Hellings-Downs curve

The Hellings-Downs curve (also known as the Hellings and Downs curve) is an analytical tool that helps to find patterns in pulsar timing data in an effort to detect long wavelength ${\displaystyle (\lambda =1-10ly)}$ gravitational waves.^[2][3][4] More precisely, the Hellings-Downs curve refers to the wave-like shape predicted to appear in a plot of timing residual correlations versus the angle of separation between pairs of pulsars.^[5][6] This theoretical correlation function assumes a gravitational wave background that is isotropic and Einsteinian.^[7][8]

Pulsar timing array residuals

Albert Einstein's theory of general relativity predicts that a mass will deform spacetime causing gravitational waves to emanate outward from the source.^[10] These gravitational waves will affect the travel time of any light that interacts with them. A pulsar timing residual is the difference between the expected time of arrival and the observed time of arrival of light from pulsars.^[2] Because pulsars flash with such a consistent rhythm, it is hypothesised that if a gravitational wave is present, a pattern may be observed in the changing arrival times of these pulsar emissions. The Hellings-Downs curve is used to infer the presence gravitational waves by finding patterns in the pulsar residual data. Pulsar timing residuals are measured using pulsar timing arrays.^[11]

History

Not long after the first suggestions of pulsars being used for gravitational wave detection in the late 1970’s,^[12][13] Donald Backer discovered the first millisecond pulsar in 1982.^[14] 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][14]

Examples in the scientific literature

In 2023, NANOGrav used pulsar timing array data collected over 15 years in their latest publications supporting the existence of a gravitational wave background.^[1] A total of 2,211 millisecond pulsar pair combinations (67 individual pulsars) were used by the NANOGrav team to construct their Hellings-Downs plot comparison.^[15] The NANOGrav team wrote that "The observation of Hellings–Downs correlations points to the gravitational-wave origin of this signal."^[4] This highlights the role that the Hellings-Downs curve plays in contemporary gravitational wave research.

Equation of the Hellings-Downs curve

Reardon et al. (2023) from the Parkes pulsar timing array team give the following equation for the Hellings-Downs curve:^[16]

${\displaystyle \Gamma _{ab}={\frac {1}{2}}\delta _{ab}+{\frac {1}{2}}-{\frac {x_{ab}}{4}}+{\frac {3}{2}}x_{ab}lnx_{ab}}$

Where:

${\displaystyle x_{ab}=(1-cos\zeta _{ab})/2}$,

${\displaystyle \delta _{ab}}$ is the kronecker delta function

${\displaystyle \zeta _{ab}}$ represents the angle of separation between the two pulsars as seen from earth

${\displaystyle \Gamma _{ab}}$ is the angular correlation function.

This curve assumes an isotropic gravitational wave background from a supermassive black hole binary system.

References