in the particle mass (seeTable 2). As can be seen, convergence is achieved atabout h8. The arrows in Figs. 2 and 3 indicate h8 forthe corresponding simulations. In Fig.3 , the solid–linearrow is for the highest–resolution simulation (10243 ef-fective number of particles), while the dashed–line arrowis for the 5123 simulation; the NFW fit is shown only forthese two cases.

The inner density profiles of all halos simulated hereare systematically steeper than the corresponding fittedNFW law. The slopes of the profiles at r ≈ 1% the virialradii, Rv, span a range from −1.4 to −1.6. For compar-ison, the slope of the density profile of a typical LCDMhalo of 2 × 1012h−1M⊙ at 0.01Rv is ≈ −1.2 (a NFWprofile was used with the corresponding concentrationgiven by Bullock et al. 2001, and re-scaled to σ8 = 0.8).At radii smaller than 0.01Rv, the slopes tend to becomeshallower but the halos are still denser and slopes steeperthan the corresponding NFW fit up to the resolution lim-its (≈ h8, see Figs. 2 and 3).

The overall density profile shapes of our halos are alsosomehow different from the NFW function. For radii

6

Fig. 2.— Upper panels: Density profiles of halos formed at the scale of damping (truncation). From the left, the first two panels presentthe profiles (thick lines) of two halos each one; the lower profiles were shifted by −1 in the log for presentation purpouses. Thin short–longdashed lines show the corresponding NFW fits. The third panel shows halo K simulated with four different resolutions. The NFW fits areshowed only for the two highest resolutions. The innermost radius of each profile corresponds to ≈ 1.3 h4 and arrows in each panel indicatethe radius h8 (see text). Lower panels: Fractional residuals of the NFW fit. We observe that all our halos are denser and steeper than theNFW model at least until h8.

larger than 0.02Rv, the profiles tend to be in generalslightly less curved than the NFW model. This is why theresiduals shown in Figs. 2 and 3 indicate a systematicaldefect at intermedium radii and then an excess at theouter radii. The outer slopes are > −3.

In summary, the density profiles of the halos simulatedhere, with masses close or below the filtering mass, havea shape slightly flatter than the NFW law for radii >0.02Rv and slopes significantly steeper at radii < 0.02Rv.Nevertheless, each profile is different. The profile of haloE5120.0 has minimal deviations from the NFW function,while the profile of halo D5120.0 significantly deviatesfrom this function.

Table 1 details the main properties of the halos studiedhere. In columns 6 to 14 are reported respectively thevirial mass, Mv, the maximum circular velocity, Vmax,the c1/5 and NFW concentration parameters, the aver-age density within 1% the virial radius, and three coreradii estimated by different criteria (the latter quantitiesapply only to halos simulated with vrms, see §§4.2). TheNFW and c1/5 concentrations are defined respectively asthe ratios between Rv and the NFW scale radius, and be-tween Rv and the radius where 1/5th of Mv is contained(Avila-Reese et al. 1999). As found in previous results,halos of masses below the truncation mass in the powerspectrum tend to be less concentrated than ΛCDM ha-los of similar masses (Avila-Reese et al. 2001; Bode etal. 2001; Knebe et al. 2002). We have simulated someΛCDM (σ8 = 0.8) halos of masses ≈ 2 × 1012 h−1M⊙

and measured NFW and c1/5 concentrations around 8–15 and 6.0–11.0, respectively, to be compared with thevalues given in Table 1 for the vrms=0 cases.

4.2. The inner structure of halos simulated with rmsvelocity added to the particles

We have rerun the halos presented in the previous sub–section but now introducing a random velocity compo-nent, vrms. As explained in §2, the particle DF used

0.1

Fig. 3.— Upper panel: Density profile for halo D. This halowas simulated with four different resolutions, separated each by afactor of eight in particle mass (see Table 2). Shown with a short-long-dashed curve is the NFW fit to the density profile for the mostwell resolved case (solid line) while arrows indicate the position ofthe radii h8 for the two highest resolution simulations.

corresponds to a Fermi–Dirac function. Regarding thevrms amplitude, we use two values: vrms(0) = 0.1 and0.3 km/s. These values are for WDM particles of thermalorigin and for non–thermal sterile neutrinos, respectively,in both cases with mW=0.5 keV. Our goal is to explorewhether the inner structure of the halos becomes affectedsignificantly or not by adding vrms.

