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The Astrophysical Journal, 616:27 ­ 39, 2004 November 20
# 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.

A

FIGURE ROTATION OF COSMOLOGICAL DARK MATTER HALOS
Jeremy Bailin1 and Matthias Steinmetz
2

Steward Observatory, University of Arizona , 933 North Cherry Avenue, Tucson, AZ 85721; and Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany; jbailin@as.arizona.edu, msteinmetz@aip.de Received 2004 May 21; accepted 2004 August 6

ABSTRACT We have analyzed galaxy- and group-sized dark matter halos formed in a high-resolution öCDM numerical N-body simulation in order to study the rotation of the triaxial figure, a property in principle independent of the angular momentum of the particles themselves. Such figure rotation may have observational consequences, such as triggering spiral structure in extended gas disks. The orientation of the major and minor axes are compared at five late snapshots of the simulation. Halos with significant substructure or that appear otherwise disturbed are excluded from the sample. We detect smooth figure rotation in 288 of the 317 halos in the sample. The pattern speeds follow a lognormal distribution centered at p ¼ 0:148 h km sþ1 kpcþ1 with a width of 0.83. These speeds are an order of magnitude smaller than required to explain the spiral structure of galaxies such as NGC 2915. The axis about which the figure rotates aligns very well with the halo minor axis in 85% of the halos and with the major axis in the remaining 15% of the halos. The figure rotation axis is usually reasonably well aligned with the angular momentum vector. The pattern speed is correlated with the halo spin parameter k but shows no correlation with the halo mass. The halos with the highest pattern speeds show particularly strong alignment between their angular momentum vectors and their figure rotation axes. The figure rotation is coherent outside 0.12rvir . The measured pattern speed and degree of internal alignment of the figure rotation axis drops in the innermost region of the halo, which may be an artifact of the numerical force softening. The axis ratios show a weak tendency to become more spherical with time. Subject headings: dark matter -- galaxies: evolution -- galaxies: formation -- galaxies: individual ( NGC 2915) -- galaxies: kinematics and dynamics -- galaxies: structure Online material: color figures

1. INTRODUCTION Although there have been many theoretical studies of the shapes of cosmological dark matter halos (e.g., Dubinski & Carlberg 1991; Warren et al. 1992; Cole & Lacey 1996; Jing & Suto 2002), there has been relatively little work done on how those figure shapes evolve with time, in particular whether the orientation of a triaxial halo stays fixed or whether the figure rotates. While the orientation of the halo can clearly change during a major merger, it is not known whether the orientation changes between cataclysmic events. Absent any theoretical prediction one way or the other, it is usually assumed that the figure orientation of triaxial halos remains fixed when in isolation (e.g., Subramanian 1988; Johnston et al. 1999; Lee & Suto 2003). Early work at detecting figure rotation in simulated halos was done by Dubinski (1992, hereafter D92). While examining the effect of tidal shear on halo shapes, he found that in all 14 of his (1 2) ; 1012 M halos the direction of the major axis rotated uniformly around the minor axis. The period of rotation varied from halo to halo and ranged from 4 Gyr at the fast end to 50 Gyr at the slow end, or equivalently the angular velocities ranged between 0.1 and 1.6 km sþ1 kpcþ1.3 It is difficult to
1 Current address: Centre for Astrophysics and Supercomputing, Swinburne University, Mail 31, P.O. Box 218, Hawthorn, Victoria 3122, Australia; jbailin@astro.swin.edu.au. 2 David and Lucile Packard Fellow. 3 It may be more intuitive to think of angular velocity in units of rad Gyrþ1 rather than the common unit of pattern speed, km sþ1 kpcþ1. Fortunately, the two units give almost identical numerical values.

