the linear density field within a cubic volume and then evolving the density field using the Zel’dovich approximation [@Zeldovich:1970p2023] before smoothing on to a lower resolution grid. Following this, [<span style="font-variant:small-caps;">21CMFAST</span>]{} estimates the ionization field by comparing the time-integrated number of ionizing photons to the number of baryons within spherical regions of decreasing radius, $R$, following the excursion-set approach described in @Furlanetto:2004p123. A cell at coordinates $(\boldsymbol{x},z)$ within the simulation volume is then tagged as fully ionized if, $$\begin{aligned}
\label{eq:ioncrit}
\zeta f_{\rm coll}(\boldsymbol{x},z,R,\bar{M}_{\rm min}) \geq 1,\end{aligned}$$ where $f_{\rm coll}(\boldsymbol{x},z,R,\bar{M}_{\rm min})$ is the fraction of collapsed matter within a spherical radius $R$ residing within haloes larger than $\bar{M}_{\rm min}$ [@Press:1974p2031; @Bond:1991p111; @Lacey:1993p115; @Sheth:1999p2053] and $\zeta$ is an ionizing efficiency describing the conversion of mass into ionizing photons (see Section \[sec:Zeta\]). Partial ionization is additionally included for cells not fully ionized by setting the ionized fraction of a cell smoothed at the minimum smoothing scale, $R_{\rm cell}$ to $\zeta f_{\rm coll}(\boldsymbol{x},z,R_{\rm cell},\bar{M}_{\rm min})$.

It is common to characterize the [EoR]{} with just three parameters: (i) the ionizing efficiency of high-$z$ galaxies; (ii) the mean free path of ionizing photons; (iii) the minimum virial temperature hosting star-forming galaxies. These parameters are somewhat empirical 