Gravitational-Wave Fourier Modes

• Gravitational-wave Fourier modes are discrete spectral components that capture the oscillatory dynamics from post-merger astrophysical events such as binary neutron star mergers.
• High-resolution general-relativistic simulations combined with Fourier analysis extract key frequencies, including the dominant m=2 mode and its nonlinear combination triplet with m=0.
• Accurate detection of these modes facilitates gravitational-wave asteroseismology by linking observed frequencies to the equation of state and underlying microphysics of dense matter.

Gravitational-wave Fourier modes are the discrete spectral components corresponding to oscillatory processes within astrophysical systems that source gravitational radiation, critically probing the underlying dynamics, excitation mechanisms, and matter states in events such as compact binary mergers. In the context of post-merger remnants from binary neutron star (BNS) or compact object mergers, as investigated using general-relativistic hydrodynamical simulations, the identification and interpretation of these Fourier modes is foundational for both theoretical modeling and gravitational-wave asteroseismology.

1. Simulation Framework and Fourier Decomposition

The investigation of gravitational-wave Fourier modes from post-merger remnants relies on high-resolution general-relativistic simulations. The approach combines the spatial conformal flatness approximation for the spacetime metric with smoothed-particle hydrodynamics (SPH) for matter evolution. In particular, the conformal flatness approximation simplifies the representation of the metric while ensuring retention of general-relativistic effects needed for dynamics of high compactness. SPH tracks the fluid evolution with a Lagrangian grid of $N \sim 5 \times 10^5$ particles, capturing nonlinear oscillatory dynamics. Simulation data are mapped onto a Cartesian grid to facilitate Fourier analysis.

The dynamical variables (notably pressure and density fields) are sampled over time in the equatorial plane and subjected to temporal Fourier transforms. This procedure enables direct extraction of mode frequencies and spatial eigenfunctions, determining both the character (multipole order and azimuthal structure) and temporal properties of the oscillatory modes that develop in the post-merger remnant.

2. Nonaxisymmetric Mode Content and Nonlinear Couplings

Dominant in the gravitational-wave (GW) signature of the post-merger remnant is the nonaxisymmetric $m=2$ oscillation mode, whose eigenfrequency, $f_{m=2}$, is robustly excited independent of model specifics (mass ratio, equation of state). The extraction procedure identifies this as the primary quadrupolar mode, typically associated with strong, time-dependent, nonaxisymmetric deformations—e.g., bar-mode instabilities in a rapidly, differentially rotating hypermassive neutron star.

Crucially, detailed Fourier analysis reveals not only the persistence of $m=2$, but also higher-$m$ components such as $m=1$, $m=3$, and $m=4$, especially in unequal-mass mergers. In these cases, the $m=1$ mode can manifest as a symmetry-imposed overtone and its spatial eigenfunction demonstrates the expected nodal patterns. The interaction of these modes yields a spectrum with multiple discrete features.

Significantly, the $m=2$ mode does not exist in isolation. Nonlinear hydrodynamical couplings, particularly with the quasi-radial $m=0$ mode, generate additional spectral components: a characteristic "triplet" with frequencies given by

$$f_{2-0} = f_{m=2} - f_{m=0}, \quad f_{2} = f_{m=2}, \quad f_{2+0} = f_{m=2} + f_{m=0}.$$

These “combination frequencies” are direct tracers of nonlinear effects in the remnant's dynamics and indicate a high amplitude for nonlinear mode coupling in the post-merger phase.

3. Gravitational-Wave Spectral Features and Mode-Combination Triplet

The GW emission is calculated using the quadrupole formula as derived from the simulated matter distribution. The resultant GW spectrum in the post-merger phase is found to be dominated by three main peaks: the central peak $f_2$ (aligned with $f_{m=2}$) and the two sidebands $f_-$ and $f_+$ (corresponding to $f_{2-0}$ and $f_{2+0}$). These features are robust across a range of hadronic equations of state and for the likely BNS mass range (1.2–1.35 $M_\odot$ per star).

• The central peak: $f_2 \simeq f_{m=2}$, traces the primary nonaxisymmetric oscillation.
• Lower sideband: $f_- \simeq f_{m=2} - f_{m=0}$, arises from nonlinear difference coupling.
• Upper sideband: $f_+ \simeq f_{m=2} + f_{m=0}$, from nonlinear sum coupling.

Fourier analysis of both hydrodynamical variables (e.g., pressure) and GW strains demonstrates a one-to-one correspondence between features in the fluid spectra and the GW amplitude spectrum, confirming that nonlinear fluid dynamics are cleanly mapped into the observable GW signal.

4. Implications for Gravitational-Wave Asteroseismology

The mode identification—particularly the triplet of $f_{2-0}$, $f_2$, and $f_{2+0}$—has fundamental astrophysical implications. Once GW detectors resolve these features with sufficient signal-to-noise, it becomes possible to extract both the $m=2$ and quasi-radial ($m=0$) frequencies. Given the mass measurement from the inspiral phase, these frequencies provide input for empirical relations that link the post-merger oscillation spectrum to the equation of state (EOS) of supranuclear density matter.

This enables gravitational-wave asteroseismology: the extracted oscillation frequencies can be used to infer the stiffness or softening of the EOS, the compactness of the remnant, and constraints on strong-field nuclear physics. Precision in frequency extraction leads directly to tight constraints on EOS models, providing a path to systematically probe the microphysics of dense matter inaccessible to electromagnetic observations.

5. Prospects for Detection and Constraints with Next-Generation Observatories

The identification of Fourier mode triplets in the post-merger signal is of particular importance for upcoming and planned GW observatories (e.g., Advanced LIGO, Einstein Telescope). Increased sensitivity and extended high-frequency coverage are required to resolve the closely spaced spectral features produced by nonlinear coupling, particularly since the key frequencies typically lie in the 2–4 kHz band.

• Resolution of the triplet structure ($f_-$, $f_2$, $f_+$) will enable simultaneous measurement of both $m=2$ and $m=0$ mode frequencies and thus access to the EOS via empirically calibrated relations.
• Characterization of the spectral triplet and its time evolution will provide further insight into the remnant’s dynamical state, rotation profile, and possible instabilities or collapse to a black hole.
• The ability to constrain EOS parameters—from multiple events analyzed in this way—will support the construction of a high-density matter parameter space from gravitational-wave observations.

6. Broader Significance and Future Directions

The direct identification of oscillation modes and nonlinear combination frequencies in the GW spectrum establishes a clear dynamical connection between post-merger fluid motions and gravitational radiation. This not only impacts asteroseismology but also informs the modeling and interpretation of potential electromagnetic counterparts, nucleosynthesis in ejecta, and future strategies for multi-messenger astronomy.

Relevant future directions include:

• Extending simulation campaigns to a broader range of mass ratios, spins, and realistic microphysics.
• Integrating additional observational channels (e.g., electromagnetic signatures) for comprehensive event characterization.
• Refining empirical EOS relations using multi-event GW data, with systematic uncertainties controlled via simulation-informed modeling.

The mode-oriented decomposition of gravitational-wave spectra thus remains central to extracting the fundamental properties of matter at extreme densities and understanding the post-merger physics of compact object coalescences.