The Hellings and Downs curve (Hellings-Downs correlation, HD correlation) is the graphed curve of a function that maps the angle between two pulsars as seen from Earth with the correlation between their timing residuals that would result from random gravitational wave (GW) signals from all over the sky, i.e., an isotropic, unpolarized stochastic gravitational-wave background. It has the potential usefulness for pulsar timing arrays to help identify and quantify the gravitational wave background (GWB) (random waves from many distant sources), which is the signal that one would need to "subtract" when attempting to identify a specific source. Other sources of randomness (e.g., measurement errors) would not show the same correlation.
Gravitational waves from a specific source can be recognized only if their signal can be isolated from the combined signals of other sources of waves at that frequency, presumably from sources throughout the universe. Timing residuals of pulses from pulsars (the delta between the expected time and actual received time that is attributed to the effect of gravitational waves) will show some correlation just from this background signal, and such a correlation depends upon the angle between the two pulse sources in the celestial sphere. The Hellings and Downs curve is a calculation of the expected correlation as a function of the angle, using some model simplifications: that the gravitational waves are plane waves (virtually true if the gravitational wave sources are distant), and that there are sources in all directions and with all polarizations. If reality is sufficiently close to these assumptions, the curve should yield the correlation between timing residuals that can be attributed to the background signal. This enables quantification of the background signal, and with such quantification at various frequencies, an angular power spectrum may be discerned.
The curve can also be directly sought in pulsar timing data as a means of detecting and confirming the characteristics of the GW background (e.g., that it is indeed isotropic), and in doing so, confirming the method being used to search for GWs is valid.