Eccentricity distribution of extreme mass ratio inspirals

TL;DR

This work derives realistic EMRI eccentricity distributions for the dominant two-body relaxation channel in nuclear star clusters around Schwarzschild MBHs by evolving end-state EMRIs to plunge with the FEW framework and applying an astrophysically informed weighting across initial semi-major axes. It finds that EMRIs can retain significant eccentricities at plunge, with a peak near $e_ ext{pl}\approx 0.2$ and a notable tail extending past $e_ ext{pl}>0.5$ for certain MBH masses, including contributions from wide and cliffhanger EMRIs. The study also reveals that current FEW flux grids inadequately sample the full parameter space at low MBH masses ($M_ullet<10^6 M_\odot$), with up to $ oughly 75\%$ of EMRIs lying outside the available grid two years before plunge for $M_ullet=10^5 M_\odot$, underscoring the need for extended grids and improved interpolation. These findings have important implications for accurate EMRI waveform modeling, detection strategies for LISA, and disentangling EMRI formation channels from gravitational-wave data.

Abstract

We present realistic eccentricity distributions for extreme mass ratio inspirals (EMRIs) forming via the two-body relaxation channel in nuclear star clusters, tracking their evolution up to the final plunge onto the central Schwarzschild massive black hole (MBH). We find that EMRIs can retain significant eccentricities at plunge, with a distribution peaking at $e_\mathrm{pl} \approx0.2$, and a considerable fraction reaching much higher values. In particular, up to $20\%$ of the forming EMRIs feature $e_\mathrm{pl} > 0.5$ for central MBH masses $M_\bullet$ in the range $10^5 \mathrm{M_\odot} \leq M_\bullet \leq 10^6 \mathrm{M_\odot}$, partially due to EMRIs forming at large semi-major axes and "cliffhanger EMRI", usually neglected in literature. This highlights the importance of accounting for eccentricity in waveform modeling and detection strategies for future space-based gravitational wave observatories such as the upcoming Laser Interferometer Space Antenna (LISA). Furthermore, we find that the numerical fluxes in energy and angular momentum currently implemented in the FastEMRIWaveforms (FEW) package may not adequately sample the full parameter space relevant to low-mass MBHs ($M_\bullet < 10^6 \mathrm{M_\odot}$), potentially limiting its predictive power in that regime. Specifically, for $M_\bullet=10^5 \mathrm{M_\odot}$ we find that about $75\%$ ($50 \%$) of EMRIs at 2 years (6 months) from plunge fall outside the currently available flux parameter space. Our findings motivate the development of extended flux grids and improved interpolation schemes to enable accurate modeling of EMRIs across a broader range of system parameters.

Figures (10)

• Figure 1: Two example EMRIs forming and plunging into a $10^5 \, \mathrm{M_\odot}$ MBH. The early part of the integration (blue shaded tracks) is from 2025AA...694A.272M, while the final phase (blue solid tracks) is presented in this work. The gray dotted line marks the separatrix, and the gray shaded region below it is the "relativistic" loss cone. The loss cone is usually delimited by the pericenter condition $r_\mathrm{p} = 8 R_\mathrm{g}$, while here the separatrix corresponds to $r_\mathrm{p} = 4 R_\mathrm{g}$ for $e \to 1$ (see Sect. \ref{['sec:rates']}). The green dashed curve separates the region dominated by two-body relaxation (above) from that dominated by GWs emission (below). Black crosses mark the initial conditions from 2025AA...694A.272M, as well as the initial semi-major axes $a_\mathrm{i}$ used in this work to weight EMRIs (see Sect. \ref{['sec:rates']}). Black dots indicate the stopping points of runs in 2025AA...694A.272M, which also serve as initial conditions in this work. Both an initially wide ($a_\mathrm{i} > a_\mathrm{c}$) and a classical ($a_\mathrm{i} < a_\mathrm{c}$) EMRI are shown (see Sect. \ref{['sec:results']}).
• Figure 2: Integration of some of our initial conditions with the method described in Sect. \ref{['sec:int']}. The blue dots show the steps taken, while the gray dotted line is the separatrix. The orange shaded region displays the range covered by the precomputed flux grids, inside which the KerrEccEqFlux model can be used. We switch to this model from the PN5 one when crossing the dashed orange curve.
• Figure 3: In the upper panel, time dependent density profiles for stars (blue) and stellar-mass BHs (orange) in our nuclear star cluster model with an MBH of $10^5 \, \mathrm{M_\odot}$. Both components are initially distributed with an inner slope of 1.5 but different normalizations. Profiles are shown at three different times: the initial condition (dotted lines), $t_\mathrm{peak}$ (dashed lines), and $10 \, t_\mathrm{peak}$ (solid lines). In the lower panel, time evolution of the local slope of the density profiles as a function of radius $r$.
• Figure 4: In the upper panel, time dependent formation rate of EMRIs $\dot{N}_\mathrm{tot}$ as a function of time for $M_\bullet=10^5\, \mathrm{M_\odot}$. $\dot{N}_\mathrm{tot}$ is computed at each time as the sum of $\dot{N}_i (t)$ over all energy cells. Vertical dashed lines correspond to $t/t_\mathrm{peak} = 1/3$ (gray), $1$ (purple), $3$ (pink), and $10$ (yellow). In the lower panel, we show (solid green histogram) the average relative number $\langle \dot{N}_i\rangle / \sum_j \langle\dot{N}_j\rangle$ of EMRIs produced at semi-major axis $a_i$ (green dots). We also plot the relative number $\dot{N}_i(t) / \sum_j \dot{N}_j(t)$ at the times marked in the upper panel (dashed histograms, same colors as above). The relative number has a similar shape at all times, but its peak semi-major axis changes according to two-body relaxation: it first decreases because of mass segregation, and later increases because of diffusion.
• Figure 5: EMRI eccentricity distribution at plunge, for different central MBH masses. Here we report the overall probability density function of our EMRI population (blue histograms) and display the underneath distributions of "classical" and "initially wide" EMRIs (green and red shaded histograms, see Sect. \ref{['sec:results']} for details), which are normalized to recover the overall distribution when summed together. We also show (orange dashed histogram) the distribution reported in 2017PhRvD..95j3012B, which has become a standard assumption in the literature.