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利用双星系统中超大质量黑洞透镜的引力波源检验精确衍射公式

Testing an exact diffraction formula with gravitational wave source lensed by a supermassive black hole in binary systems

Xiao Guo, Zhoujian Cao

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17 pages, 7 figures, comments are welcome
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AI总结：

针对长波长引力波经近距超大质量黑洞透镜时大角度衍射不可忽略的问题，本研究对比了精确衍射公式与传统近似公式的差异，明确了二者差异显著的参数范围，为未来通过引力波探测验证精确衍射公式提供了依据。

摘要：

对于长波长引力波（GWs）而言，当其被天体透镜化时，衍射效应会变得十分显著。传统衍射积分公式通常忽略大角度衍射，这在大多数情况下是适用的。但在一些特定场景中，例如引力波源被双星系统内的超大质量黑洞透镜化、且透镜与源距离很近时，大角度衍射会发挥关键作用。在我们此前的研究中，提出了一个同时考虑大角度衍射的精确通用衍射积分公式。本文探究了该精确衍射公式与传统近似公式在多种特殊条件下的差异。研究结果表明，在特定参数下，例如透镜-源距离$D_{\rm LS}=0.1$\text{AU}、透镜质量$M_{\rm L}=4\times10^{6}M_\text{☉}$时，精确衍射公式的放大因子明显小于近似公式的放大因子，二者的比例因子约为$r_F\backsimeq0.806$，该差异足够显著，具备可探测性。此外，研究还发现，比例因子$r_F$随$D_{\rm LS}$的增大从0.5逐渐上升至1，随$M_{\rm L}$的增大而减小。当$D_{\rm LS}\lesssim0.2$ AU（假设$M_{\rm L}=4\times10^{6}M_\text{☉}$）或$M_{\rm L}\gtrsim2\times10^6M_\text{☉}$（假设$D_{\rm LS}=0.1$ AU）时，精确公式与近似公式之间会出现显著差异。这些结果表明，未来有望通过引力波探测来验证我们提出的通用衍射公式。

英文摘要：

When it comes to long-wavelength gravitational waves (GWs), diffraction effect becomes significant when these waves are lensed by celestial bodies. Typically, the traditional diffraction integral formula neglects large-angle diffraction, which is often adequate for most of cases. Nonetheless, there are specific scenarios, such as when a GW source is lensed by a supermassive black hole in a binary system, where the lens and source are in close proximity, where large-angle diffraction can play a crucial role. In our prior research, we have introduced an exact, general diffraction integral formula that accounts for large-angle diffraction as well. This paper explores the disparities between this exact diffraction formula and the traditional, approximate one under various special conditions. Our findings indicate that, under specific parameters, such as a lens-source distance of $D_{\rm LS}=0.1$\,AU and a lens mass of $M_{\rm L}=4\times10^{6}M_\odot$, the amplification factor for the exact diffraction formula is notably smaller than that of the approximate formula, differing by a factor of approximately $r_F\simeq0.806$. This difference is substantial enough to be detectable. Furthermore, our study reveals that the proportionality factor $r_F$ gradually increases from 0.5 to 1 as $D_{\rm LS}$ increases, and decreases as $M_{\rm L}$ increases. Significant differences between the exact and approximate formulas are observable when $D_{\rm LS}\lesssim0.2$ AU (assuming $M_{\rm L}=4\times10^{6}M_\odot$) or when $M_{\rm L}\gtrsim2\times10^6M_\odot$ (assuming $D_{\rm LS}=0.1$ AU). These findings suggest that there is potential to validate our general diffraction formula through future GW detections.

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