---
title: LISA Eccentricity Astrophysics Package (LEAP)
url: https://www.emergentmind.com/topics/lisa-eccentricity-astrophysics-package-leap
type: topic
---

# LISA Eccentricity Astrophysics Package (LEAP)

The **LISA Eccentricity Astrophysics Package (LEAP)** is a public, open-source Python package introduced to make eccentric compact-binary astrophysics in the LISA band practical, with a primary focus on dynamically formed stellar-mass binary black holes (BBHs) in the Milky Way and in cosmological extensions [2605.15265]. In the formulation associated with its release, LEAP combines a simulated catalog with binary orbital evolution, post-Newtonian time-domain waveform generation, signal-to-noise-ratio estimation in the mHz band, and data-analysis tools tailored for eccentric systems; the simulated catalog and code are publicly available at `https://github.com/zeyuanxuan/lisa-leap/` [2605.15265]. It is presented as a simulation/catalog and waveform-detectability toolkit rather than as a generic first-principles population-synthesis framework [2605.15265].

## 1. Definition and scientific remit

LEAP is situated in a branch of LISA science where eccentricity is not a perturbative nuisance but a primary astrophysical observable. Earlier LISA studies established that orbital eccentricity in the space-based band can discriminate among compact-binary formation channels, including binaries formed in isolation and those assembled dynamically in dense stellar environments, because the low-frequency inspiral retains information that is erased by the time the system reaches ground-based detectors [1606.09558]. Related work further showed that eccentricity affects not only individual-source inference but also the observable source-count distribution as a function of frequency, so that formation channels can be constrained even without explicit eccentricity measurements for every system [1907.02283].

Within that landscape, LEAP is explicitly aimed at the stellar-mass BBH sector of LISA astrophysics. The package addresses the practical difficulty that highly eccentric stellar-mass binaries in the mHz band are not well described by slowly chirping, quasi-circular templates: they radiate in repeated periapsis bursts, with power distributed over many harmonics, and can remain in wide, long-lived eccentric configurations [2605.15265]. This suggests a division of labor in LISA eccentricity studies: broad population and phenomenology on one side, and package-level tools for catalogs, waveforms, and detectability on the other.

## 2. Astrophysical populations and catalog construction

The catalog released with LEAP includes three dynamically formed stellar-mass BBH channels: **Galactic field fly-by interactions**, **Galactic nucleus binaries driven by eccentric Kozai–Lidov evolution**, and **globular-cluster binaries drawn from cluster dynamical simulations** [2605.15265]. The package paper states that these channels are used first for the local Milky Way and then extrapolated to the local or cosmological universe [2605.15265].

For the **field fly-by channel**, the adopted ingredients are wide BBHs in the Galactic disk with initial separations \(a\sim10^2-10^4\,\mathrm{au}\), a fixed perturber mass \(0.6\,M_\odot\), stellar velocity dispersion \(50\,\mathrm{km\,s^{-1}}\), a starburst \(10\,\mathrm{Gyr}\) ago, and a wide-BBH fraction \(f_{\rm BBH}=7\times10^{-4}\) [2605.15265]. The resulting present-day merger rate is \(\sim 3\times10^{-7}\,\mathrm{yr}^{-1}\) per Milky-Way-like galaxy, corresponding to \(\sim 3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}\) [2605.15265].

For the **Galactic nucleus channel**, the package uses hierarchical-triple evolution around the Galactic-center SMBH, modeled with octupole-order secular dynamics, GR precession, GW emission, isotropic initial inner-binary orientations, and hierarchical-stability cuts [2605.15265]. In the main nuclear population, the model assumes continuous steady-state replenishment with \(\Gamma_{\rm rep}\sim3\times10^{-6}\,\mathrm{yr}^{-1}\), radial distribution \(\rho\propto r^{-2}\), and a Milky Way BBH merger rate \(2\times10^{-7}\,\mathrm{yr}^{-1}\), equivalent to \(\sim 2~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}\) under the adopted galaxy-density conversion [2605.15265]. A distinct young nuclear cluster component is also included and is expected to contribute roughly \(\sim 1-4\) LISA-detectable BBHs [2605.15265].