draw statistics from this small sample size, especially since the initial conditions of this simulation were not drawn from cosmological models but were performed in a small isolated box with the linear tidal field of the external matter prescriptively superposed ( Dubinski & Carlberg 1991). Given that the main result of D92 is that the tidal shear may have a significant impact on the shapes of halos, it is clearly important to do such studies using self-consistent cosmological initial conditions. Recent studies of figure rotation come from Bureau et al. (1999, hereafter BFPM99) and Pfitzner (1999, hereafter P99). P99 compared the orientation of cluster mass halos in two snapshots spaced 500 Myr apart in an SCDM simulation ( ¼ 1, ö ¼ 0, h ¼ 0:5). He detected rotation of the major axis in $5% of them and argued that the true fraction with figure rotation is probably higher. BFPM99 presented one of these halos, which was extracted from its cosmological surroundings and left to evolve in isolation for 5 Gyr. During that time, the major axis rotated around the minor axis uniformly at all radii at a rate of 60 Gyrþ1, or about 1 km sþ1 kpcþ1. There may be observational consequences to a dark matter halo whose figure rotates. BFPM99 suggested that triaxial figure rotation is responsible for the spiral structure of the blue compact dwarf galaxy NGC 2915. Outside of the optical radius, NGC 2915 has a large H i disk extending to over 22 optical disk scale lengths (Meurer et al. 1996). The gas disk shows clear evidence of a bar and a spiral pattern extending over the entire radial extent of the disk. BFPM99 argue that the observed gas surface density is too low for a bar or spiral structure to form by gravitational instability and that there is no evidence of an interacting companion to trigger the pattern. They propose 27


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that the pattern may instead be triggered by a rotating triaxial halo. Bekki & Freeman (2002) followed this up with smoothed particle hydrodynamics (SPH ) simulations of a disk inside a halo whose figure rotates and showed that a triaxial halo with a flattening of b=a ¼ 0:8 and a pattern speed of 3.84 km sþ1 kpcþ1 could trigger spiral patterns in the disk, or warps when the figure rotation axis is inclined to the disk symmetry axis. Masset & Bureau (2003, hereafter MB03) found that, in detail, the observations of NGC 2915 are better fitted by increasing the disk mass by an order of magnitude ( for example, if most of the hydrogen is molecular, e.g., Pfenniger et al. 1994), but that a triaxial halo with b=a % 0:85 and a pattern speed of between 6.5 and 8.0 km sþ1 kpcþ1 also provides an acceptable fit. MB03 concluded that if the halo were undergoing solid body rotation at this rate, its spin parameter would be k % 0:157, which is extremely large (only 5 ; 10þ3 of all halos have spin parameters at least that large). However, this argument may be flawed because the figure rotation is a pattern speed, not the speed of the individual particles that constitute the halo, and so it is in principle independent of the angular momentum; in some cases, the figure may even rotate retrograde to the particle orbits ( Freeman 1966). Schwarzschild (1982) discusses in detail the orbits inside elliptical galaxies with figure rotation. He finds that models can be constructed from box and X-tube orbits, which have no net streaming of particles with respect to the figure (although they have prograde streaming at small radius and retrograde streaming at large radius) and so result in figures and particles with the same net rotation. He also constructs models that include prograde-streaming Z-tube orbits, which result in a figure that rotates slower than the particles. Stable retrograde Z-tube orbits also exist, but Schwarzschild (1982) did not attempt to include them in his models, so it may also be possible for the figure to rotate faster than the particles. While these results demonstrate the independence of the figure and particle rotation, it is not clear whether they can be translated directly to dark matter halos. Dark matter halos may have different formation mechanisms and may be subject to different tidal forces than elliptical galaxies, and the different density profile may also have a large effect on the viable orbital families (Gerhard & Binney 1985). There are other consequences of triaxial figure rotation. A rotating potential introduces an oscillating force on particles moving within the potential. Disk stars that have orbital frequencies in resonance with this oscillating force may experience very large changes in their orbit due to the figure rotation. For instance, Tremaine & Yu (2000) examined the behavior of disks in halos with retrograde figure rotation. In these disks, stars can get trapped in the Binney resonance, where 3 þ 2 ¼ p , for vertical and azimuthal frequencies 2 and 3 , respectively, and a halo pattern speed of p ( Binney 1981). If the pattern speed falls slowly toward zero, stars trapped in this resonance are pulled out of the disk and into polar orbits, while if the figure rotation smoothly proceeds from retrograde to prograde, the stars trapped in this resonance are flipped 180 and end up on retrograde orbits. Figure rotation may also erase or modify any intrinsic alignments between the orientation of neighboring halos (J. Bailin & M. Steinmetz 2004, in preparation). If there are observational consequences to dark halo figure rotation, such as those found by Bekki & Freeman (2002) and Tremaine & Yu (2000), they can be used as a direct method to distinguish between dark matter and models such as modi-

