Scalar-Induced Stochastic Gravitational Waves

• Scalar-induced stochastic gravitational waves are second-order tensor perturbations generated when enhanced primordial curvature fluctuations re-enter the Hubble radius during radiation domination.
• They provide a probe linking early-universe inflation, primordial black hole formation, and modified gravity with signatures observable across nanohertz to megahertz frequency bands.
• The resulting spectra can show narrow peaks or extended plateaus depending on the underlying inflationary models and cosmic expansion history.

Scalar-induced stochastic gravitational waves (SIGWs) are second-order tensor perturbations generated when primordial curvature perturbations re-enter the Hubble radius and source tensor modes through nonlinear scalar–tensor coupling. In the standard treatment, the sourcing occurs during radiation domination, and the resulting background is stochastic because many incoherent scalar modes contribute, producing an isotropic, stationary SGWB whose shape is inherited from the small-scale curvature power spectrum. Because the same enhanced scalar fluctuations can also seed primordial black holes (PBHs), SIGWs link inflationary dynamics, post-inflationary expansion history, modified gravity, and gravitational-wave observations from pulsar timing arrays to high-frequency proposals (Arya et al., 2022, Chen et al., 2024, Kumar et al., 2024).

1. Second-order origin and stochastic character

In conformal Newtonian gauge, with negligible anisotropic stress so that $\Phi=\Psi$, the perturbed metric contains a transverse-traceless tensor perturbation $h_{ij}$ sourced at second order by scalar perturbations. The tensor mode equation takes the standard inhomogeneous form

$$h''_{k}(\eta)+2\mathcal{H}h'_{k}(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$

where $\mathcal{H}=a'/a$ and the source $S_k$ is quadratic in the scalar sector. This structure is common to the radiation-era treatments used for warm inflation, PTA analyses, localized inflationary features, and box-shaped curvature spectra (Arya et al., 2022, Chen et al., 2024, Chen et al., 12 Jul 2026).

The stochastic character follows from the fact that the source is a convolution over many scalar modes. In the PTA literature this is emphasized as a random superposition of waves sourced by enhanced primordial curvature perturbations on small scales, yielding an isotropic, stationary SGWB in the nanohertz band (Chen et al., 2024). In the standard radiation-era picture, first-order primordial tensor modes obey a homogeneous equation, while SIGWs obey an inhomogeneous one. This distinction is central in $\Lambda$CDM as well: in radiation domination the source decays after horizon entry, but in matter and $\Lambda$ domination subhorizon tensor modes are not generically free, so source and gauge issues become nontrivial (Sipp et al., 2022).

A recurrent misconception is that all stochastic backgrounds associated with scalars are the same object. Standard SIGWs are specifically second-order tensor modes induced by primordial curvature perturbations. By contrast, some papers study SGWBs produced by scalar-polarized radiation in scalar–tensor gravity, or by non-spherical collapse driven by a scalar fifth force. Those are related but distinct mechanisms, with different sources, polarizations, and observables (Flores et al., 2022, Du, 2018).

2. Radiation-era formalism and spectral observables

Assuming Gaussian scalar perturbations, the induced tensor power spectrum can be written as the standard double convolution

$$\mathcal{P}_{h}(\eta,k)=4\int_{0}^{\infty}dv\int_{|1-v|}^{1+v}du \left[\frac{4v^2-(1+v^2-u^2)^2}{4vu}\right]^2 I^2(v,u,x)\,\mathcal{P}_{\zeta}(kv)\,\mathcal{P}_{\zeta}(ku),$$

with $u=|k-\tilde k|/k$, $v=\tilde k/k$, and $h_{ij}$0 (Arya et al., 2022). The observable energy density per logarithmic interval is

$h_{ij}$1

and present-day spectra follow from the usual radiation redshifting and $h_{ij}$2 factors. In general one uses $h_{ij}$3; some works write this explicitly as $h_{ij}$4 when $h_{ij}$5 (Arya et al., 2022, Cai et al., 2023).

The standard radiation-era kernel contains a resonance at $h_{ij}$6, which fixes much of the detailed spectral shape. For monochromatic or narrow scalar spectra this produces a sharply peaked SIGW spectrum; for extended scalar spectra it generates broader signals. A recent analytic treatment for a box-shaped curvature spectrum, flat in $h_{ij}$7 over $h_{ij}$8, makes the geometry of this convolution explicit. In the narrow-box regime the result factorizes into the monochromatic spectrum times a purely geometric overlap factor $h_{ij}$9, with a break at $$h''_{k}(\eta)+2\mathcal{H}h'_{k}(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$0. In the broad-box regime the SIGW spectrum separates into a lower-edge infrared rise proportional to $$h''_{k}(\eta)+2\mathcal{H}h'_{k}(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$1, a scale-invariant plateau, and a quartic cutoff at the hard endpoint $$h''_{k}(\eta)+2\mathcal{H}h'_{k}(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$2 (Chen et al., 12 Jul 2026).

