Gravitational Wave Signatures

• Gravitational wave signatures are detailed spectral and temporal patterns that encode the dynamics, microphysics, and symmetry of astrophysical sources.
• They are modeled using multipole expansions, relativistic MHD, and neutrino transport methods to quantify amplitudes, frequencies, and angular variations.
• These signatures guide the understanding of explosion mechanisms and progenitor properties, aiding detector design and multi-messenger astronomy.

Gravitational wave signatures are the detailed spectral and temporal patterns encoded in the gravitational-wave (GW) signals produced by astrophysical processes or early-universe phenomena. Each source of GWs—ranging from core-collapse supernovae, binary black hole mergers, and early-universe phase transitions to quantum corrections in strong gravity—imprints its dynamics, microphysics, symmetry structure, and even fundamental degrees of freedom onto the observable GW strain. Rigorous extraction and interpretation of these signatures is central to current and future GW astronomy, cosmology, and the search for new physics.

1. Extracting Gravitational Wave Signatures: Formulations and Modeling

Identification and extraction of GW signatures require precise theoretical formulations that account for the relevant source physics and spacetime environment. For dynamical astrophysical sources, the leading approach relies on the multipolar expansion of the metric perturbations, where the dominant GW is classically given by the quadrupole formula: $$h_{ij}^{\mathrm{TT}}(X, t) = \frac{4G}{c^4 R} P_{ijkl}(N) \int d^3x\, T_{kl}$$ where $T_{kl}$ is the energy-momentum tensor, $P_{ijkl}$ the transverse-traceless (TT) projection, and $R$ the distance to the observer.

For strong-field scenarios such as magnetohydrodynamically (MHD) driven core-collapse supernovae, this formalism must be extended to include magnetic fields and special relativistic corrections. In axisymmetric systems, the waveform simplifies to

$$h_{ij}^{\mathrm{TT}}(X,t) = \frac{1}{R} A_{20}^{E2}(t-R/c) T_{ij}^{E2,20}(\theta, \phi)$$

with $A_{20}^{E2}$ decomposed into hydrodynamic, magnetic, and gravitational contributions. For instance, the magnetic field terms are explicit in the amplitude: $A_{20}^{E2}_{\mathrm{mag}} = -\frac{G}{c^4} 32\sqrt{15\pi} \int d\mu \int dr\, r^2 [b_r^2(3\mu^2-1) + b_\theta^2(2-3\mu^2) - b_\phi^2 - 6 b_r b_\theta \mu \sqrt{1-\mu^2}]$ Quantitative GW signal predictions require fully relativistic MHD simulations with microphysical inputs such as state-of-the-art equations of state and neutrino transport models (Takiwaki et al., 2010).

For collapsar disks, aspherical neutrino emission and matter (magnetized) motions are both tackled via stress formulae that include relativistic corrections, and for anisotropic neutrino emission, ray-tracing transfer calculations in general relativity are employed. The neutrino-induced GW strain is computed as

$$h_\nu = \frac{4G}{c^4 R} \int_0^t dt' \int_0^\pi d\theta'\, \Phi(\theta')\left[\frac{dl_\nu(\theta', t')}{d\Omega'}\right]$$

where the anisotropy function $\Phi(\theta')$ is pivotal in capturing directional emission effects (Kotake et al., 2012).

2. Source Classes and Their Gravitational Wave Signatures

(a) Core-Collapse Supernovae and MHD Explosions

Magnetohydrodynamically-driven core-collapse supernovae produce GW signals whose detailed signatures are highly sensitive to the presupernova magnetic field, rotation, and explosion dynamics. Key distinguishing features include:

• Secular GW amplitude drift: For strong precollapse magnetic fields ($B_0 \sim 10^{12}\,\mathrm{G}$) and rapid rotation, total GW amplitudes show a monotonic post-bounce increase, driven by kinetic outflows and toroidal magnetic field contributions.
• Spectral features: Such models manifest a pronounced low-frequency (%%%%6%%%%) enhancement, along with bounce-associated peaks at $\sim 1\,\mathrm{kHz}$.
• Cancellation-type waveforms: With weaker magnetic fields, hydrodynamic and magnetic contributions nearly cancel, leaving negligible post-bounce GW signals (Takiwaki et al., 2010).

Computational details show that the GW energy is predominantly sourced from regions outside $\sim 1000\,\mathrm{km}$, where the growing kinetic and magnetic pressures drive explosive jets observable in the GW signal.

(b) Hyperaccreting Collapsar Disks

In the collapsar scenario for long-duration gamma-ray bursts, RGW signatures are dominated by cumulative anisotropic neutrino emission:

• Monotonic GW amplitude growth: Neutrino-induced signals increase steadily with time due to persistently axis-symmetric emission along the disk spin axis, often surpassing matter contributions by orders of magnitude.
• Spectral content: The GW spectrum is concentrated at low frequencies (1–100 Hz), which are inaccessible to current ground-based detectors but within reach for planned space-based interferometers (e.g., DECIGO) (Kotake et al., 2012).

This signature has been proposed as a direct diagnostic for the central engine of long gamma-ray bursts and is expected to be detectable out to $\sim 100\,\mathrm{Mpc}$.