fied Newtonian dynamics ( MOND) that propose to change the strength of the force of gravity ( Milgrom 1983; Sanders & McGaugh 2002). Many of the traditional methods of deducing dark matter cannot easily distinguish between the presence of a roughly spherical dark matter halo and a modified force or inertia law. However, a major difference between dark matter and MOND is that dark matter is dynamical, and so tests that detect the presence of dark matter in motion are an effective tool to discriminate between these possibilities. Among the tests that can make this distinction are the ellipticities of dark matter halos as measured using X-ray isophotes, the SunyaevZeldovich effect, and weak lensing ( Buote et al. 2002; Lee & Suto 2003, 2004; Hoekstra et al. 2004), the presence of bars with parameters consistent with being stimulated by their angular momentum exchange with the halo (Athanassoula 2002; Valenzuela & Klypin 2003), and spatial offset between the baryons and the mass in infalling substructure measured using weak lensing (Clowe et al. 2004). Rotation of the halo figure requires that dark matter is dynamic, and therefore observable structure triggered by figure rotation potentially provides another test of the dark matter paradigm. In this paper, we use cosmological simulations to determine how the figures of öCDM halos rotate. The organization of the paper is as follows. Section 2 presents the cosmological simulations. Section 3 describes the method used to calculate the figure rotations, which are presented in x 4. Finally, we discuss our conclusions in x 5. 2. THE SIMULATIONS The halos are drawn from a large high-resolution cosmological N-body simulation performed using the GADGET2 code (Springel et al. 2001). We adopt a ``concordance'' cosmology (e.g., Spergel et al. 2003) with 0 ¼ 0:3, ö ¼ 0:7, bar ¼ 0:045, h ¼ 0:7, and 8 ¼ 0:9. The only effect of bar is on the initial power spectrum, since no baryonic physics is included in the simulation. The simulation contains 5123 ¼ 134,217,728 particles in a periodic volume 50 hþ1 Mpc comoving on a side. All results are scaled into h-independent units when possible. The full list of parameters is given in Table 1. A friends-of-friends algorithm is used to identify halos ( Davis et al. 1985). We use the standard linking length of ¯ b ¼ 0: 2n
þ1=3

;

Ï 1÷

¯ where n ¼ N =V is the global number density. Measuring the figure rotation requires comparing the same halo at different times during the simulation. We analyze
TABLE 1 Pa rameters of th e Cosmological S imulat ion Parameter N .................................................................................... Box size (hþ1 Mpc comoving) ..................................... Particle mass (107 hþ1 M ) .......................................... Force softening length (hþ1 kpc) ................................. Hubble parameter h (H0 ¼ 100 h km sþ1 Mpcþ1 ) ...... 0 .................................................................................. ö ................................................................................. 8 ................................................................................... bar ................................................................................ Value 5123 50 7.757 5 0.7 0.3 0.7 0.9 0.045


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TABLE 2 Snapshots Used t o C alculate Figure R otations Look-back Time (hþ1 Myr) 1108 496 296 98 0

Snapshot Name b090........................... b096........................... b098........................... b100........................... b102...........................

Scale Factor (a) 0.89 0.95 0.97 0.99 1.0

Redshift (z ) 0.1236 0.0526 0.0309 0.0101 0.0

the overall shape of the ellipsoid from which particles are chosen for the remainder of the calculation and therefore bias the results even when other measures are adopted to minimize their effect. The choice of a spherical region biases the derived axis ratios toward spherical values but does not affect the orientation. Second, the particles are weighted by 1=r 2 so that each mass unit contributes equally regardless of radius (Gerhard 1983). Both D92 and P99 take similar approaches, but using radii based on ellipsoidal shells. Therefore, we base our analysis on the principal axes of the reduced inertia tensor: ~ Iij ¼ X mk rk ; i r 2 rk k
k; j

