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While galaxies at $6 \lesssim z \lesssim 10$ are believed to dominate theepoch of cosmic reionization, the escape fraction of ionizing flux$f_\mathrm{esc}$ and the photon production rate $\dot n_\gamma$ from thesegalaxies must vary with redshift to simultaneously match CMB and low-redshiftobservations. We constrain $f_\mathrm{esc}(z)$ and $\dot n_\gamma(z)$ withPlanck 2015 measurements of the Thomson optical depth $\tau$, recent lowmultipole E-mode polarization measurements from Planck 2016, SDSS BAO data, and$3 \lesssim z \lesssim 10$ galaxy observations. We compare different galaxyluminosity functions that are calibrated to HST observations, using bothparametric and non-parametric statistical methods that marginalize over theeffective 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 $f_\mathrm{esc} \lesssim0.5$ at $z=8$ with slope $\beta \gtrsim 2.0$ at $68\%$ confidence with littledependence on the galaxy luminosity function or data, although there isnon-negligible probability for no redshift evolution $\beta \sim 0$ or smallescape 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, amodel-independent reconstruction of $\dot n_\gamma(z)$ predicts a suppressedescaped photon production rate at $z=9$ for the latest Planck data compared tothe other models, implying a quicker period of reionization. We find evidencefor redshift evolution in the limiting magnitude of the galaxy luminosityfunction for empirical models of the galaxy luminosity function.

Reconstructing the redshift evolution of escaped ionizing flux from early galaxies with Planck and HST observations