Studying the formation and evolution of galaxies at the earliest cosmic
times, and their role in reionization, requires the deepest imaging possible.
Ultra-deep surveys like the HUDF and HFF have pushed to mag \mAB$\,\sim\,$30,
revealing galaxies at the faint end of the LF to $z$$\,\sim\,$9$\,-\,$11 and
constraining their role in reionization. However, a key limitation of these
fields is their size, only a few arcminutes (less than a Mpc at these
redshifts), too small to probe large-scale environments or clustering
properties of these galaxies, crucial for advancing our understanding of
reionization. Achieving HUDF-quality depth over areas $\sim$100 times larger
becomes possible with a mission like the Wide Field Infrared Survey Telescope
(WFIRST), a 2.4-m telescope with similar optical properties to HST, with a
field of view of $\sim$1000 arcmin$^2$, $\sim$100$\times$ the area of the
HST/ACS HUDF.
This whitepaper motivates an Ultra-Deep Field survey with WFIRST, covering
$\sim$100$\,-\,$300$\times$ the area of the HUDF, or up to $\sim$1 deg$^2$, to
\mAB$\,\sim\,$30, potentially revealing thousands of galaxies and AGN at the
faint end of the LF, at or beyond $z$\,$\sim$\,9$\,-\,$10 in the epoch of
reionization, and tracing their LSS environments, dramatically increasing the
discovery potential at these redshifts.
(Note: This paper is a somewhat expanded version of one that was submitted as
input to the Astro2020 Decadal Survey, with this version including an Appendix
(which exceeded the Astro2020 page limits), describing how the science drivers
for a WFIRST Ultra Deep Field might map into a notional observing program,
including the filters used and exposure times needed to achieve these depths.)
