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. © 2025 IOP Publishing Ltd and Sissa Medialab. All rights, including for text and data mining, AI training, and similar technologies, are reserved.