Scalar Induced Gravitational Waves (SIGWs)

• SIGWs are gravitational waves generated at second order by quadratic scalar perturbations during the early universe's horizon re-entry.
• Their spectrum exhibits a universal infrared behavior and complex near-peak structures that reveal detailed features of the primordial curvature perturbations.
• SIGWs serve as a crucial probe for early-universe models, modified gravity scenarios, and primordial black hole formation, with observable signatures in PTA data.

Scalar induced gravitational waves (SIGWs) are gravitational waves generated at second order in cosmological perturbation theory by first-order scalar perturbations. In the early universe, sufficiently large primordial scalar or curvature perturbations source a stochastic gravitational wave background through nonlinear couplings, especially during horizon re-entry in the radiation-dominated era. They are therefore closely tied to enhanced small-scale primordial power, primordial black hole (PBH) formation, and stochastic background searches from pulsar timing arrays (PTAs) to space-based interferometers (Yuan et al., 2019, Yi et al., 2023).

1. Perturbative origin and basic observables

The defining feature of SIGWs is that the tensor sector is sourced by quadratic combinations of scalar perturbations. In a general perturbed metric with scalars and tensors, the second-order tensor perturbation $h_{ij}$ obeys a sourced wave equation of the form

$$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = -4\mathcal{T}_{ij}^{\ell m} S_{\ell m},$$

where $\mathcal{H}=a'/a$, $\mathcal{T}_{ij}^{\ell m}$ is the transverse-traceless projector, and $S_{\ell m}$ is quadratic in first-order scalar perturbations (Yuan et al., 2019). In Fourier space the corresponding Green-function solution is

$$h(\eta,\mathbf{k})=\frac{1}{a(\eta)}\int_0^\eta g_k(\eta;\eta')\,a(\eta')\,S(\eta',\mathbf{k})\,d\eta',$$

with $g_k(\eta;\eta')=\frac{1}{k}\sin[k(\eta-\eta')]$ in the radiation-dominated case (Yuan et al., 2019).

The standard observable is the gravitational-wave energy density per logarithmic interval in wavenumber, normalized to the critical density. A commonly used expression is

$$\Omega_\mathrm{GW}(\eta_0,k)=\frac{1}{24}\left(\frac{k}{\mathcal{H}(\eta_0)}\right)^2\overline{\mathcal{P}_h(k,\eta_0)},$$

where $\mathcal{P}_h$ is the induced tensor power spectrum (Yi et al., 2023). In radiation domination, the SIGW spectrum is written as a convolution over the scalar spectrum. A representative form is

$$\Omega_{\mathrm{GW}}(k)=\int_0^{+\infty} dv \int_{|1-v|}^{1+v} du\,\mathcal{T}(u,v)\,\mathcal{P}_\mathcal{R}(uk)\mathcal{P}_\mathcal{R}(vk),$$

with the kernel $$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = -4\mathcal{T}_{ij}^{\ell m} S_{\ell m},$$0 encoding mode coupling and source evolution (Li et al., 2024).

This structure makes SIGWs an unavoidable stochastic background whenever the scalar sector is enhanced. In that sense, the SIGW background is a guaranteed stochastic gravitational wave background whose amplitude and frequency shape are controlled by the small-scale primordial perturbations, rather than by primordial tensor production alone (Perna et al., 2024).

2. Spectral structure, infrared universality, and near-peak behavior

A central result in the SIGW literature is that the infrared part of the spectrum has a universal behavior that is largely independent of the details of the inflation model. In PTA-oriented analyses, the low-frequency spectrum is often parameterized as

$$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = -4\mathcal{T}_{ij}^{\ell m} S_{\ell m},$$1

with amplitude $$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = -4\mathcal{T}_{ij}^{\ell m} S_{\ell m},$$2 at a reference scale and spectral index $$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = -4\mathcal{T}_{ij}^{\ell m} S_{\ell m},$$3 (Yi et al., 2023). For a peaked scalar spectrum with pre-peak scalar index $$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = -4\mathcal{T}_{ij}^{\ell m} S_{\ell m},$$4, the infrared behavior is

$$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = -4\mathcal{T}_{ij}^{\ell m} S_{\ell m},$$5

Within this framework, supermassive black hole binaries correspond to $$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = -4\mathcal{T}_{ij}^{\ell m} S_{\ell m},$$6, and domain walls to $$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = -4\mathcal{T}_{ij}^{\ell m} S_{\ell m},$$7 (Yi et al., 2023).

