WARM DARK MATTER MODEL OF GALAXY FORMATION Y.P. JING Shanghai Astronomical Observatory, Partner Group of MPI fu¨r Astrophysik, Nandan RD 80, Shanghai 200030, China E-mail: [email protected] 2 0 Cold Dark Matter (CDM) models of galaxy formation had been remarkably suc- 0 cessfultoexplainanumberofobservationsinthepastdecade. However,withboth 2 thetheoreticalmodelingandtheobservationsbeingimproved,CDMmodelshave n been very recently shown to have excessive clustering on the sub-galactic scale. a Here I discuss a solution, based on our high-resolution numerical simulations, to J thisoutstandingproblembyconsideringWarmDarkMatter(WDM).Ourresults show that the over-clustering problem on sub-galactic scales can be overcome by 0 WDM models, and all the advantages of CDM models are preserved by WDM 1 models. Therefore,theWDMmodelwillbecomeaninterestingalternativetothe well-studiedCDMmodels 1 v 6 1 Introduction 4 1 ColdDarkMatter(CDM)modelshavebeenshownverysuccessfultoexplainmany 1 observations of galaxies on scales of about one h−1Mpc to a few hundred h−1Mpc. 0 But such models probably predict over-clusteringonsmaller scales,as recenthigh- 2 0 resolution simulations (Moore et al. 1999, Klypin et al. 1999, Jing & Suto 2000) / have shown. The halo density profiles in these simulations are steeper than those h p inferredfromtherotationcurvesoflowsurfacebrightness(LSB)galaxies,andthere - aretoo manysub-haloswithin galactichaloswhencomparedto the observednum- o berofsatellitegalaxiesaroundtheMilkyWay. Thereisalsoadditionalevidencefor r t such overclustering from, e.g. the luminosity function of dwarf galaxies. Although s a some of these discrepancies may be resolved by introducing additional astrophys- : ical processes (Bullock et al. 2000) and some others by properly interpreting the v i observations (van den Bosch & Swaters 2000), there are attempts to resolve the X discrepancies by revisiting the assumption about the dark matter (DM). A list of r the candidatesforreplacingCDMproposedsincethe summerof1999includesself- a interactingDM,warmdarkmatter(WDM), repulsiveDM,fuzzyDM,annihilating DM etc (see Dav`e et al. 2000 for references). In this talk, I will present an exten- sivestudyforawarmdarkmattermodelusinghighresolutionN-bodysimulations. Our results will show that the WDM model is in good agreement with the obser- vational data without resorting to not-well-understood astrophysical processes. A complete description of the study appeared in our recent paper submitted to the Astrophyical Journal (Jing 2000). 2 Model and Simulations We consider a model dominated by WDM with the matter density Ω0 = 0.3, the cosmological constant λ0 = 0.7, and the Hubble constant H0 = 100h = 67kms−1Mpc−1. The primordial power spectrum is ∝k, and the transfer function istakenfromBardeenetal. (1986)withazerobaryoncontent. Thefree-streaming ms: submitted to World Scientific on February 1, 2008 1 cutoff parameter R = 0.1h−1Mpc is adopted, which is also consistent with the f Ly-α forest observations (Narayanan et al. 2000). The linear power spectrum is normalized so that the current rms linear density perturbation within a sphere of radius 8h−1Mpc is 1. Because all the parameters except R have usually been as- f sumedforthelow-densityflatCDM(LCDM)modelwhichbestfitsobservationson scales of 1h−1Mpc and up, this WDM is expected to fit these observations as well since the free-steaming of the warm dark matter has little effect on these scales. Thus our study will focus on the properties on galactic and sub-galactic scales, where the free-streaming effect of the WDM is expected to become significant. To single out the free-streaming effect, we compare the results of the WDM to those ofthe LCDM,whichinmostcasescaneffectivelyeliminatethenumericalartifacts. We have run a large set of cosmological simulations with box sizes of 12.5h−1Mpc, 25h−1Mpc and 50h−1Mpc respectively. For each box size, three 3 realizations are produced and 128 particles are adopted for each model. The same initial phases are used for the WDM model and for the LCDM model. We have selected five WDM halos of ∼ 1000 particles for each boxsize as well as the corresponding halos in the LCDM simulations. The virial mass of these halos is 7×1010M⊙, 6×1011M⊙, and 5×1012M⊙ respectively from the small to the large boxsizes. We then use the Nested-Grid-P3M (Jing & Suto 2000) to simulate these halos with a much higher resolution. A total of ∼ 7×105 particles are used for simulatingeachhalo,with∼5×105 particles(small)forthe high-resolutionregion and ∼ 2×105 (massive) for the coarse-resolution region. About 3×105 particles from the high-resolutionregionwill end up in the virializedregionof the halo. Us- ing the coarse (massive) particles can properly account for the tidal force which is important for forming the internal structures of the halos. A detailed account about the simulation technique can be found in Jing & Suto (2000). 