---
title: Scalar-Induced Stochastic Gravitational Waves
url: https://www.emergentmind.com/topics/scalar-induced-stochastic-gravitational-waves
type: topic
---

# Scalar-Induced Stochastic Gravitational Waves

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 [2204.02896] [2401.09818] [2410.17291].

## 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 [2204.02896] [2401.09818] [2607.10730].

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 [2401.09818]. 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 [2212.01190].

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 [2209.04970] [1812.06068].

## 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 $x=k\eta$ [2204.02896]. The observable energy density per logarithmic interval is
\[
\Omega_{\rm GW}(\eta,k)=\frac{1}{24}\left(\frac{k}{a(\eta)H(\eta)}\right)^2\overline{\mathcal{P}_h(\eta,k)},
\]
and present-day spectra follow from the usual radiation redshifting and $g_*$ factors. In general one uses $f=k/(2\pi a_0)$; some works write this explicitly as $f=k/(2\pi)$ when $a_0=1$ [2204.02896] [2306.17822].

The standard radiation-era kernel contains a resonance at $u+v=\sqrt{3}$, 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 $\ln k$ over $[k_-,k_+]$, 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 $\Phi_{\rm box}(\kappa,\Delta)$, with a break at $k_b=2k_*\sinh\Delta$. In the broad-box regime the SIGW spectrum separates into a lower-edge infrared rise proportional to $k^3\ln^2 k$, a scale-invariant plateau, and a quartic cutoff at the hard endpoint $k=2k_+$ [2607.10730].

A separate formal subtlety concerns late-time evolution. In $\Lambda$CDM, matter- and $\Lambda$-era SIGWs cannot be evaluated by the common free-wave substitution $h'\simeq ikh$ 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 [2212.01190].

## 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,
\[
\ddot\phi+3H\dot\phi+\Upsilon\dot\phi+V'(\phi)=0,\qquad Q\equiv \frac{\Upsilon}{3H},
\]
and in the model studied with $V(\phi)=\lambda\phi^4$ and $\Upsilon=C_\phi T^3/\phi^2$, the growth factor
\[
G(Q_k)=1+4.981\,Q_k^{1.946}+0.127\,Q_k^{4.330}
\]
drives a blue-tilted enhancement of $\mathcal{P}_\zeta(k)$ toward the end of inflation. The resulting SIGW background spans $f\simeq 1\,{\rm Hz}$ to $10^6\,{\rm Hz}$, with peaks in the $10^5$–$10^6\,{\rm Hz}$ range for the displayed parameter sets [2204.02896].

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 $\mathcal{P}_\zeta(k)$ with $\mathcal{P}_\zeta^{\rm pk}\sim 10^{-2}$. In the KKLT-inspired benchmarks, this yields $\Omega_{\rm GW,0}h^2\sim 10^{-8}$ with peaks ranging from the PTA band to ground-based interferometer bands. The benchmark pairs B1/D1 peak at $f_{\rm pk}\sim 10^{-7}\,{\rm Hz}$, B2/D2 at $10^{-3}\,{\rm Hz}$, B3/D3 at $10^{-1}\,{\rm Hz}$, and B4/D4 at $10$–$10^2\,{\rm Hz}$ [2602.07951].

Spectator-field scenarios provide broader classes of blue small-scale spectra. In inflation with stochastic spectator scalar fluctuations, a light spectator with
\[
V(\sigma)=\frac{1}{2}m^2\sigma^2+\frac{1}{4}\lambda\sigma^4
\]
acquires a blue-tilted power spectrum through stochastic effects, leading to induced signals with $\Omega_{\rm GW}h^2\simeq 10^{-20}-10^{-15}$ in the $10^{-5}\,{\rm Hz}$ to $10\,{\rm Hz}$ range [2307.01248]. 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 [2510.05967].

These examples illustrate a persistent structural distinction. Warm inflation tends to generate an extended high-frequency enhancement through the continuous growth of $Q$, 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 [2204.02896] [2602.07951] [2307.01248].

## 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 $v_{\rm rel}\equiv v_m-v_r$. The tensor source then contains an additional term proportional to
\[
\frac{12\,\mathcal{H}^2\,w(1-3w)}{1+w}\,v_{\rm rel}(\mathbf q,\eta)\,v_{\rm rel}(\mathbf k-\mathbf q,\eta),
\]
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 [2410.17291].

