  We study the spherical collapse model for several dark energy scenarios using
the fully nonlinear differential equation for the evolution of the density
contrast within homogeneous spherical overdensities derived from Newtonian
hydrodynamics. While mathematically equivalent to the more common approach
based on the differential equation for the radius of the perturbation, this
approach has substantial conceptual as well as numerical advantages. Among the
most important are that no singularities at early times appear, which avoids
numerical problems in particular in applications to cosmologies with dynamical
and early dark energy, and that the assumption of time-reversal symmetry can
easily be dropped where it is not strictly satisfied. We use this approach to
derive the two parameters characterising the spherical-collapse model, i.e.~the
linear density threshold for collapse $\delta_\mathrm{c}$ and the virial
overdensity $\Delta_\mathrm{V}$, for a broad variety of dark-energy models and
to reconsider these parameters in cosmologies with early dark energy. We find
that, independently of the model under investigation, $\delta_\mathrm{c}$ and
$\Delta_\mathrm{V}$ are always very close to the values obtained for the
standard $\Lambda$CDM model, arguing that the abundance of and the mean density
within non-linear structures are quite insensitive to the differences between
dark-energy cosmologies. Regarding early dark energy, we thus arrive at a
different conclusion than some earlier papers, including one from our group,
and we explain why.
