Impact of massive neutrinos on the abundance of massive clusters

We study the spherical, top-hat collapse model for a mixed dark matter model including cold dark matter (CDM) and massive neutrinos of mass scales ranging from m(v) similar or equal to 0: 05 to a few 0.1 eV, the range of lower and upper bounds implied from the neutrino oscillation experiments and the cosmological constraints. To develop this model, we properly take into account relative differences between the density perturbation amplitudes of different components (radiation, baryon, CDM, and neutrinos) around the top-hat CDM overdensity region assuming the adiabatic initial conditions. Furthermore, we solve the linearized Boltzmann hierarchy equations to obtain time evolution of the linearized neutrino perturbations, yet including the effect of nonlinear gravitational potential due to the nonlinear CDM and baryon overdensities in the late stage. We find that the presence of massive neutrinos slows down the collapse of CDM (plus baryon) overdensity; however, the neutrinos cannot fully catch up with the nonlinear CDM perturbation due to its large free-streaming velocity for the ranges of neutrino masses and halo masses we consider. We find that, just like CDM models, the collapse time of CDM overdensity is well monitored by the linear theory extrapolated overdensity of CDM plus baryon perturbation, smoothed with a given halo mass scale, if taking into account the suppression effect of the massive neutrinos on the linear growth rate. Using these findings, we argue that the presence of massive neutrinos of mass scales 0.05 or 0.1 eV may cause a significant decrease in the abundance of massive halos compared to the model without the massive neutrinos, e.g., by 25% or factor 2, respectively, for halos with 10(15)M(circle dot) and at z = 1.