PASJ: Publ. Astron. Soc. Japan , 1–??, c 2018. Astronomical Society of Japan. Missing Dwarf Problem in Galaxy Clusters Hiroyuki Kase [email protected] Junichiro Makino Department of Astronomy, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033 and Yoko Funato Department of General Systems Studies, College of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902 (Received 2006 March 1; accepted 2000 0) Abstract We investigated the formation and evolution of CDM subhalos in galaxy-sized and cluster-sized halos by means of N-body simulations. Our aim is to make clear what the “dwarf galaxy problem” is. It has been argued that the number of subhalos in simulated galaxy-sized halos is too large compared to the observed number of dwarfs in the local group, while that in cluster-sized halos is consistent with observed number of galaxies in clusters such as the Virgo cluster. We simulated nine halos with several different mass resolutions and physical scales. We found that the dependence of the cumulative number of subhalos −3 Nc on their maximum circular velocity Vc is given by Nc ∝ Vc , down to the reliability limit, independent of the mass of the main halo. This implies that simulations for cluster-sized halos give too many halos with Vc ∼ 140km/s or less. Previous comparisons of cluster-sized halos gave much smaller number of subhalos in this regime simply because of their limited resolution. Our result implies that any theory which attempts to resolve the missing dwarf problem should also explain the discrepancy of the simulation and observation in cluster-sized halos. Key words: cosmology:dark matter — cosmology:theory — galaxies:clusters:general — galaxies:dwarf — methods:n-body simulations 1. Introduction This similarity, which looks quite natural from the the- oretical side, showed serious discrepancy with observa- The Cold Dark Matter(CDM) scenario(White & tion. The number of subhalos in the simulated CDM halo Rees 1978) has been the standard theory of the formation agreed well with the observation of the cluster of galaxies, and evolution of the structures in the Universe. In this sce- while the simulation of a galaxy-sized halo predicted too nario, galaxies and clusters of galaxies are formed bottom- many dwarf-sized halos. This discrepancy is now known up. It has been remarkably successful in explaining the as the “dwarf galaxy problem”, or “missing-dwarf prob- arXiv:astro-ph/0603074v1 3 Mar 2006 large scale structures (Davis et al. 1985) and numerous lem”, and has been one of the main topics in the study of observational results (e.g. Springel et al. 2005). Until re- galaxy formation and structure formation in the Universe. cently, however, the small-scale structures of CDM, like Solutions so far proposed for this problem can be clas- subhalos in a galaxy-sized halo, could not be studied by sified into the following two categories: numerical simulation because of the lack of the computa- • Dark matter is different from the CDM model in tional power. Recent improvement of computational pow- small scales. ers made it possible to study not only dark matter halos, • Only a small fraction of subhalos is observed as but subhalos in a parent halo. Usually these subhalos are dwarf galaxies. interpreted as corresponding to galaxies in cluster-sized halos and satellite dwarf galaxies in galaxy-sized halos. Many modified dark matter models had been proposed. Klypin et al. (1999b) and Moore et al. (1999, here- Self Interacting Dark Matter(Spergel & Steinhardt 2000) after M99) reported that about 1000 subhalos formed in a is a CDM with small self-interaction cross section, simulated galaxy-sized halo. The number distribution of which dumps small-scale fluctuation. Warm Dark subhalos as the function of the circular velocity normal- Matter(Bardeen et al. 1986) has a power spectrum which ized by that of the parent halo turned out to be remark- has cut off at small scale. Some simulations with ably similar for galaxy-sized and cluster-sized halos. This these dark matter models have been done. (e.g. Dav´e result, of course, is the direct outcome of the scale-free et al. 2001, Col´ınet al. 2000) nature of the gravity and almost power-law amplitude of If we assume that the standard CDM is correct and the density fluctuation. consider the possibility that not all subhalos are observed 2 KASE AND MAKINO [Vol. , as dwarf galaxies, the question is which subhalos are ob- calculations. Our result indicates that an analogue to the served as dwarf galaxies. Stoehr et al. (2002) argued that “missing dwarf problem” exists also in clusters of galaxies. observed dwarf galaxies are presently most massive sub- Therefore, the “solutions” for the missing dwarf problem halos, by comparing numerical