TY - JOUR

T1 - Hellings and Downs correlation of an arbitrary set of pulsars

AU - Allen, Bruce

AU - Romano, Joseph D.

N1 - Funding Information: The authors thank Curt Cutler for helpful discussions about the variance in the estimator of the squared strain, and EPTA, NANOGrav, and PPTA for providing lists of their pulsar names and sky positions. B. A. is grateful to the members of the IPTA Detection Committee for their help in understanding many aspects of PTA data analysis, and J. D. R. acknowledges support from NSF Physics Frontiers Center Award PFC-2020265 and start-up funds from Texas Tech University.

PY - 2023/8/24

Y1 - 2023/8/24

N2 - Pulsar timing arrays (PTAs) detect gravitational waves (GWs) via the correlations they induce in the arrival times of pulses from different pulsars. We assume that the GWs are described by a Gaussian ensemble, which models the confusion noise produced by expected PTA sources. The mean correlation h2μu(γ) as a function of the angle γ between the directions to two pulsars was predicted by Hellings and Downs in 1983. The variance σtot2(γ) in this correlation was recently calculated [B. Allen, Variance of the Hellings-Downs correlation, Phys. Rev. D 107, 043018 (2023)PRVDAQ2470-001010.1103/PhysRevD.107.043018] for a single noise-free pulsar pair at angle γ, which shows that after averaging over many pairs, the variance reduces to an intrinsic cosmic variance σcos2(γ). Here, we extend this to an arbitrary set of pulsars at specific sky locations, with pulsar pairs binned by γ. We derive the linear combination of pulsar-pair correlations which is the optimal estimator of the Hellings and Downs correlation for each bin, illustrating our methods with plots of the expected range of variation away from the Hellings and Downs curve, for the sets of pulsars monitored by three active PTA collaborations. We compute the variance of and the covariance between these binned estimates, and show that these reduce to the cosmic variance and covariance s(γ,γ′) respectively, in the many-pulsar limit. The likely fluctuations away from the Hellings and Downs curve μu(γ) are strongly correlated/anticorrelated in the three angular regions where μu(γ) is successively positive, negative, and positive. We also construct the optimal estimator of the squared strain h2 from pulsar-pair correlation data. Remarkably, when there are very many pulsar pairs, this determines h2 with arbitrary precision because (in contrast to LIGO-like GW detectors) PTAs probe an infinite set of GW modes. To assess if observed deviations away from the Hellings and Downs curve are consistent with predictions, we propose and characterize several χ2 goodness-of-fit statistics. While our main focus is ideal noise-free data, we also show how pulsar noise and measurement noise can be included. Our methods can also be applied to future PTAs, where the improved telescopes will provide larger pulsar populations and higher-precision timing.

AB - Pulsar timing arrays (PTAs) detect gravitational waves (GWs) via the correlations they induce in the arrival times of pulses from different pulsars. We assume that the GWs are described by a Gaussian ensemble, which models the confusion noise produced by expected PTA sources. The mean correlation h2μu(γ) as a function of the angle γ between the directions to two pulsars was predicted by Hellings and Downs in 1983. The variance σtot2(γ) in this correlation was recently calculated [B. Allen, Variance of the Hellings-Downs correlation, Phys. Rev. D 107, 043018 (2023)PRVDAQ2470-001010.1103/PhysRevD.107.043018] for a single noise-free pulsar pair at angle γ, which shows that after averaging over many pairs, the variance reduces to an intrinsic cosmic variance σcos2(γ). Here, we extend this to an arbitrary set of pulsars at specific sky locations, with pulsar pairs binned by γ. We derive the linear combination of pulsar-pair correlations which is the optimal estimator of the Hellings and Downs correlation for each bin, illustrating our methods with plots of the expected range of variation away from the Hellings and Downs curve, for the sets of pulsars monitored by three active PTA collaborations. We compute the variance of and the covariance between these binned estimates, and show that these reduce to the cosmic variance and covariance s(γ,γ′) respectively, in the many-pulsar limit. The likely fluctuations away from the Hellings and Downs curve μu(γ) are strongly correlated/anticorrelated in the three angular regions where μu(γ) is successively positive, negative, and positive. We also construct the optimal estimator of the squared strain h2 from pulsar-pair correlation data. Remarkably, when there are very many pulsar pairs, this determines h2 with arbitrary precision because (in contrast to LIGO-like GW detectors) PTAs probe an infinite set of GW modes. To assess if observed deviations away from the Hellings and Downs curve are consistent with predictions, we propose and characterize several χ2 goodness-of-fit statistics. While our main focus is ideal noise-free data, we also show how pulsar noise and measurement noise can be included. Our methods can also be applied to future PTAs, where the improved telescopes will provide larger pulsar populations and higher-precision timing.

UR - http://www.scopus.com/inward/record.url?scp=85172803277&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2208.07230

DO - 10.48550/arXiv.2208.07230

M3 - Article

AN - SCOPUS:85172803277

VL - 108

JO - Physical Review D

JF - Physical Review D

SN - 2470-0010

IS - 4

M1 - 043026

ER -