发表机构
School of Science, Huzhou University（湖州学院理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对随机引力波背景的 Fisher 预报，指出复合两阶段估计器在长基线（11.5天）下会显著放大振幅误差棒，并校准了偏差向量和均方误差矩阵，推荐使用联合最大化的 gauss_D 似然作为基线。
AI 中文摘要
Whittle 似然预报随机引力波背景支撑了计划中的毫赫兹干涉仪（LISA、太极、天琴）。一种复合两阶段估计器，即先进行每频段振幅估计，再进行未加权的对数空间模板拟合，在 11.5 天基线长度下，将已分辨频段计数上的边缘 Fisher 振幅误差棒最多放大约一个数量级。这种放大源于复合实现方式，而非其所近似的行列式近似似然 \texttt{gauss\\_D}。联合最大化时，\texttt{gauss\\_D} 是无偏的（在 $10^{-4}$ 精度内 $R=1.000$），并建议作为采用精确 Whittle 似然的基线。以精确 Whittle 作为对照、\texttt{gauss\\_D} 作为工作模型，我们校准了幂律和平滑破幂律模板的偏差向量和均方误差矩阵。首先，对于 32 点声壳扫描的尖锐信号，复合估计器在 11.5 天基线下将振幅误差棒放大 $R=10.2$（中位数 $\simeq3$），在较短分段下遵循 $R^2-1\propto T_{\mathrm{seg}}^2$，并在 $\alpha_{\mathrm{out}}$ 中存在高达 18% 的不可移除效率差距 $\mathbf{C}_b$。其次，通常的单参数 Whittle 偏差诊断无法跟踪多参数放大，且误差方向双向。第三，在 30 分钟分段上放大可忽略，在测量的偏差指数范围内 $R-1\le1.5\times10^{-3}$，在灵敏度下限以下消失，并与分辨信号强度相关，$\mathrm{corr}[\ln \mathrm{SNR},\ln(R-1)]=0.91$。结果为频率学性质；区间诊断仅在局部高斯区域可转移到平坦先验可信区间。该校准是强信号、平稳状态的陈述，仅可转移到匹配的噪声模型。
英文摘要
Whittle-likelihood forecasting of the stochastic gravitational-wave background underpins planned mHz interferometers (LISA, Taiji, TianQin). A composite two-stage estimator, per-bin amplitude estimation then unweighted log-space template fitting, inflates the marginal-Fisher amplitude error bar on resolved-bin count by up to an order of magnitude at \(11.5\)-d baseline. This inflation arises from the composite implementation, not the determinant-approximate \texttt{gauss\_D} likelihood it approximates. Jointly maximised, \texttt{gauss\_D} is unbiased (\(R=1.000\) within \(10^{-4}\)) and is recommended baseline with exact-Whittle likelihood. With exact Whittle as control and \texttt{gauss\_D} as working model, we calibrate bias vector and mean-squared-error matrix for power-law and smooth broken-power-law templates. First, for sharp signals of a \(32\)-point sound-shell scan, the composite estimator inflates amplitude error bar by \(R=10.2\) at \(11.5\)-d baseline (median \(\simeq3\)), follows \(R^2-1\propto T_{\mathrm{seg}}^2\) at shorter segments, and has a non-removable efficiency gap \(\mathbf{C}_b\) up to \(18\%\) in \(α_{\mathrm{out}}\). Second, the usual single-parameter Whittle-bias diagnostic does not track multi-parameter inflation and errs both ways. Third, inflation is negligible on \(30\)-min segments, \(R-1\le1.5\times10^{-3}\) over measured bias-exponent range, vanishes below sensitivity floor, and tracks resolved signal strength with \(\mathrm{corr}[\ln \mathrm{SNR},\ln(R-1)]=0.91\). Results are frequentist; interval diagnostics transfer to flat-prior credible intervals only in the locally Gaussian regime. The calibration is a strong-signal, stationary statement transferable only to a matching noise model.
Comments44 pages, 17 tables, 7 figures, comments welcome
