Dark matter halo's and self similarity

Abstract

This papers explores the self similar solutions of the Vlasov-Poisson system and their relation to the gravitational collapse of dynamically cold systems. Analytic solutions are derived for power law potential in one dimension, and extensions of these solutions in three dimensions are proposed. Next the self similarity of the collapse of cold dynamical systems is investigated numerically. The fold system in phase space is consistent with analytic self similar solutions, the solutions present all the proper self-similar scalings. An additional point is the appearance of an $x^{-(1/2)}$ law at the center of the system for initial conditions with power law index larger than $-(1/2)$. It is found that the first appearance of the $x^{-(1/2)}$ law corresponds to the formation of a singularity very close to the center. Finally the general properties of self similar multi dimensional solutions near equilibrium are investigated. Smooth and continuous self similar solutions have power law behavior at equilibrium. However cold initial conditions result in discontinuous phase space solutions, and the smoothed phase space density looses its auto similar properties. This problem is easily solved by observing that the probability distribution of the phase space density $P$ is identical except for scaling parameters to the probability distribution of the smoothed phase space density $P_S$. As a consequence $P_S$ inherit the self similar properties of $P$. This particular property is at the origin of the universal power law observed in numerical simulation for $\rho /{\sigma }^3$. The self similar properties of $P_S$ implies that other quantities should have also an universal power law behavior with predictable exponents. This hypothesis is tested using a numerical model of the phase space density of cold dark matter halo's, an excellent agreement is obtained.

Final version

References in corpus (8)

• The Universal Rotation Curve of Spiral Galaxies. II The Dark Matter Distribution out to the Virial Radius
• The fine-grained phase-space structure of Cold Dark Matter halos
• The Caustic Ring Model of the Milky Way Halo
• The distribution function of dark matter in massive haloes
• Self-Similar Solutions of Triaxial Dark Matter Halos
• Self-Similar Spherical Collapse with Tidal Torque
• Dark Matter Caustics
• Velocity Structure of Self-Similar Spherically Collapsed Halos

Cited by in corpus (5)

• Phase-space structure of protohalos: Vlasov versus Particle-Mesh
• Vlasov-Poisson in 1D for initially cold systems: post-collapse Lagrangian perturbation theory
• Phase-space structure of cold dark matter halos inside splashback: multi-stream flows and self-similar solution
• The gravitational force field of proto-pancakes
• Dark matter halo dynamics in 2D Vlasov Simulations: a self-similar approach