The degeneracy between microlensing wave effect and precession in strongly lensed gravitational wave

TL;DR

This paper investigates the degeneracy between microlensing effects caused by stellar fields in strong gravitational lensing and intrinsic spin precession in gravitational waves, assessing how microlensing can mimic precession signals and affect parameter estimation.

Contribution

It provides a quantitative analysis of the degeneracy between stellar-field microlensing and spin precession in strongly lensed gravitational waves, highlighting conditions under which microlensing can produce false precession evidence.

Findings

Microlensing-induced false precession evidence is generally weak at current sensitivities.

Highly magnified events show a high likelihood (72%) of significant precession evidence due to microlensing.

A positive correlation exists between microlensing strength and precession evidence, especially in Type II SLGWs.

Abstract

Microlensing induced by the stellar field within a strong lensing galaxy can introduce fluctuations in the waveforms of strongly-lensed gravitational waves. When fitting these signals with templates that do not account for microlensing,possible degeneracies can lead to false evidence of certain intrinsic parameters,resulting in a misinterpretation of the properties of the underlying system. For example,the wave effect of microlensing may mimic spin precessions,as both effects generically induce periodic waveform modulations. Although previous studies suggest that lensing-induced modulations can be distinguished from precession using parameter estimation under a geometric-optic approximation,it does not directly apply for lensed image through stellar fields due to large number of stars involved an wave-optic effects. This study aims to evaluate the degree of degeneracy between the stellar-field microlensing and spin precession,investigating whether microlensing leads false evidence of precession. In other words,to what extent observed SLGWs are identified with false evidence of precession. The main findings are as follows. First,assuming O5 sensitivity and parallel spins for the underlying binary black holes,microlensing-induced false evidence of precession is generally weak(15% of the events show significant evidence,30% if the signal-to-noise ratio doubles). Second,for highly magnified events,about 72% of the population show significant evidence of precession,which could serve as an identification criterion for microlensing. This implies the possibility of strong lensing for the recent event GW231123,which is in the mass gap and also shows evidence of precession. Finally,a moderate to strong positive correlation exists between microlensing strength and precession evidence,more pronounced in Type II SLGWs.

Equations

$$F(\omega ,\mathbf{y})=\frac{2G⟨{\mathrm{M}}_L⟩(1+z_L)\omega }{\pi c^{3}i}{\int }_{-\infty }^{\infty }{d}^{2}x\exp \left[i\omega t(\mathbf{x},\mathbf{y})\right].$$

$$\begin{array}{rl}t(\mathbf{x},{\mathbf{x}}^i,\mathbf{y}=0)& =\underset{t_{\text{smooth}}(\kappa ,\gamma ,\mathbf{x})}{\underbrace{\frac{k}{2}\left((1-\kappa +\gamma )x_1^2+(1-\kappa -\gamma )x_2^2\right)}}-\underset{t_{\text{micro}}(\mathbf{x},{\mathbf{x}}^i)}{\underbrace{\left[\frac{k}{2}\sum _i^{N_{\ast }}\frac{{\mathrm{M}}_{L,i}}{⟨{\mathrm{M}}_L⟩}\ln {\left({\mathbf{x}}^i-\mathbf{x}\right)}^2+k{\varphi }_{-}(\mathbf{x})\right]}}.\end{array}$$

$$\ln B={\rho }^2(1-{\mathrm{FF}}^2)/2+\mathrm{const},$$

$$\mathrm{FF}=\underset{\theta }{\max }\frac{⟨h_1\mid h_2⟩}{\sqrt{⟨h_1\mid h_1⟩⟨h_2\mid h_2⟩}},$$

$$⟨h_1\mid h_2⟩=4\mathrm{Re}{\int }_{f_{\text{low}}}^{f_{\text{high}}}\mathrm{d}f\frac{h_1^{\ast }(f)h_2(f)}{S_{\mathrm{n}}(f)},$$

$$B_{\mathrm{non}\text{mbox}-\mathrm{prec}/\mathrm{prec}}=\frac{p(\varphi =0|d,\mathrm{prec})}{p(\varphi =0|\mathrm{prec})}$$

$$B_{\mathrm{prec}/\mathrm{non}\text{mbox}-\mathrm{prec}}=B_{\mathrm{non}\text{mbox}-\mathrm{prec}/\mathrm{prec}}^{-1}.$$

