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    Brandon Allgood1, Ricardo Flores2, Andrey V. Kravtsov3, Joel R. Primack1, Risa H. Wechsler3,4, James S. Bullock5

    draft version; May 27, 2005

    ABSTRACT We study shape statistics of dark matter halos with masses from a few times Mvir = 1014h−1 M" to

    a few times Mvir = 1011h−1 M" from various N-body simulations in a flat ΛCDM universe. We present results from two high resolution simulations with the same initial density fluctuations but different σ8’s (= 0.9 and = 0.75). We exam the mean value of the axis ratios of halos verses their virial mass and find that less massive halos are rounder than more massive halos. We also find that there is an exponential relation between mass and mean axis ratio which remains constant with σ8 and redshift only changing in amplitude. We find that the distribution about the mean axis ratio for galaxy sized halos is roughly Gaussian. We also find that the minor axis of the halo is most often aligned with the orbital angular momentum and that the more aspherical the halo the greater the coincidence of the alignment. This indicates that angular momentum and shapes may have a similar origin. To investigate this we look at the merger history and find that both the time since the last major merger and the cumulative merger history are correlated with the axis ratios. Subject headings: cosmology: theory — galaxies: formation — galaxies: halos — large-scale structure

    of universe

    1. introduction

    A prediction of cold dark matter (CDM) is the process of bottom up halo formation, where large halos form from the merger of smaller halos which are in turn formed from even smaller halos. This is a violent process and it vio- lates most of the assumptions that go into the spherical top-hat collapse model of halo formation. Since mass falls onto halos in lumps, one would not expect halos to be spherical if the relaxation time of the halos were shorter than the time between mergers and/or the in-falling halos were come along a preferential direction such as along a filament. It fact we find in both, theoretical modeling of CDM and observations, that halos are not perfectly spher- ical. Therefore, the analysis of the shapes of halos could give us another clue into the nature of the dark matter and the process of galaxy and halo formation.

    Measurements of the shapes of both cluster and galaxy mass halos through varied observational techniques have recently become available. In the area of cluster shapes there have been studies of the X-ray morphology (McMil- lan et al. 1989; Kolokotronis et al. 2001) which can be directly related to the shape of the inner part of the halo (Flores et al. 2004; Jing & Suto 2002; Buote & Xu 1997). The shape in turn can be related to the fundamental pa- rameters of the universe as we will so and as others have shown (). There has also been a lot of activity in the area of the shape of our own Milky Way halo recently. The major focus has been on the tidal stream of the Sagittarius dwarf 1 Physics Department, University of California, Santa Cruz, CA 95064; allgood@physics.ucsc.edu, joel@scipp.ucsc.edu 2 Dept. of Physics and Astronomy, University of Missouri-St. Louis, St. Louis, MO 93121-4499; ricardo.flores@umsl.edu 3 Dept. of Astronomy and Astrophysics, Kavli Institute for Cos- mological Physics, The University of Chicago, Chicago, IL 60637; andrey@oddjob.uchicago.edu, risa@cfcp.uchicago.edu 4 Hubble Fellow, Enrico Fermi Fellow 5 Physics Department, University of California, Irvine, CA 92697; bullock@uci.edu

    spheroidal galaxy with studies implying a minor-to-major axis ratio c/a ! 0.8 (Ibata et al. 2001; Majewski et al. 2003; Law et al. 2003, but see Helmi 2004 and Mart́ınez- Delgado et al. 2004).

    The currently published galaxy-galaxy weak lensing stud- ies (Hoekstra et al. 2004) and ongoing studies of galaxy- galaxy weak lensing

    There have been many theoretical papers published re- cently on the subject of halo shapes but they have either concentrated on cluster sized halos only and/or have ana- lyzed only a small number of halos.

    This paper is organized as follows: In section 2 we will describe the simulations, halo finding and halo property determination used in this study. In section 3 we being by discussing the different methods employed to deter- mine the shapes of halos and the difference in there de- termination. We then describe the method we use and the motivation behind it. In section 4 we examine the mean axis ratios and their dependence on mass and σ8. We also examine the relationship of the angular momen- tum and velocity ellipsoid to the halo ellipsoid. In section 5 we examine the redshift dependence of the relationships examined in the previous section. In section 7 we exam- ine the relationship between the formation history of halos and their present day shapes. Finally, section 8 is devoted to summary and conclusions.

