Did We Just Find a Way to āHearā Dark Matter?
The study investigates gravity-mode (g-mode) oscillations in dark-matter-admixed neutron stars (DM-NSs) to determine how dark matter affects stellar vibrations and what this reveals about the starās interior. The authors employ a relativistic mean-field (RMF) model for hadronic matter, paired with a self-interacting fermionic dark-matter component inspired by the neutron-decay anomaly, which suggests that some neutrons may decay into dark fermions.
They construct equilibrium configurations of neutron stars containing both nuclear and dark matter by solving the TolmanāOppenheimerāVolkoff (TOV) equations. Using these solutions, they derive both equilibrium and adiabatic sound speeds, whose difference governs the BruntāVƤisƤlƤ frequency (N) responsible for g-mode oscillations. The oscillation equations are solved under the Cowling approximation, which neglects perturbations in the gravitational potential but retains relativistic corrections.
Results show that the fundamental (n = 1) and first overtone (n = 2) g-mode frequencies increase with the dark-matter fraction (f_DM), while the dependence on the DM self-interaction parameter ( G ) or the underlying nuclear equation of state (EoS) is comparatively weak. The inclusion of dark matter enhances the buoyancy within the star by increasing the difference between adiabatic and equilibrium sound speeds.
The authors identify an EoS-independent empirical relation linking the change in g-mode frequency to ( f_{DM} ). Because g-modes can couple to gravitational-wave (GW) tidal interactions during neutron-star mergers, these frequency shifts might be observable with future GW detectors. The work thus establishes g-modes as a promising asteroseismic diagnostic for probing dark matter inside neutron stars.
Introduction
Understanding the nature of dark matter (DM) is among the greatest challenges in contemporary physics. Observational evidenceāfrom galaxy rotation curves to the cosmic microwave backgroundādemonstrates that DM constitutes most of the universeās matter, yet its particle identity remains unknown. Compact stellar remnants such as neutron stars (NSs), with central densities several times nuclear saturation, offer an extreme environment where DM effects may manifest observably.
Dark matter can accumulate in neutron stars through gravitational capture or through neutron dark decay, motivated by the neutron-lifetime anomalyāthe discrepancy between ābottleā and ābeamā experimental results. If some neutrons decay to dark fermions (( \chi )) and light bosons (( \phi )), then NS cores may contain a stable dark-matter population, producing DM-admixed neutron stars.
NSs exhibit distinct families of oscillation modesāf- (fundamental), p- (pressure), r- (rotational), and g- (buoyancy) modesāeach sensitive to different physical properties. While f- and r-modes have been studied extensively in the presence of DM, g-modes, which depend on internal composition gradients and chemical stratification, have not. A g-mode occurs when a displaced fluid element in a stable, stratified medium experiences a buoyant restoring force, characterized by the BruntāVƤisƤlƤ frequency (N).
Because g-modes couple to tidal forces during binary inspiral, they can affect the emitted gravitational-wave spectrum. Detecting such frequency shifts would directly reveal the interior composition of NSs. This paper therefore examines how DM admixture influences g-mode oscillationsāparticularly the quadrupolar (( \ell = 2 )) fundamental and first overtoneāusing a relativistic mean-field EoS and the Cowling approximation. The goal is to quantify how DMās fraction and self-interaction alter g-mode spectra and to identify potential EoS-independent observables of dark matter in neutron stars.
Formalism
The authors develop a detailed theoretical framework for calculating g-mode oscillations in dark-matter-admixed neutron stars (DM-NSs), combining relativistic nuclear physics, dark-sector modeling, and perturbation theory.
