We use cosmological simulations to assess how the explosion of the first stars in supernovae (SNe) influences early cosmic history. Specifically, we investigate the impact by SNe on the host systems for Population~III (Pop~III) star formation and explore its dependence on halo environment and Pop~III progenitor mass. We then trace the evolution of the enriched gas until conditions are met to trigger second-generation star formation. To this extent, we quantify the recovery timescale, which measures the time delay between a Pop~III SN explosion and the appearance of cold, dense gas, out of which second-generation stars can form. We find that this timescale is highly sensitive to the Pop~III progenitor mass, and less so to the halo environment. For more massive progenitors, including those exploding in pair instability SNe, second-generation star formation is delayed significantly, for up to a Hubble time. The dependence of the recovery time on the mass of the SN progenitor is mainly due to the ionizing impact of the progenitor star. Photoionization heating increases the gas pressure and initiates a hydrodynamical response that reduces the central gas density, an effect that is stronger in more massive. The gas around lower mass Pop~III stars remains denser and hence the SN remnants cool more rapidly, facilitating the subsequent re-condensation of the gas and formation of a second generation of stars. In most cases, the second-generation stars are already metal-enriched to ~2-5 X 10^{-4}zsun, thus belonging to Population~II. The recovery timescale is a key quantity governing the nature of the first galaxies, able to host low-mass, long-lived stellar systems. These in turn are the target of future deep-field campaigns with the James Webb Space Telescope.
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