  In recent years, the rise in the number of Lyman Break Galaxies detected at
high redshifts z >= 6 has opened up the possibility of understanding early
galaxy formation physics in great detail. In particular, the faint-end slope
(alpha) of the Ultra-violet luminosity function (UV LF) of these galaxies is a
potential probe of feedback effects that suppress star formation in low mass
haloes. In this work, we propose a proof-of-concept calculation for
constraining the fluctuating UV background during reionization by constraining
alpha in different volumes of the Universe. Because of patchy reionization,
different volumes will experience different amount of photo-heating which
should lead to a scatter in the measured alpha. Our approach is based on a
simple model of the UV LF that is a scaled version of the halo mass function
combined with an exponential suppression in the galaxy luminosity at the
faint-end because of UV feedback. Although current data is not sufficient to
constrain alpha in different fields, we expect that, in the near future,
observations of the six lensed Hubble Frontier Fields with the James Webb Space
Telescope (JWST) will offer an ideal test of our concept.
