  As part of the Panchromatic Hubble Andromeda Treasury (PHAT) multi-cycle
program, we observed a 12' \times 6.5' area of the bulge of M31 with the
WFC3/UVIS filters F275W and F336W. From these data we have assembled a sample
of \sim4000 UV-bright, old stars, vastly larger than previously available. We
use updated Padova stellar evolutionary tracks to classify these hot stars into
three classes: Post-AGB stars (P-AGB), Post-Early AGB (PE-AGB) stars and
AGB-manqu\'e stars. P-AGB stars are the end result of the asymptotic giant
branch (AGB) phase and are expected in a wide range of stellar populations,
whereas PE-AGB and AGB-manqu\'e (together referred to as the hot
post-horizontal branch; HP-HB) stars are the result of insufficient envelope
masses to allow a full AGB phase, and are expected to be particularly prominent
at high helium or {\alpha} abundances when the mass loss on the RGB is high.
Our data support previous claims that most UV-bright sources in the bulge are
likely hot (extreme) horizontal branch stars (EHB) and their progeny. We
construct the first radial profiles of these stellar populations, and show that
they are highly centrally concentrated, even more so than the integrated UV or
optical light. However, we find that this UV-bright population does not
dominate the total UV luminosity at any radius, as we are detecting only the
progeny of the EHB stars that are the likely source of the UVX. We calculate
that only a few percent of MS stars in the central bulge can have gone through
the HP-HB phase and that this percentage decreases strongly with distance from
the center. We also find that the surface density of hot UV-bright stars has
the same radial variation as that of low-mass X-ray binaries. We discuss age,
metallicity, and abundance variations as possible explanations for the observed
radial variation in the UV-bright population.