We first present results for the case vrms(0) = 0.1 km/s,and then explore how the inner halo structures changewhen vrms(0) is increased from 0.1 to 0.3 km/s, a more

7

Fig. 4.— Upper panels: Comparing the density profiles of simulated halos without (solid lines) and with (dot–dashed lines) addingvrms to the particles. In the left panel, halos B and Bv have been shifted by −1 in the log. The innermost plotted radii are at ≈ 1.3 h4 andthe arrows indicate the corresponding h8 radii. For halo D, the profile of the 10243 particles simulation (vrms=0) is also plotted (dottedline). Medium panels: The corresponding 3–D velocity dispersion profiles of the halos shown in the upper panels. Halos B and Bv havebeen shifted by −0.5 in the log. Lower panels: Coarse–grained phase–space density profiles corresponding to the halos shown in upper andlower panels.

appropiate value for the 0.5 keV sterile neutrino modelused to generate the initial power spectrum of the sim-ulations (see §2). For halo D, we have simulated withvrms > 0 the 5123 case. Unfortunately, due to limita-tions in our computational resources, it was not possibleto run the simulation with 10243 particles for vrms > 0.

In Fig. 4 we present spherically–averaged profiles ofdifferent properties for all the simulated halos withoutand with vrms added. In the first and third columnstwo halos (A and B, and Db and D, respectively) arepresented but the latter ones are down shifted for clar-ity. The upper panels of Fig. 4 show the density pro-

files of our halos without (black solid lines), and withvrms(0) = 0.1 km/s (red dot–dashed line) and 0.3km/s (magenta dashed line). In the medium panels areplotted the corresponding three–dimensional velocity dis-persion profiles, σ3D(r), and the lower panels show thecorresponding coarse–grained phase–space density pro-files, Q(r) = ρ(r)/σ3

3D. As in Fig. 2, the arrows indicatethe strong resolution limit radius h8 of the simulations.For halo D (third column), the simulations with vrms > 0were carried out with 5123 particles and using two dif-ferent random seeds for the particle DF calculation (seebelow).

8

To some degree, all the simulated halos have been af-fected in their inner regions by the injection of initial ran-dom velocities to their particles. For the less resolved ha-los A, B (left panels), the deviations at ∼ h8 of the innerdensity profiles from the profiles obtained in the simula-tions with vrms=0 are yet marginal, but in the expecteddirection. For the halos C, D, Db, and E simulated with5123 particles, the deviations down to h8 are significant:the density profiles systematically flatten with respect tothe corresponding vrms=0 cases. The radii at which thedensity profiles of halos with vrms start to deviate (flat-ten) from the ones without vrms, are ∼ 0.010− 0.015Rv,well above the resolution limit of the simulations.

In a second series of experiments, we have resimulatedhalos C, Db and E (5123 particles) with vrms three timeslarger, i.e. vrms(0) = 0.3 km/s. The profiles correspond-ing to these simulations are plotted in Fig. 4 with dashed(magenta) lines. The flattening of the inner density pro-files is clearly more pronounced than in the simulationswith vrms(0) = 0.1 km/s.

The innermost density profiles actually vary from haloto halo. Again, halo E is the less affected not only bythe damping in the power spectrum, but also by theinjection of vrms, and halo D is the most affected byboth effects (lower dot–dashed curve in the correspond-ing panel). The latter actually shows a “true” flat corealready at h8. Since the core of this halo is too differentwith respect to the other ones, we decided to explore iflarge difference can be explained by a rare fluctuationin the random procedure of particle velocity assignment.Thus, the same halo D5120.1 was resimulated with adifferent seed in the random number generator used todraw the particle velocities. The upper curves in thethird column panels of Fig. 4 correspond to the profilesfor this halo, called D5120.1b. The inner density profileof this halo is not too different from the profiles of theother halos, though it remains as the flatest one amongall the simulated halos with vrms(0) = 0.1 km/s.

