One-point probability distribution function from spherical collapse:
  Early Dark Energy (EDE) vs. $\Lambda$CDM
  We compute the one-point PDF of an initially Gaussian dark matter density
field using spherical collapse (SC). We compare the results to other forms
available in the literature and also compare the PDFs in the $\Lambda$CDM model
with an early dark energy (EDE) model. We find that the skewed log-normal
distribution provides the best fit to the non-linear PDF from SC for both
cosmologies, from $a=0.1$ to 1 and for scales characterized by the comoving
width of the Gaussian: $\sigma_G = 0.5, 1, 2$. To elucidate the effect of
cosmology, we examine the linear and non-linear growth rates through test
cases. For overdensities, when the two models have the same initial density
contrast, the differences due to cosmology are amplified in the non-linear
regime, whereas, if the two models have the same linear density contrast today,
then the differences in cosmology are damped in the non-linear regime. This
behaviour is in contrast with voids, where the non-linear growth becomes
`self-regulatory' and is less sensitive to cosmology and initial conditions. To
compare the PDFs, we examine the difference of the PDFs and evolution of the
width of the PDF. The trends with scale and redshift are as expected. A
tertiary aim of this paper was to check if the fitting form for the non-linear
density-velocity divergence relation, derived for constant equation of state
($w$) models by Nadkarni-Ghosh holds for the EDE model. We find that it does
with an accuracy of 4\%, thus increasing its range of validity.
