Scalar Induced Gravitational Waves (SIGW)

• Scalar Induced Gravitational Waves (SIGW) are second-order tensor perturbations from quadratic scalar couplings, providing a reliable signature of early Universe fluctuations.
• Their energy density spectrum is derived via rigorous Green's function techniques and reflects key properties of the primordial scalar power spectrum and PBH formation.
• Observational targets from space and ground-based detectors make SIGWs critical for testing inflationary models and exploring modifications to General Relativity.

Scalar Induced Gravitational Waves (SIGW) are a theoretically robust component of the stochastic gravitational wave background generated during the early Universe. They arise from the quadratic coupling of first-order scalar (curvature) perturbations, a mechanism rigorously formalized at second order in cosmological perturbation theory. SIGWs are of particular interest because they inherently probe small-scale primordial fluctuations, are intimately connected to the formation of primordial black holes (PBHs), and their predicted energy density spectrum is a model-independent observable—provided the calculations account for gauge subtleties and nonlinearities. The precision paper of SIGWs plays a central role in decoding cosmological initial conditions, inflationary physics, and the validity of General Relativity on cosmological scales.

1. Theoretical Foundations and Mechanism of SIGW Generation

SIGWs are generated through the non-linear coupling in Einstein's equations, involving first-order scalar (curvature/density) perturbations from inflation. Specifically, during the radiation-dominated era, elevated curvature perturbations on small scales re-enter the horizon and drive the formation of PBHs via gravitational collapse. At the same time, the quadratic (second-order) terms in the perturbative Einstein equations source tensor (gravitational wave) modes. Unlike the linear (primordial) tensor spectrum, the SIGW component is therefore a guaranteed background—its amplitude and spectrum are set by the statistics of the primordial scalar perturbations.

The fundamental equation of motion for the tensor (SIGW) modes, $h_{ij}$, in Fourier space is

$$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = S_{ij},$$

where primes denote conformal time derivatives, $\mathcal{H}$ is the conformal Hubble parameter, and $S_{ij}$ is the second-order source term quadratic in the first-order scalar perturbations. The formal solution, for subhorizon modes after the scalar source shuts off, is expressed via a Green's function integral that determines the subsequent evolution as free gravitational waves.

A key observable is the energy density spectrum per logarithmic interval,

$$\Omega_\mathrm{GW}(k) = \frac{1}{6} \int_0^\infty du \int_{|1-u|}^{1+u} dv\, \frac{v^2}{u^2} \bigg(1 - \frac{(1+v^2-u^2)^2}{4v^2}\bigg)^2 P_\Phi(uk)\, P_\Phi(vk)\, \overline{I^2(u,v,x\to\infty)},$$

where $P_\Phi(k)$ is the primordial scalar (Bardeen) potential power spectrum, $u$ and $v$ are dimensionless momentum ratios, and $I(u,v,x)$ is the integration kernel obtained via Green's function methods, averaged over oscillations (Yuan et al., 2019).

2. Gauge Issues and Physical Observables

Though linear tensor perturbations are gauge-invariant, second-order tensors (such as SIGWs) mix with scalar and vector modes under coordinate transformations, potentially introducing spurious gauge artifacts in observable predictions. The correct treatment of second-order perturbations involves:

• Computation of the source term and transfer kernel in distinct gauges: synchronous, Newton (Poisson), and uniform curvature gauges.
• The use of a general gauge transformation formula for the induced tensor kernel, $I^G(u,v,x) = I_N(u,v,x) + I_\chi(u,v,x)$, with $I_\chi(u,v,x)$ encoding corrections from the first-order coordinate change (Lu et al., 2020).
• In synchronous, Newtonian, uniform curvature, and uniform expansion gauges, the oscillation-averaged kernel $\overline{I^2(u,v,x\to\infty)}$ and hence the resulting $\Omega_\mathrm{GW}(k)$ are found to be identical. In contrast, in total matter (comoving) and uniform density gauges, the energy density grows with time (as $\eta^2$ or $\eta^6$), indicating unphysical gauge contamination unless pure gauge modes are removed (Lu et al., 2020, Kugarajh, 28 Feb 2025).
• Numerical integration of kernel functions across gauges shows near-identical results for subhorizon modes ($k\tau \gg 1$), pointing to the gauge invariance of physical SIGW observables (Yuan et al., 2019, Kugarajh, 28 Feb 2025).

This establishes that SIGW predictions for the stochastic gravitational wave background are robust and physically meaningful when computed with proper attention to gauge-invariant structure.

