D A R K M A T T E R H A L O S H A P E S A ... - Risa...

DARK MATTER HALO SHAPES AND THEIR EVOLUTION WITH REDSHIFT

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

draft version; May 27, 2005

ABSTRACTWe 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 !CDM universe. We presentresults from two high resolution simulations with the same initial density fluctuations but di"erent !8’s(= 0.9 and = 0.75). We exam the mean value of the axis ratios of halos verses their virial mass and findthat less massive halos are rounder than more massive halos. We also find that there is an exponentialrelation between mass and mean axis ratio which remains constant with !8 and redshift only changingin amplitude. We find that the distribution about the mean axis ratio for galaxy sized halos is roughlyGaussian. We also find that the minor axis of the halo is most often aligned with the orbital angularmomentum and that the more aspherical the halo the greater the coincidence of the alignment. Thisindicates that angular momentum and shapes may have a similar origin. To investigate this we look atthe merger history and find that both the time since the last major merger and the cumulative mergerhistory 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 processof bottom up halo formation, where large halos form fromthe merger of smaller halos which are in turn formed fromeven smaller halos. This is a violent process and it vio-lates most of the assumptions that go into the sphericaltop-hat collapse model of halo formation. Since mass fallsonto halos in lumps, one would not expect halos to bespherical if the relaxation time of the halos were shorterthan the time between mergers and/or the in-falling haloswere come along a preferential direction such as along afilament. It fact we find in both, theoretical modeling ofCDM and observations, that halos are not perfectly spher-ical. Therefore, the analysis of the shapes of halos couldgive us another clue into the nature of the dark matterand the process of galaxy and halo formation.

Measurements of the shapes of both cluster and galaxymass halos through varied observational techniques haverecently become available. In the area of cluster shapesthere have been studies of the X-ray morphology (McMil-lan et al. 1989; Kolokotronis et al. 2001) which can bedirectly 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 haveshown (). There has also been a lot of activity in the area ofthe shape of our own Milky Way halo recently. The majorfocus has been on the tidal stream of the Sagittarius dwarf1 Physics Department, University of California, Santa Cruz, CA95064; [email protected], [email protected] Dept. of Physics and Astronomy, University of Missouri-St. Louis,St. Louis, MO 93121-4499; [email protected] Dept. of Astronomy and Astrophysics, Kavli Institute for Cos-mological Physics, The University of Chicago, Chicago, IL 60637;[email protected], [email protected] Hubble Fellow, Enrico Fermi Fellow5 Physics Department, University of California, Irvine, CA 92697;[email protected]

spheroidal galaxy with studies implying a minor-to-majoraxis ratio c/a ! 0.8 (Ibata et al. 2001; Majewski et al.2003; Law et al. 2003, but see Helmi 2004 and Martınez-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 eitherconcentrated 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 willdescribe the simulations, halo finding and halo propertydetermination used in this study. In section 3 we beingby discussing the di"erent methods employed to deter-mine the shapes of halos and the di"erence in there de-termination. We then describe the method we use andthe motivation behind it. In section 4 we examine themean axis ratios and their dependence on mass and !8.We also examine the relationship of the angular momen-tum and velocity ellipsoid to the halo ellipsoid. In section5 we examine the redshift dependence of the relationshipsexamined in the previous section. In section 7 we exam-ine the relationship between the formation history of halosand their present day shapes. Finally, section 8 is devotedto 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 simulationsof a concordance flat !CDM cosmological model (#0 =1 ! #!, h = 0.7, where #0 and #! are the present-daymatter and vacuum energy densities in units of critical andh is the Hubble parameter in units of 100km s!1 Mpc!1).

