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.