From Stars to Waves: Non-deterministic Inference of Microlensed Gravitational Waves

Abstract

Strongly lensed gravitational waves may pass through the stellar field of a lensing galaxy with additional modulations (on both phase and amplitude) due to gravitational microlensing effect of stars/remnants near the line of sight. These microlensed waveforms depend on the mass and location of thousands or more most relevant stars, so that their deterministic reconstruction from the data is computationally prohibitive. We classify the detection and parameter estimation of such events as non-deterministic inference problem and propose a solution with the implementation of normalizing flows. As a first step, we show that $8\%$ of microlensed events can be detected with significance $\ge 3 σ$ in the third generation era, with the chosen microlensing parameters correlated with the density of the underlying stellar field. This approach opens the door to probing microlensing effects and the properties of the underlying stellar fields. A similar construction may also be applied to other non-deterministic inference problems, such as detecting post-merger gravitational waves from binary neutron star coalescence and signals from core-collapse supernovae.

Figures (11)

• Figure 1: Microlensed BBHs and the detection method with neural networks.
• Figure 2: Microlensing surrogate parameters for network inference. The left panel shows the time-domain magnification factor (defined in Eq. (\ref{['eq:TimeDomainMag']})) and the top 10 highest peak values, $l_p$. The right panel shows the time-domain gravitational wave waveform with the microlensing effect (blue) and without the microlensing effect (yellow). One can see the deviations ($l_{d[M]}$) caused by microlensing.
• Figure 3: Detection of the microlensing signal. Left: a sampled parameter estimation (PE) result from a microlensing and null-microlensing injection. The $l_p$, $l_{d10}$, $l_{d60}$, and $l_{d120}$ are surrogate parameters, which stand for the peak values in the time-domain magnification factor and microlensing-induced deviations, as illustrated in FIG. \ref{['fig:parameters']}. Right: the modified Mahalanobis distance distribution with respect to the redshift and the chirp mass of the BBH.
• Figure 4: Modified Mahalanobis distance distribution $D_M$ and microlensing surrogate parameter $l_p$ distribution with $\kappa$ and $\kappa_s$
• Figure 5: The posterior distributions of the mass ratio $q$, aligned spins of the primary and secondary BHs ($\chi_1$ and $\chi_2$, respectively), luminosity distance $d_\mathrm{L}$, inclination angle $\iota$, right ascension $\alpha$, declination $\delta$, reference merger time $\delta t_c$, and polarization $\psi$ are shown. The orange curve labeled "$P_{\rm AI}(\theta_{\rm GR-0})$" shows the Neural Posterior Estimation (NPE) result without including chirp mass as an inference parameter. The blue curve labeled "$\int d \theta_0 P_{\rm AI}(\theta_{\rm GR})$" represents the NPE result with chirp mass included as an inference parameter. The green curve labeled "$\int d \theta_0 P(\theta_{\rm GR})$" corresponds to the result obtained from a traditional MCMC simulation using the emcee_pt sampler embedded in the PyCBC package alex_nitz_2022_6324278. The solid curves in the 2D contour plots represent the 90% credible regions, while the vertical dashed lines indicate the two-sided 90% confidence intervals. The solid red lines mark the injected (true) parameter values.