Structure formation provides a strong test of any cosmic acceleration model
because a successful dark energy model must not inhibit {\black or overpredict}
the development of observed large-scale structures. Traditional approaches to
studies of structure formation in the presence of dark energy or a modified
gravity implement a modified Press-Schechter formalism, which relates the
linear overdensities to the abundance of dark matter haloes it at the same
time. We critically examine the universality of the Press-Schechter formalism
for different cosmologies, and show that the halo abundance is best correlated
with spherical linear overdensity at 94% of collapse (or observation) time. We
then extend this argument to ellipsoidal collapse (which decreases the
fractional time of best correlation for small haloes, and show that our results
agree with deviations from modified Press-Schechter formalism seen in simulated
mass functions. This provides a novel universal prescription to measure linear
density evolution, based on current and future observations of cluster (or dark
matter) halo mass function. In particular, even observations of cluster
abundance in a single epoch will constrain the entire history of linear growth
of cosmological of perturbations.