In Fig. 5 we attempt to fit different functions to thedensity profiles of halos C, D, and E (5123 particles) withvrms(0) = 0.1 and 0.3 km/s. A general function to de-scribe density profiles of cosmic objects was proposed byZhao (1996):

ρ(r) =ρ0

(r/r0)γ [1 + (r/r0)α](β−γ)/α(5)

The NFW profile corresponds to (α, β, γ) = (1, 3, 1).This function does not provide a good description forthe profiles of our halos with vrms > 0, in particular inthe inner regions. We have fitted the halo profiles tothe NFW in order to obtain an estimate of the cNFW

concentration given in Table 1.Strigari et al. (2006) suggested a “cored” density

profile in order to derive constraints on the size of apossible shallow core in the halo of the Fornax dwarfspheroidal galaxy (Goerdt et al. 2006; Sanchez-Salcedoet al. 2006). This profile is described by eq. (5) with(α, β, γ) = (1.5, 3, 0), and they define the core radius,rc as the radius where the inner slope, g, reaches thevalue of −0.1. Thus, rc = r0/(−3/g − 1)1/α ≈ 0.1r0.The dot–dashed curves in Fig. 5 show the best fits us-ing the S2006 profile. As can be seen, this profile doesnot describe well the density profiles of the WDM halosin the simulations. Note that α characterizes the sharp-

Fig. 5.— Density profiles of simulated (5123 particles) halos withadded vrms and fitted to different model profiles. Long–dashedline: (α, β, γ) = (0.7, 3, 0); dot–dashed line: the profile proposedin S2006, (α, β, γ) = (1.5, 3, 0); dotted line: the same S2006 profilebut fitted only to the inner 30 kpc.

ness of the change in logarithmic slope. As already seenin Figs. 2 and 4, the profiles of our halos tend to beless curved than the usual NFW profile. Therefore, val-ues of α smaller than 1 should be used instead of largerthan 1. We have obtained a reasonable description of ourWDM profiles with (α, β, γ) = (0.7, 3, 0). The (magenta)dashed curves in Fig. 5 are the best fits with these pro-files. The core radius, as defined above, is in this caserc ≈ 0.0064r0. Finally, we have also tried fits to theS2006 function but taking into account only the centralhalo regions, up to ≈ 30h−1kpc. The fits are shown inFig. 5 with (green) dotted curves. The fit is speciallygood for the halo D5120.1 and those with vrms(0)=0.3km/s. Columns 12 to 14 in Table 1 report the values ofrc obtained with the three different fits.

As to the 3–D velocity dispersion profiles of the simu-lated halos, they do not differ significantly between thecases with and without vrms, the exception being halo D.In the innermost regions, σ3D(r) is similar or higher forhalos with vrms than for those without vrms. The largerdifferences are for halo D, which after introducing ran-dom velocities to the particles produce a relatively hotcore.

Finally, from the measured density and dispersion ve-locity profiles, we calculate the coarse–grained phase–space density profiles, Q(r). As seen in the lower pan-els of Fig. 4, excepting the innermost regions, the Q(r)profiles are well described by a power law Q ∝ r−α withα ∼ −1.9 close to that obtained for CDM halos by Taylor& Navarro (2001). For the inner regions, the Q(r) pro-file of the halos simulated without vrms tends to steepen,

9

specially in halos D5120.0 and C5120.0. The oppositehappens for the halos simulated with adding vrms, theinner Q(r) profile tends to be flatter as vrms is larger.

5. DISCUSSION

5.1. Robustness of the results

The halos studied here are on one hand among the firstvirialized structures to form in our simulations, and onthe other their assembling process started relatively latein the universe, between z ≈ 0.6 and 1 (for discussionsabout the mass assembling process of halos with massesclose to the damping scale in the power spectrum see e.g.,Moore et al. 2001; Avila-Reese et al. 2001; Bode et al.2001; Knebe et al. 2002; Busha et al. 2006). Because ofthe late collapse, one might argue that the halos studiedhere are not relaxed and therefore is not suprising to havethe deviations from the NFW profile, as reported in §4.1for the experiments with vrms=0. We have followed theevolution of some of our halos for more than a Hubbletime (scale factor a > 1), finding a negligible evolutionin the density profiles since a = 1. For example, haloD5120.0 was run until a = 1.4 (18.4 Gyr). The densityprofile of the halo at this epoch is practically the sameas at a = 1 (13.7 Gyr, see Fig. 4). As discussed inprevios studies (see the references above), the collapse ofhalos of scales close to the damping scale seems to bequasi–monolithic (though highly non spherical). Thus,in regions that remain relatively isolated, as in the caseof the halos selected for our study, halos suffer low massaccretion, and their structures remain almost unalteredsince the initial collapse.