For the **globular-cluster channel**, LEAP uses the **CMC Cluster Catalog** and calibrates Monte Carlo \(N\)-body models to observed Milky Way globular-cluster properties, sampling late-time snapshots and extracting both in-cluster and ejected BBH populations [2605.15265]. This channel gives a Milky Way merger rate \(4\times10^{-7}\,\mathrm{yr}^{-1}\), or \(\sim 4~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}\) [2605.15265].

The package paper summarizes the characteristic mHz-band properties of these populations as follows.

| Channel | Merger rate | Typical mHz properties |
|---|---:|---|
| Field fly-by | \(3\times10^{-7}\,\mathrm{yr}^{-1}\) MW; \(\sim 3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}\) | \(a_{\rm mid}\sim 7\,\mathrm{au}\), \(e_{\rm mid}\sim 0.999\) |
| Galactic nucleus EKL | \(2\times10^{-7}\,\mathrm{yr}^{-1}\) MW; \(\sim 2~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}\) | \(a_{\rm mid}\sim 0.1\,\mathrm{au}\), \(e_{\rm mid}\sim 0.7\) |
| Globular cluster | \(4\times10^{-7}\,\mathrm{yr}^{-1}\) MW; \(\sim 4~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}\) | in-cluster: \(a_{\rm mid}\sim 0.2\,\mathrm{au}\), \(e_{\rm mid}\sim 0.9\); ejected: \(a_{\rm mid}\sim 0.09\,\mathrm{au}\), \(e_{\rm mid}\sim 0.6\) |

Summed over the three principal channels, the package paper gives an overall volumetric merger rate \(\Gamma\sim 9~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}\) [2605.15265]. It also states that this catalog is conservative because it omits isolated binary evolution, field triples, AGN channels, and other possible dynamical channels [2605.15265].

## 3. Detectability in the LISA band

LEAP’s detectability layer is built around the fact that eccentric stellar-mass BBHs in the mHz band are long-lived yet intrinsically weak, with the Milky Way therefore dominating the resolvable population [2605.15265]. The package paper adopts a 10-year LISA observation for its headline source counts and distinguishes between simple burst-based estimates and the full harmonic-summed SNR used in the actual forecasts [2605.15265]. In the broader literature, the dependence of source counts on eccentricity is traced to three coupled effects: eccentricity shifts the peak GW frequency, increases the source occupancy needed to sustain a fixed merger rate, and reduces SNR at fixed peak frequency [1907.02283].

For a 10-year LISA mission, the package paper predicts the following Milky Way counts above total SNR threshold [2605.15265].

| Threshold | \(N_{\rm MW}\) |
|---|---:|
| \({\rm SNR}>1\) | \(36.38\) |
| \({\rm SNR}>3\) | \(12.95\) |
| \({\rm SNR}>8\) | \(4.74\) |
| \({\rm SNR}>20\) | \(2.26\) |
| \({\rm SNR}>50\) | \(1.04\) |

The same study predicts \(\sim 490\) extragalactic mHz BBHs with \({\rm SNR}>1\), \(\sim 18\) with \({\rm SNR}>3\), and \(\sim 1\) with \({\rm SNR}>8\) [2605.15265]. It also gives mean counts of individually detectable Milky Way harmonics of \(\sim 240\) for \({\rm SNR}>1\), \(\sim 78\) for \({\rm SNR}>3\), \(\sim 29\) for \({\rm SNR}>8\), and \(\sim 6.7\) for \({\rm SNR}>20\) [2605.15265].