$$V_{\mathrm{post}}(\varphi )\propto {\rho }^{-k_{\mathrm{eff}}},$$

$$p(\varphi =0\mid d,\mathrm{prec})\propto {\rho }^{k_{\mathrm{eff}}},$$

$$B_{\mathrm{non}\text{mbox}-\mathrm{prec}/\mathrm{prec}}^{\mathrm{bg}}\propto {\rho }^{k_{\mathrm{eff}}},B_{\mathrm{prec}/\mathrm{non}\text{mbox}-\mathrm{prec}}^{\mathrm{bg}}\propto {\rho }^{-k_{\mathrm{eff}}},$$

$$\ln B_{\mathrm{bg}}\equiv \ln B_{\mathrm{prec}/\mathrm{non}\text{mbox}-\mathrm{prec}}^{\mathrm{bg}}\simeq -k_{\mathrm{eff}}\ln \rho +\mathrm{const}.$$

$$\mathrm{mismatch}=1-\underset{{\varphi }_0,t_0}{\max }\frac{⟨h_1\mid h_2⟩}{\sqrt{⟨h_1\mid h_1⟩⟨h_2\mid h_2⟩}},$$

Full text

Gravitational wave — Gravitational lensing — Microlensing — Strong lensing

I Introduction

Strongly-lensed gravitational waves (SLGWs) offer a new method for probing the Universe (Li et al., 2018; Oguri, 2019; Hannuksela et al., 2020; Cao et al., 2022; Liao et al., 2022; Jana et al., 2023; Seo et al., 2024; Poon et al., 2025). Unlike strongly lensed quasars or supernovae, GWs from binary black hole coalescences can have wavelengths comparable to the Schwarzschild radius of the lens. When these waves pass through a stellar field in the lensing galaxy, additional interference and diffraction effects can arise (Nakamura and Deguchi, 1999; Nakamura, 1998; Takahashi and Nakamura, 2003; Diego et al., 2019; Cheung et al., 2021; Mishra et al., 2021; Meena et al., 2022; Yeung et al., 2023; Seo et al., 2025). By extracting the interference imprints within GWs, it may be possible to reconstruct substructure information of the lens galaxy, such as the properties of intermediate-mass black holes (Lai et al., 2018) and dark matter substructures (Liu et al., 2023).

The most critical step in the application of SLGWs is the accurate and efficient identification of such events. However, the two main methods currently available, the posterior-overlap method (Haris et al., 2018; Barsode et al., 2024) and the joint parameter estimation (joint PE) method (Lo and Magaña Hernandez, 2021; Liu et al., 2021; Janquart et al., 2021, 2023), are affected to varying degrees by issues related to false-positive rates (Çalışkan et al., 2023; Wierda et al., 2021b). Introducing a novel additional criterion would be a valuable contribution to these detection strategies. On the other hand, neglecting such wave-optics effects can lead to missed signals (Chan et al., 2024) and biased parameter estimation for SLGWs, particularly when an event is highly magnified or passes through regions with a high density of compact lenses (Meena et al., 2022; Mishra et al., 2024; Shan et al., 2024). However, in such cases, significant wave-optics effects can also provide additional evidence for strong lensing. Therefore, understanding wave optics is important not only for probing small-scale lenses but also for ensuring the robust detection of strongly lensed events.

As research into microlensing wave optics progresses, it has been observed that the effects of microlensing on GW waveforms bear similarities to intrinsic precession effects, with both producing beat-pattern-like modulations in the waveform (note that overlapping signals can also introduce a similar beat-pattern-like effect, which is beyond the scope of this study (Hu, 2025)). Although the study by Liu and Kim (2024) showed that these two effects can be disentangled using template matching, this result mainly applies to cases where template construction is relatively straightforward. One example is the eikonal limit, in which the GW wavelength is much smaller than the characteristic size of the lens, so one only needs to consider the superposition of waveforms. This mainly applies to lenses above about 100 solar masses. For a microlensing field composed of thousands to millions of stars with masses below about 10 solar masses, constructing such templates becomes extremely challenging. As a result, the “similarity” between the two effects cannot be easily disentangled, leading to potential degeneracies. This means that microlensing effects could be mistakenly interpreted as intrinsic precessional effects.

