Although N-body studies of dark matter halos show that the density profiles, ρ(r), are not simple power-laws, the quantity ρ/σ 3 , where σ(r) is the velocity dispersion, is in fact a featureless power-law over ∼ 3 decades in radius. In the first part of the paper we demonstrate, using the semi-analytic Extended Secondary Infall Model (ESIM), that the nearly scale-free nature of ρ/σ 3 is a robust feature of virialized halos in equilibrium. By examining the processes in common between numerical N-body and semi-analytic approaches, we argue that the scale-free nature of ρ/σ 3 cannot be the result of hierarchical merging, rather it must be an outcome of violent relaxation. The empirical results of the first part of the paper motivate the analytical work of the second part of the paper, where we use ρ/σ 3 ∝ r −α as an additional constraint in the isotropic Jeans equation of hydrostatic equilibrium. Our analysis shows that the constrained Jeans equation has different types of solutions, and in particular, it admits a unique "periodic" solution with α = 1.9444. We derive the analytic expression for this density profile, which asymptotes to inner and outer profiles of ρ ∼ r −0.78 , and ρ ∼ r −3.44 , respectively.

For a decade, N-body simulations have revealed a nearly universal dark matter density profile, which appears to be robust to changes in the overall density of the universe and the underlying power spectrum. Despite its universality, the physical origin of this profile has not yet been well understood. Semi-analytic models by Barnes et al. (2005) have suggested that the density structure of dark matter halos is determined by the onset of the radial orbit instability (ROI). We have tested this hypothesis using N-body simulations of collapsing dark matter halos with a variety of initial conditions. For dynamically cold initial conditions, the resulting halo structures are triaxial in shape, due to the mild aspect of the instability. We examine how variations in initial velocity dispersion affect the onset of the instability, and find that an isotropic velocity dispersion can suppress the ROI entirely, while a purely radial dispersion does not. The quantity σ 2 /v 2 c is a criterion for instability, where regions with σ 2 /v 2 c 1 become triaxial due to the ROI or other perturbations. We also find that the radial orbit instability sets a scale length at which the velocity dispersion changes rapidly from isotropic to radially anisotropic. This scale length is proportional to the radius at which the density profile changes shape, as is the case in the semi-analytic models; however, the coefficient of proportionality is different by a factor of ∼2.5. We conclude that the radial orbit instability is likely to be a key physical mechanism responsible for the nearly universal profiles of simulated dark matter halos.

Abstract. For nearly a decade, N-body simulations have revealed a nearly universal dark matter density profile. This density profile appears to be robust to changes in the overall density of the universe and the underlying power spectrum. Despite its universality, however, the physical origin of this profile has not yet been well understood. Semi-analytic models have suggested that scale lengths in dark matter halos may be determined by the onset of the radial orbit instability. We have tested this theory using N-body simulations of collapsing dark matter halos. The resulting halo structures are prolate in shape, due to the mild aspect of the instability. We find that the radial orbit instability sets a scale length at which the velocity dispersion changes rapidly from isotropic to radially anisotropic. Preliminary analysis suggests that this scale length is proportional to the radius at which the density profile changes shape, as is the case in the semi-analytic models; however, the coefficient of proportionality is different by a factor of ∼2. We conclude that the radial orbit instability may be a key physical mechanism responsible for the nearly universal profiles of simulated dark matter halos.Keywords. methods: n-body simulations, cosmology: dark matter, galaxies: formation Simulations were done using PKDGRAV (Stadel 2001), a parallel KD Tree gravity solver. The initial system of over 570,000 particles is spherical with a Gaussian density distribution. It starts at a redshift of 15 with only Hubble flow velocities. During the collapse, the radial orbit instability (ROI) sets in and creates a prolate structure.The resulting anisotropy profile, demonstrated by β = 1 − σ 2 t /2σ 2 r , exhibits isotropic orbits in the core of the halo, which may prevent a central buildup of mass, resulting in a shallower density profile. The density profile is well fit by Navarro et al. (2004) and less well by NFW (1996). The final shape is prolate within the anisotropy radius and becomes triaxial with increasing radius.The semi-analytic model described in Barnes et al. (2005) shows a direct correlation between the anisotropy radius (r a ), defined as the radius where β = 0.5, and the density scale length (r s , here defined as the radius where the slope = −2). While our N-body simulations do not confirm this exact result, the two scale lengths are clearly proportional to each other. We suggest a relation of r a ∼ 2.5r s rather than equality, and conclude that the ROI may be a critical factor in the determination of the universal profiles of dark matter halos.

High resolution N-body simulations have all but converged on a common empirical form for the shape of the density profiles of halos, but the full understanding of the underlying physics of halo formation has eluded them so far. We investigate the formation and structure of dark matter halos using analytical and semi-analytical techniques. Our halos are formed via an extended secondary infall model (ESIM); they contain secondary perturbations and hence random tangential and radial motions which affect the halo's evolution at it undergoes shell-crossing and virialization. Even though the density profiles of NFW and ESIM halos are different their phase-space density distributions are the same: ρ 1¤ 875. However, this relies on assumptions of spherical symmetry and slow accretion. While for ESIM halos these assumptions are justified, they most certainly break down for simulated halos which forms hierarchically. We speculate that our argument may apply to an "on-average" formation scenario of halos within merger-driven numerical simulations, and thereby explain why α £ 1¤ 875 for NFW halos. Thus, ρ ¡ σ 3 ∝ r¢ 1¦ 875 may be a generic feature of violent relaxation.

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