snapshots of the simulation at look-back times of approximately 1000, 500, 300, and 100 hþ1 Myr with respect to the z ¼ 0 snapshot. The scale factor a of each snapshot, along with its corresponding redshift and look-back time, is listed in Table 2. 3. METHODOLOGY 3.1. Introduction The basic method is to identify individual halos in the final z ¼ 0 snapshot of the simulation, to find their respective progenitors in slightly earlier snapshots, and to measure the rotation of the axes through their common plane as a function of time. Precisely determining the direction of the axes is crucial and difficult. When merely calculating axial ratios or internal alignment, errors on the order of a few degrees are tolerable. However, if a pattern speed of 1 km sþ1 kpcþ1, as observed in the halo of BFPM99, is typical, then a typical halo will only rotate by 4 between the penultimate and final snapshots of the simulation. Therefore, the axes of a halo must be determined more precisely than this in order for the figure rotation to be detectable. In fact, we should strive for even smaller errors to see whether slower rotating halos exist. It would have been difficult for P99 to detect halos rotating much slower than the halo presented in BFPM99; although the error varies from halo to halo ( for reasons discussed in x 3.3), Figure 5.23 of P99 shows that most of his halos had orientation errors of between 8 and 15, corresponding to a minimum resolvable figure rotation of $0.6 km sþ1 kpcþ1 for a 2 detection in snapshots spaced 500 Myr apart. A major difficulty in determining the principal axes so precisely is substructure. The orientation of a mass distribution is usually found by calculating the moment of inertia tensor P Iij ¼ k mk rk ; i rk ; j and then diagonalizing Iij to find the principal axes. However, this procedure weights particles by r 2. Therefore, substructure near the edge of the halo (or of the subregion of the halo used to calculate the shape) can exert a large influence on the shape of nearly spherical halos, especially if a particular subhalo is part of the calculation in one snapshot but not in another, such as when it has just fallen in. This is particularly problematic because subhalos are preferentially found at large radii (Ghigna et al. 2000; De Lucia et al. 2004; Gill et al. 2004; Gao et al. 2004). Moving substructures can also induce a false measurement of figure rotation due to their motion within the main halo at approximately the circular velocity. To mitigate this, we first use particles in a spherical region of radius 0.6rvir surrounding the center of the halo, rather than picking the particles from a density-dependent ellipsoid, as in Warren et al. (1992) or Jing & Suto (2002). We find that those methods allow substructure at one particular radius to influence

:

Ï 2÷

In the majority of halos, the substructure is a small fraction of the total mass, usually less than 5% of the total mass within 60% of the virial radius ( De Lucia et al. 2004, their Fig. 8), so its effect is much reduced. There are still some halos that have not yet relaxed from a recent major merger, in which case the ``substructure'' constitutes a significant fraction of the halo. To find these cases, the reduced inertia tensor is separately calculated enclosing spheres of radius 0.6, 0.4, 0.25, 0.12, and 0.06 times the virial radius to look for deviations as a function of radius (see x 3.4.1 for details). These radii are always with respect to the z ¼ 0 value of rvir . We find that only halos with at least 4 ; 103 particles, or masses of at least $3 ; 1011 hþ1 M , have sufficient resolution for the orientation of the principal axis to be determined at sufficient precision (see x 3.3). There are 1432 halos in the z ¼ 0 snapshot satisfying this criterion, with masses extending up to 2:8 ; 1014 hþ1 M . 3.2. Halo Matching To match up the halos at z ¼ 0 with their earlier counterparts, we use the individual particle numbers provided by GADGET, which are invariant from snapshot to snapshot, and find which halo each particle belongs to in each snapshot. The progenitor of each z ¼ 0 halo in a given z > 0 snapshot is the halo that contributes !90% of the final halo mass. Sometimes no such halo exists; in these cases, the halo has only just formed or underwent a major merger and so is not useful for our purposes. Figure 1 shows a histogram of the fraction of the final halo mass that comes from the b096 (z % 0:05) halo that contributes the most mass. There are also some cases in which two nearby objects are identified as a single halo in an earlier snapshot but as distinct objects in the final snapshot. We therefore impose the additional constraint that the mass contributed to the final halo must also be !90% of the progenitor 's mass. In the longer time between the earliest snapshot b090 and the final snapshot b102, a halo typically accretes a greater fraction of its mass, and so a more liberal cut of 85% is used for this snapshot (see the dashed histogram in Fig. 1); 492 of the halos that satisfied the mass cut did not have a progenitor that satisfied these criteria in at least one of the z > 0 snapshots and so were eliminated from the analysis, leaving a sample of 940 matched halos. 3.3. Error in Axis Orientation There are two sources of errors that enter into the determination of the axes: how well the principal axes of the particle distribution can be determined, and whether that particle distribution has a smooth triaxial figure. Here we estimate the error assuming that it is not biased by substructure. The halos


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Fig. 1.--Histogram of the fraction of the final mass that comes from the b096 (z % 0:05; solid line) and b090 (z % 0:12; dashed line) halo that contributes the most mass.

