Mitigating cosmic variance in the Hellings-Downs curve: A cosmic microwave background analogy

The Hellings-Downs (HD) correlation, which characterizes the signature of a stochastic gravitational wave background measured via pulsar timing arrays (PTA), is derived using a harmonic formalism. This approach closely follows the theoretical framework traditionally employed to compute correlations of temperature fluctuations in the cosmic microwave background (CMB). This parallel enables a direct comparison between the correlations observed in PTA and those in CMB experiments. After providing analytic estimates of the transmission functions due to the model-fitting procedure, we show that the covariance matrix in frequency space becomes very nondiagonal. We then build formally the quadratic estimator for the HD correlation in multipolar space, for both a perfect experiment, and for a realistic pulsar noise model. For a perfect experiment, we show that the signal to noise ratio grows with the observation time and the number of frequency bins, in turn determined by the cadence of observation. For an imperfect experiment, the behavior is similar, with an effective multipole-dependent number of frequency, obtained after weighting with noise. We predict that with $\ensuremath{\sim}200$ pulsars monitored for 25 years, multipoles of the HD correlation up to $\ensuremath{\ell}=4$ can be measured. Our findings clarify that what has been referred to as cosmic variance in some of the previous literature is not an intrinsic limitation for PTA measurements. Instead, when an optimal estimator is used, this variance can be mitigated by accumulating more observation time or improving the cadence of pulsar monitoring. Therefore, unlike CMB angular correlations, where cosmic variance represents an irreducible constraint, it can be reduced in PTA measurements and will continue to diminish in future experiments. Finally, we show that if the primordial power spectrum of tensor fluctuations was very blue with ${n}_{T}>4$, the CMB angular correlation due to these tensor modes would also exhibit a HD correlation. While our results are derived for the case of an isotropic bath of gravitons, we also discuss the case in which the graviton distribution function is anisotropic.