A separate formal subtlety concerns late-time evolution. In $$h''_{k}(\eta)+2\mathcal{H}h'_{k}(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$3CDM, matter- and $$h''_{k}(\eta)+2\mathcal{H}h'_{k}(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$4-era SIGWs cannot be evaluated by the common free-wave substitution $$h''_{k}(\eta)+2\mathcal{H}h'_{k}(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$5 once the scalar source remains active. The corrected treatment uses the derivative of the kernel integral, not the free-wave approximation, and removes the spurious growth and sharp maximum near equality found in earlier analyses (Sipp et al., 2022).

3. Primordial sources of the scalar spectrum

Warm inflation provides one concrete realization in which dissipation amplifies the small-scale curvature spectrum. The inflaton equation contains a dissipative term,

$$h''_{k}(\eta)+2\mathcal{H}h'_{k}(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$6

and in the model studied with $$h''_{k}(\eta)+2\mathcal{H}h'_{k}(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$7 and $$h''_{k}(\eta)+2\mathcal{H}h'_{k}(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$8, the growth factor

$$h''_{k}(\eta)+2\mathcal{H}h'_{k}(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$9

drives a blue-tilted enhancement of $\mathcal{H}=a'/a$0 toward the end of inflation. The resulting SIGW background spans $\mathcal{H}=a'/a$1 to $\mathcal{H}=a'/a$2, with peaks in the $\mathcal{H}=a'/a$3–$\mathcal{H}=a'/a$4 range for the displayed parameter sets (Arya et al., 2022).

Single-field inflation with a localized bump or dip in the potential produces a different morphology: a temporary deceleration of the inflaton generates a narrow peak in $\mathcal{H}=a'/a$5 with $\mathcal{H}=a'/a$6. In the KKLT-inspired benchmarks, this yields $\mathcal{H}=a'/a$7 with peaks ranging from the PTA band to ground-based interferometer bands. The benchmark pairs B1/D1 peak at $\mathcal{H}=a'/a$8, B2/D2 at $\mathcal{H}=a'/a$9, B3/D3 at $S_k$0, and B4/D4 at $S_k$1–$S_k$2 (Zhang et al., 8 Feb 2026).

Spectator-field scenarios provide broader classes of blue small-scale spectra. In inflation with stochastic spectator scalar fluctuations, a light spectator with

$S_k$3

acquires a blue-tilted power spectrum through stochastic effects, leading to induced signals with $S_k$4 in the $S_k$5 to $S_k$6 range (Ebadi et al., 2023). In modulated reheating, a spectator field sources blue-tilted, strongly non-Gaussian curvature perturbations, and the induced GW signal can be testable by BBO and DECIGO only for large coupling values not expected in low-energy particle physics setups that can be perturbatively extrapolated up to the inflationary scale (Benaco et al., 7 Oct 2025).

These examples illustrate a persistent structural distinction. Warm inflation tends to generate an extended high-frequency enhancement through the continuous growth of $S_k$7, while localized features in cold single-field inflation generate narrow peaks, and spectator mechanisms can interpolate between the two depending on whether the small-scale enhancement is narrow, blue, or plateau-like (Arya et al., 2022, Zhang et al., 8 Feb 2026, Ebadi et al., 2023).

4. Nonstandard histories, propagation, and modified gravity

If the post-inflationary universe includes an early matter-dominated phase, the standard radiation-era picture is incomplete. A detailed treatment with decay-driven early matter domination evolves the full perturbation system through the onset and end of the EMD epoch and includes the relative velocity perturbation $S_k$8. The tensor source then contains an additional term proportional to

$S_k$9

which vanishes in pure RD and pure MD but becomes important near the transitions. The resulting SIGW spectra show enhanced mid-band power around the EMD scales, a suppressed high-frequency tail due to dilution, and, in one spectator-field scenario, a flattened low-frequency plateau (Kumar et al., 2024).

A related case arises when reheating is driven by evaporating ultra-light PBHs. In the monochromatic approximation, a simultaneous matter-to-radiation transition produces the so-called Poltergeist signal. Once the irreducible critical-collapse mass spread is included, however, the PBH mass function develops an infrared tail

$\Lambda$0

which smooths reheating and suppresses the Poltergeist background by orders of magnitude. In the eight-channel decomposition of the scalar-induced signal, none of the post-formation channels reaches either the $\Lambda$1 bound or projected detector sensitivity; only the formation channel can be relevant in tuned regions (Gouttenoire et al., 20 May 2026).