1

2 ALLGOOD, FLORES, KRAVTSOV, PRIMACK, WECHSLER & BULLOCK

Table 1

Simulation parameters

Name !8 Lbox Np mp hpeak

h!1Mpc h!1 M" h!1 kpc

L800.75 0.75 80 5123 3.16 ! 108 1.2L800.9a 0.9 80 5123 3.16 ! 108 1.2L800.9b 0.9 80 5123 3.16 ! 108 1.2L1200.9 0.9 120 5123 1.07 ! 109 1.8L1200.9r 0.9 20 sphere " 2563 1.33 ! 108 1.8

Di"erent 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 aspossible. Table 2.1 presents the key parameters whichvary between simulations. For more information aboutthe L80 simulations see Tasitsiomi et al. (2004b). TheL1200.9 simulation is very similar to L80 simulations. TheL1200.9r is a resimulated spherical subregion with a radiusof R = 20h!1 Mpc at z = 0 of the L1200.9 box. Thesubregion was chosen to not contain many halos aboveM > 1013h!1 M" in order to concentrate on galaxy masshalos. The mass resolution in the subregion is eight timesthat of the mass resolution of the original L1200.9 box. Theentire L120 box was resimulated, to preserve the long rangeforces in the box, but with the mass resolution around thehigh resolution region stepped down gradually to a massresolution 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 ofthe algorithm and parameters used in the halo finder canbe 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 separatedby a minimum distances of rmin = 50h!1 kpc, definingthe minimum distinguishable separation of halos and sub-halos. Using the maxima as centers, profiles in circularvelocity and density are calculated in spherical bins. Un-bound particles are then removed from the fits iterativelyas describe in Klypin et al. (1999). We then construct fi-nal profiles using only bound particles and fit them to anNFW profile. The halo finder is complete for halos with# 50 particles. This corresponds to a mass below whichthe cumulative mass and velocity functions begin to flattenout.

The halo density profiles are fit with the NFW fitting6 To calculate the density we use the publicly available code smooth:http://www-hpcc.astro.washington.edu/tools/tools.html

formula,"NFW (r) =

"s

(r/rs)(1 + r/rs)2, (1)

using a chi-squared minimization algorithm. We comparedthese fits to others which used di"erent merit functions,such as the maximum deviation from the fit as describedin Tasitsiomi et al. (2004a) and found that they gave verysimilar results on individual halos. We define the virialradius 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). Herewe limit our analysis to isolated halos with Np $ 3000within Rvir . This corresponds to Mvir $ 9.48 " 1011 forthe 80h!1Mpc box simulation, Mvir $ 3.16" 1012 for the120h!1Mpc box simulation and Mvir $ 4"1011. A halo isdetermined to be isolated if its center is not contain withinthe outer radius of another more massive halo.

3. methods of determining shapes

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

Mij % %xixj . (2)The sum is over all particles being considered. The ratiosof eigenvalues, s = c/a and q = b/a (s < q < 1), are usedin our analysis as measures of the shape. The simplestof the four methods is to calculate the inertia tensor forall particles within a sphere with r = Rvir (Tormen 1997;Bullock 2002; Kasun & Evrard 2004) or smaller. Otherauthors have included all particles with an over-densitylarger than some threshold (Jing & Suto 2002). The fi-nal two worth mentioning are very similar and have beenshown to be equivalent to the Jing & Suto (2002) method.They involves beginning, as in the first method, with asphere of r & Rvir and then diagonalizing the inertia ten-sor, subsequently deforming the initial sphere into a triax-ial ellipsoid with axis ratios equal to the eigenvalues of theinertia tensor and pointing in the direction of the eigen-vectors. The inertia tensor is then recalculated for theparticles found within this ellipsoid and the ellipsoid is de-formed further and reoriented accordingly. This processiterates until converges is achieved. The di"erence in thefinal two methods is in the weighting of the inertia tensor.

SHAPES OF DARK MATTER HALOS 3

Fig. 1.— Isodensity shells for a !CDM cluster simulated by Nagai& Kravtsov (2003), projected in the xy–plane of the simulation box.A density is associated with each particle following the method of JS.The thick dots are particles with density within 3% (the thicknessused by JS) of the fractional level 0.01 of peak density. The smallcrosses (dots) are particles with density at a 0.004(0.001) fraction ofpeak density. The dashed curves (and solid white, for clarity) showthe projection, on the same plane, of triaxial ellipsoids of axial ratiosgiven by the values calculated from the moment of inertia tensor.Iteration starts with 10 h!1 kpc–thick spherical shells centered atradii of 105(185, 335) h!1 kpc.

