gwModel_kick_q200: GW Recoil Model

• gwModel_kick_q200 is an analytic model that predicts gravitational-wave recoil velocities from binary black hole mergers, defined for mass ratios from unity to 200.
• It employs a hybrid post-Newtonian and numerical relativity approach with 21 calibrated parameters, achieving residuals below 25 km/s for accurate extreme-mass-ratio predictions.
• The model underpins astrophysical applications such as black hole retention in star clusters and AGN disks, outperforming older models with enhanced physical consistency.

The term gwModel_kick_q200 denotes an analytic model for computing the gravitational-wave recoil (kick) velocity imparted to the remnant of a binary black hole merger, focused on binaries with mass ratios ranging from unity to extreme values ($q \sim 200$), and with aligned component spins. This model is specifically designed to remain accurate and physically well-behaved in the extreme-mass-ratio regime relevant for astrophysical applications, including black hole retention in star clusters and hierarchical mergers in dense stellar environments (Islam et al., 14 Nov 2025).

1. Physical Origin of Gravitational-Wave Kicks

When two black holes merge, the anisotropic emission of gravitational waves results in the transfer of linear momentum from the system to the gravitational field, imparting a recoil or “kick” to the remnant. The leading-order contribution to the recoil is governed by the Fitchett term, scaling as $\eta^2 \delta m$, where $\eta = q/(1+q)^2$ is the symmetric mass ratio, $q = m_2/m_1$, and $\delta m = (q-1)/(q+1)$ measures asymmetry between the component masses. Spin-orbit and spin-spin couplings, particularly for spins aligned or anti-aligned with the orbital angular momentum, contribute additional terms at higher post-Newtonian orders. Explicitly, the recoil vanishes by symmetry as $q \to 1$ and decays as $q^{-2}$ for $q \gg 1$ (Islam et al., 14 Nov 2025, Morawski et al., 2018).

2. Mathematical Structure of gwModel_kick_q200

The gwModel_kick_q200 model is built on the classical post-Newtonian ansatz for the kick, refined and extended via comprehensive calibration to both numerical relativity (NR) simulations and black hole perturbation theory (BHPT) results:

$$v_\text{kick}(q, \chi_{1z}, \chi_{2z}) = \sqrt{V_m^2 + V_\perp^2 + 2V_m V_\perp \cos \xi}$$

where

$V_m = A \eta^2 \delta m (1 + B\eta + C\eta^2)$

$$V_\perp = H \eta^2 \Big[ \Delta_\parallel + H_{2a}\tilde S_\parallel \delta m + H_{2b}\Delta_\parallel \tilde S_\parallel + H_{3a}\Delta_\parallel^2\delta m + \dots + H_{4f}\tilde S_\parallel \Delta_\parallel^3 \Big]$$

$$\xi = a + b\tilde S_\parallel + c\delta m\Delta_\parallel$$

with auxiliary variables:

• $\eta = q/(1+q)^2$
• $\delta m = (q-1)/(q+1)$
• $$\tilde S_\parallel = (\chi_{1z} + q^2 \chi_{2z})/(1+q)^2$$
• $\Delta_\parallel = (\chi_{1z} - q\chi_{2z})/(1+q)$

All coefficients ($A$, $B$, $C$, $H$, $H_{2a}$, $a$, $b$, $c$, etc.) are determined by non-linear least squares fitting to NR and BHPT data, with best-fit values and uncertainties reported in (Islam et al., 14 Nov 2025).

3. Training Data, Domain of Applicability, and Calibration

The model is calibrated using:

• 494 aligned-spin SXS NR simulations ($q \leq 20$)
• Additional RIT NR simulations at $q = 16$, $32$ (aligned) and $q \in [1.08, 128]$ (nonspinning)
• $\sim$100 Teukolsky-based BHPT runs (for $q = 200$ and several representative spin values)

This extensive dataset spans symmetric mass ratios from $\eta \sim 0.25$ (equal masses) down to the extreme-mass-ratio $\eta \sim 0.005$ ($q = 200$), and dimensionless spins $\chi_{1z}, \chi_{2z} \in [-1,1]$. All 21 model parameters are fitted with 5-fold cross-validation, yielding residuals typically below $25\,\text{km\,s}^{-1}$ and outperforming previous analytic and surrogate models in the covered domain (Islam et al., 14 Nov 2025).

