-1cm
While the final stages of the collapse appear to be somewhat chaotic, with protostellar fragmentation driven primarily by small-scale turbulence rather than the CR background, the typical mass of the sinks formed remains quite stable across all simulations at \(10-40{\,{\rm M}_{\odot}}\). This is very much in line with the results of previous simulations performed in absence of any radiative feedback \citep{Bromm2013}. As noted in Section \ref{sec:initial_collapse}, the thermodynamic behaviour of the gas displays a remarkable similarity as it approaches sink formation densities, regardless of \({{u}_{\rm \small CR}}\). This suggests that the influence of the CR background is restricted to lower densities, with little to no effect on the protostellar cores from which the first stars ultimately form.
The universality of this characteristic mass is further supported by Figure \ref{fig:Mbe}, where we show the average gas temperature as a function of number density, as well as the approximate fragmentation mass scale, as estimated by the Bonnor-Ebert mass \citep[e.g.,][]{StacyBromm2007}: \[M_{\rm \small BE} = 700{\,{\rm M}_{\odot}}\left(\frac{T}{200{\,{\rm K}}}\right)^{3/2} \left(\frac{n}{10^4{\,{\rm cm}^{-3}}} \right)^{-1/2},\] where \(n\) and \(T\) are the number density and corresponding average temperature. Approaching sink formation densities \(M_{\rm \small BE}\) is nearly independent of \({{u}_{\rm \small CR}}\), in agreement with the observed lack of evolution in the sink mass.
Also shown in Figure \ref{fig:Mbe} are the one-zone calculations from \citet{StacyBromm2007}. Investigating the impact of a CR background on Pop III star formation in a \(z=21\) minihalo using the CR background strengths shown in Figure \ref{fig:ucr}, their models suggested that a sufficiently strong background might decrease the fragmentation mass scale by an order of magnitude as a result of the cooler temperatures experienced during the loitering phase. While we observe similar behaviour from gas in the loitering phase, once the gas proceeds to runaway collapse its thermodynamic state quickly converges with that of the \({{u}_{\rm \small CR}}=0\) case, such that the fragmentation mass scale remains unaltered. Unfortunately the models of \citet{StacyBromm2007} only followed the gas evolution up to densities of \(10^6{\,{\rm cm}^{-3}}\), insufficient to observe this convergent behaviour.