  We examine the impact of two alternative dark matter models that possess
distinct non-zero equations of state, one constant and the other
time-dependent, on the nonlinear regime using the spherical collapse approach.
Specifically, we compare these models to standard cold dark matter (CDM) by
analyzing their influence on the linear density threshold for nonrelativistic
component collapse and virial overdensity. Additionally, we explore the number
count of collapsed objects, or dark matter halos, which is analogous to the
number count of galaxy clusters. Finally, in light of recent discoveries by the
James Webb Space Telescope (JWST), which indicate the potential for more
efficient early galaxy formation at higher redshifts, we have been
investigating how alternative dark matter assumptions can enhance structure
formation efficiency during the early times.
