  We present late-time Hubble Space Telescope imaging of the fields of six
Swift GRBs lying at 5.0<z<9.5. Our data includes very deep observations of the
field of the most distant spectroscopically confirmed burst, GRB 090423, at
z=8.2. Using the precise positions afforded by their afterglows we can place
stringent limits on the luminosities of their host galaxies. In one case, that
of GRB 060522 at z=5.11, there is a marginal excess of flux close to the GRB
position which may be a detection of a host at a magnitude J(AB)=28.5. None of
the others are significantly detected meaning that all the hosts lie below
L\star at their respective redshifts, with star formation rates SFR<4Mo/yr in
all cases. Indeed, stacking the five fields with WFC3-IR data we conclude a
mean SFR<0.17Mo/yr per galaxy. These results support the proposition that the
bulk of star formation, and hence integrated UV luminosity, at high redshifts
arises in galaxies below the detection limits of deep-field observations.
Making the reasonable assumption that GRB rate is proportional to UV luminosity
at early times allows us to compare our limits with expectations based on
galaxy luminosity functions derived from the Hubble Ultra-Deep Field (HUDF) and
other deep fields. We infer that a luminosity function which is evolving
rapidly towards steeper faint-end slope (alpha) and decreasing characteristic
luminosity (L\star), as suggested by some other studies, is consistent with our
observations, whereas a non-evolving LF shape is ruled out at >90% confidence.
Although it is not yet possible to make stronger statements, in the future,
with larger samples and a fuller understanding of the conditions required for
GRB production, studies like this hold great potential for probing the nature
of star formation, the shape of the galaxy luminosity function, and the supply
of ionizing photons in the early universe.
