Scalar-Induced Gravitational Waves (SIGW)

• Scalar-induced gravitational waves (SIGWs) are second-order tensor perturbations generated by quadratic scalar fluctuations, representing a key non-linear signature of early-universe physics.
• They link the primordial curvature spectrum, reheating dynamics, and non-Gaussianity to observables such as energy-density spectra, angular anisotropies, and polarization features.
• Analytical, numerical, and simulation methods provide multi-observable probes that test General Relativity and modified gravity scenarios through detectable SIGW imprints.

Searching arXiv for recent SIGW literature to ground the article in published work. Scalar-induced gravitational waves (SIGWs) are second-order tensor perturbations generated by quadratic combinations of first-order scalar fluctuations in the early Universe. In General Relativity (GR), they arise because the Einstein equations are nonlinear, so scalar and tensor sectors no longer evolve independently beyond linear order; in this sense they constitute an unavoidable contribution to the stochastic gravitational-wave background when sufficiently large scalar perturbations are present (Kugarajh et al., 27 Feb 2025). Their phenomenology connects the small-scale primordial curvature spectrum, primordial non-Gaussianity, reheating dynamics, isocurvature, statistical anisotropy, and modified gravity to observables such as the energy-density spectrum, angular anisotropies, polarization structure, and higher-point functions of the gravitational-wave background (Li et al., 22 May 2025).

1. Generation mechanism and standard formulation

In the standard radiation-dominated treatment, the perturbed metric is written as

$$ds^2 = a^2(\tau) \left\{ -(1 + 2\phi)d\tau^2 + [(1 - 2\psi)\delta_{ij} + \tfrac{1}{2} h_{ij}]dx^i dx^j \right\},$$

with $h_{ij}$ the transverse-traceless tensor perturbation. At linear order, scalar, vector, and tensor perturbations evolve independently. At second order, scalar perturbations source tensor modes through a quadratic source term (Kugarajh et al., 27 Feb 2025).

The tensor equation in configuration space is

$$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = S_{ij}[\phi^{(1)}, \psi^{(1)}],$$

and in Fourier space, for polarization $\lambda=+,\times$,

$$h_{\lambda}''(\mathbf{k}, \tau) + 2\mathcal{H} h_{\lambda}'(\mathbf{k}, \tau) + k^2 h_{\lambda}(\mathbf{k},\tau) = S_{\lambda}(\mathbf{k}, \tau).$$

During radiation domination, the first-order scalar potential is commonly written as

$\phi^{(1)}(k, \tau) = T_\phi(k\tau)\zeta_k,$

so the sourcing of SIGWs is ultimately determined by the primordial curvature perturbation $\zeta$, its statistics, and the relevant transfer functions (Kugarajh et al., 27 Feb 2025).

A generic Green-function solution has the form

$$h_\lambda(\mathbf{k}, \tau) = \frac{1}{a(\tau)} \int d\tilde{\tau} \, G_\mathbf{k}(\tau,\tilde{\tau}) a(\tilde{\tau}) S_\lambda(\mathbf{k}, \tilde{\tau}),$$

and the late-time tensor power spectrum is obtained from the two-point function of $h_\lambda$. This basic structure persists in many extensions of the standard scenario, although the source term, scalar transfer functions, tensor dispersion relation, or gauge interpretation may change (Kugarajh, 28 Feb 2025).

2. Spectral observables and morphology

The central observable is the fractional gravitational-wave energy density. In one standard convention,

$$\Omega_{\rm GW}(k, \tau) = \frac{1}{48} \left(\frac{k}{a(\tau)H(\tau)}\right)^2 \overline{\Delta_h^2(k, \tau)},$$

where the overbar denotes oscillation averaging (Kugarajh et al., 27 Feb 2025). A related form used in other analyses is

$h_{ij}$0

with $h_{ij}$1 the primordial curvature power spectrum and $h_{ij}$2 the transfer kernel (Li et al., 22 May 2025).

The spectral morphology of SIGWs is closely tied to the shape of the scalar source spectrum. For sharp or broad enhancements in $h_{ij}$3, the SIGW waveform tracks the scalar spectrum rather closely, and away from the peak one finds the general relation

$h_{ij}$4

independent of the detailed functional form of the scalar spectrum (Zhang et al., 2020). For broken power-law scalar spectra, both the asymptotic power-law tails and the intermediate peak contribute distinct features to the SIGW spectrum, and analytic approximations can capture the near-peak structure across a wide range of models (Li et al., 2024).