At the same time, although binary black holes are generally expected to precess, the observable precession can be very weak in some formation channels. As a result, clear evidence for precession remains difficult to obtain for many events with current detectors. This makes it important to ask whether some apparent evidence for precession could instead be caused by microlensing. If so, such degeneracies could bias population studies of binary black holes and lead to incorrect conclusions about their formation channels. This is because precession measurements can help distinguish among different formation scenarios, at least in a statistical sense. In particular, one may incorrectly favor black holes formed through dynamical interactions in dense clusters (the dynamical channel) (Sigurdsson and Hernquist, 1993; Portegies Zwart and McMillan, 2000; Rodriguez et al., 2016; Mapelli et al., 2022), which are more likely to produce stronger precession, over those formed through the evolution of massive binary stars (the EMBS channel) (Abbott and others, 2016; Belczynski et al., 2016; Giacobbo and Mapelli, 2018), which often predicts weaker observable precession.

In addition, in the context of strong-lensing identification, these degeneracies could serve as a new clue for strongly lensed events. In other words, one may further filter candidate strong-lensing events by identifying those that show evidence for precession, thereby helping to reduce the false-positive rate. The main goal of this work is to study the possible degeneracy between microlensing and precession in GW waveforms, and to ask whether microlensing can produce observable evidence for precession. If so, evidence for precession may not only suggest a dynamical formation channel, but may also point to a microlensing candidate that deserves further investigation. We find that microlensing can indeed produce evidence for precession, and that this effect depends on both the strength of the microlensing and the detector sensitivity. In future lower-noise detectors, events with strong microlensing effects, for example highly magnified events, are also more likely to be identified as precessing binaries. Therefore, evidence for precession may serve as an additional criterion for identifying SLGWs.

The structure of the paper is as follows: in Section II, we introduce the fundamental theories of strong lensing and microlensing, along with the simulation processes used for the data. Section III presents our results, and finally, we provide a summary and discussion in Section IV.

II Basic theory and Mock Data simulation

The GW lensing effect induced by a microlensing field embedded within a strong lensing galaxy can be described through the diffraction integral (Schneider et al., 1992b; Nakamura and Deguchi, 1999; Takahashi and Nakamura, 2003), as described in the following equation:

[TABLE]

Here, $F(\omega,\boldsymbol{y})$ represents the amplification factor, while $\omega$ and $\boldsymbol{y}$ denote the GW angular frequency and its position in the source plane (normalized by the Einstein radius of average microlens mass $\langle\mathrm{M}_{L}\rangle$), respectively. $z_{L}$ refers to the microlens redshift and $\boldsymbol{x}$ corresponds to the coordinates in the lens plane (normalized by the Einstein radius of $\langle\mathrm{M}_{L}\rangle$).

In Eq. (1), the term $t(\boldsymbol{x},\boldsymbol{y})$ represents the time delay function for the microlensing field embedded in the lens galaxy or galaxy cluster, which can be expressed as (Wambsganss, 1990; Schneider et al., 1992a; Chen et al., 2021):

[TABLE]

In this equation, $k=4G\langle\mathrm{M}_{L}\rangle(1+z_{L})/c^{3}$, $\mathrm{M}_{L,i}$ and $\boldsymbol{x^{i}}$ refer to the mass and position of the $i$th microlens, respectively, and $N_{*}$ indicates the number of microlenses. The parameters $\kappa$ and $\gamma$ denote the convergence and shear of the macro lens, respectively. Additionally, $t_{\text{smooth}}(\kappa,\gamma,\boldsymbol{x})$ and $t_{\text{micro}}(\boldsymbol{x},\boldsymbol{x}^{i})$ represent the macro and microlensing time delays, respectively. For simplicity, the macro image position is set to the origin ($\boldsymbol{y}=0$). A negative mass sheet, represented by $\phi_{-}(\boldsymbol{x})$, is included to ensure that the total convergence $\kappa$ remains unchanged when microlenses are added (Wambsganss, 1990; Chen et al., 2021; Zheng et al., 2022).

Figure 1 shows an illustrative sketch of the strong lensing GW influenced by a microlensing stellar field in the lens galaxy. One can see that the strongly-lensed GW waveform is further modulated by the microlensing effect. Note that the number of stars depicted in the lens galaxy in this figure is purely illustrative and not intended to represent realistic values. Typically, the number of stars can range from $10^{3}$ to $10^{6}$ (Katz et al., 1986; Shan et al., 2023b).

