AI 中文总结
该研究以扭秤为探针，推导了半经典引力的匹配滤波界限、扭矩扩散约束等，通过室温卡文迪什基准及 Yan 等人的实验数据给出相关噪声与系数的保守界限，搭建了扭秤数据与半经典引力搜索的连接接口。
AI 中文摘要
校准后的扭秤谱可约束标准牛顿引力的确定性与随机偏差。利用单侧谱，我们推导了已知扭矩模板的校准角等效噪声预算及有限时间匹配滤波/克拉美罗界。对于随机模型，从观测角谱中减去校准后的标准噪声预算（热噪声、牛顿噪声、环境噪声、不精确噪声、反作用噪声），得到的残差谱可通过校准扭转 susceptibility 转换为等效残差扭矩谱，以此对额外的平稳随机扭矩噪声设定界限。该频率分辨界限仅在马尔可夫白噪声极限下可压缩为单一扭矩扩散系数 $D_\tau$；非马尔可夫或有色模型则需要完整残差谱。Page–Geilker 分支鉴别与 Fedida–Kent 混合等价性检验针对不同物理问题，但在投影到扭矩模板后，二者均简化为相同的统计匹配滤波鉴别问题。对于室温卡文迪什基准，共振热角 ASD 为 $1.36\times10^{-4}\\,\mathrm{rad}/\sqrt{\mathrm{Hz}}$，测量附加的 SQL 为 $3.25\times10^{-12}\\,\mathrm{rad}/\sqrt{\mathrm{Hz}}$。对于 Yan 等人的搜索，在 $2.5\\,\mathrm{mHz}$ 处报告的 $0.3\\,\mu\mathrm{rad}/\sqrt{\mathrm{Hz}}$ 灵敏度，在假设白扭矩噪声的情况下给出保守界限 $D_\tau \lesssim 2.4\times10^{-23}\\,\mathrm{N^2\\,m^2\\,s}$。这些公式提供了连接校准扭秤数据、确定性检验与随机半经典引力搜索的接口，且未断言对完整相对论半经典爱因斯坦方程的直接检验。
英文摘要
Calibrated torsion-balance spectra constrain deterministic and stochastic deviations from standard Newtonian gravity. Using one-sided spectra, we derive a calibrated angle-equivalent noise budget and finite-time matched-filter/Cramér-Rao bounds for known torque templates. For stochastic models, subtracting the calibrated standard noise budget (thermal, Newtonian, environmental, imprecision, backaction) from the observed angle spectrum yields a residual spectrum, convertible to an equivalent residual torque spectrum via calibrated torsional susceptibility. This bounds additional stationary stochastic torque noise. This frequency-resolved bound compresses to a single torque-diffusion coefficient, $D_τ$, only in the Markovian white-noise limit; non-Markovian or colored models require the full residual spectrum. Page--Geilker branch discrimination and Fedida-Kent mixture-equivalence tests address distinct physical questions, but upon projection onto torque templates, both reduce to the same statistical matched-filter discrimination problem. For a room-temperature Cavendish benchmark, resonant thermal angle ASD is $1.36\times10^{-4}\,\mathrm{rad}/\sqrt{\mathrm{Hz}}$, while measurement-added SQL is $3.25\times10^{-12}\,\mathrm{rad}/\sqrt{\mathrm{Hz}}$. For the Yan \emph{et al.} search, the reported $0.3\,μ\mathrm{rad}/\sqrt{\mathrm{Hz}}$ sensitivity at $2.5\,\mathrm{mHz}$ yields a conservative bound $D_τ\lesssim 2.4\times10^{-23}\,\mathrm{N^2\,m^2\,s}$, assuming white torque noise. These formulations provide an interface linking calibrated torsion-balance data, deterministic tests, and stochastic semiclassical-gravity searches, without asserting a direct test of the full relativistic semiclassical Einstein equation.
Comments16 pages, 2 figures
