The influence of considering a generalized dark matter (GDM) model, which allows for a non-pressure-less dark matter and a non-vanishing sound speed in the non-linear spherical collapse model is discussed for the Einstein-de Sitter-like (EdSGDM) and $Lambda$GDM models. By assuming that the vacuum component responsible for the accelerated expansion of the Universe is not clustering and therefore behaving similarly to the cosmological constant $Lambda$, we show how the change in the GDM characteristic parameters affects the linear density threshold for collapse of the non-relativistic component ($delta_{rm c}$) and its virial overdensity ($Delta_{rm V}$). We found that the generalized dark matter equation of state parameter $w_{rm gdm}$ is responsible for lower values of the linear overdensity parameter as compared to the standard spherical collapse model and that this effect is much stronger than the one induced by a change in the generalized dark matter sound speed $c^2_{rm s, gdm}$. We also found that the virial overdensity is only slightly affected and mostly sensitive to the generalized dark matter equation of state parameter $w_{rm gdm}$. These effects could be relatively enhanced for lower values of the matter density. Finally, we found that the effects of the additional physics on $delta_{rm c}$ and $Delta_{rm V}$, when translated to non-linear observables such as the halo mass function, induce an overall deviation of about 40% with respect to the standard $Lambda$CDM model at late times for high mass objects. However, within the current linear constraints for $c^2_{rm s, gdm}$ and $w_{rm gdm}$, we found that these changes are the consequence of properly taking into account the correct linear matter power spectrum for the GDM model while the effects coming from modifications in the spherical collapse model remain negligible.
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