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Register a new userAnalytical derivation of the radial distribution function in spherical dark matter halos
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The velocity distribution of dark matter near the Earth is important for an accurate analysis of the signals in terrestrial detectors. This distribution is typically extracted from numerical simulations. Here we address the possibility of deriving the velocity distribution function analytically. We derive a differential equation which is a function of radius and the radial component of the velocity. Under various assumptions this can be solved, and we compare the solution with the results from controlled numerical simulations. Our findings complement the previously derived tangential velocity distribution. We hereby demonstrate that the entire distribution function, below 0.7 v_esc, can be derived analytically for spherical and equilibrated dark matter structures.
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All dark matter structures appear to follow a set of universalities, such as phase-space density or velocity anisotropy profiles, however, the origin of these universalities remains a mystery. Any equilibrated dark matter structure can be fully described by two functions, namely the radial and the tangential velocity distribution functions (VDF), and when we will understand these two then we will understand all the observed universalities. Here we demonstrate that if we know the radial VDF, then we can derive and understand the tangential VDF. This is based on simple dynamical arguments about properties of collisionless systems. We use a range of controlled numerical simulations to demonstrate the accuracy of this result. We therefore boil the question of the dark matter structural properties down to understanding the radial VDF.
(Abridged) We apply a very general statistical theorem introduced by Cramer (1936) to study the origin of the deviations of the halo spin PDF from the reference lognormal shape. We find that these deviations originate from correlations between two quantities entering the definition of spin, namely the ratio $J/M^{5/2}$ (which depends only on mass) and the total gravitational binding energy $E$. To reach this conclusion, we have made usage of the results deduced from two high spatial- and mass resolution simulations. Our simulations cover a relatively small volume and produce a sample of more than 16.000 gravitationally bound halos, each traced by at least 300 particles. We verify that our results are stable to different systematics, by comparing our results with those derived by the GIF2 and by a more recent simulation performed by Maccio et al. We find that the spin probability distribution function shows systematic deviations from a lognormal, at all redshifts z <= 1. These deviations depend on mass and redshift: at small masses they change little with redshift, and also the best lognormal fits are more stable. The J-M relationship is well described by a power law of exponent $alpha$ very near to the linear theory prediction (alpha=5/3), but systematically lower than this at z<= 0.3. We argue that the fact that deviations from a lognormal PDF are present only for high-spin halos could point to a role of large-scale tidal fields in the evolution of the spin PDF.
Although N-body studies of dark matter halos show that the density profiles, rho(r), are not simple power-laws, the quantity rho/sigma^3, where sigma(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 rho/sigma^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 rho/sigma^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 rho/sigma^3 proportional to r^{-alpha} 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 alpha=1.9444. We derive the analytic expression for this density profile, which asymptotes to inner and outer profiles of rho ~ r^{-0.78}, and rho ~ r^{-3.44}, respectively.
We present a wave generalization of the classic Schwarzschild method for constructing self-consistent halos -- such a halo consists of a suitable superposition of waves instead of particle orbits, chosen to yield a desired mean density profile. As an illustration, the method is applied to spherically symmetric halos. We derive an analytic relation between the particle distribution function and the wave superposition amplitudes, and show how it simplifies in the high energy (WKB) limit. We verify the stability of such constructed halos by numerically evolving the Schrodinger-Poisson system. The algorithm provides an efficient and accurate way to simulate the time-dependent halo substructures from wave interference. We use this method to construct halos with a variety of density profiles, all of which have a core from the ground-state wave function, though the core-halo relation need not be the standard one.
We have analyzed high resolution N-body simulations of dark matter halos, focusing specifically on the evolution of angular momentum. We find that not only is individual particle angular momentum not conserved, but the angular momentum of radial shells also varies over the age of the Universe by up to factors of a few. We find that torques from external structure are the most likely cause for this distribution shift. Since the model of adiabatic contraction that is often applied to model the effects of galaxy evolution on the dark-matter density profile in a halo assumes angular momentum conservation, this variation implies that there is a fundamental limit on the possible accuracy of the adiabatic contraction model in modeling the response of DM halos to the growth of galaxies.
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