Resolving The Peak Of The Black Hole Mass Spectrum

Gravitational wave (GW) detections of binary black hole (BH) mergers have begun to sample the cosmic BH mass distribution. The evolution of single stellar cores predicts a gap in the BH mass distribution due to pair-instability supernova (PISN). Determining the upper and lower edges of the BH mass gap can be useful for interpreting GW detections from merging BHs. We use \MESA\ to evolve single, non-rotating, massive helium cores with a metallicity of $Z = 10^{-5}$ until they either collapse to form a BH or explode as a PISN without leaving a compact remnant. We calculate the boundaries of the lower BH mass gap for S-factors in the range S(300 keV) = (77,203) keV b, corresponding to the $\pm 3\sigma$ uncertainty in our high resolution tabulated $^{12}$C($\alpha$,$\gamma$)$^{16}$O reaction rate probability distribution function. We extensively test the temporal and mass resolution to resolve the theoretical peak of the BH mass spectrum across the BH mass gap. We explore the convergence with respect to convective mixing and nuclear burning, finding that significant time resolution is needed to achieve convergence. We also test adopting a minimum diffusion coefficient to help lower resolution models reach convergence. We establish a new lower edge of the upper mass gap as M\textsubscript{lower} $\simeq$\,60$^{+32}_{-14}$\,\Msun\ from the $\pm 3\sigma$ uncertainty in the $^{12}\text{C}(\alpha, \gamma) ^{16}\text{O}$ rate. We explore the effect of a larger 3-$\alpha$ rate on the lower edge of the upper mass gap, finding M\textsubscript{lower} $\simeq$\,69$^{+34}_{-18}$\,\Msun. We compare our results with BHs reported in the Gravitational-Wave Transient Catalog.