Model	Domain Covered	Max $|q|$	Sub-30 km/s Residuals?	Limiting Behavior
gwModel_kick_q200	$q\in[1,200]$, aligned spins	200	Yes	Physically correct
HLZ (Lousto–Zlochower)	$q$ up to 16, older fit	16	No	Poor for $q>8$
NRSur3dq8Remnant	$q\in[1,8]$, aligned spins	8	Yes (within domain)	Diverges for $q>8$

4. Behavior in Extreme-Mass-Ratio and Spin Limits

gwModel_kick_q200 enforces physical limiting behavior:

• For $q \to 1$, $\delta m \to 0,\, \eta \to \frac{1}{4}$, so $V_m \to 0$ and thus $v_\text{kick} \to 0$ by symmetry.
• For $q \to \infty$, $\eta \sim 1/q,\; \delta m \to 1$, so $V_m \sim A\,q^{-2}$, $V_\perp \sim H\,q^{-2}$, $v_\text{kick} \sim q^{-2}\to 0$.
• The nonspinning limit correctly reduces to the Fitchett formula, with kicks scaling as $\eta^2 \delta m$.
• The analytic form prevents spurious oscillations or growth outside the calibration region, unlike GPR surrogate models (Islam et al., 14 Nov 2025).

5. Numerical Performance and Implementation

gwModel_kick_q200 achieves $R^2=0.9919$ across all aligned-spin training data up to $q=200$, with RMS errors between $10$–$15$ km/s. For applications, a python implementation is provided in the gwModels package:

from gwModels.kick import gwModel_kick_q200 q = 3.4 chi1 = 0.8 chi2 = -0.3 v_kick = gwModel_kick_q200(q, chi1, chi2) print(f"Remnant kick: {v_kick:.1f} km/s")

The model is optimized for speed, allowing its direct use in large $N$-body or Monte Carlo simulations and population-synthesis pipelines. Calls outside $q>200$ or $|\chi|>1$ are not permitted (Islam et al., 14 Nov 2025).

6. Astrophysical Applications and Retention Probability

In semi-analytic and cluster modeling, gwModel_kick_q200 enables rapid evaluation of black hole retention in globular clusters or AGN disks. Given the cluster escape velocity $v_\mathrm{esc}$, the retention probability is

$$P_\text{retain} = \langle \Theta(v_\mathrm{esc} - v_\text{kick}(q, \chi_{1}, \chi_{2})) \rangle$$

where the angle brackets denote averaging over spin magnitude/orientation distributions. Use of gwModel_kick_q200 has demonstrated increased remnant retention by 5–10% in low-mass clusters ($v_\mathrm{esc} \sim 50\,\text{km\,s}^{-1}$) compared to older HLZ-based prescriptions, with consequential impacts on projected rates for hierarchical mergers and intermediate mass black hole formation (Islam et al., 14 Nov 2025, Morawski et al., 2018).

7. Context Within the Broader Modeling Landscape

Comparison with alternative approaches reveals major improvements:

• HLZ models exhibit significant systematic errors and fail for $q > 8$.
• Surrogates such as NRSur3dq8Remnant provide high accuracy within their calibration regime ($q \le 8$) but extrapolate poorly.
• The semi-analytic PN+NR hybrid approach in gwModel_kick_q200 delivers consistent, accurate, and physically constrained kick predictions up to $q = 200$.
• For general spin orientations (precessing binaries), the related gwModel_kick_prec_flow normalizing flow model is available for full three-dimensional recoil distributions (Islam et al., 14 Nov 2025).

Crucially, the rigorous enforcement of physical limiting behavior allows gwModel_kick_q200 to be reliably utilized in settings (e.g., AGN disks, globular clusters) where mass ratio and spin exceed the reach of prior NR-calibrated surrogates. Integrating this model supports more accurate predictions for remnant locations, potential electromagnetic counterparts, and hierarchical black hole growth scenarios.