Beyond the isotropic background, the energy-density anisotropy map can be decomposed as

$h_{ij}$5

This angular power spectrum, together with higher angular correlators, extends SIGW phenomenology from a one-dimensional spectrum to a genuinely stochastic field on the sky (Li et al., 22 May 2025).

3. Primordial non-Gaussianity and higher-order statistics

A major recent development is the explicit treatment of primordial non-Gaussianity (PNG) beyond the usual quadratic or cubic truncations. For local-type PNG, the curvature perturbation is expanded as

$h_{ij}$6

and the induced GW observables can be organized diagrammatically, with each $h_{ij}$7-point correlator expanded in connected diagrams analogous to Feynman diagrams (Li et al., 22 May 2025). This formalism yields semi-analytic expressions for the isotropic energy-density spectrum, the angular power spectrum of anisotropies, and the angular bispectrum and trispectrum of the energy-density field.

Within that framework, PNG can increase or decrease the amplitude of $h_{ij}$8, generate large-scale anisotropies through long-short mode coupling, and induce nonzero angular bispectra and trispectra. In particular, for local PNG the angular bispectrum and trispectrum of the energy-density anisotropies vanish when the primordial curvature perturbations are Gaussian and become nonzero otherwise, with schematic dependences $h_{ij}$9 and $$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = S_{ij}[\phi^{(1)}, \psi^{(1)}],$$0 (Li et al., 22 May 2025).

This should be distinguished from the intrinsic non-Gaussianity of the tensor field itself. Because the SIGW source is quadratic in scalars, the induced tensor background has a nonzero bispectrum even when the primordial curvature perturbations are Gaussian. An oscillation-average scheme that preserves the exact skewness shows that the oscillation of SIGWs suppresses the bispectrum amplitude and leads to a flattened-type bispectrum; narrower primordial curvature spectra enhance this intrinsic non-Gaussianity (Zhu, 2024). A common misconception is therefore to identify “Gaussian primordial scalars” with “Gaussian SIGWs”; the recent literature treats these as separate statements referring to different observables.

Computationally, order-by-order semi-analytic expansions become increasingly difficult at high PNG order. Lattice simulations have therefore been proposed to compute SIGW spectra with non-Gaussianity up to all orders directly in real space, and have been verified against existing semi-analytic results before being applied to higher-order primordial non-Gaussianities, logarithmic mappings in the curvature perturbation, the curvaton model, and the ultra slow-roll model (Zeng et al., 14 Aug 2025). These simulations indicate that even modest non-Gaussianity can significantly alter ultraviolet behaviors in SIGW spectra.

A complementary fully non-Gaussian treatment expands the local curvature perturbation up to fifth order in the scalar seed and evaluates all relevant contributions to $$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = S_{ij}[\phi^{(1)}, \psi^{(1)}],$$1. In that analysis, neglecting astrophysical foregrounds, LISA is forecast to measure the amplitude, width, and peak of the spectrum with an accuracy up to $$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = S_{ij}[\phi^{(1)}, \psi^{(1)}],$$2, while non-Gaussianity can be measured up to $$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = S_{ij}[\phi^{(1)}, \psi^{(1)}],$$3 (Perna et al., 2024).

4. Gauge dependence, reheating, and the definition of the physical signal

Gauge dependence is a persistent structural issue because SIGWs are second-order tensors produced by first-order scalar modes. A gauge-unfixed derivation of the background, first-order, and second-order Einstein equations shows that the tensor equation can be written in a generic gauge as

$$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = S_{ij}[\phi^{(1)}, \psi^{(1)}],$$4

with the source built entirely from products of first-order scalar perturbations (Kugarajh, 28 Feb 2025). Explicit source terms have been studied in synchronous, Poisson, and uniform-curvature gauges, and numerical kernel calculations in those gauges behave closely with minimal discrepancy. For subhorizon modes, $$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = S_{ij}[\phi^{(1)}, \psi^{(1)}],$$5, the discrepancy decreases and the behavior matches, pointing to a gauge-invariant observable (Kugarajh, 28 Feb 2025).

