Hierarchical modeling of gravitational-wave populations for disentangling environmental and modified-gravity effects

TL;DR

This work develops a hierarchical Bayesian framework to disentangle environmental (local) and modified-gravity (global) effects in EMRI populations observed by LISA. By modeling distinct hypotheses—Vacuum-GR $\mathcal{H}_v$, local $\mathcal{H}_\ell$, and global $\mathcal{H}_g$—and leveraging population-level information, the approach can identify which effect class dominates and whether both are present. The authors derive analytic, approximate hyperlikelihoods using the linear-signal approximation and Fisher information, validate them against Monte Carlo integrals, and demonstrate robust hypothesis recovery across simulated populations with as few as ~20 detected sources. They find that the global effect strength $\dot{G}$ can be constrained to good precision, while the local fraction $f$ may exhibit biases due to correlations with $\dot{G}$ in mixed-population cases; a rare cancellation scenario can obscure disentanglement. The framework offers a versatile tool for beyond-vacuum-GR tests in GW populations and can be extended to more complex EMRI models and broader beyond-GR scenarios.

Abstract

The upcoming Laser Interferometer Space Antenna (LISA) will detect up to thousands of extreme-mass-ratio inspirals (EMRIs). These sources will spend $\sim 10^5$ cycles in band, and are therefore sensitive to tiny changes in the general-relativistic dynamics, potentially induced by astrophysical environments or modifications of general relativity (GR). Previous studies have shown that these effects can be highly degenerate for a single source. However, it may be possible to distinguish between them at the population level, because environmental effects should impact only a fraction of the sources, while modifications of GR would affect all. We therefore introduce a population-based hierarchical framework to disentangle the two hypotheses. Using simulated EMRI populations, we perform tests of the null vacuum-GR hypothesis and two alternative beyond-vacuum-GR hypotheses, namely migration torques (environmental effects) and time-varying $G$ (modified gravity). We find that with as few as $\approx 20$ detected sources, our framework can statistically distinguish between these three hypotheses, and even indicate if both environmental and modified gravity effects are simultaneously present in the population. Our framework can be applied to other models of beyond-vacuum-GR effects available in the literature.

Figures (6)

• Figure 1: The distribution of optimal SNR $\rho_{\rm opt}$ in the vacuum-GR population $P_v$ for the $N_{\rm pop} = 1000$ case. SNRs $\rho_{\rm opt} \geq 20$ are highlighted in orange, representing $\approx 17\%$ of all sources.
• Figure 2: Marginalized hyperposteriors on $f$ (top panels) and $\dot{G}$ (bottom panels) constructed from $N = 5000$ samples from the analysis priors weighted by the hyperposterior densities. We plot four populations, namely, $P_v$, $P_\ell$, $P_g$, and $P_{\rm mix}$ (left to right) with population size $N_{\rm pop} = 1000$. The solid vertical line in each plot represents the true value of $f$ or $\dot{G}$ while the dashed vertical line represents the sample expectation. $f$ is found to be systematically biased by $\sim 3.5\sigma$ in $P_g$ and $\sim 1.5\sigma$ in $P_{\rm mix}$, while $\dot{G}$ is recovered consistently with the truth at $\lesssim 0.01\sigma$ in all cases.
• Figure 3: Bayes factors $\mathcal{B}^v_\ell, \mathcal{B}^v_g,$ and $\mathcal{B}^g_\ell$ in the considered populations $P_v$ (left panel), $P_\ell$ (left-center panel), $P_g$ (right-center panel), and $P_{\rm mix}$ (right panel). For brevity, we notate, e.g., $\log_{10}\mathcal{B}^v_\ell$ as $v/\ell$, which is to be understood as the preference for the hypothesis $\mathcal{H}_v$ over $\mathcal{H}_\ell$. The left, center, and right vertical bars in each panel represent the Bayes factors for the three different population sizes, $N_{\rm pop} = 100$, $500$, and $1000$, respectively. The horizontal dashed line in each panel represents a log Bayes factor of zero.
• Figure 4: The log-Bayes factor $\log_{10}\mathcal{B}^g_\ell$ as a function of $f^*$ (top panel) and $\dot{G}^*$ (bottom panel) in $11$ different populations of size $N_{\rm pop} = 500$. In the top panel, $\dot{G}^* = 10^{-12} [\rm yr^{-1}]$ is fixed. In the bottom panel, we fix $f^* = 0.5$ instead. In the bottom panel, the dashed horizontal line represents $\log_{10}\mathcal{B}^g_\ell = 0.0$, i.e., for which the data does not favour either of the hypotheses.
• Figure 5: 1-dimensional visualization of the product of a normal likelihood $\mathcal{L}(x) = \mathcal{N}(x|5,\sigma)$ with a power-law prior on $x$, $p(x) = x^\alpha$ where $\alpha = 2$ is the slope hyperparameter. In both panels, the green solid curve represents this product in its exact form, while the orange dash-dotted curve represents the approximate product, where $p(x)$ is Taylor-expanded to the second order in $x - 5$ about $5$. In the top panel, the normal likelihood has a large standard deviation $\sigma = 3.0$, such that its rate of change is similar to that of the power-law prior, and the Taylor approximation fails. When we set $\sigma = 0.3$ as in the bottom panel, the normal likelihood varies at much smaller scales (as expected for GW data analysis), and the Taylor approximation holds well.