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We derive, using the spherical collapse model, a generalized Layzer-Irvineequation which can be used to describe the gravitational collapse of cold darkmatter in a dark energy background. We show that the usual Layzer-Irvineequation is valid if the dark matter and the dark energy are minimally coupledto each other and the dark energy distribution is homogeneous, independently ofits equation of state. We compute the corrections to the standard Layzer-Irvineequation which arise in the presence of dark energy inhomogeneities. We showthat, in the case of a dark energy component with a constant equation of stateparameter consistent with the latest observational constraints, thesecorrections are expected to be small, even if the dark energy has a negligiblesound speed. However, we find that, in more general models, the impact of darkenergy perturbations on the dynamics of clusters of galaxies, which will beconstrained by ESA's Euclid mission with unprecedented precision, might besignificant.

Generalized Layzer-Irvine equation: the role of dark energy perturbations in cosmic structure formation