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 framework traditionally employed to compute correlations of temperature fluctuations in the CMB. This parallel enables a direct comparison between the correlations observed in PTA and those in CMB. After providing analytic estimates of the transmission functions, we show that the covariance matrix in frequency space becomes very non-diagonal. 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 SNR grows with the observation time and the number of frequency bins, in turn determined by the cadence of observation. For an imperfect experiment, the behaviour is similar, with an effective multipole-dependent number of frequency, obtained after weighting with noise. We predict that with $\sim 200$ pulsars monitored for $25$ years, multipoles of the HD correlation up to $\ell=4$ can be measured. Our findings clarify that is called \emph{cosmic variance} in previous literature is not an intrinsic limitation for PTA measurements. Instead, with optimal estimators, it 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. 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. We also discuss the case in which the graviton distribution function is anisotropic.