We checked that the effects on the structure of halosreported in §4 are systematic by running for a WDMmodel –not shown in Table 1– the corresponding CDMsimulation, using the same random phases and changingonly the initial power spectrum. We found that the den-sity profile of the CDM halo is well fitted by the NFWfunction, while the corresponding WDM halo present thesystematic deviation already seen in Fig. 2. Figure 6compares the density and circular velocities profiles forthe halo in question in its two versions, CDM and WDMwithout random velocities. Since the WDM halo ends upat z = 0 with a slightly lower mass than the CDM halowe correct the profiles so as to make the comparison at afixed mass (= 3.0 × 1012 h−1M⊙). Notice, in particular,that the inner density profile of the WDM halo is indeedsteeper than their CDM counterpart.

Concerning the resolution limit in our simulations,based on the convergence study carried out for halo D(see Fig. 2), we find that a strong limit is h8 = 8hfor,but resolution might be still acceptable for radii slightlylarger than h4 = 4hfor, a value suggested previously forCDM halos in “equilibrium” simulated with the ARTcode (Klypin et al. 2001). With a resolution limit ath8, our simulations allow to resolve the inner structure ofhalos down to 1.2h−1kpc for the 5123 runs and down to0.61h−1kpc for the 10243 run. These radii correspondrespectively to ∼ 0.5% and 0.25% virial radius in ourhalos.

Finally, it is important to recall that the masses of thethe WDM halos analyzed in our simulations, are wellabove from the mass scale affected by discreteness ef-fects, like the spurious formation of structures and sub-

structures due to the initial grid pattern (see §3.1).

5.2. Do soft cores form in WDM halos?

Early structure formation studies based on a WDMcosmology considered particles originated in thermalequilibrium. For this case, both vrms and λf dependonly on the particle mass mW. The smaller mW, thelarger λf and vrms. Controlled numerical simulations ofisolated halos showed that in order to produce “observ-able” soft cores, the amplitude of vrms should be severaltimes higher than the values corresponding to thermalWDM particles of masses mW & 1 keV (Avila-Reese etal. 2001). Particle masses smaller than ∼ 1 keV are notallowed by the constraints on satellite galaxy abundancesas well as by the Ly–α power spectrum alone or combinedwith CMBR and large scale structure data. The Ly–αpower spectrum is the strongest of the constraints. Forthe non–thermal sterile neutrino, it places a limit on itmass at mW & 2 keV (Seljak et al. 2005; Viel et al. 2006;Abazajian 2006b). For thermal WDM particles, the ob-servational constraints give a limit of mW & 0.5 keV(Narayanan et al. 2000; Viel et al. 2005; Abazajian2006b), while a distinct analysis, using different simula-tions provide a stronger limit, mW & 2.5 keV (Seljak etal. 2006).

We may estimate the expected flat core radii of 2 ×1012h−1M⊙ WDM halos for thermal particles in the massrange mW = (2 − 0.5) keV by using the approximationgiven in Avila-Reese et al. (2001, their eq. 13). This ap-proximation is based on the monolithic collapse of haloswith non–negligible particle random velocities before thecollapse. For thermal WDM particles of masses 2 and 0.5keV, the zM of a 2× 1012h−1M⊙ perturbation are ≈ 3.4and 1.9, respectively, while the vrms(0) corresponding tothese masses are 0.1 and 0.015 km/s. Therefore, the ex-pected core radii are rc ≈ 30 and 210 pc.

The value of vrms for a non–thermal sterile neutrinoof 0.5 keV is approximately three times larger than thecorresponding to the thermal particle of the same mass.Therefore, for this case rc ≈ 630 pc. So, the resolutionsthat we may attain in our simulations of WDM halosfor the mW=0.5 keV sterile neutrino are already close tothese estimates of the flat core radii.

Recently, alternative particle models, like super–WIMPS, were proposed. The dark particles in thesemodels may acquire random velocities non–thermally,for example, through the decay process of NLSP par-ticles (e.g., charged Sleptons into gravitinos, Feng, Ra-jaraman & Takayama 2003, and see for more referencesSteffen 2006). In these cases, vrms does not depend di-rectly on the damping scale of the linear power spectrum.However, an extra parameter is introduced, the decayingepoch. Strategies to constraint this parameter using as-trophysical observations have been proposed (Feng, Ra-jaraman & Takayama 2003; Strigari et al. 2007).

