  We derive a semi-analytical extension of the spherical collapse model of
structure formation that takes account of the effects of deviations from
spherical symmetry and shell crossing which are important in the non-linear
regime. Our model is designed so that it predicts a relation between the
peculiar velocity and density contrast that agrees with the results of N-body
simulations in the region where such a comparison can sensibly be made. Prior
to turnaround, when the unmodified spherical collapse model is expect to be a
good approximation, the predictions of the two models coincide almost exactly.
The effects of a late time dominating dark energy component are also taken into
account. The improved spherical collapse model is a useful tool when one
requires a good approximation not just to the evolution of the density contrast
but also its trajectory. Moreover, the analytical fitting formulae presented is
simple enough to be used anywhere where the standard spherical collapse might
be used but with the advantage that it includes a realistic model of the
effects of virialisation.
