Evolution of spherical overdensities in holographic dark energy models
  In this work we investigate the spherical collapse model in flat FRW dark
energy universes. We consider the Holographic Dark Energy (HDE) model as a
dynamical dark energy scenario with a slowly time-varying equation-of-state
(EoS) parameter $w_{\rm de}$ in order to evaluate the effects of the dark
energy component on structure formation in the universe. We first calculate the
evolution of density perturbations in the linear regime for both phantom and
quintessence behavior of the HDE model and compare the results with standard
Einstein-de Sitter (EdS) and $\Lambda$CDM models. We then calculate the
evolution of two characterizing parameters in the spherical collapse model,
i.e., the linear density threshold $\delta_{\rm c}$ and the virial overdensity
parameter $\Delta_{\rm vir}$. We show that in HDE cosmologies the growth factor
$g(a)$ and the linear overdensity parameter $\delta_{\rm c}$ fall behind the
values for a $\Lambda$CDM universe while the virial overdensity $\Delta_{\rm
vir}$ is larger in HDE models than in the $\Lambda$CDM model. We also show that
the ratio between the radius of the spherical perturbations at the
virialization and turn-around time is smaller in HDE cosmologies than that
predicted in a $\Lambda$CDM universe. Hence the growth of structures starts
earlier in HDE models than in $\Lambda$CDM cosmologies and more concentrated
objects can form in this case. It has been shown that the non-vanishing surface
pressure leads to smaller virial radius and larger virial overdensity
$\Delta_{\rm vir}$. We compare the predicted number of halos in HDE cosmologies
and find out that in general this value is smaller than for $\Lambda$CDM models
at higher redshifts and we compare different mass function prescriptions.
Finally, we compare the results of the HDE models with observations.
