Second-Order Tensor Induced Gravitational Waves

• Second-order tensor induced gravitational waves (TIGWs) are tensor perturbations arising at nonlinear order from quadratic combinations of first-order fluctuations in a cosmological context.
• The methodology involves decomposing source terms into scalar–scalar, scalar–tensor, and tensor–tensor contributions and solving the tensor equation with Green's functions under various epochs.
• Observable implications include contributions to stochastic gravitational-wave backgrounds, primordial black-hole scenarios, and gauge-invariant strain measurements influenced by neutrino damping and non-Gaussian corrections.

Second-order tensor induced gravitational waves are tensor perturbations generated at nonlinear order in cosmological perturbation theory by quadratic combinations of first-order perturbations. In the broad cosmological usage, they include scalar–scalar, scalar–tensor, and tensor–tensor sources for the second-order tensor sector $h_{ij}^{(2)}$; in narrower usages, “scalar induced gravitational waves” denotes the scalar–scalar subclass, while some recent work reserves “tensor-scalar induced gravitational waves” for the mixed scalar–tensor sector and “tensor-induced gravitational waves” for tensor–tensor terms alone (Arya et al., 2022, Wu et al., 10 Jul 2025). At the level of observables, these secondary tensor perturbations are central to stochastic gravitational-wave background phenomenology, primordial black-hole scenarios, and the interpretation of second-order gravitational-wave strain in cosmology (Domènech et al., 17 Dec 2025).

1. Definition and perturbative scope

The standard starting point is a flat FLRW background with perturbative expansion

$$g_{\mu\nu}=g_{\mu\nu}^{(0)}+g_{\mu\nu}^{(1)}+\frac12 g_{\mu\nu}^{(2)}+\cdots,$$

with scalar, vector, and tensor perturbations decoupled at linear order but mixed at second order (Domènech et al., 17 Dec 2025, Chang et al., 2020). In this setting, first-order tensor perturbations are the usual primordial gravitational waves, whereas second-order tensor perturbations arise from nonlinear couplings of first-order modes. For the scalar–scalar case, the source is schematically

$$h_{ij}^{(2)} \sim \mathrm{TT}\big[\Phi^{(1)}\Phi^{(1)},\, \partial_i\Phi^{(1)}\partial_j\Phi^{(1)},\dots\big],$$

and these are the induced gravitational waves studied in the TIGW literature (Domènech et al., 17 Dec 2025).

A systematic classification of sources was given by Gong, who presented “all the possible second-order source terms” and explicit analytic solutions for the induced gravitational waves generated by linear scalar and tensor perturbations during matter- and radiation-dominated epochs (Gong, 2019). In the notation of more recent tensor-scalar studies, the second-order tensor source is naturally decomposed into

$$\mathcal{S}^{(2)}_{lm,\phi\phi},\qquad \mathcal{S}^{(2)}_{lm,\phi h},\qquad \mathcal{S}^{(2)}_{lm,hh},$$

corresponding respectively to scalar–scalar, scalar–tensor, and tensor–tensor couplings (Wu et al., 10 Jul 2025).

The terminology is therefore not uniform. A broad usage identifies second-order tensor induced gravitational waves with the whole $h_{ij}^{(2)}$ sector sourced by quadratic perturbations. A narrower usage separates SIGWs, TSIGWs, and TIGWs according to whether the source is scalar–scalar, scalar–tensor, or tensor–tensor. This suggests that precise source specification is more informative than the acronym alone (Wu et al., 10 Jul 2025).

2. Tensor equation, Green functions, and spectral construction

For scalar-induced gravitational waves in radiation domination, a standard equation is

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

with $S_k$ built from quadratic combinations of the Bardeen potential $\Phi$ and its derivatives (Arya et al., 2022). In the more general mixed-source formulation,

$$h_{ij}^{(2)''}(\eta,\mathbf{x})+2\mathcal H h_{ij}^{(2)'}(\eta,\mathbf{x})-\Delta h_{ij}^{(2)}(\eta,\mathbf{x}) = -4\Lambda_{ij}^{lm}\left(\mathcal{S}^{(2)}_{lm,\phi\phi}+\mathcal{S}^{(2)}_{lm,\phi h}+\mathcal{S}^{(2)}_{lm,hh}\right),$$

where $\Lambda_{ij}^{lm}$ is the transverse-traceless projector (Wu et al., 10 Jul 2025).

