GW231123: A Case for Binary Microlensing in a Strong Lensing Field

TL;DR

GW231123 presents an embedded binary microlens model within a strong-lensing galaxy to explain its unusual waveform features. A Transformer-based neural network accelerates the wave-optics diffraction integral, enabling Bayesian inference that favors the embedded binary model over single-lens and unlensed scenarios, with Bayes factors $\log_{10}B^{Binary}_{Single}\approx1.34$ and $\log_{10}B^{Binary}_{Unlensed}\approx2.67$. The inferred lens masses are $M^z_{L,1}\approx714^{+239.1}_{-309.0}$ M$_\odot$ and $M^z_{L,2}\approx87.1^{+139.3}_{-72.7}$ M$_\odot$, magnification $\mu\approx5.56^{+2.78}_{-1.98}$, and external shear $\gamma\approx0.41^{+0.03}_{-0.05}$; the corresponding source-frame masses are $m_1^{src}\approx80.0^{+21.3}_{-14.4}$ M$_\odot$ and $m_2^{src}\approx62.0^{+19.8}_{-29.4}$ M$_\odot$ with spins $a_1\approx0.37^{+0.51}_{-0.33}$, $a_2\approx0.40^{+0.52}_{-0.35}$, bringing the event into consistency with the O1–O3 BH population. The work underscores the importance of multi-body and environmental propagation effects in microlensing analyses of gravitational-wave events.

Abstract

The unusual properties of GW231123, including component masses within the pair-instability mass gap ($137^{+22}_{-17}\mathrm{M}_\odot$ and $103^{+20}_{-52}\mathrm{M}_\odot$ at 90\% credible intervals) and extremely large spins near the Kerr limit, have challenged standard formation scenarios. While gravitational lensing has been proposed as an explanation, current millilensing studies suggest the signal consists of three overlapping images, a configuration that exceeds the predictions of the isolated point-mass lens model. In this work, we investigate a binary lens model embedded within a strong lensing galaxy. This is the simplest model that not only naturally produces the observed number of images but also aligns with the fact that microlensing objects usually reside in galaxies. To overcome the high computational cost of the diffraction integral required for wave optics, we constructed a Transformer-based neural network that accurately generates lensing waveforms within milliseconds per waveform. Using the NRSur7dq4 waveform model, we find primary and secondary lens masses of $714^{+239}_{-309} \mathrm{M}_\odot$ and $87^{+139}_{-73} \mathrm{M}_\odot$, respectively. We also find a strong lensing magnification of $5.56^{+2.78}_{-1.98}$ (at 90\% credible intervals) and a Bayes factor of $\log_{10}B^\mathrm{Binary}_\mathrm{Single}\simeq1.34$. This result underscores the necessity of considering multi-body and environmental effects in microlensing studies. More crucially, under this embedded binary lens interpretation, the inferred source-frame binary black hole masses ($80.0^{+21.3}_{-14.4} \mathrm{M}_\odot$ and $62.0^{+19.8}_{-29.4} \mathrm{M}_\odot$) and spins ($0.37^{+0.51}_{-0.33}$ and $0.40^{+0.52}_{-0.35}$) shift to values consistent with the current population constrained from O1--O3.

Figures (5)

• Figure 1: Histogram of mismatches for the geometric optics approximation ($\mathrm{MM_{Geo}}$) and the neural network predictions ($\mathrm{MM_{NN}}$) across the test dataset defined in Table \ref{['tab:test']}. The grey vertical dashed line represents the criterion (0.0028) for GW231123. Values below this line indicate that waveform uncertainties do not significantly affect parameter measurements.
• Figure 2: The time-domain whitened strain data, in units of noise standard deviation ($\sigma_\mathrm{noise}$), for the observed data (grey), the coherent wave burst (cWB) reconstruction (black), and the maximum-likelihood waveforms for the unlensed (blue), point-mass lens (green), and embedded binary lens (orange) models. The red box highlights the region where the embedded binary lens model provides a better fit to the cWB reconstruction.
• Figure 3: Posterior distributions for the source properties, including source-frame component masses ($m^\mathrm{src}_1$, $m^\mathrm{src}_2$), dimensionless spins ($a_1$ and $a_2$), and precession parameters ($\chi_p$ and $\chi_\mathrm{eff}$), under different hypotheses, as shown in the legend.
• Figure 4: Posterior distributions for the lens properties: external shear ($\gamma$), the redshifted primary lens mass ($M_{L,1}^z$), the redshifted secondary lens mass ($M_{L,2}^z$), and the coordinates of the primary ($x_{11}, x_{12}$) and secondary ($x_{21}, x_{22}$) lens components. The blue and orange curves represent the results obtained using the NRSur7dq4 and IMRPhenomXPHM-SpinTaylor waveform templates, respectively.
• Figure 5: The architecture of the neural network used for predicting the residual waveform, $F_\mathrm{residual} (f)$. Two independent models are trained to predict the amplitude and phase, respectively. Both models take the 7 lensing parameters as input and use a Transformer-based structure, which consists of an embedding layer, a Transformer Encoder block, and an output head.