Katz (1991) among others have used the form of the iner-tia tensor in equation 2 without any weighting. Dubinski& Carlberg (1991) used a modified version of equation 2where each term in the sum is weighted by 1/R2, where

R =!

x2 + y2/q2 + z2/s2, (3)is the elliptical distance to the particles from the center.This weighting was used in an attempt to give all particlesequal weight, instead of biasing toward the particle furtherfrom the center as is the case when using the unweightedinertia tensor.

The first method, the spherical method, gives the im-pression that halos are “rounder” (gives axis ratios whichare closer to 1) than the isodensity methods, to which theother three methods belong. Despite this, mean trendswith mass are seen with this method. The second method(Jing & Suto 2002) uses a Friends of Friends (FOF) halofinder with two di"erent linking lengths, a standard link-ing length of 0.2 times the mean interparticle spacing isused to identify halos and a shorter linking length, tunedto a particular over-density is used within each identifiedhalo to extract an inner region. We performed a test onhigh resolution cluster from Nagai & Kravtsov (2003) inwhich we compared the method of Jing & Suto (2002) tothe method of Dubinski & Carlberg (1991), iterative in-ertia tensor method, and found them to give very similarresults as is demonstrated in fig. 1. We then comparedthe last two methods (weighted and non-weighted inertiatensor methods) and found that on a halo by halo basis

Fig. 2.— Axial ratios for halos in a cosmological simulationare divided into mass bins and the shapes are calculated using theweighted (sw) and non-weighted (sn) iterative inertia tensor meth-ods. The two methods agree within ! 10% and give the same valuewhen averaged over a given mass bin (inset graph).

the two methods di"ered by s & 0.15, but when averagedover all halos in a given mass bin the two methods gavethe same results (fig. 2).

If one plots the cumulative s = a/c as a function ofradius for a very high resolution cluster you find that thechanges in s as a function of radius are smoother for theweighted method. This is not surprising because in theweighted method as new particles are included in the fitthey are given a weight which biases them against havinga dramatic e"ect on the shape. Where as the unweightedmethod gives a lot of weight to the particles at larger radii.Therefore in the unweighted case s as a function of radiusis noisier even in the cumulative plot because substructurewhich has just entered the fit at that radius has a largelever arm on the inertia tensor.

Both weighted and unweighted methods give similar re-sults for average quantities (sub-figure in fig. 2) as a func-tion of mass. This was surprising at first, but after closerexamination it became clear. Halos which gave axial ratioscloser to 1 for the weighted method over the unweightedmethod had strong substructure near the center and ha-los which had axial ratios closer to 1 for the unweightedmethod had strong substructure near the outer edge ofthe fit. At very small axial ratios the unweighted methodseems to always give larger axial ratios. It is most likelythe case that these halos are very resent mergers and there-fore have a double core which would lead to the unweightedmethod giving lower axial ratios.

In our analysis we use the iterative method of Dubinski& Carlberg (1991) with a beginning sphere of R = 0.3Rvir.It is well known that halos are more spherical at largerradii. This was has been shown in Jing & Suto (2002) andBailin & M. (2004). We have chosen to begin with spheresof R = 0.3Rvir in order to be consistent with these twoprevious works allowing for comparison.

4. halo shapes today


Fig. 3.— Axial ratio s for three simulations of di"erent mass reso-lution are presented with a fit (solid black lines) of s = 3.74M!0.064

and sigma of 0.1. The triangles are the average s for a give mass bin.The open circles and error bars are the best fit (KS test) mean and68% confidence level for the parent distribution. The dashed linesconnect the raw dispersion for each point and the colored solid linesare the best fit (KS test) dispersion for the pertinent distribution.

We begin by studying the shapes of halos today. Weexamine the mean values of s as a function of mass. Wethen look at the scatter about the mean s and discuss itsimplications as to current observations. We then examinethe distribution of p = s/q given a value for s. We thenexamine the relationship between the halo shape, totalangular momentum and velocity dispersion tensor. Finallywe compare two simulations of the same initial conditionswith di"erent values of !8.