The solution is written with a Green’s function,

$$g_{\mu\nu}=g_{\mu\nu}^{(0)}+g_{\mu\nu}^{(1)}+\frac12 g_{\mu\nu}^{(2)}+\cdots,$$0

or, in the tensor-scalar notation, as a sum of convolution integrals over first-order scalar and tensor modes with kernel functions $$g_{\mu\nu}=g_{\mu\nu}^{(0)}+g_{\mu\nu}^{(1)}+\frac12 g_{\mu\nu}^{(2)}+\cdots,$$1, where $$g_{\mu\nu}=g_{\mu\nu}^{(0)}+g_{\mu\nu}^{(1)}+\frac12 g_{\mu\nu}^{(2)}+\cdots,$$2, $$g_{\mu\nu}=g_{\mu\nu}^{(0)}+g_{\mu\nu}^{(1)}+\frac12 g_{\mu\nu}^{(2)}+\cdots,$$3, and $$g_{\mu\nu}=g_{\mu\nu}^{(0)}+g_{\mu\nu}^{(1)}+\frac12 g_{\mu\nu}^{(2)}+\cdots,$$4 (Arya et al., 2022, Wu et al., 10 Jul 2025).

For scalar–scalar induced gravitational waves, the tensor power spectrum assumes the familiar convolution form

$$g_{\mu\nu}=g_{\mu\nu}^{(0)}+g_{\mu\nu}^{(1)}+\frac12 g_{\mu\nu}^{(2)}+\cdots,$$5

which makes explicit that the second-order tensor spectrum is a nonlocal functional of the primordial curvature power spectrum $$g_{\mu\nu}=g_{\mu\nu}^{(0)}+g_{\mu\nu}^{(1)}+\frac12 g_{\mu\nu}^{(2)}+\cdots,$$6 (Arya et al., 2022). In the tensor-scalar framework, the corresponding energy density is decomposed as

$$g_{\mu\nu}=g_{\mu\nu}^{(0)}+g_{\mu\nu}^{(1)}+\frac12 g_{\mu\nu}^{(2)}+\cdots,$$7

with fully analytic integral expressions for each component in radiation domination (Wu et al., 10 Jul 2025).

The subhorizon energy density is usually written as

$$g_{\mu\nu}=g_{\mu\nu}^{(0)}+g_{\mu\nu}^{(1)}+\frac12 g_{\mu\nu}^{(2)}+\cdots,$$8

and the present-day spectrum follows after the standard radiation redshifting and $$g_{\mu\nu}=g_{\mu\nu}^{(0)}+g_{\mu\nu}^{(1)}+\frac12 g_{\mu\nu}^{(2)}+\cdots,$$9 rescaling (Arya et al., 2022). This construction underlies most contemporary TIGW phenomenology.

3. Gauge dependence and observable strain

A central controversy in the subject is that the naive second-order transverse-traceless part of the spatial metric is gauge dependent. Hwang, Jeong, and Noh showed explicitly that “the induced tensor perturbation is generically gauge dependent,” and that “the gravitational wave power spectrum depends on the hypersurface (temporal gauge) condition taken for the linear scalar perturbation” (Hwang et al., 2017). In their formulation,

$$h_{ij}^{(2)} \sim \mathrm{TT}\big[\Phi^{(1)}\Phi^{(1)},\, \partial_i\Phi^{(1)}\partial_j\Phi^{(1)},\dots\big],$$0

which makes the slicing dependence explicit (Hwang et al., 2017).

A different resolution was developed in the gauge-invariant formulation of Lu, who constructed a second-order tensor variable in the synchronous frame,

$$h_{ij}^{(2)} \sim \mathrm{TT}\big[\Phi^{(1)}\Phi^{(1)},\, \partial_i\Phi^{(1)}\partial_j\Phi^{(1)},\dots\big],$$1

and showed that the gauge-invariant energy density spectrum defined from $$h_{ij}^{(2)} \sim \mathrm{TT}\big[\Phi^{(1)}\Phi^{(1)},\, \partial_i\Phi^{(1)}\partial_j\Phi^{(1)},\dots\big],$$2 coincides with the Newtonian-gauge result (Chang et al., 2020). In this formulation, fictitious tensor perturbations are removed by subtracting the TT projection of $$h_{ij}^{(2)} \sim \mathrm{TT}\big[\Phi^{(1)}\Phi^{(1)},\, \partial_i\Phi^{(1)}\partial_j\Phi^{(1)},\dots\big],$$3, which is built from first-order perturbations (Chang et al., 2020).

A more operational resolution was given by the recent analysis of geodesic clocks and electromagnetic signal exchange. There, the physically measurable second-order strain is encoded in the gauge-invariant tensor

$$h_{ij}^{(2)} \sim \mathrm{TT}\big[\Phi^{(1)}\Phi^{(1)},\, \partial_i\Phi^{(1)}\partial_j\Phi^{(1)},\dots\big],$$4

and the measured second-order redshift is

$$h_{ij}^{(2)} \sim \mathrm{TT}\big[\Phi^{(1)}\Phi^{(1)},\, \partial_i\Phi^{(1)}\partial_j\Phi^{(1)},\dots\big],$$5

The central result is that “the measured gravitational wave strain coincides with the transverse-traceless components in the Newton gauge” (Domènech et al., 17 Dec 2025).

