Scalar-Induced Gravitational Waves

• Scalar-induced gravitational waves are secondary tensor modes generated by the quadratic coupling of primordial curvature perturbations.
• They create a stochastic gravitational-wave background with distinct spectral morphologies, such as a k³ infrared tail and model-dependent ultraviolet slopes.
• SIGWs offer practical insights into early-universe physics including inflation, preheating, modified gravity, and the connection with primordial black holes.

Scalar-induced gravitational waves (SIGWs) are tensor perturbations of the metric generated at second order in cosmological perturbation theory by primordial scalar, or curvature, perturbations $\zeta$. They are “secondary” because they are not primordial vacuum tensor fluctuations created directly during inflation; instead they are generated later by nonlinear gravitational coupling of the scalar sector. In the standard treatment, the SIGW background is a classical stationary, isotropic stochastic gravitational-wave background characterized by its power spectrum $P_h(k)$ or energy-density spectrum $\Omega_{\rm GW}(f)$, but the same system also admits gauge-invariant reformulations and, more recently, a covariance-matrix quantum description in which residual scalar coherence can be transferred to the tensor sector (Ahmed, 20 Jun 2026, Domènech, 2021).

1. Definition and perturbative origin

On a flat FLRW background, and in conformal time $\tau$, each tensor polarization mode $h_k^\lambda$ obeys

$$h_{k}^{\lambda\prime\prime} + 2H h_{k}^{\lambda\prime} + k^2h_{k}^{\lambda} = S_{k}^{\lambda},$$

where $H=a'/a$ is the conformal Hubble parameter and the source $S_k^\lambda$ is quadratic in $\zeta$. A representative Fourier-space expression is

$$S_{k}^{\lambda}(\tau) = 4 \int \frac{d^3p}{(2\pi)^3} e_{ij}^{\lambda}(k)p^ip^j\, F(p,q,\tau)\,\zeta_{p}\,\zeta_{k-p}, \qquad q\equiv|k-p|.$$

This quadratic structure is the defining feature of SIGWs: at linear order scalar and tensor modes decouple, whereas at second order products of first-order scalar perturbations source tensor perturbations (Ahmed, 20 Jun 2026).

In Newtonian gauge, a commonly used metric is

$P_h(k)$0

with $P_h(k)$1 transverse and traceless. The induced background is therefore controlled by the scalar transfer functions, the background equation of state, and the primordial scalar statistics. In the usual classical description, $P_h(k)$2 is treated as a classical Gaussian random field with completely random phases and power spectrum $P_h(k)$3, and the SIGW spectrum is then computed by inserting this field into the quadratic source and using Wick’s theorem (Fu et al., 2019, Domènech, 2021).

2. Standard analytical formulation

The tensor field is decomposed as

$P_h(k)$4

and the sourced solution is obtained with the retarded Green function. In radiation domination one may write

$P_h(k)$5

or, equivalently in the operator language,

$P_h(k)$6

For Gaussian scalar perturbations, tensor two-point functions are therefore determined by scalar four-point functions, which reduce to two-point contractions (Domènech, 2021, Ahmed, 20 Jun 2026).

The induced tensor power spectrum takes the standard convolution form

$P_h(k)$7

with $P_h(k)$8, $P_h(k)$9, $\Omega_{\rm GW}(f)$0, and a kernel $\Omega_{\rm GW}(f)$1 built from the Green function and scalar transfer functions. In radiation domination, the GW energy density per logarithmic interval is

$\Omega_{\rm GW}(f)$2

and the present-day spectrum is obtained by standard redshifting. A frequently used explicit radiation-era expression is

$\Omega_{\rm GW}(f)$3

with the detailed kernel depending on the radiation transfer function of the scalar potential (Fu et al., 2019, Domènech, 2021).

3. Spectral morphology and dependence on the scalar sector

Because the source is quadratic, the induced tensor amplitude scales schematically as $\Omega_{\rm GW}(f)$4. For a localized feature in $\Omega_{\rm GW}(f)$5, the IR behavior in a constant-$\Omega_{\rm GW}(f)$6 era is governed by the review result

$\Omega_{\rm GW}(f)$7

with $\Omega_{\rm GW}(f)$8. In radiation domination, this gives the familiar $\Omega_{\rm GW}(f)$9 tail. The same formalism shows that efficient production is associated with a resonance near

$\tau$0

for a localized scalar peak at $\tau$1, with the sharpness and strength of the feature controlled by the background equation of state and sound speed (Domènech, 2021).