Hadronic Matter
The nuclear component is modeled using the Relativistic Mean-Field (RMF) approach, in which nucleons interact via scalar ((Ļ)), vector ((Ļ)), and isovector ((Ļ)) mesons. The energy density is expressed as:
\[Īµ_{nuc} = \sum_N \frac{1}{8Ļ^2}\left[k_{FN}E_{FN}^{*3}+k_{FN}^3E_{FN}^*ām^{*4}\ln\left(\frac{k_{FN}+E_{FN}^*}{m^*}\right)\right] + \text{meson terms}\]where ( m^* = m - g_Ļ Ļ ) is the effective nucleon mass and ( E_F^* = \sqrt{k_F^2 + m^{*2}} ) the effective Fermi energy. The system satisfies Ī²-equilibrium and charge neutrality:
\[Ī¼_n = Ī¼_p + Ī¼_e, \quad Ī¼_Ī¼ = Ī¼_e, \quad n_p = n_e + n_Ī¼\]Dark Matter Model
Dark matter is modeled as a fermion ( \chi ) produced by neutron dark decay:
\[n \rightarrow \chi + \phi\]where ( \phi ) is a light boson that escapes. The DM particle interacts via a vector self-interaction with strength ( G = (g_V/m_V)^2 ). Its energy density and pressure are:
\[Īµ_Ļ = \frac{1}{Ļ^2}\int_0^{k_{FĻ}} k^2\sqrt{k^2+m_Ļ^2}\,dk + \frac{1}{2}G n_Ļ^2\] \[P_Ļ = \frac{1}{3Ļ^2}\int_0^{k_{FĻ}} \frac{k^4}{\sqrt{k^2+m_Ļ^2}}\,dk + \frac{1}{2}G n_Ļ^2\]The total pressure and energy density are:
\[P = \sum_N Ī¼_N n_N + Ī¼_Ļ n_Ļ - Īµ_{tot}\]and the dark-matter fraction:
\[f_{DM} = \frac{\int_V Īµ_Ļ dV}{\int_V Īµ_{tot} dV}\]Structure and Metric
A single-fluid approximation allows solving the TolmanāOppenheimerāVolkoff (TOV) equations:
\[\frac{dm}{dr} = 4Ļr^2Īµ(r), \quad \frac{dP}{dr} = ā(Īµ+P)\frac{m+4Ļr^3P}{r(rā2m)}\]g-mode Equations
Within the Cowling approximation, perturbations satisfy:
\[\frac{dU}{dr}= \frac{g}{c_s^2}U + e^{Ī»/2}\left[\frac{ā(ā+1)e^{Ī½}}{Ļ^2r^2} - \frac{r^2}{c_s^2}\right]V\] \[\frac{dV}{dr}=e^{Ī»/2āĪ½}\left(\frac{Ļ^2āN^2}{r^2}\right)U + gĪ(c^{-2})V\]where the BruntāVƤisƤlƤ frequency is:
\[N^2 = g^2Ī(c^{-2})e^{Ī½āĪ»}\]Sound Speeds
Equilibrium and adiabatic sound speeds are:
\[c_e^2 = \left(\frac{dP}{dĪµ}\right)_Ī², \quad c_s^2 = \left(\frac{āP}{āĪµ}\right)_{x_i}\]Their difference ( c_s^2 - c_e^2 ) drives buoyancy. DM adds a term proportional to ( G n_Ļ^2 ), enhancing buoyancy and raising g-mode frequencies.
Results
Equation of State
Adding DM softens the EoS and increases ( c_s^2 - c_e^2 ), strengthening stratification. The peak of this difference shifts to higher densities with DM fraction.
3.2 Self-Interaction and Mass Dependence
The fundamental (gā) and first-overtone (gā) frequencies increase with stellar mass and DM content. For instance, ( Ī½_{g1} ) rises from 200 Hz to nearly 800 Hz as mass approaches ( 2.5 M_\odot ).
The frequency shift:
\[ĪĪ½_g(G,M) = Ī½_g(G,M) - Ī½_g(Gāā,M)\]is positive and scales with both ( M ) and ( f_{DM} ).
Universal Relation
Across all EoSs, the frequency enhancement follows:
\[ĪĪ½_g \propto f_{DM}^{1/2}\]This near-universal scaling allows direct inference of dark-matter content from measured g-mode frequencies, largely independent of the nuclear EoS.
Astrophysical Significance
Since g-modes couple to gravitational-wave tidal forces during binary mergers, such frequency shifts may be observable by next-generation detectors like Einstein Telescope and Cosmic Explorer, enabling gravitational-wave asteroseismology of dark-matter-rich neutron stars.
Conclusion
Dark matter increases neutron-star buoyancy by enhancing ( c_s^2 - c_e^2 ), producing higher g-mode frequencies. This correlation, approximately ( ĪĪ½_g ā f_{DM}^{1/2} ), is largely EoS-independent and can serve as an observable seismic signature of dark matter.
Future gravitational-wave detectors capable of resolving such modes could verify or constrain dark-matter presence in neutron stars. The work thus bridges nuclear astrophysics and dark-sector physics, establishing asteroseismology as a potential probe of invisible matter in the cosmos.
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