First star formation in extremely early epochs

First stars play crucial roles in the development of the Universe, influencing events like cosmic reionization and the chemcal enrichment of the intergalactic medium. While first stars are conventionally thought to form at around $z \sim 20$-30 in the standard $\Lambda$ cold dark matter ($\Lambda$CDM) cosmology, observational constraints on small-scale ($\\lt $Mpc) density fluctuations remain limited, possibly differing significantly from the scale-invariant fluctuations assumed in the $\Lambda $CDM model. Should this be the case, the formation of first stars could occur much earlier than typically predicted. In this study, we investigate the formation process of first stars in the extremely early epochs of $z \gtrsim 100$ in the post-recombination Universe. At such early times, the effects of the warm cosmic microwave background (CMB) become significant. We calculate the collapse of primordial star-forming clouds using a one-zone thermo-chemical model that accounts for CMB influences on radiative heating, Compton cooling, and photodissociation reactions. We found that the impact of the CMB on the evolution is limited at $z \lesssim 100 $, with the temperature evolution closely resembling the conventional model. However, within the range $100 \lesssim z \lesssim 400$, the formation of H$_{2}$ via the H$^{-}$ channel is impeded by H$^{-}$ photo-detachment induced by the CMB, leading to higher temperatures compared to standard thermal evolution. Consequently, first stars with masses exceeding $1000\, {M}_{\odot }$ can emerge at $z \gtrsim 100$. Furthermore, at $z \gtrsim 500$, the temperature evolution becomes nearly isothermal at several thousand Kelvin solely due to atomic cooling, as H$_{2}$ formation is entirely suppressed, including the less-efficient H$_2^{+}$ channel, which is blocked by H$_2^{+}$ photodissociation. In such cases, supermassive stars with masses around $\sim 10^{5}\, {M}_{\odot }$ are expected to form solely via atomic cooling. These findings emphasize the significant variation in the typical mass of the first stars depending on the epoch of formation.