Gravitational-Wave Energy Spectrum

• Gravitational-wave energy spectrum is a measure of energy density per logarithmic frequency interval, reflecting both microphysical emission and cosmic background evolution.
• It employs gauge invariant techniques and analytic models—such as the Isaacson tensor and Peters–Mathews approximation—to accurately capture source dynamics and asymptotic behavior.
• Key spectral features, like peak frequencies and scaling laws, provide actionable insights for calibrating detectors and testing models of astrophysical and cosmological processes.

The gravitational-wave energy spectrum quantifies the energy carried by gravitational waves (GWs) as a function of frequency, serving as a rigorous tool for characterizing the strength and physical properties of GW sources, calibrating detector sensitivities, and probing astrophysical/cosmological processes. Its definition, computation, and physical interpretation are tightly connected to both the microphysics of GW emission mechanisms and the broad-scale properties of the underlying spacetime and source evolution.

1. Formal Definition and Gauge Invariance

The gravitational-wave energy spectrum is defined as the energy density of GWs per unit volume per logarithmic frequency interval, normalized to the critical density of the Universe. The standard form is

$$\Omega_\mathrm{GW}(f) = \frac{1}{\rho_c}\frac{d\rho_\mathrm{GW}}{d \ln f}$$

where $\rho_\mathrm{GW}$ is the GW energy density (often derived from the Isaacson energy–momentum tensor in the linear regime) and $\rho_c$ is the closure density. In the linear perturbation theory regime, this quantity is gauge-invariant. At nonlinear order, however, naive application of the Isaacson tensor introduces gauge ambiguities due to the nonlocal nature of gravitational energy in General Relativity. To resolve this, a quasilocal energy construction is needed, in which the GW energy is obtained from the transverse traceless component of the metric perturbations, extracted via an SVT (scalar–vector–tensor) decomposition, and referenced to a fiducial background spacetime. With this prescription, the spectrum becomes

$$\Omega_\mathrm{GW}(f; V, t^\mu) = \frac{d[E_\mathrm{GW}(V, t^\mu)/V]}{d\ln f} \frac{1}{\rho_c}$$

where $E_\mathrm{GW}(V, t^\mu)$ is the GW quasilocal gravitational energy computed over a spacelike region $V$ with timelike observer $t^\mu$ (Cai et al., 2021).

2. Analytical Models and Asymptotics

The precise analytical form of the GW energy spectrum depends on the source class and emission dynamics. For example, for a burst from a parabolic encounter of compact objects, the energy spectrum is given by

$$\frac{dE}{df} = \frac{4\pi^2}{5} \frac{G^3}{c^5} \frac{M_1^2 M_2^2}{r_p^2} \ell(f/f_c)$$

with $f_c$ the circular frequency at periapse, $r_p$ the periapse radius, and $\ell$ a function derived from modified Bessel functions reflecting the harmonics of the motion in the large-eccentricity (parabolic, $e\to1$) limit. The spectrum’s peak is determined predominantly by the orbital periapse and not the eccentricity, with the maximum at $f \approx 1.637 f_c$ for the parabolic case. This analytic formula—derived from the Peters and Mathews weak-field approximation—is found to be accurate within 10% for periapse radii $r_p \gtrsim 20M$ in strong-field scenarios (Berry et al., 2010).

For hyperbolic encounters, the spectrum is instead written in terms of Hankel functions, with the analytic result validated against known parabolic and highly eccentric limits (Vittori et al., 2012, Vittori et al., 2012, Gröbner et al., 2020). These provide comprehensive, frequency-resolved spectra for burst-like events and unbound compact binary interactions.

3. Physical Dependencies and Spectral Features

The spectral shape and location of the energy peak are controlled by the physical scales of the source:

• Bound/Unbound Encounters: For parabolic/hyperbolic orbits, the spectrum peaks at a frequency set by the closest approach ($r_p$ or impact parameter), with detailed scaling governed by harmonic or Bessel/Hankel structure.
• Continuous Sources (Long-Lasting): For sources whose energy density redshifts as $\rho_s \propto 1/a^\beta$, the GW energy spectrum scales as

$\Omega_\mathrm{GW}(f) \propto f^{2\beta-8}$

so the source redshift behavior (i.e., $\beta$) sets the spectral index (Ramazanov, 2023). For instance, melting domain walls ($\beta=5$) yield $\Omega_\mathrm{GW} \propto f^2$, matching recent PTA observations.

• Equation of State (EOS) and Speed of Sound: In stochastic backgrounds generated by sound waves from first-order phase transitions, the EOS affects both the amplitude and the peak position of the spectrum. A softer EOS (lower speed of sound $c_s$, lower $\omega=p/e$) results in
  • a scale-independent suppression of amplitude,
  • a shift of the spectral peak to lower frequencies (with $k_\ast \propto c_s$),
  • unchanged power-law slopes in the asymptotic regions (Giombi et al., 2 Sep 2024).
• Microphysical Effects and Anisotropies: Gauge-invariant handling of tensor perturbations, as well as the effects of nonstandard physics (dark energy within neutron stars, extra dimensions, or chiral parity-violating extensions such as Chern–Simons gravity), can introduce suppressed or enhanced features, peaks, periodic modulations, or overall amplification in the spectrum, depending on the respective transport, source, or damping properties (Yazadjiev et al., 2011, Ghayour et al., 2012, Liu et al., 14 Jan 2025).

