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#### The Hellings and Downs correlation of an arbitrary set of pulsars

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2208.07230.pdf

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PhysRevD.108.043026.pdf

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##### Citation

Allen, B., & Romano, J. D. (2023). The Hellings and Downs correlation of an arbitrary
set of pulsars.* Physical Review D,* *108*(4): 043026. doi:10.1103/PhysRevD.108.043026.

Cite as: https://hdl.handle.net/21.11116/0000-000A-EB8C-0

##### Abstract

Pulsar timing arrays (PTAs) detect gravitational waves (GWs) via the

correlations that the waves induce in the arrival times of pulses from

different pulsars. The mean correlation $\mu_{\rm u}(\gamma)$ as a function of

the angle $\gamma$ between the directions to two pulsars was calculated by

Hellings and Downs in 1983. The variance $\sigma^2_{\rm tot}(\gamma)$ in this

correlation was recently calculated for a single pulsar pair at angle $\gamma$.

Averaging over many such pairs, uniformly distributed on the sky, reduces this

to an intrinsic cosmic variance $\sigma^2_{\rm cos}(\gamma)$. We extend that

analysis to an arbitrary finite set of pulsars, distributed at specific sky

locations, for which the pulsar pairs are grouped into finite-width bins in

$\gamma$. Given (measurements or calculations of) the correlations for any set

of pulsars, we find the best way to estimate the mean in each bin. The optimal

estimator of the correlation takes into account correlations among all of the

pulsars that contribute to that angular bin. We also compute the variance in

the binned estimate. For narrow bins, as the number of pulsar pairs grows, the

variance drops to the cosmic variance. For wider bins, by sacrificing angular

resolution in $\gamma$, the variance can even be reduced below the cosmic

variance. Our calculations assume that the GW signals are described by a

Gaussian ensemble, which provides a good description of the confusion noise

produced by expected PTA sources. We illustrate our methods with plots of the

GW variance for the sets of pulsars currently monitored by several PTA

collaborations. The methods can also be applied to future PTAs, where the

improved telescopes will provide larger pulsar populations and higher-precision

timing.

correlations that the waves induce in the arrival times of pulses from

different pulsars. The mean correlation $\mu_{\rm u}(\gamma)$ as a function of

the angle $\gamma$ between the directions to two pulsars was calculated by

Hellings and Downs in 1983. The variance $\sigma^2_{\rm tot}(\gamma)$ in this

correlation was recently calculated for a single pulsar pair at angle $\gamma$.

Averaging over many such pairs, uniformly distributed on the sky, reduces this

to an intrinsic cosmic variance $\sigma^2_{\rm cos}(\gamma)$. We extend that

analysis to an arbitrary finite set of pulsars, distributed at specific sky

locations, for which the pulsar pairs are grouped into finite-width bins in

$\gamma$. Given (measurements or calculations of) the correlations for any set

of pulsars, we find the best way to estimate the mean in each bin. The optimal

estimator of the correlation takes into account correlations among all of the

pulsars that contribute to that angular bin. We also compute the variance in

the binned estimate. For narrow bins, as the number of pulsar pairs grows, the

variance drops to the cosmic variance. For wider bins, by sacrificing angular

resolution in $\gamma$, the variance can even be reduced below the cosmic

variance. Our calculations assume that the GW signals are described by a

Gaussian ensemble, which provides a good description of the confusion noise

produced by expected PTA sources. We illustrate our methods with plots of the

GW variance for the sets of pulsars currently monitored by several PTA

collaborations. The methods can also be applied to future PTAs, where the

improved telescopes will provide larger pulsar populations and higher-precision

timing.