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Detectability of Massive Boson Stars using Gravitational Waves from Fundamental Oscillations

Swarnim Shirke, Bikram Keshari Pradhan, Debarati Chatterjee, Laura Sagunski, Jürgen Schaffner-Bielich

TL;DR

This work develops analytical fits for scaling relations governing massive boson stars in the strong-interaction limit ($\\Lambda \\gg 1$) and demonstrates that $f$-mode frequencies and damping times, when expressed in scaled coordinates, obey universal relations independent of the microscopic DM parameters $m$ and $\\lambda$. It provides practical formulas for static observables ($M',R',C'$) and $f$-mode properties, enabling GW asteroseismology and inference of DM properties from future detections. The authors map the observable DM parameter space to current and planned GW detectors (LISA, LIGO, CE, ET, NEMO) and quantify detectability under a burst GW model, showing potential probing depths from ~1 Mpc (advanced LIGO) to ~300 Mpc (LISA). These results establish concrete, parameter-space-aware benchmarks for identifying BSs as GW sources and for constraining dark-matter microphysics via gravitational-wave observations.

Abstract

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.

Detectability of Massive Boson Stars using Gravitational Waves from Fundamental Oscillations

TL;DR

This work develops analytical fits for scaling relations governing massive boson stars in the strong-interaction limit () and demonstrates that -mode frequencies and damping times, when expressed in scaled coordinates, obey universal relations independent of the microscopic DM parameters and . It provides practical formulas for static observables () and -mode properties, enabling GW asteroseismology and inference of DM properties from future detections. The authors map the observable DM parameter space to current and planned GW detectors (LISA, LIGO, CE, ET, NEMO) and quantify detectability under a burst GW model, showing potential probing depths from ~1 Mpc (advanced LIGO) to ~300 Mpc (LISA). These results establish concrete, parameter-space-aware benchmarks for identifying BSs as GW sources and for constraining dark-matter microphysics via gravitational-wave observations.

Abstract

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 (-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 -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 -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.

Paper Structure

This paper contains 19 sections, 31 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Parameter Space
  4. Constraints
  5. Results
  6. Scaled Relations for static observables
  7. Scaled Relations for fundamental modes
  8. Universal Relations
  9. Empirical Fits
  10. $f-$Love$-C$
  11. Observable DM parameter space
  12. Detectability
  13. Discussions
  14. Summary
  15. Comparison with other works
  16. ...and 4 more sections

Figures (10)

  • Figure 1: Figure shows the available parameter space in the $\lambda-m$ space. The color indicates the logarithm of the maximum mass of a stable BS for a given pair of ($\lambda$,m). We show select contour lines (black) to get an idea of masses: $\log_{10}M =$ -18 (mountain mass; this is also the lower bound on primordial BHs set by Hawking radiation as lighter PBHs would evaporate within Hubble time); -9 (asteroid belt mass); -6 (Earth-like planet mass); -3 (giant planet mass); 0 (stellar mass); 2 (giant massive star mass); 6 and 9 (SMBH mass); 12 (galaxy mass). The yellow contours represent values of $\Lambda$. We restrict to the strong-interaction limit $\Lambda > 1000$ in this work. All the results in this work, including the colorbar in this figure, are applicable only to this region. The orange and purple patch denotes the parameters used in earlier works on massive BSs in the strong-interaction regime Maselli2017Flores2019Celato2025. See text for details on various constraints.
  • Figure 2: Quadratic (blue) and quartic fits (green) as described in Eqn. \ref{['eqn:quadratic_quartic_relations']} for (a) $M'-C'$, and (b) $M'-R'$ relations for massive BSs in the strong-interaction limit. The dashed black line is the actual calculated numerical solution. The yellow curve is taken from Pacilio2020, where the fit is derived for non-spinning massive BSs. The percent errors for both fits are shown in the lower panels of each figure. The star marks the point beyond which the BS configurations become unstable. The stable solutions are valid only for compactness (radius) below (above) the star. The fit coefficients are reported in Table. \ref{['tab:quadratic_quartic_fit_coefficients']}.
  • Figure 3: The scaled dimensionless $f$-mode frequency ($f'$) as a function of a) scaled mass and b) compactness of BSs. The blue curve in the left panel is the fitted hyperbola as given in Eq. \ref{['eqn:f_m_fit']}. The blue and the green curves in the right panel are the fits for the quadratic and quartic functions, respectively, as given in Eqn. \ref{['eqn:quadratic_quartic_relations']}. The fit coefficients are given in Table. \ref{['tab:quadratic_quartic_fit_coefficients_fmodes']}. The lower panel shows the percent error for each fit.
  • Figure 4: Universal relations connecting (a) $f$-mode frequency in primed coordinates to the average density of BS and (b) scaled damping time $(R'^4/M'^3\tau')$ to compactness as suggested in AnderssonKokkotas1998. The grey band is the uncertainty in NSs with and without hyperonic matter. The band spans the fit relations reported in AnderssonKokkotas1998Benhar2004Pradhan2022. The blue curves are fits given in Eqs. \ref{['eqn:f_density_prime']} and \ref{['eqn:tau_c_fit']}. The errors in BS fits are shown in the lower panels.
  • Figure 5: Universal relations for mass-scaled $f$-mode frequency as a function of the (a) logarithm of the dimensionless tidal deformability and (b) compactness. The relations followed by NSs and hyperonic NSs are shown in green Pradhan2022. (c) Number of cycles ($N=f\tau=N'\tau'$) completed by BSs within the damping time. BSs complete a minimum of $N_{\rm min}=406$ oscillations for the case of the highest compactness configuration. The $N-C$ curves for pure neutron stars Pradhan2022 are shown in grey for comparison. The analytical fits to the numerical solutions are shown in blue, whose fitting coefficients are provided in Table. \ref{['tab:quadratic_quartic_fit_coefficients_UR']}. The corresponding percentage errors are shown in the bottom panels.
  • ...and 5 more figures