AI 中文总结
本研究通过Fisher矩阵预测，评估未来Sgr A*附近脉冲星计时对Buchdahl启发$R^2$引力参数$\epsilon$的约束能力，发现短周期高偏心率轨道可显著提高灵敏度，为未来观测设计提供定量指导。
AI 中文摘要
未来在Sgr~A*附近的脉冲星计时观测为探测超大质量黑洞附近的引力物理提供了独特手段。我们预测了此类测量约束一种Buchdahl启发的$R^2$引力的能力，该引力由单一偏差参数$\epsilon$参数化，使用一个自洽整合轨道动力学与光传播延迟并在整个观测跨度内保持完整计时解的计时框架。通过对代表性脉冲星进行Fisher矩阵预测，我们系统地分离了$\epsilon$精度对轨道几何的依赖关系。我们发现较短的轨道周期和较高的偏心率显著增强灵敏度，这与近心点附近观测的显著贡献一致。作为基准比较，我们进一步考虑一颗位于S2类似轨道（$P_b=16~{\rm yr}$，$e=0.88$）上的假设脉冲星，并在所采用的弱场、静态和球对称计时模型内获得$\sigma_\epsilon\sim 10^{-4}$的统计灵敏度。该灵敏度与1PN展开的自然数量级截断尺度相当，不应被解释为对真实Sgr~A*系统的完整预测。在所采用的理想化假设下，特征统计尺度比当前S2 95%置信区间半宽（$|\epsilon|_{\rm S2}^{\rm 95\\\\%}\approx 0.56$）低几个数量级，尽管鉴于置信水平不同，此比较是启发性的。这些趋势为未来银河系中心脉冲星搜索中的目标选择和活动设计提供了定量指导。
英文摘要
Future pulsar timing observations near Sgr~A* offer a unique probe of gravitational physics in the vicinity of a supermassive black hole. We forecast the ability of such measurements to constrain a Buchdahl-inspired $R^2$ gravity, parameterized by a single deviation parameter $ε$, using a timing framework that self-consistently integrates orbital dynamics with light-propagation delays and preserves the full timing solution across the observing span. Through Fisher-matrix forecasts for a representative pulsar, we systematically isolate how the precision on $ε$ depends on orbital geometry. We find that shorter orbital periods and higher eccentricities significantly enhance sensitivity, consistent with a substantial contribution from observations near periastron. As a benchmark comparison, we further consider a hypothetical pulsar on an S2-like orbit ($P_b=16~{\rm yr}$, $e=0.88$) and obtain a statistical sensitivity of $σ_ε\sim 10^{-4}$ within the adopted weak-field, static, and spherically symmetric timing model. This sensitivity is comparable to the natural order-of-magnitude truncation scale of the 1PN expansion and should not be interpreted as a complete forecast for the real Sgr~A* system. Under the adopted idealized assumptions, the characteristic statistical scale is several orders of magnitude below the current S2 95\% confidence interval half-width ($|ε|_{\rm S2}^{\rm 95\%}\approx 0.56$), though this comparison is heuristic given the differing confidence levels. These trends provide quantitative guidance for target selection and campaign design in future Galactic-center pulsar searches.
