The matter density PDF for modified gravity and dark energy with Large
  Deviations Theory
  We present an analytical description of the probability distribution function
(PDF) of the smoothed three-dimensional matter density field for modified
gravity and dark energy. Our approach, based on the principles of Large
Deviations Theory, is applicable to general extensions of the standard
$\Lambda$CDM cosmology. We show that late-time changes to the law of gravity
and background expansion can be included through Einstein-de Sitter spherical
collapse dynamics combined with linear theory calculations and a calibration
measurement of the non-linear variance of the smoothed density field from a
simple numerical simulation. In a comparison to $N$-body simulations for
$f(R)$, DGP and evolving dark energy theories, we find percent level accuracy
around the peak of the distribution for predictions in the mildly non-linear
regime. A Fisher forecast of an idealised experiment with a Euclid-like survey
volume demonstrates the power of combining measurements of the 3D matter PDF
with the 3D matter power spectrum. This combination is shown to halve the
uncertainty on parameters for an evolving dark energy model, relative to a
power spectrum analysis on its own. The PDF is also found to substantially
increase the detection significance for small departures from General
Relativity, with improvements of up to six times compared to the power spectrum
alone. This analysis is therefore very promising for future studies including
non-Gaussian statistics, as it has the potential to alleviate the reliance of
these analyses on expensive high resolution simulations and emulators.