result with observation of should give a distribution consistent with observations in velocity dispersion profile of dwarf galaxies. On the other both the galaxy scale and the cluster scale. hand, Kravtsov et al. (2004, hereafter K04) tracked each The contents of this paper is as follows. In section 2, subhalo’s evolution and showed many low-mass subhalos we describe the models and method used in our simula- were massive in the past. Additionally, they introduced tions. We give detailed explanation of the subhalo detec- the model of star formation condition, and claimed that tion method we developed. In section 3, we discuss the the number and distribution of satellite dSph galaxies was reliability limit of the number count of subhalos. Finally in agreement with that of such previously massive subha- we summarize our result and discuss possible solutions to los. the missing dwarf problem in section 4. Before jumping to such a solution, however, we should understand what is really the problem. In previous stud- 2. Model and Method ies, in order to determine the number of subhalos with 2.1. Cosmological N-Body simulation circular velocity ∼ 10km/s, the detection limit was set to a few tens of particles. It is not clear whether or not we 2.1.1. Initial condition can rely on the result obtained with such a small num- We adopted the Standard CDM model(SCDM), so ber of particles. Both softening parameters and the two- that we can directly compare our result with that of body relaxation might significantly affect mass and den- M99 who adopted SCDM. Cosmological parameters are sity profiles of subhalos (e.g. Moore et al. 1996, Diemand ΩM =1.0, ΩΛ =0.0, H0 = 50km/s, σ8 =0.7. To generate et al. 2004b). Ghigna et al. (2000) showed that the low the initial condition, we used the standard re-simulation mass end of the circular velocity distribution of subhalos is method(Navarro et al. 1996). First we simulated 15 co- affected by numerical resolution by comparing the results moving Mpc radius using 1.1 × 106 particles(m = 8.9 × 8 6 of two cluster-scale simulations with different resolutions. 10 M⊙) and 150 comoving Mpc radius using 1.1×10 par- 11 Same tendency can be seen in the more recent studies ticles (m =8.9 × 10 M⊙) to select candidates for galac- (e.g. Diemand et al. 2004a, Gao et al. 2004 and Reed tic and cluster halos from the distribution of particles at et al. 2005). This means that the agreement in a cluster z = 0. To select halo candidates, we used the standard scale described in M99 might be a numerical artifact. Friends of Friends method (FoF, Davis et al. 1985) with In the present paper, as a first step to understand the the linking length which is 0.2 times the mean interparti- formation and evolution of subhalos, we investigated the cle distance. Then we selected particles inside the radius reliability of size distribution of subhalos in simulation four times larger than radius enclosing particles detected results. We performed large-scale N-body simulations of by FoF as the region of high resolution calculation, and formation of CDM halos. We simulated both galaxy-sized traced these particles back to the initial condition. We and cluster-sized halos. replaced these particles with high resolution particles and We carried out N-body simulations of nine halos of dif- re-ran the simulation. In this way, the external tidal force ferent masses and resolutions and analyzed the distribu- from outside the high resolution region was correctly taken tion of subhalos. From the results of these simulations, into account. we estimated the effect of the mass resolution on the dis- To generate initial conditions, we used the grafic2 pack- tribution of subhalos. When the number of particles in age (Bertschinger 2001). The calculation was done on one subhalo is small, both the mass and the circular ve- an IBM pSeries690 of the Data Reservoir Project1 of the locity of that halo are affected, resulting in the deviation University of Tokyo, and on a workstation with AMD from simple power-law dependence. We formulated the Opteron 242 processors and 16GB memory. reliability criterion for mass and circular velocity. We prepared initial conditions of one region with three We then compared the velocity distribution function ob- different resolutions for both galaxy-scale and cluster-scale tained by N-body simulations with that of observed galax- runs. These runs are named C1-H, C1-M, and C1-L and ies in the Virgo Cluster. We found that the observational G1-H, G1-M, G1-L, for cluster scales and galaxy scales. result is in good agreement with our medium-resolution Here, postfixes H, L and M denotes high, medium and low- result, which is affected by small-N effect around Vc ∼ resolution runs, respectively.