  If the dark matter (DM) were composed of axions, then structure formation in
the Universe would be suppressed below the axion Jeans scale. Using an analytic
model for the halo mass function of a mixed DM model with axions and cold dark
matter, combined with the abundance-matching technique, we construct the
UV-luminosity function. Axions suppress high-$z$ galaxy formation and the
UV-luminosity function is truncated at a faintest limiting magnitude. From the
UV-luminosity function, we predict the reionization history of the universe and
find that axion DM causes reionization to occur at lower redshift. We search
for evidence of axions using the Hubble Ultra Deep Field UV-luminosity function
in the redshift range $z=6$-$10$, and the optical depth to reionization,
$\tau$, as measured from cosmic microwave background polarization. All probes
we consider consistently exclude $m_a\lesssim 10^{-23}\text{ eV}$ from
contributing more than half of the DM, with our strongest constraint ruling
this model out at more than $8\sigma$ significance. In conservative models of
reionization a dominant component of DM with $m_a=10^{-22}\text{ eV}$ is in
$3\sigma$ tension with the measured value of $\tau$, putting pressure on an
axion solution to the cusp-core problem. Tension is reduced to $2\sigma$ for
the axion contributing only half of the DM. A future measurement of the
UV-luminosity function in the range $z=10$-$13$ by JWST would provide further
evidence for or against $m_a=10^{-22}\text{ eV}$. Probing still higher masses
of $m_a=10^{-21}\text{ eV}$ will be possible using future measurements of the
kinetic Sunyaev-Zel'dovich effect by Advanced ACTPol to constrain the time and
duration of reionization.
