Towards A Universal Analytical Model of Population III Star Formation: A Bridge Between Cosmological Scales and Protostars

We construct an analytical model of Population III star formation that connects the cosmological radiation background to sub-AU protostellar disk fragmentation, a dynamic range inaccessible to any single simulation. Our approach is based on combining separate models of the disparate relevant scales: from the cosmological environment to the host-halo scale, from the halo scale to the scale of the star-forming cloud, and from the cloud scale to the fragmenting, accreting protostellar disk. Individually and collectively, the models agree well with the predictions of state of the art simulations, while remaining computationally inexpensive and physically transparent. As an example of the applicability of the model, we study the effects of varying the Lyman-Werner flux on the Pop. III star formation efficiency. We show that depending on the halo properties and the strength of the dissociating radiation field, the halo-scale Pop. III star formation efficiency varies by more than two orders of magnitude from $\varepsilon_{\rm SFE,H} \approx 10^{-3}$ to $\varepsilon_{\rm SFE, H} \approx 0.5$. The abrupt transitions between hydrogen-deuteride cooling (in low virial temperature mini-halos subjected to low radiation backgrounds), molecular hydrogen cooling (at intermediate temperatures and radiation intensities), and atomic cooling (in higher temperature halos exposed to strong radiation fields) produces sharp features in the halo-scale star formation efficiency as a function of the halo properties. Meanwhile, at the scale of individual star-forming clouds, the star formation efficiency is $\varepsilon_{\rm SFE,c} \gtrsim 0.2$. That is, pristine gas in a halo is converted into unstable clouds at a wide range of efficiencies, and these unstable clouds are efficiently converted into Pop. III stars.