4.1. Median Relationships for Distinct Halos

As has been found previously (Jing & Suto 2002; Bul-lock 2002; Kasun & Evrard 2004) there is a relationshipbetween the mean value of the axial ratio s = a/c and themass of a halo. As we show in fig. 3 the less massive thehalo the higher the mean value of s or the more sphericala halo tends to be. This is in agreement with others find-ings. What is not in agreement is the exact dependenceof < s > on mass. We use three di"erent simulations(L800.9b, L1200.9, and L1200.9r) with varying mass resolu-tion to determine the < s > vs. mass relationship and finda very good fit to the form

< s >= #M! (4)with best fit values at z = 0.0

# = 3.74 ± 0.02, $ = !0.064 ± 0.002. (5)The fit was preformed using a chi-squared minimizationon the bins from the L800.9b box, due to the fact that ithas the highest mass resolution of our simulations. Onlyhalos above our previously stated lower bound resolutionlimit were used and only mass bins with at least 20 haloswere included in the fit. This fit was then compared to theL1200.9 and the L1200.9r for validation with high statisticsand better mass resolution. It was found to be a good fitto all of the simulation we had on hand.

Fig. 4.— The distribution about the mean value for the massbin 1.6" 1012h!1 M" < M < 3.2 " 1012h!1 M" from the L800.9bsimulations. For each mass bin s has a Gaussian shape with a slighttail to lower s.

Our findings di"er from what other authors have foundin the past. Kasun & Evrard (2004) found the halos to bemuch rounder but this is due to the fact that they calcu-late the shapes of the halos at Rvir using only a sphericalwindow. As pointed out in section 3 both using a spher-ical window and using such a large radius will give onea more spherical shape for the halo so it is not surpris-ing that there is a disagreement. Bullock (2002) also useda similar method but found a steeper relationship withmass than Kasun & Evrard (2004). We suspect this hassomething to do with the resolution of the simulations.The more concerning of the di"erences is that with Jing& Suto (2002). This was somewhat unexpected due to thefact that we tested our method against their method andfound good agreement. In Jing & Suto (2002) the authorsrestrict their analysis to high mass halos which we havealso found are starting to show a flattening in the < s >vs M relationship. They also do not include halos whichwhere found to have multiple disjoint cores which shouldcase their mean to have a higher value of < s >.

In Jing & Suto (2002) the authors show that the dis-tribution of s within a given mass bin is Gaussian. Wefind that within a given bin the distribution is Gaussianwith a tail to low < s > (fig. 4), as also found in Bailin &M. (2004). As with Bailin & M. (2004) we found that thetail contains halos that may have been thrown out of theanalysis by Jing & Suto (2002) due to the fact that theywere actively merging. This tail to lower values of s canalso be seen in fig. 3 in the fact that the average s for eachbin is often slightly lower than the best fit mean.

Jing & Suto (2002) measure a dependence of the disper-sion of s on the mean value. We show a possible similartrend for halos at z = 0.0 (left graph in fig. 5, but thisrelationship does not seem to hold beyond z ! 0.5 as isevident in the right graph in fig. 5. Due the fact that thetrend is weak and that the dispersion is consistent with avalue of 0.1 at all redshifts we assume this constant value

Isn't it true that K&E used r200 background and B02 used rvir crit?

Can you make it clear on the figure (3) where J & S actually made their measurements?

ie, they didn't go down to 1e11, right?


Fig. 5.— < s > vs. M . We seem to find some evidence of adependence of the dispersion on < s > at z = 0.0 if we preforma weighted linear least squares fit to the best values of < s > and!<s>. By z = 1.0 this relationship seems to have disappeared. Sincewe are unable to predict a relationship and since even the z = 0.0points are consistent with a constant dispersion of 0.1, we assumethis.