for which this assumption does not hold will become apparent later in the calculation. For a smooth triaxial ellipsoid represented by N particles, the error is a function of N and of the intrinsic shape: as the axis ratio b=a or c=b approaches unity, the axes become degenerate. To quantify this, we have performed a bootstrap analysis of the particles in a sphere of radius 0.6rvir of each z ¼ 0 halo ( Heyl et al. 1994). If the sphere contains N particles, then we resample the structure by randomly selecting N particles from that set allowing for duplication and determine the axes from this bootstrap set. We do this 100 times for each halo. The dispersion of these estimates around the calculated axis is taken formally as the ``1 '' angular error and is labeled boot . As expected, the two important parameters are N and the axis ratio. We focus here on the major axis, for which the important axis ratio is b=a (the results for the minor axis are identical with the minor-to-intermediate axis ratio c=b replacing b=a). The top panels of Figures 2 and 3 show the dependence of the bootstrap error on N and b=a, respectively, for all halos with M > 1011 hþ1 M . The solid lines are empirical fits: err and
err; b=a ;N

Fig. 2.--Angular bootstrap error boot as a function of the number of particles N within the central 0.6rvir of each halo. Points are the cosmological halos, and asterisks are randomly sampled smooth NFW halos. Top: Angular error boot . The solid line is the fit err; N from eq. (3). Middle: Ratio between the angular error and the error expected for the halo given its axis ratio b=a, i.e., boot =err; b=a . The solid line is err; N from eq. (3) renormalized to the typical error of 0.02 rad. Bottom: Ratio between the angular error and the analytic estimate err from eq. (5). [See the electronic edition of the Journal for a color version of this figure.]

x 3.1 biases axis ratios toward spherical, the recovered b=a of these randomly sampled halos is larger than the input value and ranges from 0.65 to 0.95. The errors for these randomly sampled smooth halos are shown as asterisks in Figures 2 and 3. The top panel of Figure 2 shows a rise in the dispersion of the error for N P 4000, with many halos having errors greater than the 0.1 rad necessary to detect the figure rotation of the halo presented in BFPM99. Therefore, we only use halos with N > 4000. The bootstrap error appears to be completely determined by N and b=a. The residuals of boot with respect to err; N are due to err; b=a and vice versa. This is shown in the middle panels of Figures 2 and 3. In the middle panel of Figure 2 we have divided out the dependence of boot on the axis ratio, making

2 ¼ pffiffiffiffi ; N

Ï 3÷

pffiffiffiffiffiffiffiffi b=a : ¼ 0:005 1 þ b=a

Ï 4÷

The form of equation (3) is not surprising; if a smooth halo was randomly sampled, we would expect the errors to be Poissonian with an N þ1=2 dependence. However, the cosmological halos are not randomly sampled. Individual particles ``know'' where the other particles are, because they have acquired their positions by reacting in the potential defined by those other particles. Therefore, the errors may be less than expected from a randomly sampled halo. To test this, we construct a series of smooth prolate NFW halos ( Navarro et al. 1996) with b=a axis ratios ranging from 0.5 to 0.9, randomly sampled with between 3 ; 103 and 3 ; 105 particles, and perform the bootstrap analysis identically for each of these halos as for the cosmological halos. Because the method for calculating axis ratios outlined in

Fig. 3.--Angular bootstrap error boot as a function of the axis ratio b=a of each halo. Points are the cosmological halos, and asterisks are randomly sampled smooth NFW halos. Top: Angular error boot . The solid line is the fit err; b=a from eq. (4). Middle : Ratio between the angular error and the error expected for the halo given the number of particles N, i.e., boot =err; N . The solid line is err; b=a from eq. (4) renormalized to the typical error of 0.02 rad. Bottom:Ratio between the angular error and the analytic estimate err from eq. (5). [See the electronic edition of the Journal for a color version of this figure.]


