Simulating dust collapse: the impact of gauge choices on primordial black hole formation

We investigate why numerical relativity simulations of dust collapse in a matter-dominated era frequently fail before primordial black hole formation due to numerical limitations and inappropriate gauge choices. Using 3-dimensional fully non-linear simulations of a dust fluid, we detail the advantages of advanced numerical methods and the impact of the lapse and shift evolution equations. The challenge is twofold: 1) conventional lapse slicing evolution equations, highly effective in typical black hole simulations, must be adapted in a cosmological context. We demonstrate how different proposed lapse evolution equations perform, enabling fair comparison, and propose our own. 2) The shift evolution equation's role is to spread out the numerical grid observers for a stable simulation. However, the Gamma-driver evolution fails when confronted with shell-focussing of the dust fluid. We show how the shift damping choice depends on the resolution of the shell-focusing region, suggesting that future work should have spatially varying damping. Our improved gauge choices allow simulations to evolve significantly further and potentially reach black hole formation. The best-performing simulations reach close proximity to a black hole; however, further work on the gauge choice is required for this to be stable.