In this study, we adopt the same strategy as in Shan et al. (2023a, 2024) to simulate SLGW signals. Specifically, we assume that the merger rate of binary black holes is proportional to the star formation rate (Haris et al., 2018), and determine whether a GW event is a strong lensing event based on the multi-image optical depth of the singular isothermal sphere (SIS) model. We then calculate the microlensing stellar field density at the positions of the strong lensing images, which follow the Sérsic light profile (Vernardos, 2018).

The microlensing stellar field consists of two components: stars and remnants. Here, we assume that the stellar mass function follows the Chabrier initial mass function (Chabrier, 2003), while the remnant mass function is determined using the initial-final mass relation described in Spera et al. (2015). Finally, we use the TAAH (Trapezoid Approximation-based Adaptive Hierarchical) method proposed by Shan et al. (2025) to evaluate the microlensing wave effects. For further details, please refer to Appendix A.

Based on the previous procedure, we build two SLGW data sets: one containing events detectable at the O5 noise level (Abbott et al., 2020), and the other containing events with the same GW waveform (i.e., the same strong-lensing and microlensing effects) but at a lower noise level, with the noise power spectrum reduced by a factor of $4$ compared to O5. Hereafter, we refer to this lower-noise configuration as O5 Plus. Here, we assume Gaussian noise and do not include any non-Gaussian features such as glitches. Each data set contains 125 binary black hole merger signals. The reason we also choose a lower-noise detector is that we want to investigate the influence of the signal-to-noise ratio (SNR) on the detectability of microlensing-induced degeneracy and to see whether the precession evidence induced by microlensing could be a useful criterion for SLGWs in the future. Here, we use the IMRPhenomXP (Pratten et al., 2021) waveform model with aligned-spin parameters to generate GW signals. We do not include intrinsic precession in the mock data. This is because our main goal is to study the degeneracy between microlensing and precession. To do this clearly, it is better to start with signals that have no true precession, so that any recovered evidence for precession can be directly attributed to microlensing. If the signal already contains intrinsic precession, then precession evidence would be easier to obtain, because the precessing part of the signal cannot be fully captured by aligned-spin templates. In that case, it would be harder to isolate the specific effect caused by microlensing.

III Result

III.1 Microlensing induced precession evidence

In this section, we characterize the degree of degeneracy between microlensing and precession by quantifying the precession Bayes factor induced by microlensing effects in SLGW signals. Specifically, in the parameter-estimation procedure, we recover the GW parameters using the Dynesty (Higson et al., 2019) sampler under two different assumptions, while using the same waveform template, IMRPhenomXP. In one case, the template includes precession effects, and in the other, it assumes parallel spins. We then compute the Bayes factor for precession by comparing the detection evidences under these two assumptions.

To quantify the strength of the precession evidence induced by microlensing, we also simulate a background sample. This sample consists of 125 standard GW events under O5 sensitivity, together with 125 additional standard GW events with the noise power spectrum reduced by a factor of 4 relative to O5. Here, “standard” refers to events drawn from the same BBH population model but without strong lensing or spin precession. We then define two background-calibrated thresholds based on the ranking statistic. Specifically, for a reference significance level $s$, we define the threshold as the $\Phi(s)$ quantile of the unlensed-background distribution, where $\Phi$ is the cumulative distribution function of the standard normal distribution. In this work, we use $s=1$ and $s=3$, corresponding to one-sided Gaussian-equivalent reference levels of $1\sigma$ and $3\sigma$, or equivalently to the 84.1st and 99.9th percentiles of the background distribution, respectively.

Figure 2 shows the complementary cumulative distribution function (CCDF), defined as $P(\ln B>x)$, of the logarithmic Bayes factor for precession for both microlensed events (red curve) and normal unlensed events (blue curve). The left panel shows the results for O5 sensitivity, and the right panel shows the results for O5 Plus sensitivity. We use two dashed vertical lines to mark the $1\sigma$- and $3\sigma$-equivalent background thresholds. We find that, in the lower-SNR scenario (O5), 16.25% and 4.88% of events lie above the $1\sigma$- and $3\sigma$-equivalent background thresholds, respectively. In the higher-SNR scenario (O5 Plus), these fractions increase to 34.21% and 14.91%, respectively.