Modified gravity changes the scalar sector and therefore the induced signal. In metric $\Lambda$2 gravity, viable models keep $\Lambda$3, but the scalar source is modified by a gravitational slip and extra curvature terms. The effect found in the perturbative regime is primarily a damping of the low-frequency tail, most visible in the PTA band (Kugarajh et al., 27 Feb 2025). In a complementary R$\Lambda$4 analysis of PBH Poisson fluctuations, the requirement that SIGWs are not overproduced gives

$\Lambda$5

which is reported as $\Lambda$6 tighter than the corresponding GR bound (Papanikolaou et al., 2021).

Propagation effects can also be constrained phenomenologically. A PTA analysis allowing a nontrivial SIGW propagation speed finds

$\Lambda$7

with $\Lambda$8 consistent at the $\Lambda$9 credible level, for a lognormal curvature spectrum (Chen et al., 2024).

5. Primordial black holes and observational windows

The connection to PBHs is immediate: enhanced small-scale curvature perturbations collapse at horizon re-entry and simultaneously source SIGWs. In radiation domination the standard relation is $\Lambda$0, so the GW peak frequency and the characteristic PBH mass probe the same scale. Warm inflation provides an explicit example in which the amplified small-scale spectrum produces PBHs with $\Lambda$1 and a SIGW peak at very high frequency (Arya et al., 2022). In the localized bump/dip model, the benchmark PBH masses range from $\Lambda$2 down to evaporated populations near $\Lambda$3, while the associated SIGW peaks sweep from PTA to ground-based bands (Zhang et al., 8 Feb 2026).

PTA phenomenology has become especially prominent. A broken-power-law fit to the recent PTA common-spectrum signals favors a blue-tilted rise with $\Lambda$4–$\Lambda$5, a peak frequency $\Lambda$6, and $\Lambda$7 of order a few $\Lambda$8 (Cai et al., 2023). Within explicit inflationary models, the B1 and D1 benchmarks of the localized-feature construction yield $\Lambda$9 and $$\mathcal{P}_{h}(\eta,k)=4\int_{0}^{\infty}dv\int_{|1-v|}^{1+v}du \left[\frac{4v^2-(1+v^2-u^2)^2}{4vu}\right]^2 I^2(v,u,x)\,\mathcal{P}_{\zeta}(kv)\,\mathcal{P}_{\zeta}(ku),$$0, consistent with the NANOGrav and EPTA posterior reconstructions shown in that work (Zhang et al., 8 Feb 2026).

At higher frequencies, the observational situation depends strongly on the production mechanism. Warm inflation predicts peaks at $$\mathcal{P}_{h}(\eta,k)=4\int_{0}^{\infty}dv\int_{|1-v|}^{1+v}du \left[\frac{4v^2-(1+v^2-u^2)^2}{4vu}\right]^2 I^2(v,u,x)\,\mathcal{P}_{\zeta}(kv)\,\mathcal{P}_{\zeta}(ku),$$1–$$\mathcal{P}_{h}(\eta,k)=4\int_{0}^{\infty}dv\int_{|1-v|}^{1+v}du \left[\frac{4v^2-(1+v^2-u^2)^2}{4vu}\right]^2 I^2(v,u,x)\,\mathcal{P}_{\zeta}(kv)\,\mathcal{P}_{\zeta}(ku),$$2, outside the bands of Advanced LIGO/Virgo/KAGRA, Cosmic Explorer, and Einstein Telescope, and therefore motivates levitated-sensor detectors, microwave cavities, decameter Michelson interferometers, and resonant mass detectors (Arya et al., 2022). By contrast, localized feature models can place the peak directly in the LISA/TianQin/Taiji band, the DECIGO/BBO band, or the $$\mathcal{P}_{h}(\eta,k)=4\int_{0}^{\infty}dv\int_{|1-v|}^{1+v}du \left[\frac{4v^2-(1+v^2-u^2)^2}{4vu}\right]^2 I^2(v,u,x)\,\mathcal{P}_{\zeta}(kv)\,\mathcal{P}_{\zeta}(ku),$$3–$$\mathcal{P}_{h}(\eta,k)=4\int_{0}^{\infty}dv\int_{|1-v|}^{1+v}du \left[\frac{4v^2-(1+v^2-u^2)^2}{4vu}\right]^2 I^2(v,u,x)\,\mathcal{P}_{\zeta}(kv)\,\mathcal{P}_{\zeta}(ku),$$4 ground-based band (Zhang et al., 8 Feb 2026). Spectator-driven models populate the $$\mathcal{P}_{h}(\eta,k)=4\int_{0}^{\infty}dv\int_{|1-v|}^{1+v}du \left[\frac{4v^2-(1+v^2-u^2)^2}{4vu}\right]^2 I^2(v,u,x)\,\mathcal{P}_{\zeta}(kv)\,\mathcal{P}_{\zeta}(ku),$$5–$$\mathcal{P}_{h}(\eta,k)=4\int_{0}^{\infty}dv\int_{|1-v|}^{1+v}du \left[\frac{4v^2-(1+v^2-u^2)^2}{4vu}\right]^2 I^2(v,u,x)\,\mathcal{P}_{\zeta}(kv)\,\mathcal{P}_{\zeta}(ku),$$6 interval and are therefore naturally discussed in relation to future space interferometers (Ebadi et al., 2023).