Fig. 6.— The distribution of p = a/b in given bins of s show avery similar to that found in Jing & Suto (2002).

for the dispersion throughout our analysis.One of the main reasons an analysis of the shapes of

halos is so interesting is the amount of attention resentestimates of the Milky Way’s host halo have been getting.Most estimates using di"erent methods estimate the MilkyWay’s host halo to have an s $ 0.8 which given a disper-sion of 0.1 makes it at least 2 sigma o" the mean predictedby Jing & Suto (2002). Compared to our findings herethe Milky Way’s host halo still seems to be 1.5 sigma o"the mean relationship, which is a bit uncomfortable by aspointed out in Kazantzidis et al. (2004) the presence of gascooling will make the halo more spherical. The extent ofthe e"ect is not yet understood because no group has yetsimulated a statistically large sample of galaxy mast haloswith gas cooling, not to mention there is still an uncertainover-cooling problem.

s does not uniquely determine an ellipsoidal shape. Thereis still the question of the relationship of the middle axisto the minor or major axis. We found in our analysis thatthe value of q = b/c does not have a nice form. Instead wefind as does Jing & Suto (2002) that the relationship be-tween the minor and the middle axis (p = a/b) does have

Fig. 7.— The cos of the angle between the major, middle, andminor axis and the angular momentum vector are plotted. If theangular momentum was randomly oriented the graph would be aflat line at a value of 1.

a nice form. This can be seen in fig. 6, which contains sixplots of the q for di"erent ranges in s. The curves are a fitproposed in Jing & Suto (2002),

P (p|s) =3

2(1 ! rmin)

"1 !

#2s ! 1 ! rmin

1 ! rmin

$2%

(6)

with rmin = 0.55 for s < 0.55 and rmin = s for s $ 0.55.In Jing & Suto (2002) they find a fit with a cut-o" ofrmin = 0.5, but otherwise we find agreement with theirresults.

4.2. Alignment with Velocity and Angular Momentum

Now that we know the distribution of shapes with masswe would like to understand the possible origins of theshape. One immediate possibility is that the angular mo-mentum and velocity dispersion within each halo may berelated to the shapes. In order to investigate this welooked at the alignment of the angular momentum andthe velocity anisotropy of the halo with the shape. Theangular momentum used here is calculated using the sameparticles found in the resulted ellipsoid from our iterativemethod for determining the shape. We find, as was origi-nally pointed out in Tormen (1997) and subsequently seenby others, that the angular momentum is highly corre-lated with the minor axis of the halo. In fig. 7 we plotthe cosine of the angle between the indicated axis and theangular momentum direction. If the relationship was ran-dom the plot would be of a straight line at a value of 1.A peak at cos(%) = 1 means that the axis is most oftenaligned and a peak at cos(%) = 0 means that the axis ismost often at and angle of &/2 from the angular momen-tum. As one can see the minor axis is most often alignedand the major axis is most often at an angle of &/2 fromthe angular momentum.

Another dynamical measurement of the halo is the ve-locity anisotropy tensor. This was first examined in Tor-men (1997) as well. This is defined in the same way as the

run-on sentences here...

this is actually p, yes?


Fig. 8.— The cos of the angle between the major, middle, andminor axis and the angular momentum vector are plotted. If theangular momentum was randomly oriented the graph would be aflat line at a value of 1.

inertia tensor used to measure the shape,Vij % %vivj . (7)

We don’t not use a weighted version of this and we do notiterate, because either of these don’t make much physicalsense. We calculate the velocity anisotropy tensor againusing the particles found within the ellipsoidal shell definedby the shape analysis. We then determine the angle be-tween the respective axis (i.e. minor-minor, middle-middle,and major-major). The results are presented in fig. 8. Inthe figure, one can see that all three of the axis are highlycorrelated. This means that the velocity structure of thehalo is intimately connected to the shapes and may evenbe the driver of shape. A look at the evolution of theshapes may give us further clues as to the origin. This iswhat we explore in the next section.

5. evolution of shapes

One would like to understand how the relationships wefound in section 4 evolve with time, because our measure-ments of shape don’t tend to be at z = 0.0 except for themeasurement of the shape of our own halo. So we exam-ine in this section how halos of di"erent masses evolve withredshift. In order to do this we analyzed about 50 ! 100snapshots from all five of the simulations in table 2.1 andexamined the < s > vs. M relationship (fig. 9). What wefind is that at any given redshift the relationship between< s > and M can be described by equation 4 with a con-stant value for $ = !0.064 and time varying value of $.The solid lines in figure are the best fit lines of equation4 with a fixed power law. When all of the 64 snapshotsfor the L800.9b are fit in this way one sees a vary tightrelationship with scale factor (fig. 10). The line in fig. 10is power law fit in scale factor with the following form

