The Hellings and Downs curve (Hellings-Downs correlation, HD correlation) is the graphed curve of a function that maps the angle between pairs of pulsars as seen from Earth with the correlation between such pairs' 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. The curve has 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, such as 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 throughout the universe. Timing residuals of pulses from pulsars (the delta between the expected time and actual received time that can be the effect of gravitational waves) will show some correlation merely 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 this 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 a correlation to the curve is found in the data, characteristics of the GW background are confirmed (e.g., whether it is isotropic), its quantification is obtained, and with such quantification at various frequencies, an angular power spectrum may be discerned. Furthermore, use of PTAs to search for GWs is proved valid and means is provided to isolate any specific GW signal, e.g., from those of some specific binary SMBH.