Effect of supersonic relative motion between baryons and dark matter on
  collapsed objects
  Great attention is given to the first star formation and the epoch of
reionization as main targets of planned large radio interferometries (e.g.
Square Kilometre Array). Recently, it is claimed that the supersonic relative
velocity between baryons and cold dark matter can suppress the abundance of
first stars and impact the cosmological reionization process. Therefore, in
order to compare observed results with theoretical predictions it is important
to examine the effect of the supersonic relative motion on the small-scale
structure formation. In this paper, we investigate this effect on the nonlinear
structure formation in the context of the spherical collapse model in order to
understand the fundamental physics in a simple configuration. We show the
evolution of the dark matter sphere with the relative velocity by both using
N-body simulations and numerically calculating the equation of motion for the
dark matter mass shell. The effects of the relative motion in the spherical
collapse model appear as the delay of the collapse time of dark matter halos
and the decrease of the baryon mass fraction within the dark matter sphere.
Based on these results, we provide the fitting formula of the critical density
contrast for collapses with the relative motion effect and calculate the mass
function of dark matter halos in the Press-Schechter formalism. As a result,
the relative velocity decreases the abundance of dark matter halos whose mass
is smaller than $10^8~M_\odot/h$.