Some cosmological models with super–WIMP particlesmay be in agreement with constraints based on structureformation, specially the Ly-α forest, and still allow forrelatively large velocity dispersions, able to ameloriatethe potential problems of ΛCDM at small scales. Thisis the case of the so called Meta–dark matter modelsthat consider the late decay of neutralinos into graviti-nos. These models preserve a power spectrum similar toΛCDM models and at the same time set a phase space

10

limit in the innermost structure of dark halos due to theinjection of random velocities to the particles (Strigariet al. 2007; Kaplinghat 2005). However, as mentionedin the Introduction, it was still an open question of howefficient the introduction of vrms is for producing sig-nificant effects on the inner structure of simulated darkhalos. One of the goals of the present paper was just toexpore this question by means of numerical simulationsable to resolve the WDM halos down to ∼ 0.005 Rv.

The results presented in §4.2 show definitively thatthe addition of vrms flattens the inner density profilesof WDM halos, and more as vrms is higher. Differ-ences in density profiles of halos simulated with vrms > 0and those with vrms = 0, start to be evident at radiiof 2–3 h−1kpc(∼ 0.010Rv − 0.015Rv) for our ≈ 2 − 4 ×1012h−1M⊙ halos. The halo average inner density mea-sured at 0.01Rv, ¯ρ1% decreases on average by factors 1.5and 2.5 for the models with vrms(0) = 0.1 and 0.3 km/s,respectively (column 11 in Table 1). Nevertheless, wenotice that the concentration of the halos seems not tochange significantly by the introduction of vrms.

In general, the NFW function does not describe wellthe inner density profiles of our simulated halos; theirprofiles are much shallower than γ = −1 as can be seenin Fig. 5. However, is the size of the random velocityeffect as theoretically expected? The theoretical predic-tions are based on the existence of an upper limit in thefine–grained phase–space density due to the collisionlessnature and finite relict velocity dispersion of particles.This upper limit, Q0,max, implies that the halo densityprofile must saturate and form a constant–density core(Hogan & Dalcanton 2000).

For the random velocities used in this study (vrms(0) =0.1 and 0.3 km/s, corresponding respectively to thermaland non–thermal 0.5 keV sterile neutrinos), Q0,max =3 × 10−5 and 1.1 × 10−6 M⊙pc−3/(km/s)3. Accordingto Hogan & Dalcanton (2000), the core radius produced

by the phase–space packing scales as m−2X v

−1/2c,∞ , where

vc,∞ is the asymptotic circular velocity for the assumednon–singular isothermal sphere (their eq. 18). This im-plies that more massive halos have smaller, more tightlybound cores. Applying the same equation for mX = 0.5keV and vc∞ = 200 km/s, the core radius for the ther-mal particle (Q0,max = 3 × 10−5 M⊙pc−3/(km/s)3) is85 pc, while for the sterile neutrino (Q = 1.1 × 10−6

M⊙pc−3/(km/s)3) is 450 pc.In any case, our results can only marginally test these

estimates. The resolution limit in our simulations withnon–zero vrms is in between ≈ 1 and 1.7 kpc. If the innerdensity profile response to the introduction of vrms isgradual, one expect to see yet some effects at these radii,and this happen to be the case.

A way to attempt to infer (extrapolate) the sizes ofpossible flat cores in the simulated halos is by fitting themeasured density profiles to an analytical function thatimplies a flat core. Results of these fits were presentedin §4.2 using the Zhao (1996) profile with (α, β, γ) =(0.7, 3, 0) as well as the one suggested by S2006. The lat-ter function gives a poor description of the overall mea-sured density profiles, which tend to be significantly lesscurved than the analytical model (see Fig. 5). The ob-tained (overestimated) values for rc (= 0.1r0, see §4.2)are reported in column 13 of Table 3. When the S2006

Fig. 6.— Upper panel: Comparison of the density profile of aWDM Milky Way-like halo, without the random velocity, (squares)with the density profile of its counterpart CDM, this latter gener-ated using the same random phases (crosses). Lower panel: Sameas upper panel but for circular velocity.

function is fitted to only the inner 30 kpc, the fits im-prove and the estimated core radii become smaller byroughly a factor of two (column 14). However, even forthis case the core radii seems to be upper limits, withthe exception of halo D5120.1.

We have found that the WDM profiles are better de-scribed by the Zhao function with (α, β, γ) = (0.7, 3, 0).The best fits to the measured profiles give extrapolatedcore radii rc ≈ 5− 8 times smaller than the S2006 profilefitted to only the inner 30 kpc. Should we have the suffi-cient resolution to resolve the flat cores, their radii wouldlie in between rc and rS+

c,30. The only halo for which theflat core is patent at our resolution limit is D5120.1; avisual inspection shows that the core radius is close to 1h−1kpc.