(c) Rotational Instabilities in Supernova Cores

Rapidly rotating core-collapse supernovae exhibit GW signals reflecting both the macroscopic “bounce” and non-axisymmetric instabilities:

• Type I waveforms: Bounce transients show sharp negative spikes and ringdown features, with amplitudes scaling with the degree of rotation.
• Non-axisymmetric modes: One-armed spiral (m=1) and bar-like (m=2) instabilities inject narrow-band, quasi-periodic signals (100–250 Hz), with frequency shifts determined by acoustic propagation and Doppler shifts from rotation.
• Neutrino memory: Anisotropic emission adds a slowly varying, angular-dependent “memory” (Kuroda et al., 2013).

Detection prospects are favorable for Galactic events, with SNRs exceeding 10 for advanced detectors in the main emission bands.

3. Theoretical GW Signature Classification: Frequency, Angular, and Amplitude Characteristics

A rigorous classification of GW signatures is possible along key physical axes:

Source Type	Frequency Range	Signature Morphology
MHD Supernova Jets	$\sim 100$–$1000$ Hz	Monotonically rising post-bounce GW amp
Collapsar Neutrino Emission	$1$–$100$ Hz	Steady, linearly growing amplitude
Rotational Instabilities	$100$–$1000$ Hz	Quasi-periodic oscillations, bounce spike
Anisotropic Neutrino Memory	$<100$ Hz	Memory component; slow secular change

Angular anisotropy is present in both high-frequency (modally excited) and low-frequency (memory) components, with strain offsets and amplitudes strongly direction-dependent (Vartanyan et al., 2023).

Energy scaling and progenitor dependence: GW energy output can vary by up to three orders of magnitude depending on the progenitor’s core compactness: $$E_{\mathrm{GW}} \propto \xi^{0.73} \quad (\xi = \frac{M/M_\odot}{R(M)/1000\,\rm km})$$ with more compact cores yielding greater GW energy (Vartanyan et al., 2023).

4. Observational Implications and Detector Sensitivities

Spectral and temporal GW signatures inform detection strategies and experimental design:

• Low-frequency constraints: GW features below $\sim 10$ Hz face sensitivity limitations due to terrestrial seismic noise. This confines the detectability of low-frequency (memory and collapsar) signals to next-generation space-based observatories (e.g., DECIGO, LISA, Big Bang Observer).
• Amplitude sensitivity: Monotonically increasing GW amplitudes from MHD jet models with explosion energies $>10^{51}\,\mathrm{erg}$ and strong low-frequency components are the most promising for detection in the local Universe.
• Angular dependence: The detection probability increases significantly for the strongest (and most anisotropic) outflows, as GW amplitude and energy are maximal along the axis of explosion or mass ejection.

5. Microphysics and Simulation Frameworks: Role in GW Signatures

Accurate prediction and interpretation of GW signatures require simulation frameworks featuring:

• Microphysics: Realistic nuclear equations of state and detailed neutrino transport (leakage or moment-based schemes) are implemented to capture cooling and energy transport.
• Special Relativistic Magnetohydrodynamics (SRMHD): Special relativity is essential for modeling high-velocity flows and limiting unphysical propagation speeds (e.g., Alfvén velocities), especially near the protoneutron star.
• Parametric initial conditions: Systematic exploration of magnetic field strengths and differential rotation allows mapping variations in resulting GW signatures (Takiwaki et al., 2010, Kotake et al., 2012).

Spatial diagnostics, such as cumulative integrals of the waveform source in radius, further identify the characteristic “regions of emission.”

6. Diagnostic Utility of Gravitational Wave Signatures

The specific features of GW signatures serve as robust diagnostics for fundamental questions:

• Explosion mechanism: A secular post-bounce amplitude increase uniquely identifies MHD jet-driven explosions, with energetic thresholds tied to successful dynamical ejection.
• Angular momentum: Presence and frequency of quasi-periodic modulation components signal non-axisymmetric instabilities and the rotation profile of the core.
• Progenitor compactness: The overall GW energy and spectral content correlate strongly with progenitor structure, directly linking observables to stellar evolution parameters.
• Neutrino emission geometry: Memory and monotonic GW trends from anisotropic neutrino emission provide rare “smoking gun” evidence for asymmetric energy transport mechanisms at the core.

Planned detector capabilities will allow not only detection, but also discrimination between these distinct source classes in multi-messenger observations.

7. Summary

Gravitational wave signatures capture the dynamical, symmetry, and microphysical structure of their sources through well-defined spectral, temporal, and angular properties. Advances in multidimensional, relativistic simulation—including explicit magnetic field and neutrino dynamics—enable quantitative predictions of GW signal morphology as a function of progenitor parameters and explosion mechanism. Key features such as monotonic secular amplitude growth, low-frequency memory, directional anisotropy, and narrow-band quasi-periodic oscillations furnish empirical diagnostics for both fundamental physics and stellar evolution. The precise characterization and interpretation of these GW signatures will be central to the broader program of using gravitational wave astronomy as a high-fidelity probe of compact object formation, strong-field dynamics, and the underlying microphysics of stellar explosions (Takiwaki et al., 2010, Kotake et al., 2012, Kuroda et al., 2013, Vartanyan et al., 2023).