For peaked scalar spectra generated in concrete inflationary models, the detailed UV and IR slopes can be model dependent. In inflation with gravitationally enhanced friction, the curvature spectrum near the peak is a broken power law,

$\tau$2

and the induced GW spectrum obeys

$\tau$3

in the ultraviolet, with the bound $\tau$4 implying $\tau$5, while in the infrared the effective slope is approximately

$\tau$6

A complementary real-space interpretation has been developed in which the SIGW background is mainly produced around the peaks of the curvature perturbation field, with the aspherical part of those peaks providing the quadrupole responsible for GW emission. In that framework, a compact estimate for the emitted GW energy density is

$\tau$7

and peaks account for about $\tau$8 of the total SIGW signal in the examples studied (Fu et al., 2019, Iovino et al., 29 Sep 2025).

4. Realizations in inflation, preheating, defects, and modified gravity

A broad class of early-universe mechanisms generates the enhanced scalar power required for observable SIGWs. In warm inflation with quartic potential $\tau$9 and dissipation coefficient

$h_k^\lambda$0

the curvature power spectrum receives a thermal growth factor

$h_k^\lambda$1

and the resulting SIGWs populate the $h_k^\lambda$2 Hz range. In $h_k^\lambda$3-attractor self-resonant preheating, scalar amplification during a short matter-dominated preheating stage produces very-high-frequency SIGWs, and Big Bang nucleosynthesis yields the bounds

$h_k^\lambda$4

which translate into

$h_k^\lambda$5

Single-field inflation with a localized bump or dip feature in the potential can amplify $h_k^\lambda$6 to $h_k^\lambda$7 and generate SIGWs peaking from PTA to ground-based bands, while the corresponding PBH abundances remain compatible with current bounds in the benchmark cases studied (Arya et al., 2022, del-Corral et al., 24 Apr 2025, Zhang et al., 8 Feb 2026).

The same second-order formalism has also been generalized beyond the minimal radiation-era GR setting. In $h_k^\lambda$8 gravity, the tensor propagation equation in the early radiation era retains the GR form but the source is modified through the scalar sector and the gravitational slip, leading mainly to a suppression of the low-frequency tail of the SIGW spectrum. In chiral scalar-tensor theory with

$h_k^\lambda$9

and explicit focus on $$h_{k}^{\lambda\prime\prime} + 2H h_{k}^{\lambda\prime} + k^2h_{k}^{\lambda} = S_{k}^{\lambda},$$0 and $$h_{k}^{\lambda\prime\prime} + 2H h_{k}^{\lambda\prime} + k^2h_{k}^{\lambda} = S_{k}^{\lambda},$$1, the helicity-dependent factors

$$h_{k}^{\lambda\prime\prime} + 2H h_{k}^{\lambda\prime} + k^2h_{k}^{\lambda} = S_{k}^{\lambda},$$2

produce parity-violating modifications of the SIGW spectrum and a large circular polarization degree

$$h_{k}^{\lambda\prime\prime} + 2H h_{k}^{\lambda\prime} + k^2h_{k}^{\lambda} = S_{k}^{\lambda},$$3

A distinct defect-driven realization arises from scalar perturbations sourced by domain wall networks: the scalar power behaves as

$$h_{k}^{\lambda\prime\prime} + 2H h_{k}^{\lambda\prime} + k^2h_{k}^{\lambda} = S_{k}^{\lambda},$$4

and the resulting SIGW spectrum has a sharp resonant peak, an IR behavior $$h_{k}^{\lambda\prime\prime} + 2H h_{k}^{\lambda\prime} + k^2h_{k}^{\lambda} = S_{k}^{\lambda},$$5, and a steep UV fall-off $$h_{k}^{\lambda\prime\prime} + 2H h_{k}^{\lambda\prime} + k^2h_{k}^{\lambda} = S_{k}^{\lambda},$$6 (Kugarajh et al., 27 Feb 2025, Feng et al., 2024, Lu, 2024).

5. Relation to primordial black holes and observational bands

SIGWs are tightly linked to primordial black holes because the same enhanced small-scale scalar perturbations that can collapse into PBHs also source second-order tensor modes. In radiation domination the PBH mass is related to the horizon mass at re-entry, and representative formulas include

$$h_{k}^{\lambda\prime\prime} + 2H h_{k}^{\lambda\prime} + k^2h_{k}^{\lambda} = S_{k}^{\lambda},$$7

and the present-day frequency relation

$$h_{k}^{\lambda\prime\prime} + 2H h_{k}^{\lambda\prime} + k^2h_{k}^{\lambda} = S_{k}^{\lambda},$$8

or equivalently $$h_{k}^{\lambda\prime\prime} + 2H h_{k}^{\lambda\prime} + k^2h_{k}^{\lambda} = S_{k}^{\lambda},$$9. Consequently, stellar-mass PBHs are associated with PTA bands, Earth-mass and lighter PBHs with space-based interferometer bands, and still smaller scales with ground-based and very-high-frequency bands (Fu et al., 2019, del-Corral et al., 24 Apr 2025, Zhang et al., 8 Feb 2026).

