A pulsar timing array (PTA) uses the extraordinarily stable pulses of millisecond pulsars as a galaxy-scale array of clocks. A passing nanohertz gravitational wave stretches and compresses spacetime along the line of sight to each pulsar, adding a tiny extra delay (a "timing residual") to its arrival times. A stochastic background from many merging supermassive black-hole binaries perturbs every pulsar at once, so residuals from different pulsar pairs become statistically correlated — and General Relativity predicts exactly how that correlation must depend on the angle between the two pulsars on the sky.
x = (1 − cos θ) / 2
Γ(θ) = 1/2 + (3/2)·x·ln(x) − x/4 (Hellings & Downs, 1983)
This quadrupolar Γ(θ) curve — high correlation for nearby pulsar pairs, a dip near ~90–120°, and a partial rise back toward 180° — is the unique "smoking gun" signature that separates a real gravitational-wave background from ordinary clock or timing-model noise. NANOGrav, EPTA, PPTA and CPTA reported evidence for exactly this curve in their pulsar data in 2023.
- GWB strain amplitude — how strongly the shared gravitational-wave signal perturbs every pulsar; each epoch it is generated from a Cholesky factorisation of the theoretical Γ(θij) covariance matrix, so pulsars really are correlated by that exact formula.
- Pulsar timing noise — independent, uncorrelated noise added to every pulsar every epoch (real pulsar clocks, the interstellar medium, and instruments all contribute this).
- GWB toggle — switch the correlated signal off entirely: the scatter then has no angular structure, showing what a non-detection looks like.
- Scatter plot — each dot is the Pearson correlation, accumulated online over the whole observation run, between one pair of pulsars' residual time series, plotted against their true angular separation; the solid line is the theoretical Γ(θ) curve above.
As observation time and array size grow, noise averages down and the scattered points visibly converge onto the curve — precisely how PTAs turned decades of timing data into a gravitational-wave detection.