Eccentricity distribution of extreme mass ratio inspirals

This dataset provides eccentricity distributions for extreme mass ratio inspirals around non-spinning massive black holes. It includes distributions both at decoupling from the host nuclear star cluster and at the final plunge, evaluated for five different central black hole masses. Contents The zip file contains a folder with five CSV files within, each corresponding to a different central massive black hole (MBH) mass (in solar masses). The secondary mass is fixed at 10 M_sun. All files share the following columns: p0 [Rg] — semi-latus rectum (in units of the MBH gravitational radius) at decoupling from the nuclear star cluster e0 — eccentricity at decoupling ppl [Rg] — semi-latus rectum at plunge (where FEW stops) epl — eccentricity at plunge (where FEW stops) weight — astrophysical weight of each run (weights in each file sum to 1) Methods Overview This is a short overview of the methods used to generate these data. Further details are provided in the companion papers (Mancieri+25b [arXiv:2509.02394] and Mancieri+25a [A&A 694, A272; arXiv:2409.09122]) The initial EMRI population is taken from the Monte Carlo simulations of Mancieri et al. 2025a (A&A 694, A272; arXiv:2409.09122). In those simulations, EMRIs are followed until they reach the point where they are decoupled from the nuclear star cluster and gravitational wave (GW) emission only determines their evolution. This is defined by the condition on the GW emission timescale t_GW = 10^-3 * t_rlx , where t_rlx is the angular-momentum relaxation time due to two-body interactions in the nuclear star cluster. Parameters (p0, e0) in the dataset generally represent the EMRIs at the moment of decoupling. However, a subset of EMRIs, primarily in the high MBH mass models, had to be terminated before this condition was reached for computational cost reasons (see Sect. 3.4 of Mancieri+25a). When the condition t_GW = 10^-3 * t_rlx could not be satisfied, the parameters (p0, e0) were recorded at a stage where 10^-3 < t_GW / t_rlx < 1. In this interval, relaxation is already subdominant, although its residual influence may slightly perturb the parameters at full decoupling relative to those that would be obtained in an integration neglecting relaxation entirely. From these initial conditions, each EMRI is evolved to plunge (ppl, epl) using the FastEMRIWaveforms (FEW) package (v2.0.0). The evolution is performed in two stages: PN5 module for the early inspiral, used until the orbit reaches e = 0.8–0.9 (the limiting value depends on p). KerrEccEqFlux module from that point down to plunge. At the time these data were produced, this module is only valid below e = 0.8–0.9. The integration stops when the Schwarzschild separatrix is reached, p_sep = 6 + 2*e , but FEW returns the last point slightly before the separatrix, such that ppl - (6 + 2*epl) = 2*10^-3 . A small fraction (the tail at e > 0.82 in Fig. 5 of Mancieri+25b) of EMRIs plunge before entering the domain of validity of the KerrEccEqFlux module; in that case, the final point satisfies ppl - (6 + 2*epl) = 10^-1 . Astrophysical Weights Each EMRI carries a weight representing its expected contribution to the astrophysical EMRI population. The weight depends on the initial semi-major axis a_i used in the Monte Carlo simulations of Mancieri et al. 2025a (A&A 694, A272; arXiv:2409.09122). Two competing effects shape the EMRI formation rate as a function of a_i: relaxation more