Photon-budget reionization threshold ($f_{\mathrm{coll}}=N_{\mathrm{req}}/N_{\gamma }$)

If each collapsed baryon supplies $N_{\gamma }$ effective ionizing photons and $N_{\mathrm{req}}$ photons per cosmic baryon are required, the threshold collapsed fraction is $f_{\mathrm{coll}}=N_{\mathrm{req}}/N_{\gamma }$. The Press-Schechter formalism gives $\mathrm{erfc}[{\delta }_{\mathrm{sc}}/(\sqrt{2}\sigma (M_{\min },z))]=f_{\mathrm{coll}}$. Escape losses and recombinations must either be incorporated in the effective yield/requirement or counted separately, not twice.