Towards a universal analytical model for Population III star formation: interplay between feedback and fragmentation

JWST has brought us new insights into Cosmic Dawn with tentative detection of the unique signatures of metal-free Population III (Pop III) stars, such as strong He II emission, extremely blue ultraviolet spectrum, and enhanced nitrogen abundance. Self-consistent theoretical predictions of the formation rates, sites, and masses of Pop III stars are crucial for interpreting the observations, but are challenging due to complex physical processes operating over the large range of length-scales involved. One solution is to combine analytical models for the small-scale star formation process with cosmological simulations that capture the large-scale physics such as structure formation, radiation backgrounds, and baryon-dark matter streaming motion that regulate the conditions of Pop III star formation. We build an analytical model to predict the final masses of Pop III stars/clusters from the properties of star-forming clouds, based on the key results of small-scale star formation simulations and stellar evolution models. Our model for the first time considers the interplay between feedback and fragmentation and covers different modes of Pop III star formation ranging from ordinary small ($\sim\!{10{-}2000}\ \rm M_\odot$) clusters in molecular-cooling clouds to massive ($\gtrsim\!{10^{4}}\ \rm M_\odot$) clusters containing supermassive ($\sim\!{10^{4}{-}3}\times 10^{5}\ \rm M_\odot$) stars under violent collapse of atomic-cooling clouds with large gas accretion rates of $\gtrsim\!{0.1}\ \rm M_\odot \ yr^{-1}$. As an example, the model is applied to the Pop III star-forming clouds in the progenitors of typical haloes hosting high-z luminous quasars ($M_{\rm h}\sim 10^{12}\ \rm M_\odot$ at $z\sim 6$), which shows that formation of Pop III massive clusters is common ($\sim\!{20{-}70}{{\ \rm per\ cent}}$) in such biased ($\sim\!{4}\sigma$) regions, and the resulting heavy black hole seeds from supermassive stars can account for a significant fraction of observed luminous ($\gtrsim\!{10^{46}}\ \rm erg\ s^{-1}$) quasars at $z\sim 6$.