Title: Inferring the Distribution of the Ionising Photon Escape Fraction

Abstract: The escape fraction of ionising photons from galaxies ($f_\mathrm{esc}$) is a key parameter for understanding how intergalactic hydrogen became reionised, but it remains mostly unconstrained. Measurements have been limited to the average value in galaxy ensembles and handfuls of individual detections. To help understand which mechanisms govern ionising photon escape, here we infer the distribution of $f_\mathrm{esc}$. We develop a hierarchical Bayesian inference technique to estimate the population distribution of $f_\mathrm{esc}$ from the ratio of Lyman Continuum to non-ionising UV flux measured from broadband photometry. We apply it to a sample of 148 z ~ 3.5 star-forming galaxies from the VANDELS spectroscopic survey. We explore four physically motivated distributions: constant, log-normal, exponential and bimodal, recovering $\langle f_\mathrm{esc} \rangle \approx5\%$ for most models. We find the observations are best described by an exponential $f_\mathrm{esc}$ distribution with scale factor $\mu=0.05^{+0.01}_{-0.02}$. This indicates most galaxies in our sample exhibit very low escape fractions while predicting substantial ionising photon leakage for only a few galaxies, implying a range of optical depths in the ISM and/or time variability in ionising photon escape. We rule out a bimodal distribution at high significance, indicating that a purely bimodal model of ionising photon escape (due to very strong sightline and/or time variability) is not favoured. We compare our recovered exponential distribution with the SPHINX simulations and find that, while the simulation also predicts an exponential-like distribution, it significantly underpredicts our inferred mean. The distribution of $f_\mathrm{esc}$ can be a vital test for simulations in understanding ionising photon leakage and is important to consider to gain a complete picture of reionisation.