For broken power-law scalar power spectra, the scalar spectrum can be parameterized as

$$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = -4\mathcal{T}_{ij}^{\ell m} S_{\ell m},$$8

where $$h_{ij}'' + 2\mathcal{H} h_{ij}' - \nabla^2 h_{ij} = -4\mathcal{T}_{ij}^{\ell m} S_{\ell m},$$9 is the peak scale and $\mathcal{H}=a'/a$0 characterize the infrared and ultraviolet tails (Li et al., 2024). In this case, analytical approximations show that both asymptotic tails and the intermediate peak contribute distinct features to the SIGW spectrum. The paper on broken power-law peaks identifies the canonical infrared behavior

$\mathcal{H}=a'/a$1

the critical case

$\mathcal{H}=a'/a$2

and the very steep-spectrum limit

$\mathcal{H}=a'/a$3

for extremely narrow spectra or very steep slopes (Li et al., 2024).

Away from the peak, the waveform tracks the scalar source in a particularly simple way. For sharp or broad spikes in the scalar power spectrum, the relation

$\mathcal{H}=a'/a$4

holds away from the peak of the spike, independent of the functional form of the scalar power spectrum (Zhang et al., 2020). This makes the SIGW waveform a direct probe of the shape of $\mathcal{H}=a'/a$5 at small scales.

Near the peak, however, the convolution structure produces richer behavior. The broken power-law analysis gives a closed analytic approximation that captures the shape, height, width, asymmetry, and transitions between asymptotic tails and the internal near-peak structure, and shows that near-peak features can in principle distinguish among different models of inflation or early-universe physics (Li et al., 2024).

3. Gauge dependence and the status of the observable spectrum

SIGWs raise a nontrivial gauge issue because tensor perturbations sourced at second order are not generally gauge-invariant. A general formula valid in any gauge was derived for the second-order transverse-traceless tensor perturbation, together with the explicit gauge transformation relating kernels computed in different gauges (Lu et al., 2020).

Despite this formal gauge dependence, several results delimit the physically relevant regime. For the radiation-dominated era, the energy density spectra $\mathcal{H}=a'/a$6 calculated in synchronous gauge, Newton gauge, and uniform curvature gauge are identical (Yuan et al., 2019). A later analysis extended this comparison and found that the Newtonian gauge, the uniform curvature gauge, the synchronous gauge, and the uniform expansion gauge yield the same result for the energy density of SIGWs, provided that the pure gauge modes in the synchronous gauge are identified and eliminated (Lu et al., 2020).

The same analysis also exhibited gauges with unphysical late-time growth. In the total matter gauge and the comoving orthogonal gauge, the energy density of SIGWs increases as $\mathcal{H}=a'/a$7; in the uniform density gauge, it increases as $\mathcal{H}=a'/a$8 (Lu et al., 2020). These results were interpreted as consequences of non-decaying or growing scalar sources rather than of physical gravitational-wave energy density.

A later review revisited the kernels in synchronous, Poisson, and uniform curvature gauge and found that the numerically computed kernels behave closely with minimal discrepancy. As expected, when going in sub-horizon modes, $\mathcal{H}=a'/a$9, the discrepancy decreases and the behavior matches, pointing to a gauge-invariant observable (Kugarajh, 28 Feb 2025). In practice, the observable SIGW background is therefore treated as effectively gauge-invariant in the sub-horizon regime relevant for detection.

4. Non-Gaussianity, anisotropies, and PBH connections

Even when the primordial curvature perturbation is Gaussian, SIGWs are intrinsically non-Gaussian because they are generated by nonlinear gravitational dynamics. The basic higher-order observables are the bispectrum and the skewness. For Gaussian primordial curvature perturbations with a lognormal spectrum,

$\mathcal{T}_{ij}^{\ell m}$0

the intrinsic non-Gaussianity of SIGWs increases as the width $\mathcal{T}_{ij}^{\ell m}$1 decreases (Zhu, 2024). An oscillation average scheme that conserves the exact results of skewness shows that late-time oscillations suppress the bispectrum amplitude and lead to a flattened-type bispectrum, rather than the equilateral behavior picked out by the envelope approximation (Zhu, 2024).