The first main reason for the higher fraction at higher SNR is that, in the linear-signal approximation, the Bayes factor scales as (Cornish et al., 2011)

[TABLE]

where $\rho$ is the SNR and $\mathrm{FF}$ is the fitting factor, defined as

[TABLE]

where $h_{1}$ is the unlensed waveform and $h_{2}$ is the microlensed waveform. The ${\theta}$ denotes all GW parameters. The operator $\langle\cdot\mid\cdot\rangle$ represents the noise-weighted inner product, defined as

[TABLE]

where $S_{\mathrm{n}}(f)$ is the single-sided power spectral density of the detector noise, and “∗” denotes complex conjugation. Therefore, for a fixed $\mathrm{FF}$, the Bayes factor increases with $\rho^{2}$.

The second main reason is that false-positive support for precession from the unlensed background is suppressed at higher SNR. For nested models, the non-precessing hypothesis is a special case of the precessing hypothesis obtained at the non-precessing limit of the additional precession parameters. In this case, the Bayes factor in favour of the simpler non-precessing model over the precessing model can be written through the Savage–Dickey density ratio as (Wagenmakers et al., 2010)

[TABLE]

where $\phi$ denotes the additional precession parameters, and $\phi=0$ corresponds to the non-precessing limit. Here, the numerator is the posterior density at the non-precessing point under the precessing model, while the denominator is the corresponding prior density. Therefore, the Bayes factor in favour of precession is

[TABLE]

For a truly unlensed and non-precessing event, the data do not contain a genuine precession signal, so any preference for the precessing model can only arise from fitting random noise fluctuations. In the high-SNR limit, the Fisher matrix for the additional precession parameters scales as $\Gamma_{\phi\phi}\propto\rho^{2}$, so the corresponding covariance scales as $\Gamma_{\phi\phi}^{-1}\propto\rho^{-2}$. Therefore, the characteristic posterior width of each independently measurable additional parameter scales as $\Delta\theta\propto\rho^{-1}$. Hence, for $k_{\mathrm{eff}}$ effective additional precession degrees of freedom, the posterior volume in the precession sector shrinks as

[TABLE]

which implies

[TABLE]

while the prior density $p(\phi=0\mid\mathrm{prec})$ is independent of $\rho$. Therefore,

[TABLE]

and one can expect the scaling

[TABLE]

Therefore, as the SNR increases, the posterior volume of the additional precession parameters shrinks and the Occam penalty becomes stronger, which suppresses the false-positive precession evidence from unlensed background events. Combining the above two reasons, the overlap between the microlensed and unlensed populations is reduced, leading to a higher detectability of microlensing-induced precession evidence at higher SNR. This result also highlights the importance of waveform-model systematics for future, more sensitive GW detectors (Pürrer and Haster, 2020).

To show more clearly how well microlensing-induced precession evidence can be separated from the unlensed background, Figure 3 also shows the detection efficiency as a function of the false alarm probability. Here, the detection efficiency is defined as the fraction of microlensed events above a given threshold in $\ln B$, while the false alarm probability is defined as the fraction of unlensed background events above the same threshold. One can see that, for O5 Plus sensitivity, the detection-efficiency curve lies clearly above the dashed curve, which represents random guessing, i.e., the case in which microlensed and unlensed events cannot be distinguished and events are classified no better than by chance. However, for the lower O5 sensitivity, microlensing-induced precession events can be distinguished from the background only in the long tail.

III.2 Correlation between microlensing and precession

The preceding results indicate a degeneracy between microlensing and precession. To further explain why the precession waveform gives higher evidence, we examine the correlation between the improvement in the maximum likelihood, $\Delta\ln\mathcal{L}$, and the logarithmic Bayes factor for precession, $\ln B$, as shown in Figure 4. Under both O5 and O5 Plus sensitivities, the microlensing events show a strong positive correlation between $\Delta\ln\mathcal{L}$ and $\ln B$. This suggests that the higher evidence for precession in some microlensing events mainly comes from the better fit of the precession waveform to the data. In this sense, the result provides a waveform-fitting explanation for the degeneracy between microlensing and precession. We also note that, under O5 sensitivity, the background unlensed events still show a moderate correlation between $\Delta\ln\mathcal{L}$ and $\ln B$. This suggests that, at lower SNR, noise can more easily produce false evidence for precession. We stress that this does not imply any physical degeneracy between precession and Gaussian noise. Rather, it reflects a statistical effect in the inference, in which the additional freedom of the precessing waveform can partially absorb random noise fluctuations and lead to an artificially enhanced preference for precession in some realizations. However, under O5 Plus sensitivity, this effect is no longer significant.