    2. simulations

    2.1. The Numerical Simulations

    The simulations were performed using the Adaptive Re- finement Tree (ART) N-body code of Kravtsov & Klypin (1999) which implements successive refinements in the spa- cial and time grids in high density environments. All sim- ulation results presented here were taken from simulations of a concordance flat ΛCDM cosmological model (Ω0 = 1 − ΩΛ, h = 0.7, where Ω0 and ΩΛ are the present-day matter and vacuum energy densities in units of critical and h is the Hubble parameter in units of 100km s−1 Mpc−1).



    Table 1

    Simulation parameters

    Name σ8 Lbox Np mp hpeak h−1Mpc h−1 M" h

    −1 kpc

    L800.75 0.75 80 5123 3.16 × 108 1.2 L800.9a 0.9 80 5123 3.16 × 108 1.2 L800.9b 0.9 80 512

    3 3.16 × 108 1.2 L1200.9 0.9 120 5123 1.07 × 109 1.8 L1200.9r 0.9 20 sphere ∼ 2563 1.33 × 108 1.8

    Different simulations were analyzed in order to check nu- merical convergence of results, to test cosmological vari- ance and to test the results on as many mass scales as possible. Table 2.1 presents the key parameters which vary between simulations. For more information about the L80 simulations see Tasitsiomi et al. (2004b). The L1200.9 simulation is very similar to L80 simulations. The L1200.9r is a resimulated spherical subregion with a radius of R = 20h−1 Mpc at z = 0 of the L1200.9 box. The subregion was chosen to not contain many halos above M > 1013h−1 M" in order to concentrate on galaxy mass halos. The mass resolution in the subregion is eight times that of the mass resolution of the original L1200.9 box. The entire L120 box was resimulated, to preserve the long range forces in the box, but with the mass resolution around the high resolution region stepped down gradually to a mass resolution of M = 8.5 × 109h−1 M" at the furthest dis- tances from the high resolution region.

    2.2. Halo Identification and classification

    The halos and sub-halos in our simulations were iden- tified using a variant of the Bound Density Maximum (BDM) algorithm (Klypin et al. 1999). The details of the algorithm and parameters used in the halo finder can be found in Kravtsov et al. (2004). We will briefly de- scribe the main steps in the halo finder. First, all parti- cles are assigned a density using the smooth algorithm6, which uses a symmetric SPH (Smoothed Particle Hydro- dynamics) smoothing kernel on the 32 nearest neighbors. Density maxima are then identified which are separated by a minimum distances of rmin = 50h−1 kpc, defining the minimum distinguishable separation of halos and sub- halos. Using the maxima as centers, profiles in circular velocity and density are calculated in spherical bins. Un- bound particles are then removed from the fits iteratively as describe in Klypin et al. (1999). We then construct fi- nal profiles using only bound particles and fit them to an NFW profile. The halo finder is complete for halos with ≈ 50 particles. This corresponds to a mass below which the cumulative mass and velocity functions begin to flatten out.

    The halo density profiles are fit with the NFW fitting 6 To calculate the density we use the publicly available code smooth: http://www-hpcc.astro.washington.edu/tools/tools.html

    formula, ρNFW (r) =

    ρs (r/rs)(1 + r/rs)2

    , (1)

    using a chi-squared minimization algorithm. We compared these fits to others which used different merit functions, such as the maximum deviation from the fit as described in Tasitsiomi et al. (2004a) and found that they gave very similar results on individual halos. We define the virial radius as the radius of a sphere within which the aver- age over-density drops below a value of ∆v = 340 deter- mined using the formula of Bryan & Norman (1998). Here we limit our analysis to isolated halos with Np ≥ 3000 within Rvir . This corresponds to Mvir ≥ 9.48 × 1011 for the 80h−1Mpc box simulation, Mvir ≥ 3.16× 1012 for the 120h−1Mpc box simulation and Mvir ≥ 4×1011. A halo is determined to be isolated if its center is not contain within the outer radius of another more massive halo.

    3. methods of determining shapes

    There are a many different methods that previous au- thors have used to determine shapes of halos, of which four deserve mention here, all of which involve the digitaliza- tion of the inertia tensor in some form,

    Mij ≡ Σxixj . (2) The sum is over all particles being considered. The ratios of eigenvalues, s = c/a and q = b/a (s < q < 1), are used in our analysis as measures of the shape. The simplest of the four methods is to calculate the inertia tensor for all particles within a sphere with r = Rvir (Tormen 1997; Bullock 2002; Kasun & Evrard 2004) or smaller