3. Relation to Primordial Black Holes and the Primordial Power Spectrum

Enhanced primordial curvature perturbations required to generate PBHs unavoidably also produce SIGWs. The detailed spectrum of SIGWs encodes direct information on the primordial scalar power spectrum $P_\zeta(k)$ at small scales, far below those probed by the CMB. Notable features include:

• For a monochromatic (delta-function) curvature spectrum, the analytic computation of $\Omega_\mathrm{GW}(k)$ yields a resonance at $k \sim (2/\sqrt{3}) k_*$ (Yuan et al., 2019).
• For broader or broken power-law spectra, the SIGW energy density follows the square of the curvature power spectrum on either tail: $$\Omega_\mathrm{GW}(k) \sim [\mathcal{P}_\zeta(k)]^2$$, away from the peak (Zhang et al., 2020). For a power-law curvature spectrum $\mathcal{P}_\zeta \propto k^{n_s-1}$, this yields $\Omega_\mathrm{GW} \propto k^{2(n_s-1)}$.
• The model-dependent spectral features are preserved in the SIGW waveform, establishing SIGWs as a tool to reconstruct or constrain models of inflation, reheating, and PBH dark matter scenarios via the measurement of $\Omega_\mathrm{GW}(k)$.

Thus, simultaneous detection or non-detection of SIGWs and PBHs constrains both the amplitude and shape of small-scale primordial fluctuations.

4. Mathematical Structure and Calculation Techniques

The computation of SIGW spectra involves several rigorous steps:

• Expansion of the perturbed metric to second order, including scalar and transverse-traceless tensor modes.
• Use of Green's function techniques to solve the tensor equation, which, for the radiation-dominated Universe, yields a kernel involving sine functions of $k\eta$.
• The physical, oscillation-averaged energy density spectrum is obtained by integrating over the convolved scalar power spectra with the squared kernel, as in

$$\Omega_\mathrm{GW}(k) = \frac{1}{6} \int_0^\infty du \int_{|1-u|}^{1+u} dv\, \frac{v^2}{u^2} \bigg(1 - \frac{(1+v^2-u^2)^2}{4v^2}\bigg)^2 P_\Phi(uk) P_\Phi(vk)\, \overline{I^2(u,v,x\to\infty)}.$$

• For monochromatic input power spectra, analytic integration is often tractable. For arbitrary spectra, numerical implementation is generally required (Yuan et al., 2019).
• In each gauge under consideration, the full calculation of transfer functions, source terms, and kernels need meticulous treatment of residual gauge modes to isolate the physical result (Lu et al., 2020).

This mathematical structure assures the fidelity of SIGW predictions across cosmological models and gauge choices.

5. Implications for Observations and Theoretical Cosmology

The robust gauge-invariance of the SIGW energy density spectrum underpins its value for upcoming and future gravitational wave observations:

• The regular and universal form of $\Omega_\mathrm{GW}(k)$ enables direct comparison with observables from space-based (LISA), ground-based interferometers, and pulsar timing arrays (PTA).
• Measurement or stringent upper limits on $\Omega_\mathrm{GW}(k)$ map onto constraints on the small-scale amplitude, shape, and possible non-Gaussianity of the primordial scalar power spectrum.
• The presence of SIGWs serves as a smoking gun for early Universe scenarios characterized by enhanced small-scale fluctuations, including PBH production. Cross-correlation with other cosmological measurements may help break degeneracies among inflation, reheating, and dark matter models.
• These analyses also provide a benchmark for distinguishing modifications of General Relativity from alternative gravity models, since the detailed propagation and generation of SIGWs is theory-dependent only at a deeper level (modifying the evolution equations or introducing new degrees of freedom).

This highlights the centrality of SIGWs as cosmological observables of high diagnostic power.

6. Outlook and Directions for Future Research

Future work on SIGWs will leverage the established gauge-invariant computational framework to:

• Systematically include non-Gaussian initial conditions to capture higher-order statistics and bispectrum contributions to the SIGW background.
• Extend calculations to alternative gauge choices and general relativistic modifications, providing templates for the identification of new physics through GW background measurements.
• Develop improved theoretical models to relate observed features—such as deviations from simple power-law scaling or the presence of resonances—to specific primordial or gravitational scenarios.
• Advance methods for extracting model-dependent shapes of $\Omega_\mathrm{GW}(k)$, including infrared logarithmic dependencies, from observational data.
• Integrate multi-messenger and multi-probe strategies using SIGWs, PBH abundance, and CMB/large-scale structure constraints to maximally utilize GW backgrounds in uncovering the physics of the early Universe.

This continued progress will establish SIGWs as a critical probe at the interface of high-precision gravitational wave astronomy, early-universe cosmology, and fundamental gravitation.