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.