An intermediate mass black hole (IMBH) may have a dark matter (DM) minihalo around it and develop a spiky structure called DM minispike. Gravitational waves (GWs) can be produced if a stellar compact object, such as a black hole or neutron star, inspirals into the IMBH. This kind of systems are known as intermediate-mass-ratio-inspirals (IMRIs) and may be observed by space-based GW detectors including LISA, Taiji and Tianqin. In this paper, we lay the foundations for the construction of analytic expressions for Fourier-domain GWs produced by eccentric IMRIs with DM minispikes in a post-circular or small-eccentricity approximation (e < 0.4). We take the effect of dynamical friction from the DM as a perturbation, and decompose the dynamical equations into perturbed part and unperturbed part. The equations are solved in a series expansion form about zero initial eccentricity to eighth order. The time-dependent, ‘plus’ and ‘cross’ polarizations are expanded in Bessel functions, which are then self-consistently reexpanded in a power series about zero initial eccentricity. The stationary-phase approximation is then employed to obtain the explicit DM-modified analytic expressions for the Fourier transform of the post-circular expanded, time-domain signal. We exemplify this framework by considering a typical IMRI with a DM minispike and find the GW detectability strongly depends on the radial profile of the DM distribution. When the density of DM is large enough, the signal to noise ratio will be degraded significantly and a detection loss may occur if we use a template without the effect of DM to treat a signal including the DM effect. With the Fourier-domain gravitational waveforms we also estimate the accuracy of the measurement of the DM minispike parameters in our reference model. Our framework hold the promise to construct a ‘ready-to-use’ Fourier-domain waveforms for data analysis of eccentric IMRIs with DM minispikes.

中文翻译：

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具有暗物质小尖峰的引力波：偏心中间质量比螺旋的傅立叶域波形

中等质量黑洞 (IMBH) 周围可能有暗物质 (DM) 迷你晕，并形成称为 DM 迷你尖峰的尖峰结构。如果恒星致密天体（例如黑洞或中子星）吸入中等质量黑洞，就会产生引力波（GW）。这种系统被称为中间质量比螺旋系统（IMRI），可以通过 LISA、太极和天琴等天基引力波探测器进行观测。在本文中，我们为构建由具有 DM 小尖峰的偏心 IMRI 在后圆或小偏心率近似中产生的傅里叶域引力场的解析表达式奠定了基础（e < 0.4）。我们将DM的动摩擦效应视为扰动，并将动力学方程分解为扰动部分和未扰动部分。这些方程以关于零初始偏心率至八阶的级数展开形式求解。与时间相关的“正”和“交叉”极化在贝塞尔函数中展开，然后在关于零初始偏心率的幂级数中自洽地重新展开。然后采用平稳相位近似来获得后循环扩展的时域信号的傅立叶变换的显式 DM 修改的解析表达式。我们通过考虑具有 DM 小尖峰的典型 IMRI 来举例说明该框架，并发现 GW 可检测性很大程度上取决于 DM 分布的径向轮廓。当DM的密度足够大时，如果使用没有DM效应的模板来处理包含DM效应的信号，信噪比会显着下降，并且可能会出现检测损失。利用傅立叶域引力波形，我们还估计了参考模型中 DM 微型尖峰参数的测量精度。我们的框架有望构建“即用型”傅里叶域波形，用于具有 DM 微型尖峰的偏心 IMRI 的数据分析。