Exact results on the number of gravitons radiated during binary inspiral

Abstract

We derive an exact formula F(e) which provides a concrete estimate for the total number and angular momentum of gravitons emitted during the nonrelativistic inspiral of two black holes. We show that the function F(e) is a slowly growing monotonic function of the eccentricity 0 ≤ e ≤ 1 and F(1) = 1.0128 ⋯. We confirm and extend the results obtained by Page for the function F(e). We also get an exact result for the ratio ν (ei) = 2ħN(Li, ei)/Li where the numerator 2ħN(Li, ei) is the sum of the spin angular momentum magnitudes of the gravitons emitted and N(Li, ei) is the total number of gravitons emitted in the gravitational waves during nonrelativistic inspiral from an initial eccentricity ei down to a final eccentricity e = 0 and the denominator Li is the magnitude of the initial orbital angular momentum. If the orbit starts off with unit eccentricity ei = 1, we get the value ν(1) = 1.002 268 666 2 ± 10-10 which confirms the Page's conjecture that the true value of ν(1) will lie between 1.001⋯ and 1.003⋯. We also show that the formula F(e) for gravitons emitted, originally expressed as an infinite series, can be represented by a single function through an integral representation.