LISA Eccentricity Astrophysics Package (LEAP)

• LEAP is a Python package that simulates and analyzes eccentric stellar-mass binary black holes in the LISA band using multi-harmonic waveform techniques.
• It integrates simulated catalogs, orbital evolution models, and post-Newtonian waveform generation to differentiate dynamical formation channels.
• LEAP provides practical insights into merger rates and detectability thresholds, enhancing our understanding of Galactic and cosmological BBH populations.

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 (Xuan et al., 14 May 2026). 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/ (Xuan et al., 14 May 2026). It is presented as a simulation/catalog and waveform-detectability toolkit rather than as a generic first-principles population-synthesis framework (Xuan et al., 14 May 2026).

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 (Breivik et al., 2016). 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 (Randall et al., 2019).

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 (Xuan et al., 14 May 2026). 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 (Xuan et al., 14 May 2026). The package paper states that these channels are used first for the local Milky Way and then extrapolated to the local or cosmological universe (Xuan et al., 14 May 2026).

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 $f_{\rm BBH}=7\times10^{-4}$ (Xuan et al., 14 May 2026). The resulting present-day merger rate is $\sim 3\times10^{-7}\,\mathrm{yr}^{-1}$ per Milky-Way-like galaxy, corresponding to $\sim 3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$ (Xuan et al., 14 May 2026).

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 (Xuan et al., 14 May 2026). 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 $0.6\,M_\odot$0 under the adopted galaxy-density conversion (Xuan et al., 14 May 2026). A distinct young nuclear cluster component is also included and is expected to contribute roughly $0.6\,M_\odot$1 LISA-detectable BBHs (Xuan et al., 14 May 2026).

For the globular-cluster channel, LEAP uses the CMC Cluster Catalog and calibrates Monte Carlo $0.6\,M_\odot$2-body models to observed Milky Way globular-cluster properties, sampling late-time snapshots and extracting both in-cluster and ejected BBH populations (Xuan et al., 14 May 2026). This channel gives a Milky Way merger rate $0.6\,M_\odot$3, or $0.6\,M_\odot$4 (Xuan et al., 14 May 2026).

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

Channel	Merger rate	Typical mHz properties
Field fly-by	$0.6\,M_\odot$5 MW; $0.6\,M_\odot$6	$0.6\,M_\odot$7, $0.6\,M_\odot$8
Galactic nucleus EKL	$0.6\,M_\odot$9 MW; $50\,\mathrm{km\,s^{-1}}$0	$50\,\mathrm{km\,s^{-1}}$1, $50\,\mathrm{km\,s^{-1}}$2
Globular cluster	$50\,\mathrm{km\,s^{-1}}$3 MW; $50\,\mathrm{km\,s^{-1}}$4	in-cluster: $50\,\mathrm{km\,s^{-1}}$5, $50\,\mathrm{km\,s^{-1}}$6; ejected: $50\,\mathrm{km\,s^{-1}}$7, $50\,\mathrm{km\,s^{-1}}$8

Summed over the three principal channels, the package paper gives an overall volumetric merger rate $50\,\mathrm{km\,s^{-1}}$9 (Xuan et al., 14 May 2026). It also states that this catalog is conservative because it omits isolated binary evolution, field triples, AGN channels, and other possible dynamical channels (Xuan et al., 14 May 2026).

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 (Xuan et al., 14 May 2026). 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 (Xuan et al., 14 May 2026). 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 (Randall et al., 2019).

For a 10-year LISA mission, the package paper predicts the following Milky Way counts above total SNR threshold (Xuan et al., 14 May 2026).

Threshold	$10\,\mathrm{Gyr}$0
$10\,\mathrm{Gyr}$1	$10\,\mathrm{Gyr}$2
$10\,\mathrm{Gyr}$3	$10\,\mathrm{Gyr}$4
$10\,\mathrm{Gyr}$5	$10\,\mathrm{Gyr}$6
$10\,\mathrm{Gyr}$7	$10\,\mathrm{Gyr}$8
$10\,\mathrm{Gyr}$9	$f_{\rm BBH}=7\times10^{-4}$0

