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There is a significant possibility that astrophysical black holes withnearly-extremal spins exist. Numerical simulations of such systems requiresuitable initial data. In this paper, we examine three methods of constructingbinary-black-hole initial data, focusing on their ability to generate blackholes with nearly-extremal spins: (i) Bowen-York initial data, includingstandard puncture data (based on conformal flatness and Bowen-York extrinsiccurvature), (ii) standard quasi-equilibrium initial data (based on theextended-conformal-thin-sandwich equations, conformal flatness, and maximalslicing), and (iii) quasi-equilibrium data based on the superposition ofKerr-Schild metrics. We find that the two conformally-flat methods (i) and (ii)perform similarly, with spins up to about 0.99 obtainable at the initial time.However, in an evolution, we expect the spin to quickly relax to asignificantly smaller value around 0.93 as the initial geometry relaxes. Forquasi-equilibrium superposed Kerr-Schild (SKS) data [method (iii)], weconstruct initial data with \emph{initial} spins as large as 0.9997. We evolveSKS data sets with spins of 0.93 and 0.97 and find that the spin drops by onlya few parts in 10^4 during the initial relaxation; therefore, we expect thatSKS initial data will allow evolutions of binary black holes with relaxed spinsabove 0.99. [Abstract abbreviated; full abstract also mentions severalsecondary results.]

Binary-black-hole initial data with nearly-extremal spins