3 Results We have made a very detailed analysis both for the cosmological simulations and for the high-resolution halo simulations. These results were presented in the talk, but can not be accommodated in this proceedings paper because of the limited space. A detailed account of these results can be found in our journal paper (Jing 2000). Here we just highlight a few interesting results. In Figure 1 we show the differential mass function of dark matter halos in the WDM model as well as in the LCDM model at redshift z =4. The mass function is defined as the mean number density of halos within a unit logarithmic interval of halo mass. Because of the free-streaming motion of the warm dark matter, the halo abundance in the WDM model is smaller than in the LCDM model at sub- galactic scales. The WDM halos are about 2 times and 4 times less abundant at M =1011M⊙ andM =2×1010M⊙ respectively. Acrucialtestforthiseffectwould be the hydrogen content of the damped Ly-α systems. Since the halo density is reduced only by a factor of two on the relevant scales, the WDM is well consistent withtheobservationsofthedampedLy-αsystems(seeMo&Miralda-Escude1994, Ma et al. 1997) The density profiles of dark matter halos are presented in Figure 2. We found ms: submitted to World Scientific on February 1, 2008 2 Figure1. Thedifferentialmassfunctionofdarkmatterhalosatredshiftz=4intheWDMmodel (lower)andintheLCDMmodel(upper). that the profiles of the halos less massive than 5×1011M⊙ are significantly flatter in the WDM model. Especially the halos with the mass similar to the those of the dwarf galaxies have formed cores at their centers, which could be potentially important for explaining the slowly rising rotation curves observed for the LSB galaxies. Our comparison of the halo circular velocity with the observed rotation curves shows that the LCDM model could be ruled out for its too steep density profiles, and the WDM model is consistent with the observations of LSB galaxies. Now let’s consider sub-halos within virialized halos. The smoothed density of each dark matter particle is estimated in the way used in Smoothed Particle Hydrodynamics simulations. The averageof the smootheddensity is calculatedfor each radial shell, and the particles with the smoothed density five times above the radially averaged value are identified. These identified particles are grouped with the friend-of-friend algorithm of a bonding length equal to one tenth of the global mean particle separation. We find that the resulted catalogue of the sub-halos is quite robust against the parameters we have taken. Figure 3 shows the number of sub-haloswith the circularvelocitylargerthanv . Therearemuchfewersub-halos c in the WDM halos than in the LCDM halos. To compare with obsrevations, we plot the observed abundance of the satellite galaxies within the Milky Way which hasacircularvelocityof220kms−1. The numberofsub-haloswithin aMilkyWay like halo, which should be in between the curves of v = 270kms−1 and of c,host v =135kms−1, can be readily read out for the two dark matter models. The c,host LCDM halos have too many halos as many previous studies have pointed out, but theWDMhaloshaveacomfortableamountofsub-haloswhichisingoodagreement ms: submitted to World Scientific on February 1, 2008 3 Figure2. Thedensity profilesof thedarkmatter halos intheWDM model(squares) and inthe LCDMmodel(crosses). with the observations. We noted that the our results for the LCDM are in good agreement, for the mass range plotted in the figure, with the result of Moore et al.(1999). In summary, we have presented a first, very detailed simulation study for the WDM model. Our results show that the model predicts enough halos at high redshiftz ≈4tobeconsistentwiththeobservationsofdampedLy-αsystems. This modelisalsoconsistentwiththeclusteringoftheLy-αabsorptionlines(Narayanan et al. 2000). The density profiles of the halos are significantly flatter than in the LCDM model for halo mass less than ∼ 1011M⊙, bringing about good agreement with the recent high-resolution observation of the rotation curves of LSB galaxies by Swaters et al.(2000). In contrast, we found that the CDM halos are NOT consistent with the observation of Swaters et al.. There are significantly fewer subhalos in WDM halos than in LCDM halos, and the number of subhalos within MilkyWaylikehalosintheWDMmodelagreesverywellwiththeobservednumber of satellite galaxies around the Milky Way without resorting to poorly-understood astrophysical processes. Since there are much fewer sub-halos, we expect that the over-cooling problem on the galactic scales can be alleviated and large galactic disks can be formed, which has been a serious difficulty for CDM models. All these attractive features of the WDM model warrant further detailed studies of this model. The work is supported by the One-Hundred-Talent Program and by nkbrsf- g19990754. ms: submitted to World Scientific on February 1, 2008 4 Figure 3. The number of sub-halos with the circular velocity larger than vc as a function of vc/vc,host where vc,host is the circular velocity of the virialized host halo where the sub-halos reside. References 1. Bullock, J. S., Kravtsov, A. V., & Weinberg, D. H, 2000, ApJ, submitted (astro-ph/0002214) 2. Dav´e,R.,Spergel,D.N.,Steinhardt,P.J.,Wandelt,B.D.,2000,ApJ,submitted (astro-ph/0006218) 3. van den Bosch, F. C., Swaters, R. A. 2000, AJ, submitted, astro-ph/0006048 4. Jing, Y. P. & Suto, Y. 2000,ApJ, 529, L69 5. Jing, Y. 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