A Practical Theorem on Gravitational Wave Backgrounds

TL;DR

The paper proves a practical, cosmology-independent theorem that links the stochastic gravitational-wave background spectrum to the time-integrated energy spectrum of individual sources and the present-day comoving density of remnants. It provides a physical derivation and a rigorous mathematical proof, and demonstrates the approach with merging binaries in circular orbits to yield compact analytic expressions for Omega_gw and h_c. By applying the results to white-dwarf mergers and supermassive black-hole binaries, the work shows how background estimates can be obtained quickly and with quantified uncertainties. The findings have significant implications for planning gravitational-wave observatories like LISA and for surveying potential backgrounds without detailed cosmological modeling.

Abstract

There is an extremely simple relationship between the spectrum of the gravitational wave background produced by a cosmological distribution of discrete gravitational wave sources, the total time-integrated energy spectrum of an individual source, and the present-day comoving number density of remnants. Stated in this way, the background is entirely independent of the cosmology, and only weakly dependent on the evolutionary history of the sources. This relationship allows one easily to compute the amplitude and spectrum of cosmic gravitational wave backgrounds from a broad range of astrophysical sources, and to evaluate the uncertainties therein.