A Cosmological Kinetic Theory for the Evolution of Cold Dark Matter Halos with Substructure: Quasi-Linear Theory

We present a kinetic theory for the evolution of the phase-space distribution of dark matter particles in galaxy halos in the presence of a cosmological spectrum of fluctuations. This theory introduces a new way to model the formation and evolution of halos, which traditionally have been investigated by analytic gravitational infall models or numerical N-body methods. Unlike the collisionless Boltzmann equation, our kinetic equation contains nonzero terms on the right-hand side arising from stochastic fluctuations in the gravitational potential due to substructures in the dark matter mass distribution. Using statistics for constrained Gaussian random fields in standard cosmological models, we show that our kinetic equation to second-order in perturbation theory is of the Fokker-Planck form, with one scattering term representing drift and the other representing diffusion in velocity-space. The drift is radial, and the drift and diffusion coefficients depend only on positions and not velocities; our relaxation process in the quasilinear regime is therefore different from the standard two-body relaxation. We provide explicit expressions relating these coefficients to the linear power spectrum of mass fluctuation and present results for the currently favored cold dark matter model with a nonzero cosmological constant. Solutions to this kinetic equation will provide a complete description of the cold dark matter spatial and velocity distributions for the average halo during the early phases of galaxy halo formation.