Halo Profile Evolution and Velocity Bias
  We propose a simple model that elucidates the generation of halo velocity
bias. The fluid equation approximation is often adopted in modelling the
evolution of the halo density field. In this approach, halos are often taken to
be point particles even though in reality they are finite-sized objects. In
this paper, we generalize the fluid equation approximation to halos to include
the finite extent of halos by taking into account the halo profile. We compute
the perturbation of the halo density and velocity field to second order and
find that the profile correction gives rise to $k^2$ correction terms in
Fourier space. These corrections are more important for velocity than for
density. In particular, the profile correction generates $k^2$ correction term
in the velocity bias and the correction terms do not decay away in the long
term limit, but it is not constant. We model the halo profile evolution using
the spherical collapse model. We also measure the evolution of proto-halo
profile at various redshifts numerically. We find that the spherical collapse
model gives a reasonable description of the numerical profile evolution. Static
halo profile is often adopted in modelling halos in theories such as the
excursion set theory. Our work highlights the importance of including the
profile evolution in the calculations.
