Dark Matter Halos: Velocity Anisotropy – Density Slope Relation

Dark matter (DM) halos formed in CDM cosmologies seem to be characterized by a power law phasespace density profile. The density of the DM halos is often fitted by the NFW profile but a better fit is provided by the Sersic fitting formula. These relations are empirically derived from cosmological simulations of structure formation but have not yet been explained on a first principle basis. Here we solve the Jeans equation under the assumption of a spherical DM halo in dynamical equilibrium, that obeys a power law phase space density and either the NFW-like or the Sersic density profile. We then calculate the velocity anisotropy, β(r), analytically. Our main result is that for the NFW-like profile the β − γ relation is not a linear one (where γ is the logarithmic derivative of the density ρ[r]). The shape of β(r) depends mostly on the ratio of the gravitational to kinetic energy within the NFW scale radius Rs. For the Sersic profile a linear β − γ relation is recovered, and in particular for the Sersic index of n ≈ 6.0 case the linear fit of Hansen & Moore is reproduced. Our main result is that the phase-space density power law, the Sersic density form and the linear β − γ dependence constitute a consistent set of relations which obey the spherical Jeans equation and as such provide the framework for the dynamical modeling of DM halos. Subject headings: cosmology: dark matter — galaxies: evolution — galaxies: formation — galaxies: halos — galaxies: kinematics and dynamics — galaxies: clusters