The situation becomes sharper for isocurvature-sourced SIGWs. An analytical study across nine gauges during radiation domination finds a clear dichotomy: in uniform-density, total-matter, uniform-curvature, comoving-orthogonal, and transverse-traceless gauges the raw energy density grows polynomially in conformal time, while in longitudinal, uniform-expansion, Newtonian-motion, and N-body gauges the late-time energy spectrum converges and SIGWs behave as radiation (Ali et al., 8 Oct 2025). The proposed resolution is a kernel projection that isolates the luminal, freely propagating gravitational-wave components oscillating as $$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = S_{ij}[\phi^{(1)}, \psi^{(1)}],$$6 and $$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = S_{ij}[\phi^{(1)}, \psi^{(1)}],$$7. After this radiative projection, the kernel decays as $$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = S_{ij}[\phi^{(1)}, \psi^{(1)}],$$8 and the late-time spectrum becomes finite and gauge independent (Ali et al., 8 Oct 2025). This suggests that the physical observable is the radiative component rather than the full gauge-dependent tensor metric perturbation.

Background dependence extends beyond pure radiation domination. For reheating, a general SIGW source term has been derived in an arbitrary gauge for a smooth transition from an inflaton field to a radiation fluid, possibly via a period of matter domination. In that treatment, both inflaton and fluid curvature perturbations appear in the source with time-dependent weights set by the evolving background enthalpy, and gauge invariance is verified up to second order (Laine et al., 4 Dec 2025). A plausible implication is that accurate SIGW predictions in non-instantaneous reheating scenarios require multi-component, gauge-invariant source terms rather than single-fluid reductions.

5. Modified gravity and departures from GR

Because SIGWs are sourced by scalar dynamics and propagate as tensor modes, they are sensitive to modifications on both sides of the problem. In $$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = S_{ij}[\phi^{(1)}, \psi^{(1)}],$$9 gravity, the second-order tensor equation acquires an additional source contribution,

$\lambda=+,\times$0

and the beyond-GR correction leaves an observational imprint mainly in the low-frequency part of the spectrum. For the model $\lambda=+,\times$1, the dominant effect is a suppression in the low-frequency tail of the SIGW spectrum, potentially relevant in the PTA band (Kugarajh et al., 27 Feb 2025).

Spatially covariant gravity provides a broader Lorentz-violating framework in which only spatial diffeomorphism invariance is retained. There the usual Newtonian gauge is unavailable, and the calculation is performed in unitary gauge. General kernels have been computed for polynomial-type Lagrangians up to $\lambda=+,\times$2, with attention restricted to operators whose tensor modes propagate at the speed of light in order to avoid late-time divergences in the fractional energy density. Representative parameter choices yield scale-dependent modifications to both amplitude and spectral shape relative to GR (Jiang et al., 27 Aug 2025).

Parity-violating theories produce a more heterogeneous picture. In chiral scalar-tensor gravity, parity-violating terms generate amplitude and velocity birefringence between right- and left-handed modes, and the resulting SIGW energy density can differ from GR before and after the peak frequency, producing a large degree of circular polarization (Feng et al., 2024). By contrast, in Chern-Simons gravity during slow-roll inflation the correction from the parity-violating term is negligible on large scales, and the degree of circular polarization of SIGWs is very small (Feng et al., 2023). In parity-violating symmetric teleparallel gravity with a non-minimally coupled boundary term, the boundary term resolves a strong-coupling problem and modifies the scalar transfer functions, leading to significant deviations from GR, particularly at high frequencies (Zhang, 20 Aug 2025).

Teleparallel constructions also illustrate that not every modified-gravity sector leaves a practical SIGW signature. In symmetric teleparallel gravity with a parity-violating term, direct parity-violating contributions to SIGWs are negligible once the observational constraints on the GW speed are imposed, but connection perturbations can still generate a multipeak structure in the energy density spectrum (Zhang et al., 2023). In viable $\lambda=+,\times$3 and mono-parametric $\lambda=+,\times$4 models without matter-gravity couplings, however, both the source and propagation sectors yield a GW signal indistinguishable from GR; breaking that degeneracy requires non-minimal matter-gravity couplings (Tzerefos et al., 2023). The modified-gravity SIGW program therefore spans cases with negligible deviations, low-frequency distortions, multipeak structures, and polarization asymmetries.

6. Statistical anisotropy, inhomogeneity, and isocurvature

SIGWs need not inherit only the isotropic Gaussian statistics of the simplest adiabatic setup. If the primordial scalar sector is statistically anisotropic, the differential SIGW energy spectrum acquires multipole moments. For quadrupole anisotropy in the scalar power spectrum, the resulting SIGW spectrum contains anisotropies up to $\lambda=+,\times$5; for delta-function-like and log-normal source spectra, analytic and numerical calculations show that the monopole itself is modified, including an additional local minimum in the high-$\lambda=+,\times$6 tail, while the higher multipoles encode the preferred-direction dependence (Chen et al., 2022).