Primordial non-Gaussianity further reshapes the SIGW background. A local expansion of the curvature perturbation,

$\mathcal{T}_{ij}^{\ell m}$2

was used to study fully non-Gaussian SIGWs up to fifth order in the scalar seeds without any hierarchy (Perna et al., 2024). In that framework, LISA, neglecting the impact of astrophysical foregrounds, will be able to measure the amplitude, the width and the peak of the spectrum with an accuracy up to $\mathcal{T}_{ij}^{\ell m}$3, while non-Gaussianity can be measured up to $\mathcal{T}_{ij}^{\ell m}$4 (Perna et al., 2024).

A complementary line of work develops a diagrammatic approach for primordial non-Gaussianity up to any order and derives semi-analytic formulas for the isotropic energy-density spectrum, the angular power spectrum of energy-density anisotropies, and the angular bispectrum and trispectrum (Li et al., 22 May 2025). In this formulation, primordial non-Gaussianity can significantly alter the amplitude of the energy-density spectrum and generate substantial anisotropies through the initial inhomogeneities, while the angular bispectrum and trispectrum always vanish when the primordial curvature perturbations are Gaussian; otherwise, they do not (Li et al., 22 May 2025).

All-order lattice methods reach the same issue from a numerical direction. A lattice-simulation proposal directly computes SIGWs with non-Gaussianity up to all orders and finds that even a modest non-Gaussianity can significantly alter ultraviolet behaviors in SIGW spectra (Zeng et al., 14 Aug 2025). This is particularly relevant for PBH scenarios, because the same enhanced small-scale perturbations that source SIGWs can collapse into PBHs.

SIGWs can also be treated as an additional radiation component. Using the current observational data of $\mathcal{T}_{ij}^{\ell m}$5, the energy density spectrum of SIGWs up to third order and the abundance of PBHs can be jointly constrained (Zhou et al., 2024). In that context,

$\mathcal{T}_{ij}^{\ell m}$6

sets a bound on the total present-day SIGW energy density, tying the SIGW background, small-scale primordial curvature perturbations, and PBH abundance to CMB, BAO, and BBN constraints on dark radiation (Zhou et al., 2024).

5. PTA interpretation and the case for a cosmological origin

PTA observations have made SIGWs central to current stochastic-background phenomenology. A dedicated Bayesian analysis of the infrared part of the SIGW spectrum used combined NANOGrav 15-year data (14 frequency bins) and EPTA Data Release 2 (9 bins), for a total of 23 bins, and compared three SIGW infrared regimes with the supermassive black hole binary scenario (Yi et al., 2023). The analysis employed Bilby and the dynesty nested sampling algorithm, with the likelihood computed for the model-predicted $\mathcal{T}_{ij}^{\ell m}$7 values under the assumption of statistical independence between bins (Yi et al., 2023).

The key result is that the infrared parts of SIGWs fit the PTA data better than the stochastic background from supermassive black hole binaries (Yi et al., 2023).

Model	Best fit	$\mathcal{T}_{ij}^{\ell m}$8
SIGW ($\mathcal{T}_{ij}^{\ell m}$9)	$S_{\ell m}$0, $S_{\ell m}$1	9.1
SIGW ($S_{\ell m}$2)	$S_{\ell m}$3, $S_{\ell m}$4	8.4
SIGW ($S_{\ell m}$5)	$S_{\ell m}$6, $S_{\ell m}$7	7.5
SMBHB	$S_{\ell m}$8, $S_{\ell m}$9	0

These Bayes factors were interpreted as strong statistical preference for the SIGW interpretation over the SMBHB explanation (Yi et al., 2023). At the same time, current data is unable to distinguish between the small differences in spectral index between the universal log-dependent and more constant cases within the SIGW category (Yi et al., 2023). PBH limits also matter: the amplitude of curvature perturbations necessary for detectable SIGWs can overproduce PBHs, so only part of the PTA-favored parameter space is viable unless additional ingredients such as non-Gaussianity are invoked (Yi et al., 2023).

This PTA result does not by itself settle the origin of the nanohertz background, but it elevates the universal infrared behavior of SIGWs into a concrete model-discrimination tool.