Additionally, one may ask whether larger values of the precession evidence, $\ln B$, correspond to a stronger microlensing signature. If so, $\ln B$ could be used to infer the strength of microlensing. We therefore investigate the correlation between the microlensing strength and the precession evidence, $\ln B$. Here, we quantify the microlensing strength using the mismatch induced by microlensing in the GW waveform. The mismatch is defined as follows:

[TABLE]

where $h_{1}$ and $h_{2}$ represent the waveforms of signals 1 and 2, respectively. $\phi_{0}$ and $t_{0}$ are the initial phase and start time of signal 1. This equation accounts for the time delays caused by both macrolensing and microlensing. In this analysis, signal 1 represents the unlensed waveform, while signal 2 represents the microlensed waveform. Note that this match calculation is different from the fitting factor calculation (Eq. 4), since the former maximizes only over the initial phase and start time, while the latter maximizes over all GW parameters.

To quantify the correlation between microlensing and precession, we also use Spearman’s correlation coefficient. Specifically, we aim to determine whether stronger microlensing effects generally lead to higher evidence for precession, or conversely, whether higher evidence for precession corresponds to a stronger microlensing effect. Figure 5 shows the distribution of $\ln B$ for SLGW events as a function of microlensing mismatch. The left panel shows the events under O5 sensitivity, while the right panel shows those under O5 Plus sensitivity. Here, we use only the events above the corresponding one-sided Gaussian-equivalent $1\sigma$ background thresholds, shown as red stars, to calculate the correlation coefficient. One can find that, under O5 sensitivity, there is only a weak correlation between the mismatch and $\ln B$. This means that such events are strongly affected by noise, and a stronger microlensing effect does not necessarily lead to a higher $\ln B$. However, in the right panel, under the higher-SNR configuration, the correlation coefficient reaches 0.62 and the $p$-value is as low as the order of $10^{-5}$, showing a high level of statistical significance. This means that a stronger microlensing effect often corresponds to higher evidence for precession. Therefore, if one detects an event with very high precession evidence, then, besides the possibility that it is a true precession event, it may also be a lensing event with strong microlensing.

Finally, we investigate the correlation between microlensing strength and precession evidence for two types of SLGW images: Type I, originating from the minimum point of the time-delay surface, and Type II, originating from the saddle point of the time-delay surface. We then ask which type of event can more easily mimic a precession signature. In this analysis, we only consider events under O5 Plus sensitivity, as shown in the right panel of Figure 5. The results are shown in Figure 6.

The left panel of the figure corresponds to Type I images, while the right panel corresponds to Type II images. Here, we also restrict the analysis to events above the corresponding one-sided Gaussian-equivalent $1\sigma$ background thresholds in order to reduce the impact of false positives caused by noise.

We find that the correlation is slightly stronger for Type II images than for Type I images. The correlation coefficients for Type I and Type II images are 0.55 and 0.64, respectively, and the $p$-values are of the order of $10^{-3}$ and $10^{-2}$, showing statistical significance at this level. Therefore, we conclude that, for each strong-lensing image type, there is a moderate positive correlation between the strength of the microlensing effect and the evidence for precession. Thus, one can expect stronger evidence for precession when the microlensing effect is stronger.

These results also suggest that microlensing effects in Type II images can more easily mimic precession effects. For a better understanding, we can return to Eq. (3). This scaling relation indicates that, from a statistical point of view, Type II images will have a smaller fitting factor than Type I images at a comparable SNR. This also suggests that orbital precession is more effective at capturing the microlensing-induced fluctuations in Type II events than in Type I events. This is likely because the microlensing effects in Type II images produce a stronger modulation of the phase. The phase modulation between Type I and Type II images can be compared in the rightmost panel of Figure 7, as shown there. Therefore, for one SLGW pair, one may expect to observe one event with a precession effect and another without a precession effect. Consequently, when identifying SLGW events using the parameter-overlap method or the joint parameter-estimation method, we need to treat these precession parameters with care.

