Sensitivity of the lower-edge of the pair instability black hole mass gap to the treatment of time dependent convection

Gravitational-wave detections are now probing the black hole (BH) mass distribution, including the predicted pair-instability mass gap. These data require robust quantitative predictions, which are challenging to obtain. The most massive BH progenitors experience episodic mass ejections on timescales shorter than the convective turn-over timescale. This invalidates the steady-state assumption on which the classic mixing-length theory relies. We compare the final BH masses computed with two different versions of the stellar evolutionary code \texttt{MESA}: (i) using the default implementation of \cite{paxton:18} and (ii) solving an additional equation accounting for the timescale for convective deceleration. In the second grid, where stronger convection develops during the pulses and carries part of the energy, we find weaker pulses. This leads to lower amounts of mass being ejected and thus higher final BH masses of up to $\sim$\,$5\,M_\odot$. The differences are much smaller for the progenitors which determine the maximum mass of BHs below the gap. This prediction is robust at $M_{\rm BH, max}\simeq 48\,M_\odot$, at least within the idealized context of this study. This is an encouraging indication that current models are robust enough for comparison with the present-day gravitational-wave detections. However, the large differences between individual models emphasize the importance of improving the treatment of convection in stellar models, especially in the light of the data anticipated from the third generation of gravitational wave detectors.