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We compute the one-point PDF of an initially Gaussian dark matter densityfield using spherical collapse (SC). We compare the results to other formsavailable in the literature and also compare the PDFs in the $\Lambda$CDM modelwith an early dark energy (EDE) model. We find that the skewed log-normaldistribution provides the best fit to the non-linear PDF from SC for bothcosmologies, from $a=0.1$ to 1 and for scales characterized by the comovingwidth of the Gaussian: $\sigma_G = 0.5, 1, 2$. To elucidate the effect ofcosmology, we examine the linear and non-linear growth rates through testcases. For overdensities, when the two models have the same initial densitycontrast, the differences due to cosmology are amplified in the non-linearregime, whereas, if the two models have the same linear density contrast today,then the differences in cosmology are damped in the non-linear regime. Thisbehaviour is in contrast with voids, where the non-linear growth becomes`self-regulatory' and is less sensitive to cosmology and initial conditions. Tocompare the PDFs, we examine the difference of the PDFs and evolution of thewidth of the PDF. The trends with scale and redshift are as expected. Atertiary aim of this paper was to check if the fitting form for the non-lineardensity-velocity divergence relation, derived for constant equation of state($w$) models by Nadkarni-Ghosh holds for the EDE model. We find that it doeswith an accuracy of 4\%, thus increasing its range of validity.

One-point probability distribution function from spherical collapse: Early Dark Energy (EDE) vs. $\Lambda$CDM