Our results show that the different estimates of thecore radius increase by less than a factor of 1.5 fromthe simulations with vrms(0)=0.1 km/s to the ones withvrms(0)=0.3 km/s. The amount of this increase is lessthan the one we would predict using the monolithic col-lapse approximation of Avila-Reese et al. (2001); ac-cording to this approximation, the core radius of halosformed at the same time depends linearly on the injectedvrms at the maximum expansion of the perturbation,rc ∝ vrms,zM = vrms,0(1 + zM ). We have estimated thepredicted values of rc for our halos by using this approx-imation. From the simulations, we find that the redshiftsof maximum expansion of halos C, D, and E are roughlyzM = 1.6, 1.3 and 1.8, respectively; these redshifts arepractically the same for the different values of vrms. Thecalculated rc for vrms(0) = 0.1 (0.3) km/s are then 221(663), 240 (720), and 209 (627) pc, respectively. Thus, ingeneral these predictions give core radii just in betweenrc and rS+

c,30 (see Table 1), though the dependence on

11

vrms is much more pronounced than for the (extrapo-lated) core radii estimated from the fits to our halos.

5.3. How is the structure of halos at the dampingscale?

The simulations carried out in this paper allowed usalso to explore the structure of dark matter halos formedfrom perturbations at the scale of damping of the lin-ear power spectrum. The cutoff in the power spec-trum used here corresponds to a relatively large mass,Mf = 2.6 × 1012h−1M⊙. Therefore, the formation ofthe first structures in this model, namely those struc-tures with masses close to Mf , happens relatively late.We speculate that the formation process of the trunca-tion halos is generic. If this is true, then the structureof the late–formed truncation halos simulated here withseveral millions of particles (vrms=0) should be similar tothe structure of early–formed truncation (micro)halos inmodels with much smaller filtering masses than the oneused here, for example in the CDM models. If this is thecase, then our results may enrich the discussion about theformation and structure of the first microhalos in CDMmodels (Earth–mass scales).

We have found a clear systematic trend in the den-sity profiles of the simulated truncation halos: theyare significantly steeper than r−1 in the inner regions,r . 0.02Rv − 0.03Rv, and lie below the best NFW fits inthe intermediate region. CDM halos with an inner slopesteeper than r−1 have been reported in other contexts:recently merged group– and cluster–size halos (Knebe etal. 2002; Tasitsiomi et al. 2004) or microhalos formed atthe scale of CDM power–spectrum damping (Diemandet al. 2005). It has been argued that the recent ma-jor merger is the one to blame for the steepening of thedensity profile while subsequent secondary infall modifiesthe external region. For the halos at the damping scale,rather than a major merger, the dynamical situation cor-responds to a fast (cuasi–monolithic) collapse. However,in both cases, the process is dynamically violent. Wehave checked that the obtained density profiles do notcorrespond to a transient configuration. As mentionedin §5.1, for halo D the profile remains almost unchangeduntil a = 1.4.

Extrapolating our results to the damping scale ofCDM, we can speculate that CDM microhalos may besignificantly steeper than the NFW profile. This im-plies that the possible contribution of surviving micro-halos to the γ−ray flux originated by neutralino anni-hilation, might be comparable to the central flux fromhost halos (Diemand et al. 2006). Another implicationof our results could be related to the buildup of the innerdensity profile of dark halos in general. Dehnen (2005)and Kazantzidis et al. (2006) argued that the assembly ofhalo inner density profiles happens very early in the his-tory of the universe and specifically that the inner slopeof the cuspier progenitor survives up to the final halo. Ifthe density profiles in our simulated halos are represen-tative of objects formed at the damping of power scalesin general, then, according to the mentioned studies, thecentral slope of present day dark matter halos should bemuch steeper than the NFW profile.

In order to verify our results, a more systematic studyhalo structure at the scale of damping is required, ex-ploring any possible dependence with the shape of the

cutoff and the power spectrum slope at the scale of damp-ing. Currently the only studies discussing a similar situ-ation report different results for the scale of galaxy clus-ters(Moore et al. 1999b) and the microhalos (Diemandet al. 2005). It is unclear if this suggests a slope depen-dence on the profile of the smallest dark matter haloswith the power spectrum slope.