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
\[
\frac{df_{\rm PBH}}{d\ln M_{\rm PBH}}\propto M_{\rm PBH}^{3.78},
\]
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 $\Delta N_{\rm eff}$ bound or projected detector sensitivity; only the formation channel can be relevant in tuned regions [2605.21474].

Modified gravity changes the scalar sector and therefore the induced signal. In metric $f(R)=R+\alpha R^2$ gravity, viable models keep $c_T=1$, 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 [2502.20137]. In a complementary R$^2$ analysis of PBH Poisson fluctuations, the requirement that SIGWs are not overproduced gives
\[
\Omega_{\rm PBH,f}<5.5\times 10^{-5}\left(\frac{10^9{\rm g}}{m_{\rm PBH}}\right)^{1/4},
\]
which is reported as $45\%$ tighter than the corresponding GR bound [2112.15059].

Propagation effects can also be constrained phenomenologically. A PTA analysis allowing a nontrivial SIGW propagation speed finds
\[
c_g\gtrsim 0.61 \quad \text{(95\% credible interval)},
\]
with $c_g=1$ consistent at the $90\%$ credible level, for a lognormal curvature spectrum [2401.09818].

## 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 $M_{\rm PBH}\propto k^{-2}$, 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 $M_{\rm PBH}\sim 10^3\,{\rm g}$ and a SIGW peak at very high frequency [2204.02896]. In the localized bump/dip model, the benchmark PBH masses range from $3\times10^{-3}M_\odot$ down to evaporated populations near $10^{-20}M_\odot$, while the associated SIGW peaks sweep from PTA to ground-based bands [2602.07951].

PTA phenomenology has become especially prominent. A broken-power-law fit to the recent PTA common-spectrum signals favors a blue-tilted rise with $\alpha\sim 0.8$–$1.2$, a peak frequency $f_c\gtrsim 10^{-7}\,{\rm Hz}$, and $\Omega_{\rm GW}(f_c)h_0^2$ of order a few $\times 10^{-8}$ [2306.17822]. Within explicit inflationary models, the B1 and D1 benchmarks of the localized-feature construction yield $f_{\rm pk}\sim 10^{-7}\,{\rm Hz}$ and $\Omega_{\rm GW,0}h^2\sim 10^{-8}$, consistent with the NANOGrav and EPTA posterior reconstructions shown in that work [2602.07951].

At higher frequencies, the observational situation depends strongly on the production mechanism. Warm inflation predicts peaks at $10^5$–$10^6\,{\rm Hz}$, 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 [2204.02896]. By contrast, localized feature models can place the peak directly in the LISA/TianQin/Taiji band, the DECIGO/BBO band, or the $10$–$10^2\,{\rm Hz}$ ground-based band [2602.07951]. Spectator-driven models populate the $10^{-5}$–$10\,{\rm Hz}$ interval and are therefore naturally discussed in relation to future space interferometers [2307.01248].

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 $\Omega_{\rm GW}(f)$ and only infers qualitatively that $f_c\sim10^{-7}$–$10^{-6}\,{\rm Hz}$ corresponds to PBHs lighter than $\mathcal{O}(10^{-1})M_\odot$ [2306.17822], while other analyses emphasize degeneracies among the peak scale, width, amplitude, and propagation effects [2401.09818].

## 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” $g_{\rm NL}$ correction,
\[
\Omega_{\rm GW}\big|_{\rm new}=12\,g_{\rm NL}\,\mathcal{A}_{\mathcal R}\,\Omega_{\rm GW}\big|_{\rm Gaussian},
\]
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 $\mathcal{O}(10^{-4})$, while non-Gaussianity can be measured up to $\mathcal{O}(10^{-3})$ [2403.06962].

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
\[
\kappa(k)\propto \frac{|\gamma_k|^2}{\alpha_k^2},
\]
and the claimed robust signature is therefore not a universal shift of $\Omega_{\rm GW}$, but a correlated tensor background with nontrivial covariance and phase structure [2606.21901].

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 [2209.04970]. 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 [1812.06068] [2302.04995].

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 $\mathcal{P}_\zeta(k)$, 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.

Source: https://www.emergentmind.com/topics/scalar-induced-stochastic-gravitational-waves