IV Conclusion and discussion

Wave-optical effects in gravitational lensing can produce frequency-dependent fluctuations in GW waveforms, often described as a “beat pattern” (Diego et al., 2019; Hou et al., 2020). Similar features can also arise from intrinsic source physics, especially orbital precession in BBH systems. This similarity makes it important to understand under what conditions microlensing can mimic precession, and how this may affect the interpretation of GW signals. In previous work, Liu and Kim (2024) showed that these two effects can in principle be separated with a dedicated template that includes both lensing under geometric approximation and precession. However, that strategy is much harder to apply in a stellar microlensing field, where the number of stars and remnants can be as large as $10^{6}$ (Katz et al., 1986). In such a situation, constructing a complete template becomes very difficult. Recent work has suggested that simulation-based inference (SBI) may provide a possible way forward for this problem, because they can learn the statistical mapping between waveform features and the underlying microlensing field without requiring a fully complete template (Su et al., 2025). However, such methods are still at an early stage, and it remains valuable to study how microlensing can be confused with intrinsic binary parameters in standard MCMC-type inference settings.

In this work, we studied how often microlensing wave effects in a strongly lensed galaxy can produce precession-like signatures, and how the inferred precession evidence is related to the strength of microlensing. To do this, we analyzed two SLGW datasets with different detector sensitivities, corresponding to O5 and a lower-noise O5 Plus configuration. Our results show that microlensing can indeed generate apparent evidence for precession, and that this effect becomes more visible at higher SNR. Under O5 sensitivity, 16.25% and 4.88% of microlensed events lie above the one-sided Gaussian-equivalent $1\sigma$ and $3\sigma$ background thresholds, respectively. Under O5 Plus sensitivity, these fractions increase to 34.21% and 14.91%, respectively. The detection efficiency in the higher-sensitivity case is also clearly above the level expected from random guessing, as shown in Figure 3. This means that, in a quieter detector environment, the precession-like signature produced by microlensing is easier to identify.

The physical reason is twofold. First, for a fixed fitting factor, the Bayes factor grows with SNR, so waveform differences caused by microlensing are more easily translated into measurable evidence at higher SNR. Second, the false-positive support for precession from unlensed background events is reduced as the SNR increases. Together, these two effects decrease the overlap between microlensed and unlensed populations and improve the detectability of microlensing-induced precession-like signals. This point is important for future detectors with even higher sensitivity, where waveform-model systematics, such as microlensing modulation, will become more important and can bias the inference and produce false evidence for intrinsic parameters. However, such precession-like signatures may also provide useful supplementary information for identifying SLGW events.

We also examined whether stronger microlensing tends to produce larger precession evidence. The answer depends on the detector sensitivity. Under O5 sensitivity, the correlation between the microlensing mismatch and $\ln B$ is weak, suggesting that noise still plays a large role and can hide the underlying trend, as also shown by the unlensed events in Figure 4. Under O5 Plus sensitivity, however, the correlation becomes much clearer, with a Spearman coefficient of 0.62 and a $p$-value of order $10^{-5}$. This indicates that, in a higher-SNR regime, stronger microlensing usually leads to stronger evidence for precession. Therefore, a GW event with large precession evidence does not always have to be a truly precessing system; it may also be a lensed event with a strong microlensing wave effect.

We further compared two types of SLGW images, Type I and Type II. We found that Type II images show a slightly stronger correlation between microlensing strength and precession evidence than Type I images. This suggests that Type II images are more likely to mimic precession. A likely reason is that microlensing produces stronger phase modulation in Type II images, making them easier to fit with a precessing waveform. As a result, within one SLGW pair, it is possible that one image shows apparent precession while the other does not. This possibility should be treated carefully when SLGW candidates are identified using parameter-overlap methods or joint parameter-estimation methods.

Overall, our results show that precession evidence in GW data should be interpreted with care. Strong evidence for precession may point to a genuinely precessing BBH and may therefore support a dynamical formation channel, but it may also be produced by microlensing wave effects in an SLGW event. Since clear precession evidence appears to be rare in the present GW catalog, such events deserve special attention. In some cases, precession may serve as a useful supplementary signature of SLGWs, especially for highly magnified events with large microlensing mismatches. This also means that GW events in the mass gap that show strong precession-like features may remain consistent with the possibility of being high-magnification SLGW events.