The same study predicts $f_{\rm BBH}=7\times10^{-4}$1 extragalactic mHz BBHs with $f_{\rm BBH}=7\times10^{-4}$2, $f_{\rm BBH}=7\times10^{-4}$3 with $f_{\rm BBH}=7\times10^{-4}$4, and $f_{\rm BBH}=7\times10^{-4}$5 with $f_{\rm BBH}=7\times10^{-4}$6 (Xuan et al., 14 May 2026). It also gives mean counts of individually detectable Milky Way harmonics of $f_{\rm BBH}=7\times10^{-4}$7 for $f_{\rm BBH}=7\times10^{-4}$8, $f_{\rm BBH}=7\times10^{-4}$9 for $\sim 3\times10^{-7}\,\mathrm{yr}^{-1}$0, $\sim 3\times10^{-7}\,\mathrm{yr}^{-1}$1 for $\sim 3\times10^{-7}\,\mathrm{yr}^{-1}$2, and $\sim 3\times10^{-7}\,\mathrm{yr}^{-1}$3 for $\sim 3\times10^{-7}\,\mathrm{yr}^{-1}$4 (Xuan et al., 14 May 2026).

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 (Xuan et al., 14 May 2026). The same paper states that those harmonics may mimic circular binaries with systematically biased chirp masses (Xuan et al., 14 May 2026). This is consistent with the earlier argument that number counts as a function of frequency, not only direct $\sim 3\times10^{-7}\,\mathrm{yr}^{-1}$5-measurement, carry formation-channel information (Randall et al., 2019).

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

$\sim 3\times10^{-7}\,\mathrm{yr}^{-1}$6

chosen because it reduces smoothly to $\sim 3\times10^{-7}\,\mathrm{yr}^{-1}$7 in the circular limit (Xuan et al., 14 May 2026). 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 $\sim 3\times10^{-7}\,\mathrm{yr}^{-1}$8 and

$\sim 3\times10^{-7}\,\mathrm{yr}^{-1}$9

up to 2PN (Xuan et al., 14 May 2026).

For signal representation, the package paper distinguishes an individual-harmonic characteristic strain,

$\sim 3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$0

from a smoothed-envelope representation for highly eccentric sources,

$\sim 3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$1

with the slow-evolution relation

$\sim 3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$2

These formulas encode the fact that eccentric sources can accumulate significant total SNR even when no single harmonic dominates (Xuan et al., 14 May 2026).

The package paper also gives the chirp mapping relevant to circular-template confusion: $\sim 3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$3 with

$\sim 3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$4

If the $\sim 3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$5-th harmonic is interpreted with the circular chirp formula, the apparent chirp mass is

$\sim 3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$6

so a highly eccentric stellar-mass BBH can masquerade as a much lower chirp-mass circular binary (Xuan et al., 14 May 2026).

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

$\sim 3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$7

and remain useful up to roughly

$\sim 3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$8

in the mHz band, at lower frequencies for higher masses (Xuan et al., 14 May 2026). In adjacent waveform-model development, the EFPE_ME family extends fully precessing eccentric inspiral modeling to $\sim 3~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$9 and shows that waveform differences become significant for $$\Gamma_{\rm rep}\sim3\times10^{-6}\,\mathrm{yr}^{-1}$$0, especially below $$\Gamma_{\rm rep}\sim3\times10^{-6}\,\mathrm{yr}^{-1}$$1 in the LISA band (Arredondo et al., 2024). 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

$$\Gamma_{\rm rep}\sim3\times10^{-6}\,\mathrm{yr}^{-1}$$2

and with significant fractions of merging binaries in isolated triples and galactic centers showing observable KL behavior (Randall et al., 2019). 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 (Hoang et al., 2019).

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 $$\Gamma_{\rm rep}\sim3\times10^{-6}\,\mathrm{yr}^{-1}$$3 for favorable low-mass systems in vacuum, with thresholds degrading toward $$\Gamma_{\rm rep}\sim3\times10^{-6}\,\mathrm{yr}^{-1}$$4 for heavier systems (Garg et al., 2023). Another shows that eccentricity and gas-induced perturbations can be jointly fit only with degraded accuracy, with a vacuum threshold $$\Gamma_{\rm rep}\sim3\times10^{-6}\,\mathrm{yr}^{-1}$$5 worsening to $$\Gamma_{\rm rep}\sim3\times10^{-6}\,\mathrm{yr}^{-1}$$6 when gas is also modeled, and with weak gas perturbations to circular binaries being mimicable by eccentric vacuum inspirals (Garg et al., 2024).

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 (Xuan et al., 14 May 2026). 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 (Xuan et al., 14 May 2026).

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 (Garg et al., 2024). 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 (Key et al., 2010).

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 (Derdzinski et al., 11 Dec 2025). 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 (Chen et al., 7 May 2026). 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 (Xuan et al., 14 May 2026).