[论文解读] Estimating False Alarm Rates of Sub-Dominant Quasi-normal Modes in GW190521

C. D. Capano|arXiv (Cornell University)|Sep 1, 2022

Pulsars and Gravitational Waves Research被引用 7

一句话总结

本研究通过模拟信号的统计分析，验证了在引力波事件GW190521中检测到一个次主导的$(\ell,m,n) = (3,3,0)$准正规模态。当该模态不存在时，其误报率为$\sim 0.02$，支持其存在性，并与广义相对论一致。

ABSTRACT

A major aim of gravitational wave astronomy is to test observationally the Kerr nature of black holes. The strongest such test, with minimal additional assumptions, is provided by observations of multiple ringdown modes, also known as black hole spectroscopy. For the gravitational wave merger event GW190521, we have previously claimed the detection of two ringdown modes emitted by the remnant black hole. In this paper we provide further evidence for the detection of multiple ringdown modes from this event. We analyse the recovery of simulated gravitational wave signals designed to replicate the ringdown properties of GW190521. We quantify how often our detection statistic reports strong evidence for a sub-dominant $(\ell,m,n)=(3,3,0)$ ringdown mode, even when no such mode is present in the simulated signal. We find this only occurs with a probability $\sim 0.02$, which is consistent with a Bayes factor of $t_{ m ref} + 6\,\mathrm{ms}$ (1$σ$ uncertainty) found for GW190521. We also quantify our agnostic analysis of GW190521, in which no relationship is assumed between ringdown modes, and find that only 1 in 250 simulated signals without a $(3,3,0)$ mode yields a result as significant as GW190521. Conversely, we verify that when simulated signals do have an observable $(3,3,0)$ mode they consistently yield a strong evidence and significant agnostic results. We also find that constraints on deviations from the $(3,3,0)$ mode on GW190521-like signals with a $(3,3,0)$ mode are consistent with what was obtained from our previous analysis of GW190521. Our results support our previous conclusion that the gravitational wave signal from GW190521 contains an observable sub-dominant $(\ell,m,n)=(3,3,0)$ mode.

研究动机与目标

• 通过统计方法验证在GW190521中检测到次主导$(3,3,0)$准正规模态。
• 量化在无该模态的模拟信号中检测该模态的误报率。
• 评估不预先假设模态关系的无偏分析方法的稳健性。
• 验证当$(3,3,0)$模态存在时，对偏离Kerr预测的约束是否与广义相对论一致。
• 评估$(2,2,0) + (3,3,0)$模型是否足以在GW190521的信噪比下描述该信号。

提出的方法

• 生成模拟引力波信号以复现GW190521的环 ringing 特性，包括主导模态与次主导模态。
• 使用贝叶斯检测统计量评估$(3,3,0)$模态的证据，并通过500次不含该模态的模拟计算误报率。
• 开发无偏统计量$\zeta$，用于比较两个模态的频率与阻尼时间后验分布，检验其与Kerr预测的一致性。
• 将真实GW190521数据中$(3,3,0)$模态的后验约束与模拟广义相对论信号中的结果进行比较，以验证一致性。
• 使用$(2,2,0) + (2,2,1)$模型识别并合时间并计算贝叶斯因子，确保与预期的统计一致性。
• 通过在无$(3,3,0)$模态的零假设下从模拟集合中推导p值，评估统计显著性。

实验结果

研究问题

• RQ1 当GW190521类信号中不存在$(3,3,0)$模态时，检测该次主导准正规模态的误报率是多少？
• RQ2 无偏$\zeta$统计量在区分存在与不存在可观测$(3,3,0)$模态的信号方面表现如何？
• RQ3 GW190521中$(3,3,0)$模态偏离Kerr预测的约束是否与模拟广义相对论信号中获得的结果一致？
• RQ4 在GW190521的信噪比下，$(2,2,0) + (3,3,0)$模型是否足以充分描述其环 ringing 信号？
• RQ5 尽管不完整，是否可以可靠地使用$(2,2,0) + (2,2,1)$模型来识别并合时间并计算贝叶斯因子？

主要发现

• 在不含$(3,3,0)$模态的模拟信号中检测该模态的误报率为$\sim 0.02$，与GW190521中观察到的贝叶斯因子$56 \pm 1$一致。
• 在500次不含$(3,3,0)$模态的模拟信号中，仅有10次的贝叶斯因子高于56，对应误报率为$1/50 = 0.02$。
• 无偏$\zeta$统计量在零假设（无$(3,3,0)$模态）下，获得与GW190521结果同样极端结果的概率为$0.004$。
• GW190521中$(3,3,0)$模态偏离Kerr预测的约束与模拟广义相对论信号中获得的结果一致，支持广义相对论。
• 当分析广义相对论波形时，$(2,2,1)$泛音的后验分布明显偏离Kerr预测，表明$(2,2,0) + (2,2,1)$模型不足以完整描述信号。
• $(2,2,0) + (3,3,0)$模型在GW190521的信噪比下足够充分，可实现对无毛定理的可靠检验，精度约为10%。