Fundamental oscillations as a tool to distinguish boson stars from neutron stars and black holes

Massive boson stars are self-gravitating configurations of self-interacting scalar fields and can be modeled by a massive scalar field with a quartic self-interaction potential. It has been shown that the equation of state and static structure properties, such as mass and radius, follow scaling relations independent of microscopic dark matter properties. In this work, we demonstrate for the first time that non-radial fundamental ($f$-)mode characteristics also follow a scaling in the strong interaction limit, opening up the outstanding prospect of evaluating the mode properties for boson stars for arbitrary masses spanning the scalar dark matter parameter space allowed by current observations. We provide the scaling relations within full general relativity and obtain the mode characteristics corresponding to the maximum boson star mass configuration. We apply these to determine the $f$-mode properties for boson stars solely as a function of their mass and compactness, which allows distinguishing them from those of neutron stars and black hole quasinormal modes in comparable mass range. In particular, we show that the frequencies are always lower than those of corresponding black holes of the same mass by a factor of 4.5. This provides a smoking gun for the distinguishability of boson stars from other compact objects using gravitational wave observations.

💡 Research Summary

The paper investigates the fundamental (f‑)mode oscillations of massive boson stars (BSs) composed of a self‑interacting scalar field with a quartic potential V(ϕ)=½m²|ϕ|²+¼λ|ϕ|⁴. In the strong‑interaction regime (Λ≡λM_Pl²/4πm²≫1) the Einstein‑Klein‑Gordon system reduces to an effective perfect‑fluid description with an equation of state p=(m⁴/9λ)(√(1+3λρ/m⁴)−1)². Introducing the dimensionless combination x≡√λ/m², the macroscopic mass and radius scale as M∝x M_Pl³/m² and R∝x M_Pl/m².

By applying this scaling to the perturbation equations (both in the Cowling approximation and full general relativity), the authors derive a universal relation for the complex eigenfrequency: ω=ω′/(x M_Pl). Consequently, the observable frequency f and damping time τ obey f=f′/(x M_Pl) and τ=τ′·(x M_Pl), where the primed quantities depend only on the dimensionless mass M′=M/(x M_Pl³) and compactness C=M/R. Numerical integration yields a maximum dimensionless frequency f′_max≈0.21 (at M′≈0.062, C≈0.16) and a minimum dimensionless damping time τ′_min≈1900 (in natural units).

These results imply that for any microscopic parameters (λ,m) the f‑mode frequency can be expressed solely in terms of the star’s mass M_BS and compactness C. The authors provide an explicit formula:

 f_BS = 2.03×10² f′(C) M′(C) (M_⊙/M_BS) kHz,

and for the most compact configuration

 f_BS,max ≈ 0.013 (M_Pl²/M_BS) = 2.6 (M_⊙/M_BS) kHz.

Comparing with the fundamental l=2 quasinormal mode of a non‑rotating Schwarzschild black hole (f_BH≈0.059 (M_Pl²/M_BH) kHz), the ratio

 R_BS/BH = f_BS,max / f_BH ≈ 0.22

shows that black‑hole frequencies are at least 4.5 times higher than those of the most compact boson stars of the same mass. This factor stems from the higher compactness of black holes (C=0.5) versus the maximal BS compactness (C≈0.16).

When juxtaposed with neutron stars, which occupy the mass range 1–2.5 M_⊙ and compactness 0.1–0.3, BS f‑mode frequencies lie slightly below the typical NS band and become distinctly lower for masses above ≈2 M_⊙. Hence, a gravitational‑wave detection of an f‑mode signal in the neutron‑star mass range but outside the expected NS frequency band would be a strong indicator of a boson star.

The paper also discusses the relation between the f‑mode frequency and the innermost stable circular orbit (ISCO) frequency. For black holes, f_BH/f_ISCO≈5.5, whereas for boson stars the ratio depends on compactness and reaches ≈6.6 for the most compact case, reflecting the weaker tidal response of black holes.

In summary, the authors establish that in the strong self‑interaction limit, boson‑star f‑mode properties obey universal scaling laws independent of the underlying dark‑matter microphysics. These scaling relations enable rapid evaluation of mode frequencies and damping times across the entire allowed (λ,m) parameter space. The key observational consequence is that boson‑star f‑mode frequencies are systematically lower—by a factor of ~4.5—than those of black holes of the same mass, providing a clear “smoking‑gun” signature for distinguishing boson stars from black holes and neutron stars with forthcoming gravitational‑wave detectors. Future work should extend the analysis to rotating configurations, alternative scalar potentials, and nonlinear oscillations to fully map the gravitational‑wave phenomenology of exotic compact objects.