Dark Matter Halos: Velocity Anisotropy -- Density Slope Relation

Dark matter (DM) halos formed in CDM cosmologies seem to be characterized by a power law phase-space 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, beta(r), analytically. Our main result is that for the NFW-like profile the beta - gamma relation is not a linear one (where gamma is the logarithmic derivative of the density rho[r]). The shape of beta(r) depends mostly on the ratio of the gravitational to kinetic energy within the NFW scale radius R_s. For the Sersic profile a linear beta - gamma 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 beta - gamma 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.