AI 中文总结
研究嵌入指数球暗物质晕的类史瓦西黑洞，通过有理频率比分类周期轨道，利用数值近似框架计算EMRI波形，发现波形能从空间范围解出暗物质晕总质量，为超大质量黑洞周围暗物质分布提供强场探测。
AI 中文摘要
我们研究了嵌入指数球（ESM）暗物质晕中的类史瓦西黑洞，其特征为晕质量\(M_0\)和尺度半径\(r_0\)。首先表明晕对边缘束缚轨道（MBO）和最内稳定圆轨道（ISCO）的影响比简单的位移更丰富：特征半径和能量随\(r_0\)非单调变化，先低于史瓦西值再恢复，而角动量单调减小；作为\(M_0\)的函数，响应甚至可能改变符号，只有ISCO能量始终单调。通过有理频率比\(q=\omega_\phi/\omega_r - 1 = w + v/z\)对周期轨道进行分类，发现\(r_0\)和\(M_0\)在轨道谱上留下明显可区分的印记。使用数值近似框架，计算了相应的极端质量比旋进（EMRI）波形，表明改变\(r_0\)会产生强烈的单调效应，扩大轨道、延长径向周期并引入明显的相位偏移，而\(M_0\)的类似变化除非\(M_0\)成为黑洞质量的相当一部分，否则信号几乎不变。这些结果表明，EMRI波形原则上可以从暗物质晕的空间范围中解出其总质量，为超大质量黑洞周围暗物质分布提供强场探测。
英文摘要
We study a Schwarzschild-like black hole embedded in an exponential-sphere (ESM) dark matter halo, characterized by a halo mass $M_0$ and a scale radius $r_0$. We first show that the halo's effect on the marginally bound orbit (MBO) and innermost stable circular orbit (ISCO) is richer than a simple shift: the characteristic radii and energies vary non-monotonically with $r_0$, dipping below their Schwarzschild values before recovering, while the angular momenta decrease monotonically; as a function of $M_0$, the response can even reverse sign depending on how extended the halo is, with only the ISCO energy remaining monotonic throughout. We classify periodic orbits by the rational frequency ratio $q=ω_ϕ/ω_r-1=w+v/z$ and find that $r_0$ and $M_0$ leave clearly distinguishable imprints on the orbit spectrum. Using the numerical kludge framework, we compute the corresponding extreme-mass-ratio-inspiral (EMRI) waveforms and show that varying $r_0$ produces a strong, monotonic effect enlarging the orbits, lengthening the radial period, and introducing a clear dephasing while comparable variations in $M_0$ leave the signal nearly unchanged unless $M_0$ becomes a sizable fraction of the black hole mass. Together, these results indicate that EMRI waveforms can, in principle, disentangle the total mass of a dark matter halo from its spatial extent, offering a strong-field probe of the dark matter distribution around supermassive black holes.
Comments18 pages, 11 figures, 4 tables
