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We present an analytical description of the probability distribution function(PDF) of the smoothed three-dimensional matter density field for modifiedgravity and dark energy. Our approach, based on the principles of LargeDeviations Theory, is applicable to general extensions of the standard$\Lambda$CDM cosmology. We show that late-time changes to the law of gravityand background expansion can be included through Einstein-de Sitter sphericalcollapse dynamics combined with linear theory calculations and a calibrationmeasurement of the non-linear variance of the smoothed density field from asimple numerical simulation. In a comparison to $N$-body simulations for$f(R)$, DGP and evolving dark energy theories, we find percent level accuracyaround the peak of the distribution for predictions in the mildly non-linearregime. A Fisher forecast of an idealised experiment with a Euclid-like surveyvolume demonstrates the power of combining measurements of the 3D matter PDFwith the 3D matter power spectrum. This combination is shown to halve theuncertainty on parameters for an evolving dark energy model, relative to apower spectrum analysis on its own. The PDF is also found to substantiallyincrease the detection significance for small departures from GeneralRelativity, with improvements of up to six times compared to the power spectrumalone. This analysis is therefore very promising for future studies includingnon-Gaussian statistics, as it has the potential to alleviate the reliance ofthese analyses on expensive high resolution simulations and emulators.

The matter density PDF for modified gravity and dark energy with Large Deviations Theory