  We present a new determination of the UV galaxy luminosity function (LF) at
redshift z ~ 7 and z ~ 8, and a first estimate at z ~ 9. An accurate
determination of the form and evolution of the LF at high z is crucial for
improving our knowledge of early galaxy evolution and cosmic reionization. Our
analysis exploits fully the new, deepest WFC3/IR imaging from our HST UDF12
campaign, and includes a new, consistent analysis of all appropriate,
shallower/wider-area HST data. Our new measurement of the evolving LF at z ~
7-8 is based on a final catalogue of ~600 galaxies, and involves a step-wise
maximum likelihood determination based on the redshift probability distribution
for each object; this makes full use of the 11-band imaging now available in
the HUDF, including the new UDF12 F140W data, and the deep Spitzer IRAC
imaging. The final result is a determination of the z ~ 7 LF extending down to
M_UV = -16.75, and the z ~ 8 LF down to M_UV = -17.00. Fitting a Schechter
function, we find M* = -19.90 (+0.23/-0.28), log phi* = -2.96 (+0.18/-0.23),
and a faint-end slope alpha=-1.90 (+0.14/-0.15) at z~7, and M* = -20.12
(+0.37/-0.48), log phi* = -3.35 (+0.28/-0.47), alpha=-2.02 (+0.22/-0.23) at
z~8. These results strengthen suggestions that the evolution at z > 7 is more
akin to `density evolution' than the apparent `luminosity evolution' seen at z
~ 5-7. We also provide the first meaningful information on the LF at z ~ 9,
explore alternative extrapolations to higher z, and consider the implications
for the evolution of UV luminosity density. Finally, we provide catalogues
(including z_phot, M_UV and all photometry) for the 100 most robust z~6.5-11.9
galaxies in the HUDF used in this analysis. We discuss our results in the
context of earlier work and the results of an independent analysis of the UDF12
data based on colour-colour selection (Schenker et al. 2013).
