The one-point probability distribution function ( PDF ) of the matter density field in the universe is a fundamental property that plays an essential role in cosmology for estimates such as gravitational weak lensing , non-linear clustering , massive production of mock galaxy catalogs , and testing predictions of cosmological models .
Here we make a comprehensive analysis of the dark matter PDF using a suite of \sim 7000 N -body simulations that covers a wide range of numerical and cosmological parameters .
We find that the PDF has a simple shape : it declines with density as a power-law P \propto \rho ^ { -2 } , which is exponentially suppressed on both small and large densities .
The proposed double-exponential approximation provides an accurate fit to all our N -body results for small filtering scales R < 5 \mbox { $h ^ { -1 } $Mpc } with rms density fluctuations \sigma > 1 .
In combination with the spherical infall model that works well for small fluctuations \sigma < 1 , the PDF is now approximated with just few percent errors over the range of twelve orders of magnitude – a remarkable example of precision cosmology .
We find that at \sim 5 - 10 \% level the PDF explicitly depends on redshift ( at fixed \sigma ) and on cosmological density parameter \Omega _ { m } .
We test different existing analytical approximations and find that the often used log-normal approximation is always 3-5 times less accurate than either the double-exponential approximation or the spherical infall model .