6. CONCLUSIONS

We have studied by means of cosmological N–bodysimulations the structure of dark halos formed in the con-text of a WDM model corresponding to a non–thermalsterile neutrino particle of mass mW=0.5 keV. The firstseries of simulations did not include the injection of arandom velocity component, vrms, to the particles andwere aimed at exploring the structure of the halos formedfrom perturbations at the damping scale in the linearpower spectrum (Mf = 2.6×1012 h−1M⊙ for the concreteWDM model studied here). The second series of simula-tions included a vrms component of a (i) a mW=0.5 keVthermal neutrino (vrms(z = 0) = 0.1 km/s), and one thatroughly corresponds (ii) to a mW=0.5 keV non–thermalsterile neutrino (vrms(z = 0) ≈ 0.3 km/s). This lattermodel was used to generate the initial power spectrum.These simulations were aimed at exploring the effect thata random velocity component has on the inner structureof halos; in particular, at dilucidating whether constant–density cores are produced or not in the simulations. Theresults of our study lead us to the following two mainconclusions:• The structure of halos formed from perturbations of

scales close to Mf , and resolved with up to more than16 millions of particles (with vrms=0), is peculiar: theinner density profile (. 0.02Rv) is systematically steeperthan the best corresponding NFW fit (and the respec-tive CDM counterpart), and the overall density profile(> 0.02Rv) tends to be less curved than the best NFWfit; the outer profile slope is never steeper than −3. Ac-cording to our tests, these differences with respect to thestructure of halos assembled hierarchically, can hardlybe attributed to a peculiar dynamical state of the halossimulated here.• The effect of adding vrms to the particles pro-

duces a significant flattening of the inner density profile(r . 2−3h−1kpc corresponding to ∼ 0.010Rv−0.015Rv)of the simulated halos. The different estimated (extrap-olated) sizes of the nearly constant–density cores are ofthe order of the theoretical predictions, which give valuesbelow our resolution limit. For the halo masses simulatedhere, Mv ≈ (2 − 4) × 1012 h−1M⊙, the flat core radii es-timated from different fittings are between ∼ 0.1 − 0.8h−1kpc. An increase in vrms(0) from 0.1 to 0.3 km/s pro-duces an increase in the extrapolated core radii of a fac-tor 1.5 or less. For one of our simulations (halo D5120.1),the presence of a nearly constant–density core, of radius≈ 1 h−1kpc, is already revealed at the resolution limit;the same halo simulated with a different random velocityseed is less flattened.

Although the simulations presented here refer to aconcrete WDM model, they can be interpreted withina wide range of contexts. The density profile of darkhalos with masses close to the truncation mass in thelinear power spectrum is systematically different fromthe NFW profile; in particular, the inner regions tend

12

to be steeper. These could have important implicationsin the context of CDM models if a significant fractionof microhalos formed at the free–streaming CDM scales(∼ 10−6h−1M⊙) have survived until the present epoch.In this case, the predicted γ−ray flux from the neu-tralino annihilation in the center of these cuspy micro-halos might be comparable to the central flux from hosthalos. On the other hand, the fact that the microhalosare so cuspy, could have some interesting implications inthe building up of the next hierarchies of the halo as-sembling, as well as in the inner structure of the largerhalos.

Regarding the effects of random velocity injection tothe particles, our results show evidence of significant in-ner flattening of the halo density profile at our resolutionradii. These resolutions are not enough to test directlythe predicted core radii by phase–space constraints (e.g.,Hogan & Dalcanton 2000) or by dynamical models ofgravitational collapse with initial random velocities (e.g.,

Avila-Reese et al. 2001; Bode et al. 2001). However, theinner extrapolations of the best–fit models to our simu-lated halos are consistent with such predictions.

This work was supported by PAPIIT–UNAM grantsIN112806-2 and IN107706-3 and by a bilateralCONACyT–DFG grant. The authors gratefully acknowl-edge the hospitality extended by the ”Astrophysikalis-ches Institut Potsdam” where this paper was finished.OV acknowledges support from the NSF grant 02-05413assigned to the UW during the initial stage of the project,and a CONACyT Repatriacion fellowship. Some of thesimulations presented in this paper were performed us-ing the HP CP 4000 cluster (Kan-Balam) at DGSCA–UNAM. We acknowledge the anonymous referee, whosehelpful comments and suggestions improved some as-pects of this paper.

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