The critical average ultraviolet flux density per unit volume required to maintain the ionization balance of the Universe at z = 7, assuming C_H = 3, f_esc = 0.2 and T = 10⁴ K (see Section 5.1) and the three fiducial star formation histories assumed in Fig. 12 and Section 5.1. The dashed line indicates the critical luminosity density of Madau et al. (1999), assuming L_UV = 8 × 10²⁷ (SFR/M_⊙ yr⁻¹) erg s⁻¹ Hz⁻¹ and a Salpeter IMF (Madau et al 1998).