Beyond Jcrit: a critical curve for suppression of H2-cooling in protogalaxies

Suppression of H2-cooling in early protogalaxies has important implications for the formation of supermassive black holes seeds, the first generation of stars, and the epoch of reionization. This suppression can occur via photodissociation of H2 (by ultraviolet Lyman-Werner [LW] photons) or by photodetachment of H-, a precursor in H2 formation (by infrared [IR] photons). Previous studies have typically adopted idealised spectra, with a blackbody or a power-law shape, in modeling the chemistry of metal-free protogalaxies, and utilised a single parameter, the critical UV flux, or Jcrit, to determine whether H2-cooling is prevented. This can be misleading, and that independent of the spectral shape, there is a critical curve in the (kLW , kH^- ) plane, where kLW and kH^- are the H2-dissocation rates by LW and IR photons, which determines whether a protogalaxy can cool below ~1000 Kelvin. We use a one-zone model to follow the chemical and thermal evolution of gravitationally collapsing protogalactic gas, to compute this critical curve, and provide an accurate analytical fit for it. We improve on previous works by considering a variety of more realistic Pop III or Pop II-type spectra from population synthesis models and perform fully frequency-dependent calculations of the H2-photodissociation rates for each spectrum. We compute the ratio kLW/kH^- for each spectrum, as well as the minimum stellar mass M_star, for various IMFs and metallicities, required to prevent cooling in a neighboring halo a distance d away. We provide critical M_star/d2 values for suppression of H2-cooling, with analytic fits, which can be used in future studies.