All code and results used in this work are publicly available at https://github.com/xkshan97/Microlensing_Precession. This work is partly supported by the National Science Foundation of China (Grant No. 12133005). X.S. acknowledges support from Shuimu Tsinghua Scholar Program (No. 2024SM199) and the China Postdoctoral Science Foundation (Certificate Number: 2025M773189). O.A.H. acknowledges suport by grants from the Research Grants Council of Hong Kong (Project No. CUHK 14304622, 14307923, and 14307724), the start-up grant from the Chinese University of Hong Kong, and the Direct Grant for Research from the Research Committee of The Chinese University of Hong Kong. This material is based upon work supported by NSF’s LIGO Laboratory which is a major facility fully funded by the National Science Foundation.

Appendix A Simulation procedures

Complementing Section II, this appendix details the simulation procedures used to generate a mock dataset. Using the Monte Carlo method and following the procedure described in Haris et al. (2018); Xu et al. (2022); Shan et al. (2024), we synthesized a population of GW events. The key simulation steps were:

Source Parameterization: We use the IMRPhenomXP waveform approximation to generate a population of simulated BBH events with parameters defined as follows:

• •

The source redshift ($z_{s}$) is sampled from a star formation rate (SFR) based merger model with a 50 Myr delay (see Appendix B of Xu et al. (2022)). The component masses ($m_{1},m_{2}$) are drawn from a power-law plus peak distribution (Abbott and others, 2019).

• •

The dimensionless spin magnitudes are drawn from $a_{1},a_{2}\sim\mathrm{U}(0,0.99)$.

• •

The sky location and orientation angles are sampled from their isotropic distributions:

• –

Inclination: $p(\iota)\propto\sin(\iota)$ for $\iota\in[0,\pi]$.

• –

Declination: $p(\delta)\propto\cos(\delta)$ for $\delta\in[-\pi/2,\pi/2]$.

• –

Right Ascension: $\alpha\sim\mathrm{U}(0,2\pi)$.

• –

Polarization: $\psi\sim\mathrm{U}(0,\pi)$.

• •

The merger time ($t_{c}$) is drawn uniformly over a 1-year period.

Strong Lensing Selection: The likelihood of an sBBH at redshift $z_{s}$ undergoing strong lensing (multiple imaging) was evaluated using the SIS optical depth, $\tau(z_{s})$ (Haris et al., 2018). A stochastic selection was performed: if $\tau(z_{s})$ exceeded a random variate drawn from $U(0,1)$, the event was flagged as an SLGW; otherwise, it was discarded.

Lens Modeling for SLGWs: Lensing effects for candidates were computed using an SIE model (Kormann et al., 1994), with lens properties ($\sigma_{v},q$) drawn from SDSS galaxy survey statistics (Choi et al., 2007; Wierda et al., 2021a).

Observational Filtering: Detectability was assessed using a three-detector network (LIGO Livingston/Hanford, Virgo) and an SNR threshold of 12. The simulation proceeded until 125 detectable SLGW events were obtained.

Microlensing Environment Simulation: The final stage involved modeling the microlensing environment pertinent to each detected SLGW. This included defining a stellar mass function based on the Chabrier IMF (Chabrier, 2003) for stars in the mass range $[0.1,1.5]M_{\odot}$ (Diego et al., 2022), and assuming an elliptical Sérsic profile (Vernardos, 2018) for their density. A population of remnant objects was also incorporated, using an initial-final mass relation (Spera et al., 2015) for their masses and assuming their mass density constitutes $10\%$ of the stellar mass density.

Figure 7 presents the microlensing diffraction results for these simulated SLGW events. Different curves represent different events. The first and second rows illustrate results for Type I and Type II SLGWs, respectively. The first column shows the residual time-domain amplification factor ($F(t)-F_{\mathrm{smooth}}(t)$, where $F_{\mathrm{smooth}}(t)$ excludes microlensing effects). The second and third columns display the normalized frequency-domain amplification factor and the complex phase. The gradual convergence of each curve’s tail towards zero in the first column indicates the convergence of the diffraction integral (Shan et al., 2023b).