Several benchmark scenarios make this correspondence explicit. In the bump/dip single-field model, the benchmark sets B1–B4 and D1–D4 yield PBH masses from $H=a'/a$0 down to $H=a'/a$1, while the associated SIGW peaks range from $H=a'/a$2 to $H=a'/a$3. In the warm-inflation example, the enhanced scalar spectrum leads to PBHs of mass $H=a'/a$4 and a high-frequency SIGW background extending over $H=a'/a$5 Hz. In domain-wall-induced scalar perturbations, the peak frequency is set by the annihilation scale, and the SIGW peak amplitude scales roughly as $H=a'/a$6 in the parametric estimates quoted. Across these scenarios, future searches involve PTAs, LISA, Taiji, TianQin, DECIGO, BBO, ground-based interferometers, and a range of very-high-frequency concepts such as bulk acoustic wave devices, optically levitated dielectric sensors, Holometer-type interferometers, microwave cavity detectors, and Gertsenshtein-effect proposals (Arya et al., 2022, Zhang et al., 8 Feb 2026, Lu, 2024).

6. Gauge dependence and observable definitions

At second order, tensor perturbations are not gauge invariant, and the source term for SIGWs depends on the slicing and threading used in the perturbative expansion. Explicit gauge transformations show that while $H=a'/a$7 is invariant at linear order, it acquires scalar-dependent terms at second order. A fully gauge-unfixed derivation of the background, first-order, and second-order Einstein equations shows that the TT-projected tensor equation is always of the form

$H=a'/a$8

but with gauge-dependent scalar source functions $H=a'/a$9 in synchronous, Poisson, and uniform-curvature gauges. Numerically, however, the kernels computed in all three gauges behave closely with minimal discrepancy, and when $S_k^\lambda$0 the discrepancy decreases and the behavior matches, pointing to a gauge-invariant observable (Kugarajh, 28 Feb 2025).

A more structural resolution starts from physical observables rather than from a gauge-fixed tensor field. Two examples are the magnetic part of the Weyl tensor and the Cotton tensor of a slicing of spacetime. Both vanish in the background and do not depend linearly on scalar perturbations, so a generalized Stewart–Walker argument implies that the scalar-induced second-order contributions are automatically gauge invariant. In this framework the tensor mode entering the magnetic Weyl tensor,

$S_k^\lambda$1

and the tensor mode entering the Cotton tensor,

$S_k^\lambda$2

are both gauge-invariant objects related explicitly to the standard Newtonian-gauge tensor $S_k^\lambda$3. In radiation domination and for short wavelengths, the observable Weyl-based SIGWs coincide with the Newtonian-gauge SIGWs, thereby giving a physical underpinning to the standard phenomenological spectrum used in much of the SIGW literature (Comeau, 2023).

7. Quantum-state description, discord, and residual coherence

The usual classical treatment assumes that decoherence has erased all phase-sensitive information, so the scalar sector is specified entirely by $S_k^\lambda$4. A quantum reformulation starts instead from a decohered two-mode squeezed Gaussian state of opposite-momentum scalar modes. For one pair $S_k^\lambda$5, the quadrature vector is

$S_k^\lambda$6

and after isotropic Gaussian decoherence the covariance matrix is

$S_k^\lambda$7

with

$S_k^\lambda$8

The smallest symplectic eigenvalue of the partially transposed covariance matrix is

$S_k^\lambda$9

so entanglement disappears when $\zeta$0, but Gaussian discord and anomalous opposite-mode coherence can remain nonzero. The scalar anomalous-coherence fraction is

$\zeta$1

and the anomalous correlator is packaged as

$\zeta$2

This quantity is the part of the scalar covariance matrix not captured by the classical phase-random description (Ahmed, 20 Jun 2026).

Propagating this Gaussian scalar state through the quadratic source yields an effective tensor Gaussian state for the pair $\zeta$3,

$\zeta$4

with transfer relations

$\zeta$5

Thus ordinary tensor power is sourced by scalar power contractions, whereas opposite-mode tensor coherence is sourced by anomalous scalar-coherence contractions. The Gaussian tensor discord is

$\zeta$6

and for weak coherence

$\zeta$7

A particularly robust observable is the connected covariance of opposite-mode tensor power,

$\zeta$8

which vanishes in a phase-random classical Gaussian background but satisfies

$\zeta$9

for the induced Gaussian tensor state. The resulting phenomenological message is that residual quantum information in SIGWs is not a universal shift of the ordinary spectrum; rather, it appears as a correlated tensor background with nontrivial covariance and phase structure, potentially relevant for future space-based interferometers and pulsar timing arrays (Ahmed, 20 Jun 2026, Ahmed, 20 Jun 2026).