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

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.

💡 Research Summary

The paper presents a Bayesian hierarchical framework designed to distinguish between two broad classes of beyond‑vacuum‑General‑Relativity (GR) effects in the population of extreme‑mass‑ratio inspirals (EMRIs) that will be observed by the upcoming Laser Interferometer Space Antenna (LISA). EMRIs, consisting of a compact object (∼10 M⊙) orbiting a supermassive black hole (10⁵–10⁷ M⊙), spend of order 10⁵ orbital cycles in the LISA band, making them exquisitely sensitive to tiny deviations from the vacuum‑GR dynamics. Previous work has shown that for a single EMRI, environmental effects (e.g., migration torques from an accretion disk) and modified‑gravity effects (e.g., a time‑varying Newtonian constant G) can be highly degenerate, rendering them difficult to separate.

The authors argue that these two classes have distinct statistical signatures at the population level. Environmental (“local”) effects are expected to affect only a fraction f of the sources, whereas a modified‑gravity (“global”) effect would be present in every source. To exploit this, they construct a hierarchical model in which each source is described by parameters θ = {ln M, z, A_ℓ, n_ℓ, A_g}. The hyper‑parameters λ = {K, α, β, f, μ_Aℓ, μ_nℓ, σ_Aℓ, σ_nℓ, Ġ} govern the population‑level distributions: a mass‑redshift distribution set by a power‑law mass function (K, α, β), a bimodal distribution for the local effect (fraction f drawn from a 2‑D Gaussian in (A_ℓ, n_ℓ) or set to zero), and a Dirac‑delta prior for the global effect (all sources share the same ˙G = Ġ).

The dynamical model adds power‑law corrections to the GR angular‑momentum flux: \