A central package-level conclusion is that eccentric stellar-mass BBHs are relevant to the **LISA global fit** because a single highly eccentric BBH can appear as a cluster of individually resolvable harmonics [2605.15265]. The same paper states that those harmonics may mimic circular binaries with systematically biased chirp masses [2605.15265]. This is consistent with the earlier argument that number counts as a function of frequency, not only direct \(e\)-measurement, carry formation-channel information [1907.02283].

## 4. Waveform modeling and signal representation

The waveform side of LEAP is built around the fact that highly eccentric binaries are multi-harmonic and burst-like. The package paper adopts the peak-frequency convention
\[
f_{\rm GW}\sim 2f_{\rm orb}(1-e)^{-3/2},
\]
chosen because it reduces smoothly to \(f_{\rm GW}=2f_{\rm orb}\) in the circular limit [2605.15265]. It uses the **x-model** of Hinder et al. and Huerta et al. for time-domain eccentric waveforms, with conservative dynamics up to **3PN** and radiation reaction mapped to the evolution of \(e\) and
\[
x\equiv (\omega M)^{2/3}
\]
up to **2PN** [2605.15265].

For signal representation, the package paper distinguishes an individual-harmonic characteristic strain,
\[
h_{c,n}=\sqrt{2}\,h_n\sqrt{\min\left\{\frac{f_n^2}{\dot f_n},\,f_nT_{\rm obs}\right\}},
\]
from a smoothed-envelope representation for highly eccentric sources,
\[
h_{c,\rm env}(f)=\sqrt{2}\,h_n(f)\,\sqrt{\frac{f^2T_{\rm obs}}{f_{\rm orb}}},
\]
with the slow-evolution relation
\[
h_{c,n}=h_{c,\rm env}(f_n)\sqrt{\frac{f_{\rm orb}}{f_n}}.
\]
These formulas encode the fact that eccentric sources can accumulate significant total SNR even when no single harmonic dominates [2605.15265].

The package paper also gives the chirp mapping relevant to circular-template confusion:
\[
\dot f_{\rm orb}= \frac{96}{5}\pi^{8/3}\mathcal{M}_{c,0}^{5/3}f_{\rm orb}^{11/3} \frac{F(e)}{2^{8/3}},
\]
with
\[
F(e)=\frac{1+\frac{73}{24}e^2+\frac{37}{96}e^4}{(1-e^2)^{7/2}}.
\]
If the \(n\)-th harmonic is interpreted with the circular chirp formula, the apparent chirp mass is
\[
\mathcal{M}_c'=\mathcal{M}_{c,0}F(e)^{3/5}\left(\frac{2}{n}\right)^{8/5},
\]
so a highly eccentric stellar-mass BBH can masquerade as a much lower chirp-mass circular binary [2605.15265].

The package paper further argues that PN methods remain reliable for stellar-mass eccentric BBHs in the mHz band because the periapsis remains in the weak-field regime. Its overlap study concludes that PN waveforms converge well for stellar-mass BBHs when
\[
f_{\rm GW}\lesssim 10\,\mathrm{mHz},
\]
and remain useful up to roughly
\[
M\lesssim 10^3\,M_\odot
\]
in the mHz band, at lower frequencies for higher masses [2605.15265]. In adjacent waveform-model development, the **EFPE\_ME** family extends fully precessing eccentric inspiral modeling to \(e\le 0.8\) and shows that waveform differences become significant for \(e_0>0.5\), especially below \(10^5\,M_\odot\) in the LISA band [2402.06804]. This suggests a broader methodological context for LEAP even though the package paper itself centers on stellar-mass BBHs and the x-model.

## 5. Relation to broader eccentricity astrophysics

LEAP was introduced for dynamically formed stellar-mass BBHs, but the package sits within a wider literature in which eccentricity is used as an environmental and formation-channel tracer across several LISA source classes. For stellar-mass triples, Kozai–Lidov-driven eccentricity oscillations in the LISA band can be directly observable, with many systems populating
\[
0.1\,\mathrm{mHz}<f_p<0.1\,\mathrm{Hz},\qquad t_\Delta\lesssim O(10\,\mathrm{yr}),
\]
and with significant fractions of merging binaries in isolated triples and galactic centers showing observable KL behavior [1902.08604]. A related proof-of-concept for BBHs near supermassive black holes argues that SMBH-driven eccentricity oscillations should be detectable with LISA out to a few Mpc, thereby isolating a galactic-nucleus channel [1903.00134].