A further misconception is that a PTA-compatible SIGW signal automatically fixes the PBH abundance. The PTA-band analyses in the supplied literature do not derive a unique PBH abundance from the common-spectrum signal. One study works directly with a broken-power-law $$\mathcal{P}_{h}(\eta,k)=4\int_{0}^{\infty}dv\int_{|1-v|}^{1+v}du \left[\frac{4v^2-(1+v^2-u^2)^2}{4vu}\right]^2 I^2(v,u,x)\,\mathcal{P}_{\zeta}(kv)\,\mathcal{P}_{\zeta}(ku),$$7 and only infers qualitatively that $$\mathcal{P}_{h}(\eta,k)=4\int_{0}^{\infty}dv\int_{|1-v|}^{1+v}du \left[\frac{4v^2-(1+v^2-u^2)^2}{4vu}\right]^2 I^2(v,u,x)\,\mathcal{P}_{\zeta}(kv)\,\mathcal{P}_{\zeta}(ku),$$8–$$\mathcal{P}_{h}(\eta,k)=4\int_{0}^{\infty}dv\int_{|1-v|}^{1+v}du \left[\frac{4v^2-(1+v^2-u^2)^2}{4vu}\right]^2 I^2(v,u,x)\,\mathcal{P}_{\zeta}(kv)\,\mathcal{P}_{\zeta}(ku),$$9 corresponds to PBHs lighter than $u=|k-\tilde k|/k$0 (Cai et al., 2023), while other analyses emphasize degeneracies among the peak scale, width, amplitude, and propagation effects (Chen et al., 2024).

6. Non-Gaussianity, quantum correlations, and conceptual boundaries

Non-Gaussian scalar statistics modify both the amplitude and the shape of the induced background. A fully non-Gaussian treatment up to fifth order in the local expansion of the curvature perturbation shows that the Gaussian contribution is supplemented by connected and disconnected higher-order terms. A particularly simple result is the linear “new” $u=|k-\tilde k|/k$1 correction,

$u=|k-\tilde k|/k$2

which rescales the Gaussian baseline and introduces a degeneracy with the scalar amplitude. In the same study, a Fisher analysis for LISA, neglecting astrophysical foregrounds, finds that the amplitude, width, and peak of the spectrum can be measured with an accuracy up to $u=|k-\tilde k|/k$3, while non-Gaussianity can be measured up to $u=|k-\tilde k|/k$4 (Perna et al., 2024).

Recent work has gone beyond the classical stochastic description by keeping track of residual quantum-information content in the scalar sector. In the covariance-matrix formalism, ordinary tensor power is sourced by scalar power contractions, whereas opposite-mode tensor coherence is sourced by anomalous scalar-coherence contractions. The connected covariance obeys

$u=|k-\tilde k|/k$5

and the claimed robust signature is therefore not a universal shift of $u=|k-\tilde k|/k$6, but a correlated tensor background with nontrivial covariance and phase structure (Ahmed, 20 Jun 2026).

Finally, the term “scalar-induced” is not used uniformly across the literature. One distinct mechanism uses a very light scalar field to mediate a strong attractive force, driving rapid structure formation on microscopic scales before matter–radiation equality; the resulting SGWB is sourced by the time-varying quadrupole of collapsing overdensities and is explicitly distinguished from standard second-order SIGWs (Flores et al., 2022). Likewise, scalar-polarized stochastic backgrounds in Brans–Dicke gravity or massive scalar–tensor gravity arise from propagating scalar degrees of freedom, often dominated by monopole or memory emission from stellar collapse, rather than by second-order mode coupling of primordial curvature perturbations (Du, 2018, Rosca-Mead et al., 2023).

Taken together, the current literature defines SIGWs as a technically mature but still rapidly evolving subject. The radiation-era convolution formalism is standard, yet its phenomenology remains sensitive to the small-scale structure of $u=|k-\tilde k|/k$7, to nonstandard cosmic histories such as EMD or PBH reheating, to modified gravity and propagation effects, and to non-Gaussian or even residual quantum correlations. This combination is what makes scalar-induced stochastic gravitational waves a uniquely dense probe of early-universe dynamics across many decades in frequency.