Optimal all-angle reconstruction of the Hellings-Downs curve
Pulsar timing arrays (PTAs) detect nanohertz gravitational waves through spatial correlations between the timing residuals of different pulsars. For an isotropic, unpolarized stochastic background in general relativity, the ensemble-mean correlation follows the Hellings--Downs (HD) curve; measuring this angular pattern tests the gravitational-wave origin of the signal. Standard bin-by-bin reconstructions optimize the weights within each angular bin separately. We reconstruct the curve jointly using the full pulsar-pair covariance, retaining a free amplitude in every bin. The resulting all-angle best linear unbiased estimator minimizes the variance of every bin value and every linear combination of bins. Applied to the public NANOGrav 15 yr data products, our method reduces the bin standard deviations by up to 12.9\%, with a median reduction of 9.5\%. For a future Square Kilometre Array Observatory (SKAO)-like PTA, the predicted reduction reaches 36.4\%, with a median of 29.6\%, enabling significantly more precise measurements of the gravitational-wave background.