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apparent an extremely tight relation between the residual and N, while in the middle panel of Figure 3 we have divided out the dependence of boot on N, showing the equally tight relation between the residual and b=a. It is apparent from comparing the points and asterisks that the errors in the cosmological halos are slightly smaller than for randomly sampled smooth halos. Combining equations (3) and (4), and noting that the typical error is boot % 0:02 rad, we find the bootstrap error is well fitted by pffiffiffiffiffiffiffiffi b=a 1 : Ï 5÷ err ¼ pffiffiffiffi 1 þ b=a 2N The bottom panels of Figures 2 and 3 show the residual ratio between the bootstrap error boot and the analytic estimate err . The vast majority of points lie between 0.8 and 1.0, indicating that err overestimates the error by $10%. Equation (5) breaks down as b=a approaches unity; these halos are nearly oblate and so do not have well-defined major axes. It also becomes inaccurate at very low b=a because of low-mass, poorly resolved halos. Even in these cases, the error estimate is conservative, but to be safe we have eliminated axes with b=a < 0:35 or b=a > 0:95 from the subsequent analysis, regardless of the nominal error. The randomly sampled smooth halos follow equation (5) extremely well, so the non-Poissonianity of the sampling in simulated halos reduces the errors by 10%. Calculating the bootstraps is computationally expensive, so equation (5) is used for the error in all further computation. Because this estimate is expected to be correct for smooth ellipsoids, cases in which the error is anomalous are indications of residual substructure. 3.4. Figure Rotation Ideally one would fit the figure rotation by comparing the orientation of each of the axes at each snapshot to that of a unit vector rotating uniformly along a great circle and minimize the 2 to find the best-fit uniform great circle trajectory. However, this requires minimizing a nonlinear function in a four-dimensional parameter space, a nontrivial task. We adopt two simpler and numerically more robust methods for measuring the figure rotation. The first method, referred to as the ``plane method,'' involves fitting the major or minor axis measurements at all five snapshots to a plane and then measuring the rotation of the axis along the plane. This fully takes the errors and measurements at all snapshots into account. However, it presupposes that the figure rotation axis is perpendicular to the plane containing the major or minor axis. The second method, referred to as the ``quaternion method,'' involves comparing all of the axes at two snapshots to find the axis through which the figure has rotated. This method gives a figure rotation axis that is not constrained to have any particular relation to the major or minor axis. However, by construction it measures the rotation from a single reference frame to another single reference frame and therefore can only include information from two snapshots at a time. It is also not possible to take into account the errors in the axis determinations; in particular, for prolate halos, where the error in the determination of the intermediate and minor axes are much larger than the error in the major axis, physical rotation of the major axis can be masked by spurious fluctuations in the two degenerate axes. The strengths and weaknesses of these two methods complement each other well. We adopt the plane method as our primary method of measuring the figure rotation. The quaternion

Fig. 4.--Diagram that demonstrates how we fit a plane measurements at all snapshots (thick lines) and then find the as a function of time. The figure rotation axis is perpendic plane and defined such that increases around it counterclo

to the major axis increase of phase ular to the best-fit ckwise with time.

method is used to check for bias in the derived figure rotation axes.
3.4.1. Plane Method

For the plane method, we first solve for the plane z ¼ ax × by that fitsthe majoraxismeasurementsofthe halo best at all time steps, assuming the error is negligible. The change of the phase of the axes in this plane as a function of time are then fitted by linear regression. A schematic diagram of this process is shown in Figure 4. We follow the same procedure for the minor axes when appropriate, as discussed in x 4. The degree to which the axes are consistent with lying in a plane is checked by calculating the out-of-plane 2:
2 oop



1 X ài2 ; 2 i err; i

Ï 6÷

where is the number of degrees of freedom and ài is the minimum angular distance between the major axis at time step i and the great circle defined by the best-fit plane. Because the axes have reflection symmetry, it is impossible to measure a change in phase of more than =2. The phases are adjusted by units of such that the difference in phase between adjacent snapshots is always less than =2. If the figure were truly rotating by 90 or more between the snapshots , it would be impossible to accurately measure this rotation since the angular frequency would be larger than the Nyquist frequency of our sampling rate. Any faster pattern speeds would be aliased to lower angular velocities, with an aliased angular velocity of Nyq þ (p þ Nyq ), where p is the intrinsic angular velocity of the pattern and Nyq is the Nyquist frequency. For snapshots spaced 500 hþ1 Myr apart, the maximum time between the snapshots we analyze, the maximum detectable angular velocity is 3.8 h km sþ1 kpcþ1. We do not expect the figure to change so dramatically, as we have excluded major mergers. However, this can be checked post facto by checking whether the distribution of measured angular velocities extends up to the Nyquist frequency; if so, then there are likely even more rapidly rotating figures whose angular frequency is aliased into the detectable range, fooling us into thinking


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Fig. 5.--Log-weighted projected density of four halos with a range of subhalo fractions fs . The subhalo fractions are 0.166 (top left), 0.065 (top right), 0.045 (bottom left), and 0.016 (bottom right). Axes are in units of hþ1 kpc from the halo center. All halos have masses in the range (2 3) ; 1012 hþ1 M .

they are rotating slower. If the measured distribution does not extend to the Nyquist frequency, then it is unlikely that there are any figures rotating too rapidly to be detected (see x 4). The best-fit linear relation for the phase as a function of time is found by linear regression. Bec