# = nam (8)with

n = 3.74 ± 0.001i, m = 0.292± 0.004. (9)

Fig. 9.— < s > vs M relationship for z = 0.0, 0.5, 1.0, 2.0, 3.0.The points are the same as in fig. 3, but now for many di"erentredshifts. The solid line is a fit using the same power law relationas equation 4 with the same value for " but di"erent values for #

Fig. 10.— # (from equation 4) vs scale factor. If the powerlaw in equation 4 is fixed the " variable has a vary tight redshiftdependence as soon here. The open circle are the best fit values for" at each scale factor for the L800.9b simulation.

As one might expect at a fix mass the mean s becomessmaller as the halos are observed at earlier times. WHATABOUT THE DISTRIBUTION OF P?

6. !8 dependence

The current cosmological model seems to be on firmground with most of the parameters going into the modelfor structure formation observationally well constrained.The parameter which seems the least constrained and themost prone to change every time a new paper comes outis !8. We decided to examine the question of whether ornot mean halo shape observed through lensing and othermethods could add another probe of !8. As stated in sec-

?

does "i" this mean something, or just a typo?


Fig. 11.— < s > vs M with di"erent values of !8. Di"erent valuesof !8 predict di"erent values for the < s > vs M relationship. Hereone can see that a universe with a lower !8 produces halos which aremore elongated although the power law relationship between massremains the same.

tion 2, L800.75 and L800.9a were produced with the sameGaussian random field but di"erent values for the nor-malization. Therefore the only di"erence between the twosimulations can only be a produced of the di"erent valuesfor !8. As one can see in fig. 11 the two simulations doindeed produce di"erent relations. Not surprisingly sincethis was the case for di"erent redshifts the power law re-lationship between masses doesn’t seem to change but theproportionality factor $ does. Determining the best fitdi"erence between the $s for both z = 0.0and1.0 it seemsthat the di"erence in $ between !8 = 0.75 and 0.9 is con-stant with redshift and has a value of ' 0.2. In fig. 11 theremay be some evidence of a change in slope at z = 1.0 forthe !8 = 0.75 case. MORE REDSHIFTS SHOULD BEEXAMINED.

As mentioned above the di"erence in shape as a functionof mass may be another tool for determining !8. Therehas been a recent measurement of the shape of halos atz = 0.33 by Hoekstra et al. (2004). They determine theaverage shapes of halos by measuring the orientation of thegalaxies, then stacking the galaxy images with the orienta-tions aligned, and finally measuring the shear field aroundthe stacked image. From the determination of the shearthey were able to determine a rough measurement of theshapes of halos at z = 0.33. Their determination was hin-dered by the fact that the galaxies were stacked togetherregardless of morphological type and one would reasonablyassume that morphological type is related to merger his-tory which is in turn related to halo shapes as we show inthe next section. Another assumption that has gone intothe determination is that the orientation of the galaxy isrelated to the orientation of the halo which is currentlybeing called into question. We present predictions of thedistribution of projected ellipticities by choosing 1000 ran-dom lines of sight through the simulation box for both val-ues of !8 and present them in fig. 12. As one can see itmaybe very di&cult to constrain !8 through the determi-

Fig. 12.— PLOT IS OLD! Projected ellipticity of galaxies masshalos at z = 0.33 for !8 = 0.9 (black) and !8 = 0.75 (red).

nation of shapes but the figure still gives a prediction forthe distribution of projected shape which consistent withthe findings of Hoekstra et al. (2004). A more promisingroute is to examine the distribution of shapes of clusterswhich we address in Flores et al. (2004).

