Spherical collapse in the extended quintessence cosmological models

Abstract: We use the spherical collapse method to investigate the non-linear density perturbations of pressureless matter in the cosmological models with the extended quintessence as dark energy in the metric and Palatini formalisms. We find that for both formalisms, when the coupling constant is negative, the deviation from the $\Lambda$CDM model is the least according to the evolutionary curves of the linear density contrast $\delta_{c}$ and virial overdensity $\Delta_{v}$, and it is less than one percent. And this indicates that, in the extended quintessence cosmological models in which the coupling constant is negative, all quantities dependent on $\delta_{c}$ or $\Delta_{v}$ are essentially unaffected if the linear density contrast or the virial overdensity of the $\Lambda$CDM model is used as an approximation. Moreover, we find that the differences between different formalisms are very small in terms of structure formation, and thus can not be used to distinguish the metric and Palatini formalisms.