Scalar-Induced Gravitational Waves

• Scalar-induced gravitational waves are stochastic signals generated at second order by the nonlinear coupling of primordial scalar perturbations during the radiation-dominated era.
• Their formalism involves convolution integrals that connect the scalar power spectrum with tensor modes, producing distinctive spectral features sensitive to inflationary dynamics.
• SIGWs serve as a vital probe into small-scale cosmology and primordial black hole formation, offering constraints on inflationary models and early-universe physics.

Scalar-induced gravitational waves (SIGWs) are stochastic gravitational waves produced at second order by the nonlinear coupling of primordial scalar (curvature) perturbations, particularly during the radiation-dominated era of the early universe. They offer a unique probe of small-scale primordial perturbations, inflationary model-building, primordial black holes (PBHs), early-universe phase transitions, and departures from general relativity. The theoretical apparatus for SIGWs spans nontrivial mathematical frameworks, a variety of parameterizations for the source scalar spectrum, observational implications, and intricate connections to fundamental cosmology.

1. Second-Order Sourcing Mechanism and Formalism

SIGWs arise when first-order scalar perturbations—encoded in the curvature perturbation $\zeta$ or the Bardeen potential $\Phi$—source second-order tensor (gravitational-wave) perturbations through the quadratic nonlinearity of Einstein’s equations during radiation domination. In Fourier space, the evolution equation for each tensor mode $h_k$ is schematically

$h_k'' + 2\mathcal{J} h_k' + k^2 h_k = 4S_k,$

where $\mathcal{J} = a'/a$ is the conformal Hubble parameter and $S_k$ involves convolution integrals over two scalar modes: $$S_k \propto \int d^3q\,F(\vec{q},\vec{k}-\vec{q},\eta)\Phi_q\Phi_{k-q}.$$ The quadratic form ensures that—even if the primordial gravitational wave background is negligible—gravitational waves are inevitably generated wherever scalar perturbations are present and sufficiently large.

The SIGW power spectrum is generically given by

$$\mathcal{P}_h(k,\eta) = 4\int_0^\infty dv \int_{|1-v|}^{1+v} du\,[K(u,v)]^2\,I_\text{RD}^2(u,v,x)\,\mathcal{P}_\zeta(vk)\mathcal{P}_\zeta(uk),$$

where

$$K(u,v) = \frac{4v^2 - (1 + v^2 - u^2)^2}{4uv}, \qquad x = k\eta,$$

and $I_\text{RD}$ encodes the radiation-era transfer function with oscillatory and logarithmic dependence. The energy density in gravitational waves is then

$$\Omega_\text{GW}(k,\eta) = \frac{1}{24}\left(\frac{k}{\mathcal{J}(\eta)}\right)^2\langle\mathcal{P}_h(k,\eta)\rangle.$$

For a nearly scale-invariant primordial spectrum, this machinery produces a background with energy density peaking around $k \sim k_\text{peak}/\sqrt{3}$ and characteristic infrared and ultraviolet scaling (Zhang et al., 2020, Yuan et al., 2021).

2. Relation Between SIGW Spectrum and Scalars: Parameterizations and Scaling

The link between the frequency spectrum of SIGWs and the primordial scalar (curvature) power spectrum $\mathcal{P}_\zeta(k)$ is direct and robust when the source spectrum is smooth. Away from resonant features (e.g., sharp peaks), a nearly universal scaling emerges,

$\Omega_\text{GW}(k) \sim [\mathcal{P}_\zeta(k)]^2,$

independently of the detailed functional form of $\mathcal{P}_\zeta(k)$ (Zhang et al., 2020).

Two central parameterizations are used to capture relevant features at small scales:

• Broken Power Law ("sharp spike"): the spectrum is modeled by piecewise power laws connected at break points, capturing abrupt features from inflationary transitions or fast-roll periods.
• Broad Spike/Arched Shape: parameterized (e.g., by $1/(c \log_{10}(k/k_p) - e)$), describing wide enhancements as might arise from near-inflection points or broad instability regimes in inflation.

