A halo conducive for SMS formation must grow to the atomic-cooling limit (2.2 × 107Mʘ at z = 15; orange dotted line) without hosting star formation or being chemically enriched from nearby galaxies. Haloes with masses below minimum mass Mmin,LW (dashed green line) are suppressed by an external LW radiation field. Above this mass, haloes with sufficient dynamical heating to suppress radiative cooling grow above a critical rate (equation (1)), shown for H2 number fractions fH2 of 10−4 (blue solid line), 10−5 (orange solid line) and 10−6 (green solid line). The simulated growth rates of the MMH (circles) and LWH (triangles) are above the 10−6 rate once the halo masses pass Mmin,LW. Above a halo mass of 8 × 106Mʘ (a virial temperature of 8,000 K at z = 15), dynamical heating will not suppress cooling because the atomic-radiative cooling rates are several orders of magnitude higher than the molecular ones. Both haloes grow rapidly to Mmin,LW, causing dynamical heating and preventing collapse for a sound-crossing time. The LWH grows from 8 × 106Mʘ to the atomic-cooling limit within a dynamical time of the central core. Both conditions set a critical growth rate (thick solid grey lines). All other atomic-cooling haloes (grey points) have similar growth rates between halo masses of Mmin,LW and 8 × 106Mʘ but far short of the critical growth rate. Nearly all of these haloes cool and form stars before reaching the atomic-cooling limit.