7. merger history and shapes

We have shown how halo shapes evolve with redshiftfor two di"erent values of !8 for a given mass bin, butwhat is the origin of the shape and how does it dependon the merger history of the halo. In order to explorethis question we first examine the relationship between sand ac (scale factor of formation), which is a measure ofmass accretion history of a halo through its most massiveprogenitors developed by Wechsler et al. (2002). In brief,ac is determined by fitting the mass accretion history withthe function

M(a) = Moe!zac/2 (10)

where Mo is the mass of the halo a z = 0 and ac is the scalefactor where the slope of the mass accretion history is atits peak, which is a slope of 2. Halos with lower values ofac were formed earlier and as is shown in Wechsler et al.(2002) have a higher concentration. In fig. 13 we plot therelationship between s and ac for all halos in the L1200.9

simulation. We find that the combination of equations 10and 4, represented by the dashed line, runs through thelocust of points and predicts to a very good degree the< s > to mean ac relationship. The combination of thesetwo relationships also predicts a lower limit on < s >" 0.4.Because ac is defined independent of the redshift at whichit is measured, this value is a theoretical lower limit on <s >, which in our analysis seems to always be the case. Inorder to disentangle the mass vs ac dependence in fig.13 wereplot the relationship, but now divide the s value of eachhalo by the mean s value for that mass halo as predictedfrom equation 4. This is shown in fig. 14. One can seefrom this figure that some of the dispersion at a fixed massfor lower mass halos may come from the di"erence in thevalues of ac, but can not explain the dispersion completely.

line-type mismatch, dashed/dotted.


Fig. 13.— s vs ac for all masses in the L1200.9 box. The blackpoints are the values of s and ac for each halo. The colored pointsare the average values of s for a given bin in ac. The average pointsare well fit by a line (presented in the same color). The dashedline represents the relationship of the mean s and ac predicted bycombining equation 10 and 4. The arrows are the average ac for thegiven mass bin.

Fig. 14.— The same plot as fig. 13, but with each value of sdivided by the predicted mean for that given mass.

8. conclusions

1. Discuss main model predictions.2. Discuss galaxy-galaxy modeling3. Talk about the problem compared to Ibata and other

MW halo observations.

I want to thank my wife and New Castle Brown Ale.

REFERENCES

Bailin, J. & M., S. 2004, ApJ submitted (astro-ph/0408163)

Fig. A15.— Results of applying our shape determination pro-cedure to 450 Monte Carlo halos produced with determined axialratios. We found that the error in the recovered value of s could beas large as ! 0.1.

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APPENDIX

resolution tests

The iterative method we use in this analysis convergesfor all halos with a reasonable number of particles, but one

Can you average the points in the plot as well (left) -- do they match with the line when averaged over all masses?

Are the arrows on this plot switched, or am i misinterpreting something? it looks like they imply the massive ones are rounder.


Fig. A16.— TOP PLOT IS OLD AND SHOULD BE RE-PLACED.

would expect that there is a lower bound on the numberof particles needed in order to resolve the shape. In thisappendix we explore the resolution e"ects of our analysis.We test our method for determining axis ratios by applyingit to Monte Carlo halos built to have an ellipsoidal NFWprofile (fig.A15). We constructed approximately 450 halo,each containing ' 1000 particles. We found that the errorin the recovered value of s could be as large as ' 0.1 forindividual halos, but the distribution of s values for a givenmass range remained stable.

In order to test our ability to recover the axis ratios in acosmological simulation and to test the dependence of themeasured scatter and redshift behavior we analyzed all ofour simulation which have three di"erent mass resolutions(table 2.1). In figure A16a one can see that the mean sverses mass relationship for the low resolution simulations(60h!1Mpc and 120h!1Mpc) begins to deviate from thehigher resolution 80h!1Mpc simulation at Mvir ' 3.2 "1012h!1 M" and the relation for the 80h!1Mpc simulationbegins to deviate from the NEW simulation at Mvir '9"1011h!1 M". For both sets of simulations these massescorrespond to the same number of particles, ' 3000 (fig.A16b). What we find is that when measuring halo shapes

with an initial sphere of r = 0.3Rvir when the total haloparticles drops below ' 3000 the halos begin to appearmore spherical than in higher resolution simulations. Thiscould possibly be an e"ect of two body relaxation.

WE SHOULD MAYBE DO THIS FOR ANOTHER RED-SHIFT AS WELL.

Based on the above analysis we conclude that our resolu-tion in the two 80h!1Mpc boxes are adequate to make pre-dictions about the mean properties of galaxy mass halos,which is what is needed for understanding galaxy-galaxyweak lensing, and more massive.

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