  We report the findings of new exact analytical solutions to the cosmological
fluid equations, namely for the case where the initial conditions are
perturbatively close to a spherical top-hat profile. To do so we enable a fluid
description in a Lagrangian-coordinates approach, and prove the convergence of
the Taylor-series representation of the Lagrangian displacement field until the
time of collapse ("shell-crossing"). This allows the determination of the time
for quasi-spherical collapse, which is shown to happen generically earlier than
in the spherical case. For pedagogical reasons, calculations are first given
for a spatially flat universe that is only filled with a non-relativistic
component of cold dark matter (CDM). Then, the methodology is updated to a
$\Lambda$CDM Universe, with the inclusion of a cosmological constant
$\Lambda>0$.
