Proving the short-wavelength approximation in Pulsar Timing Array gravitational-wave background searches

TL;DR

This paper rigorously proves the validity of the short-wavelength approximation used in pulsar timing array analyses for detecting gravitational-wave backgrounds, confirming its assumptions and applicability.

Contribution

It provides the first rigorous analytical proof of the short-wavelength approximation in pulsar timing array gravitational-wave background searches.

Findings

The short-wavelength approximation is valid under certain conditions.

Analytical expression for the Hellings and Downs curve is confirmed.

Supports the use of simplified models in gravitational-wave detection strategies.

Abstract

A low-frequency gravitational-wave background (GWB) from the cosmic merger history of supermassive black holes is expected to be detected in the next few years by pulsar timing arrays. A GWB induces distinctive correlations in the pulsar residuals --- the expected arrival time of the pulse less its actual arrival time. Simplifying assumptions are made in order to write an analytic expression for this correlation function, called the Hellings and Downs curve for an isotropic GWB, which depends on the angular separation of the pulsar pairs, the gravitational-wave frequency considered, and the distance to the pulsars. This is called the short-wavelength approximation, which we prove here rigorously and analytically for the first time.