and in most cases are averaged over redshift and/or halo mass dependences. Nevertheless, they offer the flexibility to describe a wide variety of EoR signals, and can have a straightforward physical interpretation. Below, we summarize each of these three parameters and their physical origins.

### Ionizing efficiency, $\zeta$ {#sec:Zeta}

The ionizing efficiency of high-$z$ galaxies (equation \[eq:ioncrit\]) can be expressed as $$\begin{aligned}
 \label{eq:Zeta}
 \zeta = 30\left(\frac{f_{\rm esc}}{0.3}\right)\left(\frac{f_{\star}}{0.05}\right)\left(\frac{N_{\gamma}}{4000}\right)
 \left(\frac{2}{1+n_{\rm rec}}\right)
 \end{aligned}$$ where, $f_{\rm esc}$ is the fraction of ionizing photons escaping into the IGM, $f_{\star}$ is the fraction of galactic gas in stars, $N_{\gamma}$ is the number of ionizing photons produced per baryon in stars and $n_{\rm rec}$ is the typical number of times a hydrogen atom recombines. While our reionization model only depends on the product of equation (\[eq:Zeta\]), we provide reasonable estimates for the individual factors. The choice of $N_{\gamma}\approx4000$ is expected from Population II stars [e.g. @Barkana:2005p1934], while $f_{\star}$ and $f_{\rm esc}$ are extremely uncertain in high-$z$ galaxies [e.g. @Gnedin:2008p1935; @Wise:2009p1968; @Ferrara:2013p1248], though our choices are consistent with observations of galaxy luminosity functions [e.g. @Robertson:2013p2123; @Cooke:2014p2861; @Dayal:2014p2876]. Finally, $n_{\rm rec} \sim 1$ is consistent with the models of [@Sobacchi:2014p1157] which result in a ‘photon-starved’ end to reionization, consistent 