We reinvestigate the effect of annihilations on the distribution ofcollisionless dark matter (DM) in a spherical density spike around a massiveblack hole. We first construct a very simple, pedagogic, analytic model for anisotropic phase space distribution function that accounts for annihilation andreproduces the "weak cusp" found by Vasiliev for DM deep within the spike andaway from its boundaries. The DM density in the cusp varies as $r^{-1/2}$ for$s$-wave annihilation, where $r$ is the distance from the central black hole,and is not a flat "plateau" profile. We then extend this model by incorporatinga loss cone that accounts for the capture of DM particles by the hole. The losscone is implemented by a boundary condition that removes capture orbits,resulting in an anisotropic distribution function. Finally, we evolve aninitial spike distribution function by integrating the Boltzmann equation toshow how the weak cusp grows and its density decreases with time. We treat twocases, one for $s$-wave and the other for $p$-wave DM annihilation, adoptingparameters characteristic of the Milky Way nuclear core and typical WIMP modelsfor DM. The cusp density profile for $p$-wave annihilation is weaker, varyinglike $\sim r^{-0.34}$, but is still not a flat plateau.
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机译：我们重新研究了hil灭对大质量黑洞周围球形密度峰值中无碰撞暗物质（DM）分布的影响。我们首先为各向异性相空间分布函数构建一个非常简单的教学分析模型，该模型解释了ni没现象，并重现了Vasiliev发现的DM的“弱尖峰”，它位于尖峰之内并远离其边界。尖端的DM密度随$ s $-波的wave灭而变化为$ r ^ {-1/2} $，其中$ r $是距中心黑洞的距离，并且不是平坦的“高原”轮廓。然后，我们通过合并一个损失锥来扩展此模型，该损失锥用于解释孔对DM粒子的捕获。损失锥是通过消除捕获轨道的边界条件实现的，从而产生各向异性的分布函数。最后，我们通过积分Boltzmann方程来演化初始尖峰分布函数，以显示弱尖峰如何生长以及其密度随时间降低。我们处理两种情况，一种用于$ s $波，另一种用于$ p $波DM hil没，采用银河系核芯的特征参数和DM的典型WIMP模型。 $ p $波wave灭的尖端密度分布较弱，类似于$ \ sim r ^ {-0.34} $，但仍不是平稳的。
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