A different departure from the standard assumption arises for coherent initial states of inflationary fluctuations. In that case the primordial scalar perturbation acquires a nonzero space-dependent mean, violating statistical homogeneity, statistical isotropy, and parity. The resulting SIGW background has a nonzero mean tensor field, off-diagonal correlations between different wavevectors, possible parity violation, and correlations between different polarization modes (Mukherjee et al., 30 Jun 2025). Detection of these signatures would probe the statistical nature of primordial perturbations on scales inaccessible to the cosmic microwave background.

Isocurvature perturbations provide an additional source class. Lattice simulations have been developed to compute SIGWs from pure isocurvature and mixed initial conditions, and the numerical results show excellent agreement with semi-analytical predictions in the pure isocurvature case (Zeng, 2 Oct 2025). For multi-peak scalar spectra, the induced GW spectra exhibit multi-peak structures under general initial conditions, closely matching those produced in purely adiabatic scenarios. In early matter-dominated eras, the peak amplitude and spectral slope become sensitive to the microphysical properties of the dominant field, such as primordial black hole mass, abundance, or soliton decay rate (Zeng, 2 Oct 2025). This suggests that SIGWs can carry information not only about primordial initial conditions but also about the composition and decay history of the pre-radiation universe.

7. Detection, interpretation, and cosmological use

The observational motivation for SIGWs spans frequencies from the nanohertz PTA regime through the millihertz LISA band to the kilohertz range of ground-based interferometers. A common mapping is $\lambda=+,\times$7, with PTA sensitivity around $\lambda=+,\times$8, LISA around $\lambda=+,\times$9, and ground-based interferometers around $$h_{\lambda}''(\mathbf{k}, \tau) + 2\mathcal{H} h_{\lambda}'(\mathbf{k}, \tau) + k^2 h_{\lambda}(\mathbf{k},\tau) = S_{\lambda}(\mathbf{k}, \tau).$$0 (Kugarajh et al., 27 Feb 2025). Across these windows, the amplitude and shape of the SIGW spectrum can constrain the primordial scalar power spectrum on small scales, the presence of primordial non-Gaussianity, and deviations from GR.

Anisotropy and non-Gaussianity provide additional discriminants beyond the isotropic spectrum. The local-PNG analysis argues that the isotropic SIGW background, the angular power spectrum of anisotropies, and higher angular correlators can collectively distinguish SIGWs with primordial non-Gaussianity from other stochastic backgrounds (Li et al., 22 May 2025). Likewise, parity-sensitive and multipolar observables in anisotropic or parity-violating scenarios broaden the inference problem from “is there a background?” to “what tensor-statistical structure does it carry?” (Chen et al., 2022).

The link to primordial black holes (PBHs) is both an opportunity and a tension. Enhanced scalar perturbations that generate large SIGW backgrounds can also overproduce PBHs, so the mapping between a detected stochastic background and viable early-universe models is nontrivial. A reassessment of the PTA interpretation of the signal as cosmological SIGWs extends the usual second-order treatment by including third-order gravitational waves,

$$h_{\lambda}''(\mathbf{k}, \tau) + 2\mathcal{H} h_{\lambda}'(\mathbf{k}, \tau) + k^2 h_{\lambda}(\mathbf{k},\tau) = S_{\lambda}(\mathbf{k}, \tau).$$1

and finds that third-order contributions can substantially enhance the spectral amplitude (Zhao et al., 5 Mar 2026). In a combined analysis of CMB, BAO, and PTA data, the favored parameter region can to some extent alleviate the PBH overproduction problem, and the SIGW interpretation remains robust even when a background from supermassive black hole binaries is included (Zhao et al., 5 Mar 2026).

Taken together, recent work positions SIGWs as a multi-observable probe of early-universe physics rather than a single spectral template. The isotropic $$h_{\lambda}''(\mathbf{k}, \tau) + 2\mathcal{H} h_{\lambda}'(\mathbf{k}, \tau) + k^2 h_{\lambda}(\mathbf{k},\tau) = S_{\lambda}(\mathbf{k}, \tau).$$2, angular anisotropies, bispectra and trispectra, polarization content, gauge-invariant radiative kernels, and sensitivity to reheating or modified gravity now form a connected research program. This suggests that the eventual scientific return of a detection will depend as much on higher-order and anisotropic characterization as on the existence of the stochastic background itself.