A Derivation of (Half) the Dark Matter Distribution Function

Abstract

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 tangential velocity distribution functions (VDFs), and once these two are understood 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.

Erratum: Anisotropic q-Gaussian 3D velocity distributions in ΛCDM haloes

“…One may wonder whether the linear relation between q and β can be explained from first principles. Our results are in line with the combination of 1) q increasing linearly with decreasing density slope γ found for the radial VDF by (at radii where the density profile has a slope shallower than −2.5, roughly the virial radius) and 2) the wide-wing tangential VDF found by Hansen & Sparre (2012), which could be assimilated to a q-Gaussian with constant q > 1. Our linear q − β relation follows naturally from the linear β − γ relation at these radii , and our linear q − γ relation.…”

“…Moreover, simulations of both collapsing structures and cosmological haloes indicate that for both for the radial velocity and the tangential velocity distributions, the q parameter of non-Gaussianity is found to vary roughly linearly with the slope of the density profile for radii where the slopes are γ = d ln ρ/d ln r between -2.7 and -1 ). Finally, Hansen & Sparre (2012) demonstrate that the tangential VDF must scale, outside its wings, as [1 + v 2 /(3σ 2 v )] −5/2 at all radii. In fact, were the dynamical evolution of these systems just determined by two-body interactions, as is the case for ideal gases, i.e., if the two-body relaxation time were short, then the system would rapidly evolve to isotropic velocities in a short time scale, the distribution function would then depend solely on energy, f = f (E), and could be obtained from the density profile (Eddington 1916), and finally, the velocity modulus distribution function at radius r would then simply be fv(v|r) ∝ v 2 f (v 2 /2 + Φ(r)).…”

Exploring the consideration of university teachers’ basic psychological needs in the design of professional development initiatives

“…The v θ distribution also broadens as r approaches r s , but as r further increases, the distributions narrow again somewhat, especially the models with high β ∞ , which show a strong peak at large radii. It would be interesting to examine more closely whether these distributions are in agreement with those found in simulations Hansen & Sparre 2012).…”

Asymmetric velocity anisotropies in remnants of collisionless mergers

“…Several studies [39][40][41][42][43][44][45][46] have attempted to derive or characterise distribution functions of completely relaxed haloes from first principles. We have demonstrated that merger remnants have the merger history encoded in their velocity anisotropy profiles, and they are therefore not expected to follow simple distribution functions, where β is constant within spherical bins or along the isodensity contours.…”