4. Cosmological Context and Source Modeling

The spectrum’s detailed structure carries direct information on source microphysics and cosmic evolution:

• Thermal Plasmas: In the early universe or stellar contexts, thermal GW emission arises from both microscopic collisions (bremsstrahlung/photoproduction, yielding a flat spectrum at low frequency up to logarithmic corrections) and macroscopic hydrodynamic shear fluctuations (scaling as $\sim \omega^2$ at low frequencies). The amplitude in the hydrodynamic regime is set by the plasma shear viscosity ($\eta$), which may become large (notably at $T > 160$ GeV in the Standard Model, due to right-handed lepton dominance) (Ghiglieri et al., 2015, García-Cely et al., 25 Jul 2024).
• First-Order Phase Transitions: The GW power spectrum from first-order QCD or EW transitions is governed by the phase transition rate $\beta/H$. Faster transitions (large $\beta/H$) yield higher-frequency, lower-amplitude signals (peaking in the mHz range, optimal for LISA/Taiji); slower transitions (small $\beta/H$), e.g., in the presence of high baryon chemical potential or critical quark nuggets, shift the spectrum to the nHz band (within PTA sensitivity) and can significantly increase amplitude (Shao et al., 9 Oct 2024).
• Persistent Source Accumulation: For a long-lasting, stationary source (e.g., sound waves in an expanding universe), the growth of the integrated GW energy spectrum is universally governed by a factor $\Upsilon$ set by the equation-of-state parameter $w$:

$$\Upsilon = \frac{2\left[1-y^{3(w-1)/2}\right]}{3(1-w)} \qquad y = \frac{a(t)}{a(t_s)}$$

with limiting forms (e.g., $\Upsilon_{w=1/3} = 1-1/y$ for radiation domination), generalizing previous results and allowing analytic extrapolation to arbitrary cosmic backgrounds (Guo et al., 31 Oct 2024).

5. Impact of New Physics and Structural Transitions

• Extra Dimensions and Modified Gravity: If high-frequency thermal gravitons are generated due to extra-dimensional effects, the amplitude of the spectrum can be enhanced by a factor $\coth^{1/2}[k/(2T)]$ at high frequencies, but the integrated spectral energy density remains consistent with cosmological nucleosynthesis constraints. Such scenarios predict potential observational signatures in the GHz regime and encode the early universe’s effective dimensionality (Ghayour et al., 2012).
• Parity Violation and Nonstandard Tensor Evolution: In theories such as Chern–Simons gravity, right- and left-polarized GW modes evolve differently due to parity-violating terms ($\pm \Theta k h'_s$ in the tensor equations). Numerical integration yields a birefringent spectrum, with small but distinct amplitude differences, and a pattern of chirality-dependent peaks and dips—potentially measurable with high-sensitivity space-based interferometers. Oscillatory features in the summed GW spectrum (“chiral independent power spectrum”) directly encode the parity-violating coupling and reheating parameters (Liu et al., 14 Jan 2025).

6. Observational Forecasts and Model Discrimination

• Detection Strategies: The spectral energy density $\Omega_\mathrm{GW}(f)$ provides a direct metric for calibrating and optimizing GW detectors. Stochastic backgrounds are searched in frequency bands where the expected $\Omega_\mathrm{GW}(f)$—from models of compact-object encounters, cosmological phase transitions, or topological defects—intersects the sensitivity curves of observatories such as LISA (mHz), Taiji, TianQin (mHz), PTAs (nHz), or next-generation ground-based interferometers.
• Source Constraints and Inference: Deviations from canonical flat or power-law behavior (e.g., peaks, broken power laws, or scale-independent suppression) can be directly traced to underlying physical mechanisms—allowing constraints on, for example, the speed of sound, EOS, duration of the GW source, cosmological expansion history, the presence of dark energy signatures in neutron stars, or new scalar or axion fields (Oikonomou, 2023, Li et al., 17 Jul 2024, Cai et al., 2021).

7. Schematic Table: Spectrum Scaling for Characteristic Source Classes

Source/Era	Scaling of Source Energy Density	GW Spectrum Scaling ($\Omega_\mathrm{GW}(f)$)	Key Spectral Features
Parabolic (Keplerian) Enc.	Dirac at periapse	Peak at $f\approx1.637f_c$, shape from Bessel/K-functions	Peak set by periapse (Berry et al., 2010)
Hyperbolic Encounter	Burst; transient	Frequency-domain analytic form; peaks/lobes	Scales with encounter parameters
Long-lasting source ($\beta$)	$\propto 1/a^\beta$	$\propto f^{2\beta-8}$	e.g., $\beta=4: f^0$, $\beta=5: f^2$
SIGWs (broken PL)	Broken power law (α, β)	IR: $k^3\ln^2 k$; UV: $k^{-2\beta}$; intermediate peak	Peak and power-law tails (Li et al., 17 Jul 2024)
Sound waves, 1st order PT	Stationary source, EOS (w)	Accumulation ∝ $\Upsilon(w,y)$; peak $\propto c_s$	Suppression, peak shift (Giombi et al., 2 Sep 2024, Guo et al., 31 Oct 2024)

This table summarizes the analytic scaling relationships derived for archetypal sources as established across recent literature.

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The gravitational-wave energy spectrum is thus a central, physically rich observable: its analytic structure, amplitude, and features encode the astrophysical and cosmological processes that generate GWs. Its rigorous computation requires careful attention to gauge invariance, source dynamics, and cosmic background evolution, while its measurement and interpretation in different frequency bands provides a unique probe of fundamental physics at both astrophysical and cosmological scales.