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

Evolution of spherical overdensities in holographic dark energy models