[论文解读] Hellings and Downs correlation of an arbitrary set of pulsars

B. Allen, Joseph D. Romano|arXiv (Cornell University)|Aug 15, 2022

Pulsars and Gravitational Waves Research参考文献 41被引用 1

一句话总结

本文为脉冲星计时阵列（PTAs）中的Hellings和Downs（HD）相关性开发了一种最优估计器，考虑了角度分箱、宇宙涨落以及脉冲星对之间的相关性。推导了分箱HD相关性估计的方差与协方差，表明其在脉冲星数量极大时收敛至宇宙涨落水平，并提出了χ²拟合优度统计量以检验对HD曲线的偏离，从而实现在噪声数据中精确探测引力波背景。

ABSTRACT

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.

研究动机与目标

• 为具有已知天球位置的任意脉冲星集合，开发一种最优线性估计器，用于估计Hellings和Downs相关性。
• 量化分箱HD相关性估计的方差与协方差，区分脉冲星方差与宇宙方差。
• 表征由于宇宙涨落引起的HD曲线周围预期涨落，特别是在HD曲线依次为正、负、正的区域。
• 构建一个稳健的平方应变h²估计器，基于脉冲星对的相关性，证明其在脉冲星数量极大时可实现任意精度。
• 提出并验证基于χ²拟合优度的统计量，以评估观测到的对HD曲线的偏离是否在统计上与预测一致。

提出的方法

• 推导脉冲星对相关性的最优线性组合，以估计每个角度分箱中的HD相关性，从而最小化方差。
• 将总方差σ²_tot(γ)表示为脉冲星方差与宇宙方差σ²_cos(γ)之和，后者源于随机引力波背景的相位涨落。
• 利用球谐分解与Clebsch-Gordan系数，计算不同角度下分箱HD估计之间的协方差。
• 证明在脉冲星数量极大的极限下，估计器的方差收敛至固有的宇宙方差s(γ, γ′)。
• 基于逆协方差矩阵引入χ²统计量，用于检验观测相关性与HD曲线的拟合优度。
• 将框架扩展至包含脉冲星计时噪声与测量噪声，证明其适用于现实世界及未来的PTA。

实验结果

研究问题

• RQ1 当脉冲星对按角距离分箱时，Hellings和Downs相关性的最优线性估计器是什么？
• RQ2 分箱HD相关性估计的方差与协方差如何依赖于脉冲星的角分布与数量？
• RQ3 宇宙方差在限制HD相关性测量精度方面起什么作用？其与脉冲星方差相比如何？
• RQ4 基于完整协方差结构的χ²统计量能否在真实PTA数据中可靠检测对HD曲线的偏离？
• RQ5 随着脉冲星对数量的增加，平方应变h²估计器的精度如何提升？其在脉冲星数量极大时是否能达到任意精度？

主要发现

• 每个角度分箱中HD相关性的最优估计器可最小化方差，并在脉冲星数量极大时收敛至宇宙方差s(γ, γ′)。
• 当对所有脉冲星对取向平均时，分箱HD相关性估计器的方差简化为固有的宇宙方差σ²_cos(γ)。
• 在HD曲线依次为正、负、正的三个角区域中，HD曲线周围的涨落表现出强相关或反相关。
• 由于PTAs可探测无限多的引力波模态，平方应变h²估计器在脉冲星对数量极大时可实现任意精度。
• 所提出的χ²统计量考虑了脉冲星对之间的相关性，提供了一种稳健的方法，可在存在噪声的情况下检验观测数据与HD曲线的一致性。
• 该框架可扩展至真实数据，包括脉冲星计时噪声与测量误差，适用于当前及未来拥有更大脉冲星群体的PTA。