Evolution of spherical overdensities in holographic dark energy models

In this work, we investigate the spherical collapse model in flat Friedmann–Robertson–Walker (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 parameter w 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 behaviour of the HDE model and compare the results with standard Einstein–de Sitter and Λ cold dark matter (ΛCDM) models. We then calculate the evolution of two characterizing parameters in the spherical collapse model, i.e. the linear density threshold δc and the virial overdensity parameter Δvir. We show that in HDE cosmologies the growth factor g(a) and the linear overdensity parameter δc fall behind the values for a ΛCDM universe while the virial overdensity Δvir is larger in HDE models than in the ΛCDM 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 ΛCDM universe. Hence, the growth of structures starts earlier in HDE models than in ΛCDM 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 Δvir. We compare the predicted number of haloes in HDE cosmologies and find out that in general this value is smaller than for ΛCDM models at higher redshifts and we compare different mass function prescriptions. Finally, we compare the results of the HDE models with observations.