Reconstructing the redshift evolution of escaped ionizing flux from
  early galaxies with Planck and HST observations
  While galaxies at $6 \lesssim z \lesssim 10$ are believed to dominate the
epoch of cosmic reionization, the escape fraction of ionizing flux
$f_\mathrm{esc}$ and the photon production rate $\dot n_\gamma$ from these
galaxies must vary with redshift to simultaneously match CMB and low-redshift
observations. We constrain $f_\mathrm{esc}(z)$ and $\dot n_\gamma(z)$ with
Planck 2015 measurements of the Thomson optical depth $\tau$, recent low
multipole E-mode polarization measurements from Planck 2016, SDSS BAO data, and
$3 \lesssim z \lesssim 10$ galaxy observations. We compare different galaxy
luminosity functions that are calibrated to HST observations, using both
parametric and non-parametric statistical methods that marginalize over the
effective clumping factor $C_\mathrm{HII}$, the LyC production efficiency
$\xi_\mathrm{ion}$, and the time-evolution of the UV limiting magnitude
$dM_\mathrm{SF}/dz$. Using a power-law model, we find $f_\mathrm{esc} \lesssim
0.5$ at $z=8$ with slope $\beta \gtrsim 2.0$ at $68\%$ confidence with little
dependence on the galaxy luminosity function or data, although there is
non-negligible probability for no redshift evolution $\beta \sim 0$ or small
escape fraction $f_\mathrm{esc} \sim 10^{-2}$. A non-parametric form for
$f_\mathrm{esc}(z)$ evolves significantly with redshift, yielding
$f_\mathrm{esc} \sim 0.2, 0.3, 0.6$ at $z=6,9,12$, respectively. However, a
model-independent reconstruction of $\dot n_\gamma(z)$ predicts a suppressed
escaped photon production rate at $z=9$ for the latest Planck data compared to
the other models, implying a quicker period of reionization. We find evidence
for redshift evolution in the limiting magnitude of the galaxy luminosity
function for empirical models of the galaxy luminosity function.