6. Modified gravity, polarization, and numerical generalizations

A large recent literature studies how SIGWs change once the scalar sector, the tensor propagation law, or the gravitational action is modified. In slow-roll inflation with dynamical Chern-Simons gravity, SIGWs are generally polarized and characterized by the degree of circular polarization

$$h(\eta,\mathbf{k})=\frac{1}{a(\eta)}\int_0^\eta g_k(\eta;\eta')\,a(\eta')\,S(\eta',\mathbf{k})\,d\eta',$$0

but the correction from the parity-violating term is negligible on large scales, so $$h(\eta,\mathbf{k})=\frac{1}{a(\eta)}\int_0^\eta g_k(\eta;\eta')\,a(\eta')\,S(\eta',\mathbf{k})\,d\eta',$$1 and circular polarization is very small (Feng et al., 2023). By contrast, in chiral scalar-tensor gravity, semi-analytic calculations with a monochromatic curvature power spectrum yield a spectrum that behaves differently from GR before and after the peak frequency and results in a large degree of circular polarization (Feng et al., 2024).

Teleparallel and non-metricity-based models generate a broader phenomenology. In symmetric teleparallel gravity with a parity-violating term, the contribution from the parity-violating term is negligible once the observational constraints on the speed of GWs are imposed, but connection perturbations can contribute significantly and produce a multipeak structure in the SIGW energy density (Zhang et al., 2023). In metric teleparallel gravity with the Nieh–Yan term, the spectrum of the energy density of SIGWs is significantly different from that in GR and is regular everywhere, with the difference traced mainly to extra scalar degrees of freedom and modified scalar dynamics rather than to observable parity-violating propagation effects (Zhang et al., 2024). A later construction in parity-violating symmetric teleparallel gravity adds a non-minimally coupled boundary term to avoid strong coupling and finds a rightward-shifted, regular energy-density peak that differs substantially from GR, especially at high frequencies (Zhang, 20 Aug 2025).

Other modified-gravity scenarios affect different parts of the spectrum. In $$h(\eta,\mathbf{k})=\frac{1}{a(\eta)}\int_0^\eta g_k(\eta;\eta')\,a(\eta')\,S(\eta',\mathbf{k})\,d\eta',$$2 gravity, the beyond-GR correction leaves its main observational imprint in the low-frequency part of the spectrum, producing a suppression relative to GR that is most relevant in the PTA band (Kugarajh et al., 27 Feb 2025). In modified teleparallel gravity theories of the $$h(\eta,\mathbf{k})=\frac{1}{a(\eta)}\int_0^\eta g_k(\eta;\eta')\,a(\eta')\,S(\eta',\mathbf{k})\,d\eta',$$3 or mono-parametric $$h(\eta,\mathbf{k})=\frac{1}{a(\eta)}\int_0^\eta g_k(\eta;\eta')\,a(\eta')\,S(\eta',\mathbf{k})\,d\eta',$$4 type without non-minimal matter-gravity couplings, however, the SIGW signal is indistinguishable from that within GR; breaking the degeneracy requires explicit matter-gravity couplings (Tzerefos et al., 2023). In spatially covariant gravity, the breaking of time reparametrization symmetry necessitates a unitary gauge analysis, and within the subset with luminal tensor propagation the resulting SIGW spectrum shows scale-dependent modifications to both amplitude and spectral shape (Jiang et al., 27 Aug 2025).

Finally, numerical frameworks have expanded beyond purely adiabatic sources. A lattice simulation framework for adiabatic, isocurvature, and mixed initial conditions reproduces semi-analytical pure-isocurvature results, captures multi-peak structures, and shows that in early matter-dominated eras the peak amplitude and spectral slope are sensitive to the microphysical properties of the dominant field, such as the primordial black hole mass, abundance, or soliton decay rate (Zeng, 2 Oct 2025). This suggests that SIGWs are not only a probe of enhanced curvature power but also of the detailed transition history and microphysics of the early universe.

Across these extensions, one pattern recurs: the induced background is sensitive to both the scalar source statistics and the gravitational theory that mediates scalar-to-tensor conversion. In minimal scenarios that sensitivity appears chiefly through universal infrared behavior, near-peak structure, and non-Gaussian statistics; in modified gravity it can additionally appear through polarization, multipeak spectra, shifted resonances, or low-frequency suppression.