Black holes located within a dark matter cloud can create overdensity regions known as dark matter spikes. The presence of spikes modifies the gravitational-wave signals from binary systems through changes in the gravitational potential or dynamical friction effects. We assess the importance of including relativistic effects in both the dark matter distribution and the dynamical friction. As a first step we numerically calculate the particle dark matter spike distribution in full general relativity, using both Hernquist and Navarro-Frenk-White profiles in a Schwarzschild background, and we produce analytical fits to the spike profiles for a large range of scale parameters. Then we use a post-Newtonian prescription for the gravitational-wave dephasing to estimate the effect of relativistic corrections to the spike profile and to the dynamical friction. Finally we include the torques generated by dynamical friction in fast-to-generate relativistic models for circular extreme mass-ratio inspirals around a nonspinning black hole. We find that both types of relativistic corrections positively impact the detectability of dark matter effects, leading to higher dephasings and mismatches between gravitational-wave signals with and without dark matter spikes.

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相对论修正对引力波暗物质尖峰可探测性的影响

位于暗物质云中的黑洞会产生被称为暗物质尖峰的高密度区域。尖峰的存在通过引力势或动力摩擦效应的变化改变了来自二元系统的引力波信号。我们评估了在暗物质分布和动力摩擦中包含相对论效应的重要性。作为第一步，我们使用 Schwarzschild 背景中的 Hernquist 和 Navarro-Frenk-White 轮廓对全广义相对论中的粒子暗物质尖峰分布进行数值计算，并且我们对大范围的尺度参数的尖峰轮廓进行分析拟合。然后我们使用引力波相移的后牛顿公式来估计相对论修正对尖峰轮廓和动态摩擦的影响。最后，我们将动态摩擦产生的扭矩包括在快速生成的相对论模型中，用于非自旋黑洞周围的圆形极端质量比螺旋。我们发现，这两种类型的相对论校正都会对暗物质效应的可探测性产生积极影响，从而导致有和没有暗物质尖峰的引力波信号之间出现更高的相移和失配。