The convolution structure of the SIGW generation means that, for a power-law primordial spectrum $\mathcal{P}_\zeta(k) = Ak^{n_s-1}$,

$\Omega_\text{GW}(k) \propto k^{2(n_s-1)}$

so that the functional forms (spectral indices, width and asymmetry of peak, etc.) are reflected in the GW signal. Near the spike, nonlinearities and resonance cause additional bumps and detailed structure, which must be treated beyond simple scaling (Li et al., 17 Jul 2024).

3. Physical Interpretation: Resonance and the Role of Primordial Features

The kernel coupling two scalar modes to each GW mode is sharply peaked (for $u,v$ near $\sqrt{3}/2$), aligning SIGW production with resonance among horizon-reentering scalar modes. This means the GW spectrum closely traces the "square" of the structure seen in the scalar spectrum,

• The GW "waveform" mirrors that of $\mathcal{P}_\zeta(k)$ away from the peak, and
• Spectral features (steps, dips, log-enhanced tails) in the source yield corresponding secondary features in $\Omega_\text{GW}(k)$.

Physically, this enables a form of "inversion": with high-precision GW spectroscopy, one could reconstruct otherwise inaccessible aspects of the small-scale primordial curvature spectrum, potentially probing inflationary physics at scales far smaller than accessible via the cosmic microwave background (CMB) (Zhang et al., 2020).

Critically, the amplitude of SIGWs is sensitive to enhancements in $\mathcal{P}_\zeta(k)$ (e.g., $10^{-9} \to 10^{-2}$ relevant for PBH formation), so that even subdominant primordial black hole populations imply large observable backgrounds.

4. Model Constraints and Discrimination using SIGWs

SIGWs provide a sharp tool for constraining or distinguishing among early-universe models:

• Primordial black hole (PBH) scenarios: The exponential sensitivity of the PBH formation probability ($\beta \sim \sqrt{A}e^{-\delta_c^2/2A}$, $A$ is the amplitude of $\mathcal{P}_\zeta$) means that even a very modest secondary SIGW background may rule out, or be required by, PBH dark matter models (Yuan et al., 2021). Cross-correlation with PBH abundance constraints (microlensing, CMB, merger rates) intervenes directly on $\mathcal{P}_\zeta$ and thus on predicted $\Omega_\text{GW}$.
• Inflationary Model Selection: The precise spectral shape of SIGWs (position and slope of peak, fall-off, presence of broad or sharp enhancements) discriminates between classes of inflationary models, whether slow-roll, inflection-point, temporary ultra slow-roll, or with features/multifield effects.
• Cosmological Data Fitting: Bayesian analyses of PTA (Pulsar Timing Array) data show SIGWs (with their predicted "blue" infrared spectrum or model-dependent log-corrections) fit current observed stochastic signals better than the more gently sloped supermassive black hole binary (SMBHB) backgrounds, with strong Bayes factors favoring SIGWs (Yi et al., 2023).

Comparison Table: Parameterizations, Key Relations, and Observables

Parameterization	SIGW–Scalar Relation	Observational Discriminator
Broken power law	$\Omega_\text{GW}(k)\sim[\mathcal{P}_\zeta(k)]^2$ outside peak; near-peak from convolution	Asymmetry, width, power-law tails in $\Omega_\text{GW}$
Broad / arched peak	Similar $\Omega_\text{GW}(k)\sim[\mathcal{P}_\zeta(k)]^2$; broad central features	Slope, log-runnings, position and scale of enhancement
Monochromatic (delta)	Analytic expressions; sharp resonance peak (divergence in GR)	Position and divergence at $k \sim 2k_*/\sqrt{3}$

5. Observational Prospects and Data Analysis Strategies

Detection of the stochastic gravitational wave background attributable to SIGWs is a target for:

