Simulations of nearly extremal binary black holes

Astrophysical black holes could have nearly extremal spins; therefore, nearly extremal black holes could be among the binaries that current and future gravitational-wave observatories will detect. Predicting the gravitational waves emitted by merging black holes requires numerical-relativity simulations, but these simulations are especially challenging when one or both holes have mass $m$ and spin $S$ exceeding the Bowen-York limit of $S/m^2=0.93$. Using improved methods we simulate an unequal-mass, precessing binary black hole coalescence, where the larger black hole has $S/m^2=0.99$. We also use these methods to simulate a nearly extremal non-precessing binary black hole coalescence, where both black holes have $S/m^2=0.994$, nearly reaching the Novikov-Thorne upper bound for holes spun up by thin accretion disks. We demonstrate numerical convergence and estimate the numerical errors of the waveforms; we compare numerical waveforms from our simulations with post-Newtonian and effective-one-body waveforms; and we compare the evolution of the black-hole masses and spins with analytic predictions.