Exact Results On the Number of Gravitons Radiated During Binary Inspiral

TL;DR

This paper derives an exact formula for the total number and angular momentum of gravitons emitted during binary black hole inspiral, confirming previous conjectures and extending earlier results with precise calculations.

Contribution

It provides an exact integral representation for the graviton emission function and confirms the conjectured ratio of graviton spin angular momentum to initial orbital angular momentum.

Findings

Derived an exact formula for graviton emission during inspiral.

Confirmed and extended Page's results on graviton emission.

Precisely estimated the ratio of graviton spin to initial orbital angular momentum.

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 $0 \le e \le 1$ and $F(1) = 1.0128 \cdots $. We confirm and extend the results obtained by Page for the function $F(e)$. We also get an exact result for the ratio $\nu (e_i) = \frac{2\hbar N(L_i, e_i)}{L_i}$ where the numerator $2\hbar N(L_i, e_i)$ is the sum of the spin angular momentum magnitudes of the gravitons emitted and $N(L_i, e_i)$ is the total number of gravitons emitted in the gravitational waves during nonrelativistic inspiral from an initial eccentricity $e_i$ down to a final eccentricity $e = 0$ and the denominator $L_i$ is the magnitude of the initial orbital angular momentum. If the orbit starts off with unit eccentricity $e_i=1$, we get the value $\nu(1) = 1.002\, 268\, 666\, 2 \pm 10^{-10}$ which confirms the Page's conjecture that the true value of $\nu(1)$ will lie between $1.001\cdots$ and $1.003\cdots$. 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.