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Shapes of dark matter halos

I present an analysis of the density shapes of dark matter halos in LCDM and LWDM cosmologies. The main results are derived from a statistical sample of galaxy-mass halos drawn from a high resolution LCDM N-body simulation. Halo shapes show significant trends with mass and redshift: low-mass halos are rounder than high mass halos, and, for a fixed mass, halos are rounder at low z. Contrary to previous expectations, which were based on cluster-mass halos and non-COBE normalized simulations, LCDM galaxy-mass halos at z=0 are not strongly flattened, with short to long axis ratios of s = 0.70 +/- 0.17. I go on to study how the shapes of individual halos change when going from a LCDM simulation to a simulation with a warm dark matter power spectrum (LWDM). Four halos were compared, and, on average, the WDM halos are more spherical than their CDM counterparts (s =0.77 compared to s = 0.71). A larger sample of objects will be needed to test whether the trend is significant.


Introduction
A variety of observational indicators suggest that galaxies are embedded within massive, extended dark matter halos, lending support to the idea that a hierarchical, cold dark matter (CDM) based cosmology may provide a real description of the universe, especially on large scales (see, e.g., the review by Primack 11 and references therein). A useful small-scale test of CDM and variant theories may come from observations aimed at inferring the density structure of dark halos. Specifically, the quest to measure dark halo shapes has developed into a rich and complex subfield of its own. 4,12 Predictions for shapes of dark matter halos formed by dissipationless gravitational collapse are most reliably derived using numerical studies. 6,5,14 These past investigations focused on galaxy-sized halos formed from power-law or pre-COBE CDM power spectra, and found that halos were typically flattened triaxial structures, with short-to-long axis ratios of s ≃ 0.5 ± 0.15. Examinations based on currently favored cosmologies have been done, but they studied only cluster mass halos, 8,13,9 and also found significantly flattened objects, s ∼ 0.4 − 0.5. Interestingly, Thomas and collaborators 13 saw an indication that low matter density models (with early structure formation) tended to produce more spherical clusters than high density models. This result was qualitatively consistent with indications from Warren et al. 14 that high-mass halos are more flattened than (early-forming) low-mass halos. In light of these hints that formation history plays a role in setting halo shapes, it is important to reexamine the question for a currently standard cosmological model. Specifically, this work aims at characterizing halo flattening as a function of mass and redshift for the popular flat ΛCDM cosmology. I also explore how shapes of halos are affected when going to a ΛWDM model, in which the power spectrum is damped on small scales.

Simulations and shape measurement
The simulations were performed using the Adaptive Refinement Tree (ART) code. 7 The main results are derived from a ΛCDM simulation with Ω m = 0.3, Ω Λ = 0.7, h = 0.7, and σ 8 = 1.0, which followed 256 3 particles of mass 1.1× 10 9 h −1 M ⊙ within a periodic box of comoving length 60 h −1 Mpc, obtaining a formal force resolution of 1.8 h −1 kpc. A second simulation was run until z = 1.8 with the same particle number, half the box size, and thus eight times the mass resolution; it was used to check resolution issues. Dark halos were identified using a (spherical) bound density maxima method, 2 and masses were determined by counting particles within the spherical virial radius R v , inside which the mean overdensity has dropped to a value ∆ v ≃ (18π 2 + 82p − 39p 2 )/(1 + p), where p ≡ Ω m (z) − 1. This sample contains ∼ 800 halos that span the mass range 3 × 10 11 − 5 × 10 14 .
A second pair of simulations 1 were used to explore how shapes of halos are affected by damping the power spectrum. We compare four halos, each simulated from the same initial conditions, for a ΛCDM and ΛWDM model with the multiple-mass ART code and the same cosmological parameters and effective mass per particle discussed above. The filter mass scale for the ΛWDM run was 1.7 × 10 14 h −1 M ⊙ , and the four halos we compare have masses (2 − 8) × 10 13 h −1 M ⊙ . a Simulation results were kindly supplied by P. Colín.
Halo axis ratios are determined using the moments of the particle distributions within the virial radius R v . The short-to-long axis ratio, s, and intermediate-to-long axis ratio q are calculated by iteratively diagonalizing the tensor 5 where q 2 ≡ M yy /M xx and s 2 ≡ M zz /M xx .
a Although this filtering mass is much too large to be an consistent with with Ly-α forest measurements, 10 the simulation provides a useful comparison to test the effect of an imposed filtering scale, since the halo masses considered are well within the affected regime. Since

Results
The left panel in Figure 1 shows the average value of s as a function of halo mass for three redshifts, z = 0, 1, and 3. Low mass halos, on average, are rounder than high mass halos, as are halos of fixed mass at low z. The relation is wellapproximated by s ≃ 0.7(M/10 12 h −1 M ⊙ ) −0.05 (1 + z) −0.2 over the mass and redshift ranges explored. The right panel of Figure 1 shows the distribution of s parameters for galaxy-mass halos as a function of z. The average and rms dispersion in these distributions are s = 0.70 ± 0.17, 0.61 ± 0.17, and 0.55 ± 0.15, for z = 0, 1, and 3, respectively. Note that the distributions are quite non-Gaussian, with a significant tail of highly flattened halos. b How do the shapes of halos change with radius? The axial ratios presented in Figure 1 were obtained using particles within R v . The left panel of Figure 2 shows this average "virial" flattening measurement as a function of halo mass (at z = 0) compared with the average s measured within a sphere of radius 30 h −1 kpc for each halo. Although the difference is quite small for the low mass halos (since 30 h −1 kpc contains much of the halo mass), generally halos the effect should simply scale with the filtering mass, similar results would be obtained for ∼ 5 × 10 10 h −1 M ⊙ halos formed from ∼ 1keV WDM particles. b In order to check resolution effects, the z = 3 distribution was compared to the corresponding one obtained in the high resolution simulation and found to be statistically equivalent. are rounder at small radii. Interestingly, within this fixed central radius, halos typically have the same flattening, s ∼ 0.7, independent of the halo mass.
Finally in the right panel of Figure 2, we compare the measured s and q values of four halos, simulated in ΛCDM and ΛWDM. There is a tendency for the halos to be rounder (approaching the upper right corner) in WDM, although one halo (designated by the triangles) does become slightly flatter in WDM. The average flattening shifts from s ≃ 0.71 for CDM to 0.77 for WDM, but it is difficult to make strong conclusions based on four halos. Indeed, Moore 9 has simulated a single halo using CDM and WDM power spectra and finds a very similar shape for each. A larger sample of objects will be needed to test for systematic trends.

Conclusions
We find that ΛCDM galaxy-mass halos at z = 0 are more spherical than previously believed, with s = 0.70 ± 0.17. High mass halos show more substantial flattening, as do halos of fixed mass at high redshift: s ∝ (1 + z) −0.2 M −0.05 . Halos are also more spherical in their centers, and tend to become more flattened near the virial radius. These trends suggest collapsed structures become more spherical with time, perhaps because they have had more time to phase mix and to obtain isotropic orbit distributions. It is also possible that the accretion history itself plays a role. Halos formed within a ΛWDM simulation show a slight indication of being less flattened than their ΛCDM counterparts. This may be a reflection of substructure differences between the two models, but a larger number of halos will be needed to decisively test this conclusion. A more complete description of these results, and some discussion of shape correlation with other halo parameters is presented in a forthcoming paper. 3