efficiently scatters compact objects close to the MBH at large semi-major axes, but objects on wide, very eccentric orbits tend to directly plunge rather than form long-lived EMRIs. Together, these effects produce a peak EMRI formation rate around a_i = 10^-2 R_inf (where R_inf is the MBH influence radius). Because a different number of simulations n_i were performed at each a_i, the final weight for each run is w_i = (1 / n_i) * <dot{N}_i> / ( sum_j <dot{N}_j> ) , where <dot{N}_i> is the time-averaged EMRI production rate at that a_i. References Please cite the companion papers and the dataset itself if you use it Title: Eccentricity distribution of extreme mass ratio inspirals Authors: Davide Mancieri, Luca Broggi, Morgan Vinciguerra, Alberto Sesana, Matteo Bonetti DOI: https://doi.org/10.48550/arXiv.2509.02394 Title: Hanging on the cliff: Extreme mass ratio inspiral formation with local two-body relaxation and post-Newtonian dynamics Authors: Davide Mancieri, Luca Broggi, Matteo Bonetti, Alberto Sesana DOI: https://doi.org/10.1051/0004-6361/202452306 Contact: [e-mail]本数据集提供了无自转大质量黑洞（massive black hole, MBH）周围极端质量比旋近（Extreme Mass Ratio Inspiral, EMRI）的偏心率分布。数据集涵盖了从宿主核星团脱耦时，以及最终旋近坠入时的偏心率分布，共针对5种不同的中心黑洞质量进行了计算。 ## 内容说明 该压缩包内含一个文件夹，其中包含5个CSV文件，每个文件对应一种不同的中心大质量黑洞（MBH）质量（单位为太阳质量）。次级天体质量固定为10倍太阳质量（M☉）。所有文件均包含以下列： - p0 [Rg]：脱耦于核星团时的半通径（semi-latus rectum，以MBH引力半径（gravitational radius）为单位） - e0：脱耦时的偏心率 - ppl [Rg]：旋近坠入时的半通径（FastEMRIWaveforms（FEW）终止计算时的位置） - epl：旋近坠入时的偏心率（FEW终止计算时的位置） - weight：单次模拟的天体物理权重（每个文件内的权重总和为1） ## 方法概述 本部分简要概述了生成该数据集所用的方法，详细细节可参见配套论文（Mancieri+25b [arXiv:2509.02394] 与 Mancieri+25a [A&A 694, A272; arXiv:2409.09122]）。 初始EMRI样本取自Mancieri等人2025年的蒙特卡洛模拟（Monte Carlo simulation）（A&A 694, A272; arXiv:2409.09122）。在该模拟中，EMRI的演化被追踪至其从核星团脱耦、仅由引力波（gravitational wave, GW）辐射主导其轨道演化的阶段。该脱耦条件由引力波辐射时标 t_GW = 10^-3 * t_rlx 定义，其中 t_rlx 为核星团内两体相互作用（two-body interactions）导致的角动量弛豫时标（angular-momentum relaxation time）。本数据集中的(p0, e0)参数通常对应EMRI脱耦时刻的轨道参数。但出于计算成本的考量，部分EMRI（主要存在于高MBH质量模型中）未能达到该脱耦条件便提前终止了模拟（详见Mancieri+25a的第3.4节）。当无法满足 t_GW = 10^-3 * t_rlx 这一条件时，我们会在 10^-3 < t_GW / t_rlx < 1 的阶段记录(p0, e0)参数。在此区间内，弛豫过程已处于次主导地位，尽管其残余影响可能会使完全脱耦时的轨道参数与完全忽略弛豫过程的积分结果之间存在小幅偏差。 基于上述初始条件，我们使用FastEMRIWaveforms（FEW）软件包（v2.0.0）将每一个EMRI演化至旋近坠入阶段（ppl, epl）。演化过程分为两个阶段： 1. PN5模块用于早期旋近阶段，直至轨道偏心率达到0.8~0.9（临界值随半通径p变化）； 2. 此后切换至KerrEccEqFlux模块，直至轨道旋近坠入。 在本数据集生成时，KerrEccEqFlux模块仅适用于偏心率e < 0.8~0.9的轨道。 积分过程会在达到史瓦西分界面（Schwarzschild separatrix）时终止，此时 p_sep = 6 + 2e。但FEW软件包会返回分界面之前的最后一个计算点，满足 ppl - (6 + 2*epl) = 2*10^-3。仅有极少部分EMRI（对应Mancieri+25b中图5中e>0.82的尾部样本）在进入KerrEccEqFlux模块的有效计算域前便已旋近坠入，此时最终参数满足 ppl - (6 + 2*epl) = 0.1。 ## 天体物理权重 每个EMRI均带有一个权重，用于表征其在天体物理EMRI种群中的预期贡献占比。该权重取决于Mancieri等人2025年蒙特卡洛模拟中使用的初始半长轴 a_i。有两种相互竞争的效应共同决定了EMRI形成率随 a_i 的变化关系： 1. 弛豫过程在大半长轴区间内更高效地将致密天体散射至MBH附近； 2. 轨道宽且高偏心率的天体往往会直接旋近坠入，而非形成长寿命的EMRI。 这两种效应共同作用，使得EMRI形成率在 a_i = 10^-2 R_inf（其中 R_inf 为MBH的影响半径）附近达到峰值。由于不同 a_i 对应的模拟次数 n_i 各不相同，单次模拟的最终权重为：w_i = (1 / n_i) * <dot{N}_i> / (sum_j <dot{N}_j>)，其中 <dot{N}_i> 为该 a_i 对应的时间平均EMRI产生率。 ## 引用说明 若使用本数据集，请引用其配套论文与数据集本身： 1. 论文标题：*Eccentricity distribution of extreme mass ratio inspirals* 作者：Davide Mancieri、Luca Broggi、Morgan Vinciguerra、Alberto Sesana、Matteo Bonetti DOI：https://doi.org/10.48550/arXiv.2509.02394 2. 论文标题：*Hanging on the cliff: Extreme mass ratio inspiral formation with local two-body relaxation and post-Newtonian dynamics* 作者：Davide Mancieri、Luca Broggi、Matteo Bonetti、Alberto Sesana DOI：https://doi.org/10.1051/0004-6361/202452306 联系方式：[e-mail]