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We determine the binding energy, the total gravitational wave energy flux,and the gravitational wave modes for a binary of rapidly spinning black holes,working in linearized gravity and at leading orders in the orbital velocity,but to all orders in the black holes' spins. Though the spins are treatednonperturbatively, surprisingly, the binding energy and the flux are given bysimple analytical expressions which are finite (respectively third- andfifth-order) polynomials in the spins. Our final results are restricted to theimportant case of quasi-circular orbits with the black holes' spins alignedwith the orbital angular momentum.

Gravitational waves from spinning binary black holes at the leading post-Newtonian orders at all orders in spin