Detectability of Massive Boson Stars using Gravitational Waves from Fundamental Oscillations

Boson Stars are macroscopic self-gravitating configurations made of complex scalar fields. These exotic compact objects would manifest as dark Boson stars and, in the absence of electromagnetic signatures, could mimic properties of compact stars in the gravitational wave spectrum. In a recent study, using the simplest potential for massive Boson stars, we demonstrated that fundamental non-radial oscillations ($f$-modes) obey scaling relations that allow them to be distinguished from neutron stars and black holes. In this work, we provide analytical fits for these scaling relations, valid for the dark matter parameter space compatible with current astrophysical and cosmological data, that can be directly incorporated into future studies of massive Boson stars in the strong coupling regime, avoiding the need for numerical calculations. We also provide analytical fits for empirical and universal relations for gravitational wave asteroseismology, which can be used to infer microscopic dark matter properties following a successful detection. Further, we investigate the possibility of detection of $f$-modes and the dark matter parameter space that can be probed with current and future gravitational wave detectors across multiple frequency bands. Assuming a burst gravitational wave model and demanding a signal-to-noise ratio of 5, we show that the current and future detectors can, in principle, probe Boson star $f$-modes up to cosmological distances: 1 Mpc with aLIGO, 30 Mpc with Cosmic Explorer and Einstein Telescope, and in the best case scenario, about 300 Mpc with LISA.

💡 Research Summary

The paper investigates the detectability of massive boson stars (BSs) – compact objects composed of a self‑interacting complex scalar field – through their fundamental non‑radial oscillation (f‑mode) gravitational‑wave (GW) signatures. Starting from a simple quartic self‑interaction potential V(φ)=½m²|φ|²+¼λ|φ|⁴, the authors focus on the strong‑interaction regime where the dimensionless coupling Λ=λM_Pl⁴/(4πm²)≫1. In this limit the system exhibits a powerful scaling symmetry: by defining the effective parameter x≡√λ/m², the mass and radius can be rendered dimensionless as M′=M/(xM_Pl³) and R′=R/(xM_Pl). The Tolman‑Oppenheimer‑Volkoff (TOV) equations and the Einstein‑Klein‑Gordon field equations become independent of x, yielding a universal M′–R′ relation that can be rescaled to any (λ,m) pair.

The authors compute the f‑mode frequency f and damping time τ using a full general‑relativistic linear perturbation framework rather than the Cowling approximation. They demonstrate that f and τ obey simple scaling relations f=f′/(xM_Pl) and τ=τ′·(xM_Pl), where the primed quantities depend only on the dimensionless structure variables (M′,R′). Consequently, once the universal f′(M′,R′) and τ′(M′,R′) curves are known, the f‑mode characteristics for any boson‑star model in the strong‑coupling regime can be obtained analytically without additional numerical integration.

A comprehensive review of astrophysical and cosmological constraints on the scalar‑dark‑matter parameters (λ,m) is presented. Constraints include self‑interaction cross‑section per unit mass (σ/m≈0.1–100 cm²/g) from small‑scale structure problems, CMB and large‑scale‑structure limits on λ/m⁴, perturbativity bounds λ≲4π, and the requirement Λ>1000. The allowed region spans particle masses from 10⁻²⁵ eV up to 100 GeV and couplings up to λ≈15. Within this region the maximum stable boson‑star mass M_max varies from sub‑solar to super‑massive scales (≈10⁹ M_⊙), as illustrated by a colour‑coded λ–m diagram.

The detectability analysis assumes a burst‑type GW emission and adopts a signal‑to‑noise ratio threshold of 5. Using the noise power spectral densities of current and planned detectors, the authors estimate horizon distances for f‑mode detection: roughly 1 Mpc for Advanced LIGO (kHz band), about 30 Mpc for third‑generation ground‑based observatories such as Cosmic Explorer and the Einstein Telescope, and up to ∼300 Mpc for the space‑based LISA (mHz band) in the most favourable configurations. These distances imply that massive boson stars could be observable across extragalactic scales, especially if they reside in the stellar‑mass to intermediate‑mass regime where the f‑mode frequency lies within detector sensitivity windows.

Beyond detection prospects, the paper provides analytical fits for several universal relations: the f‑mode frequency–compactness (f–C) relation, the damping‑time–compactness (τ–C) relation, and the moment‑of‑inertia–Love‑number–compactness (I‑Love‑C) relation. These relations enable “gravitational‑wave asteroseismology” of boson stars: a measured f‑mode frequency and damping time can be inverted to infer the star’s compactness, moment of inertia, and ultimately the underlying particle physics parameters (m, λ). This constitutes a novel pathway to probe dark‑matter microphysics with GW observations.

In the discussion, the authors compare their work with earlier studies that examined f‑modes for a few discrete parameter choices. By exploiting the scaling symmetry, they extend the analysis to the full viable parameter space, providing simple analytical expressions that replace costly numerical scans. They acknowledge limitations: the results are strictly valid only in the Λ≫1 regime, and rotating boson stars or binary merger remnants are not treated. Future directions suggested include extending the scaling approach to the weak‑coupling (Λ≪1) regime, incorporating rotation, and studying the impact of tidal interactions in binary inspirals.

Overall, the study presents a robust theoretical framework that links the macroscopic GW signatures of massive boson stars to their microscopic scalar‑field properties, offering concrete predictions for current and next‑generation GW observatories and opening a promising avenue for probing dark‑matter physics through gravitational‑wave asteroseismology.