Hellings-Downs curve
The method entails searching for spatial correlations of the timing residuals from pairs of pulsars and comparing the data with the Hellings-Downs curve.
When the data fit exceeds the standard 5 sigma threshold, the pulsar timing array can declare detection of gravitational waves.
[2][3][4] More precisely, the Hellings-Downs curve is the expected correlations of the timing residuals from pairs of pulsars as a function of their angular separation on the sky as seen from Earth.
[5][6] This theoretical correlation function assumes Einstein's general relativity and a gravitational wave background that is isotropic.
[7][4] Albert Einstein's theory of general relativity predicts that a mass will deform spacetime causing gravitational waves to emanate outward from the source.
The Hellings-Downs curve is used to infer the presence of gravitational waves by finding patterns of angular correlations in the timing residual data of different pulsar pairings.
More precisely, the expected correlations on the vertical axis of the Hellings-Downs curve are the expected values of pulsar-pairs correlations averaged over all pulsar-pairs with the same angular separation and over gravitational-wave sources very far away with noninterfering random phases.
[13] The following year Ron Hellings and George Downs published the foundations of the Hellings-Downs curve in their 1983 paper "Upper Limits on the Isotropic Gravitational Radiation Background from Pulsar Timing Analysis".
[7] Donald Backer would later go on to become one of the founders of the North American Nanohertz Observatory for Gravitational Waves (NANOGrav).
[1][13] In 2023, NANOGrav used pulsar timing array data collected over 15 years in their latest publications supporting the existence of a gravitational wave background.
[14] The NANOGrav team wrote that "The observation of Hellings–Downs correlations points to the gravitational-wave origin of this signal.
These examples highlight the critical role that the Hellings-Downs curve plays in contemporary gravitational wave research.
Reardon et al. (2023) from the Parkes pulsar timing array team give the following equation for the Hellings-Downs curve,[15] which in the literature is also called the overlap reduction function:[16]
This curve assumes an isotropic gravitational wave background that obeys Einstein's general relativity.
