[e.g. @OMeara:2007p3360; @Prochaska:2009p3339; @Songaila:2010p3348; @McQuinn:2011p3293]. Motivated by recent sub-grid recombination models [@Sobacchi:2014p1157], here we explore the range $R_{\rm mfp}\in[5, 20]$ cMpc[^8]. While this range is relatively narrow, we point out that this parameter only becomes important once the size of the ionized regions become larger than $R_{\rm mfp}$ [e.g. @McQuinn:2007p1665; @Alvarez:2012p1930; @Mesinger:2012p1131]. Expanding the allowed range of $R_{\rm mfp}$ we consider in this work would therefore not greatly modify our conclusions.

### Minimum virial temperature of star-forming haloes, $T^{\rm Feed}_{\rm vir}$ {#sec:Mthresh}

Throughout, we choose to define the minimum threshold for a halo hosting a star-forming galaxy to be in terms of its virial temperature, which regulates processes important for star formation: gas accretion, cooling and retainment of supernovae outflows. The virial temperature is related to the halo mass via, [e.g. @Barkana:2001p1634] $$\begin{aligned}
M_{\rm min} &=& 10^{8} h^{-1} \left(\frac{\mu}{0.6}\right)^{-3/2}\left(\frac{\Omega_{\rm m}}{\Omega^{z}_{\rm m}}
\frac{\Delta_{\rm c}}{18\upi^{2}}\right)^{-1/2} \nonumber \\
& & \times \left(\frac{T_{\rm vir}}{1.98\times10^{4}~{\rm K}}\right)^{3/2}\left(\frac{1+z}{10}\right)^{-3/2}M_{\sun},\end{aligned}$$ where $\mu$ is the mean molecular weight, $\Omega^{z}_{\rm m} = \Omega_{\rm m}(1+z)^{3}/[\Omega_{\rm m}(1+z)^{3} + 
\Omega_{\Lambda}]$, and $\Delta_{\rm c} = 18\upi^{2} + 82d - 39d^{2}$ where $d = \Omega^{z}_{\rm m}-1$. Typically, $T_{\rm vir}\approx10^{4}$ K has been adopted in the literature as the minimum temperature when efficient atomic cooling occurs, corresponding 