For **massive black hole binaries**, the adjacent science case is different but conceptually related. One study of LISA MBHBs finds a minimum measurable eccentricity of roughly \(e_{\rm min}\sim10^{-2.75}\) for favorable low-mass systems in vacuum, with thresholds degrading toward \(\sim10^{-1.5}\) for heavier systems [2307.13367]. Another shows that eccentricity and gas-induced perturbations can be jointly fit only with degraded accuracy, with a vacuum threshold \(e_0\gtrsim10^{-2.75}\) worsening to \(e_0\gtrsim10^{-2}\) when gas is also modeled, and with weak gas perturbations to circular binaries being mimicable by eccentric vacuum inspirals [2402.14058].

These neighboring results do not define LEAP’s current package scope, but they delimit the scientific ecosystem in which the package operates. A plausible implication is that a stellar-mass eccentricity package naturally interfaces with broader LISA efforts on tertiary perturbations, gas-rich environments, and multimessenger interpretation, even when the released package content is more narrowly focused.

## 6. Limitations, systematics, and prospective extensions

The package paper is explicit about its limitations. The released catalog is intentionally incomplete, omitting isolated binary evolution, field triples, AGN channels, and other dynamical channels; the cosmological extrapolation assumes Milky-Way-like orbital-parameter distributions; and the Galactic-nucleus channel depends on uncertain assumptions about spatial distributions, replenishment, and orientations [2605.15265]. On the waveform side, the same paper cautions that PN tools are robust in the eccentric stellar-mass to intermediate-mass, mHz, weak-field regime, but are not universally reliable near the high-frequency edge of LISA or for very massive systems [2605.15265].

More generally, the surrounding literature shows that eccentricity inference is tightly coupled to waveform-systematics control. For LISA MBHBs, neglecting residual eccentricity or modest gas-disc effects can produce false violations of GR in several PN orders when high-SNR signals are analyzed with circular-vacuum templates [2410.02910]. Earlier end-to-end studies of spinning, eccentric massive black hole binaries likewise found that failing to include eccentricity in the waveform can bias masses and spins and lose signal power, even though LISA should measure eccentricity one year before merger to parts in a thousand for typical sources [1006.3759].

Environmental modeling remains a major uncertainty frontier. In AGN-like disc IMRI studies, code-comparison work finds that thin-disc torques can disagree in both magnitude and sign across hydrodynamical methods, especially because the Hill-sphere flow dominates the net result in the weakly nonlinear regime [2512.10893]. This suggests that any future LEAP extension toward gas-coupled eccentric evolution would require wide theory-error envelopes rather than a single deterministic prescription.

A further extension concerns unresolved populations. Recent work on the LISA stochastic signal from eccentric stellar-mass BBHs shows that the SGWB can distinguish highly eccentric populations from quasi-circular ones, can separate eccentric vacuum evolution from sufficiently dense environmental effects, and can place an upper bound on the maximum eccentricity of the sBBH population in the ground-based band [2605.05537]. That direction lies beyond the present package release, but it identifies a natural population-level continuation of the LEAP program: from individual eccentric repeated-burst sources to eccentric foregrounds and stochastic backgrounds.

In that sense, LEAP is best understood as an initial computational infrastructure for a specific but increasingly central LISA problem: how to represent, count, detect, and interpret eccentric stellar-mass BBHs whose signals are distributed over many harmonics and whose astrophysical content is inseparable from their non-circular dynamics [2605.15265].

Source: https://www.emergentmind.com/topics/lisa-eccentricity-astrophysics-package-leap