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 .