• Pulsar Timing Arrays (PTAs): NANOGrav, EPTA, PPTA, CPTA—best suited for $f\sim10^{-9}$–$10^{-7}\,\text{Hz}$. Characteristic "blue" infrared spectra (e.g., $\Omega_\text{GW} \sim f^{2n_1}$ or $\Omega_\text{GW}(f)\sim f^3/\ln(f)$) distinguish SIGW backgrounds from SMBHBs (Yi et al., 2023).
• Space-based Interferometers: LISA, DECIGO, Taiji, TianQin—milli-Hertz to deci-Hertz, sensitive to enhancements at small scales; signal-to-noise calculations depend directly on the predicted $\Omega_\text{GW}(f)$.
• Ground-based Interferometers: LIGO/Virgo/KAGRA; typically higher frequencies, less suited unless assessed for high-$k$ enhancement features. Detection strategies benefit from rapid semi-analytic approximation formulas covering a broad range of spectral shapes, including near-peak regions (Li et al., 17 Jul 2024). Analytical segmentations speed template generation and parameter inference in gravitational-wave data analysis pipelines, enabling robust constraints on the underlying scalar spectrum from stochastic signals.

Signatures in SIGW spectra, such as position-shifted peaks (relative to the source spectrum), logarithmic tails, or features corresponding to non-trivial inflationary transitions, provide discriminants for underlying microphysics.

6. Physical and Model-based Implications

The quadratic sourcing renders the SIGW amplitude sensitive, making it a decisive test not just of large primordial scalar perturbations but of the inflationary sector and early-universe dynamics:

• Microscale Cosmology: SIGWs are the only practical avenue for direct observational probes of primordial fluctuations on ultra-small scales, probing $k \gtrsim 10^5\,\text{Mpc}^{-1}$ unreachable by CMB or large-scale structure.
• Primordial Black Holes: The required scalar power to yield $O(1)$ PBH abundances necessarily saturates (or exceeds) the limits set by non-observation of SIGWs, placing strong bounds on all PBH models (Yuan et al., 2021).
• Model Discrimination: The possibility of reconstructing $\mathcal{P}_\zeta(k)$ from $\Omega_\text{GW}(k)$ (or at least constraining its features) enables exclusion or confirmation of inflationary models with specific small-scale signatures.

7. Limitations, Inversion, and Theoretical Consistency

While the relation $\Omega_\text{GW}(k)\sim[\mathcal{P}_\zeta(k)]^2$ holds robustly away from pronounced nonlinearity and near-peak resonances, subtleties arise:

• Near strong peaks, mode-coupling and interference, as captured by the full convolution kernel, produce nontrivial additional features, requiring full numerical or semi-analytic treatments (Zhang et al., 2020, Li et al., 17 Jul 2024).
• Degeneracies between broad and sharp scalar source features can yield similar GW waveforms in the "squared" region but differ in peak-to-tail structure or off-peak slopes.
• The inversion of $\Omega_\text{GW}(k)$ to infer $\mathcal{P}_\zeta(k)$ is approximate: fine features or broad peaks require careful consideration, potentially involving regularization or prior constraints from other cosmological probes.

It is fundamental that the gravitational wave spectrum produced through these mechanisms is not only generically calculable for a given scalar spectrum, but that the converse—constraining scalar physics from observed $\Omega_\text{GW}$—is informative for inflation, PBH scenarios, or the assessment of departures from canonical single-field inflation.

---

In summary, the rigorous theoretical connection between the SIGW waveform and the primordial scalar power spectrum forms the backbone of using stochastic gravitational wave observations as a probe of small-scale inflationary physics and PBH-related phenomena. This connection, organized through convolution integrals and robust scaling laws, underpins both model-building and data analysis strategies for next-generation gravitational wave cosmology (Zhang et al., 2020, Yuan et al., 2021, Li et al., 17 Jul 2024, Yi et al., 2023).