  Hydrogen in the Universe was (re)ionised between redshifts $z \approx 10$ and
$z \approx 6$. The nature of the sources of the ionising radiation is hotly
debated, with faint galaxies below current detection limits regarded as prime
candidates. Here we consider a scenario in which ionising photons escape
through channels punctured in the interstellar medium by outflows powered by
starbursts. We take account of the observation that strong outflows occur only
when the star formation density is sufficiently high, and estimate the
galaxy-averaged escape fraction as a function of redshift and luminosity from
the resolved star formation surface densities in the EAGLE cosmological
hydrodynamical simulation. We find that the fraction of ionising photons that
escape from galaxies increases rapidly with redshift, reaching values of 5-20
percent at $z > 6$, with the brighter galaxies having higher escape fractions.
Combining the dependence of escape fraction on luminosity and redshift with the
observed luminosity function, we demonstrate that galaxies emit enough ionising
photons to match the existing constraints on reionisation while also matching
the observed UV-background post-reionisation. Our findings suggest that
galaxies above the current Hubble Space Telescope detection limit emit half of
the ionising radiation required to reionise the Universe.