An observationally clean example of source construction was given by the gauge-invariant baryon–CDM relative velocity $$h_{ij}^{(2)} \sim \mathrm{TT}\big[\Phi^{(1)}\Phi^{(1)},\, \partial_i\Phi^{(1)}\partial_j\Phi^{(1)},\dots\big],$$6, whose quadratic anisotropic stress sources induced tensor perturbations and a corresponding CMB $$h_{ij}^{(2)} \sim \mathrm{TT}\big[\Phi^{(1)}\Phi^{(1)},\, \partial_i\Phi^{(1)}\partial_j\Phi^{(1)},\dots\big],$$7-mode signal. Although the resulting effect is unobservably small, it demonstrates “the importance of using observable quantities to remove the gauge ambiguity” and also the observable consequences of tensor perturbations which are not propagating gravitational waves (Gurian et al., 2021).

4. Source classes and characteristic spectral signatures

The scalar–scalar channel remains the standard benchmark. In radiation domination, its kernel produces the familiar resonant behavior and logarithmic running in the low-frequency tail for peaked primordial spectra. By contrast, scalar–tensor induced gravitational waves “do not present resonances or a logarithmic running in the low frequency tail in the case of peaked primordial spectra,” because the relevant kernel never reaches the scalar–scalar resonant configuration in the infrared (Bari et al., 2023). The scalar–tensor sector also partly inherits any primordial parity violation of tensor modes and can display chiral asymmetry in the ultraviolet region (Bari et al., 2023).

For scalar–tensor induced gravitational waves, the radiation-era tensor equation can be written as

$$h_{ij}^{(2)} \sim \mathrm{TT}\big[\Phi^{(1)}\Phi^{(1)},\, \partial_i\Phi^{(1)}\partial_j\Phi^{(1)},\dots\big],$$8

so the source is linear in $$h_{ij}^{(2)} \sim \mathrm{TT}\big[\Phi^{(1)}\Phi^{(1)},\, \partial_i\Phi^{(1)}\partial_j\Phi^{(1)},\dots\big],$$9 and corresponds to a modulation of primordial tensor modes by scalar inhomogeneities (Bari et al., 2023). The corresponding induced spectrum depends on both $$\mathcal{S}^{(2)}_{lm,\phi\phi},\qquad \mathcal{S}^{(2)}_{lm,\phi h},\qquad \mathcal{S}^{(2)}_{lm,hh},$$0 and the primordial tensor spectra $$\mathcal{S}^{(2)}_{lm,\phi\phi},\qquad \mathcal{S}^{(2)}_{lm,\phi h},\qquad \mathcal{S}^{(2)}_{lm,hh},$$1, and the induced chirality is generically smaller than the primordial chirality because the polarization mixing is smeared between right- and left-handed modes (Bari et al., 2023).

The tensor-scalar formalism of Li and collaborators makes this decomposition explicit by writing

$$\mathcal{S}^{(2)}_{lm,\phi\phi},\qquad \mathcal{S}^{(2)}_{lm,\phi h},\qquad \mathcal{S}^{(2)}_{lm,hh},$$2

and by deriving compact analytic expressions for the scalar–scalar, scalar–tensor, and tensor–tensor pieces in radiation domination (Wu et al., 10 Jul 2025). For monochromatic and log-normal primordial spectra, the mixed term scales as $$\mathcal{S}^{(2)}_{lm,\phi\phi},\qquad \mathcal{S}^{(2)}_{lm,\phi h},\qquad \mathcal{S}^{(2)}_{lm,hh},$$3, the scalar–scalar term as $$\mathcal{S}^{(2)}_{lm,\phi\phi},\qquad \mathcal{S}^{(2)}_{lm,\phi h},\qquad \mathcal{S}^{(2)}_{lm,hh},$$4, and the tensor–tensor term as $$\mathcal{S}^{(2)}_{lm,\phi\phi},\qquad \mathcal{S}^{(2)}_{lm,\phi h},\qquad \mathcal{S}^{(2)}_{lm,hh},$$5, which produces a characteristic degeneracy between small-scale curvature and tensor amplitudes in SGWB inference (Wu et al., 10 Jul 2025).

A technical complication specific to the scalar–tensor sector is the potential divergence associated with long-wavelength scalar modes. In the Poisson-gauge treatment of scalar–tensor induced gravitational waves, the ultraviolet regime can contain a contribution behaving as $$\mathcal{S}^{(2)}_{lm,\phi\phi},\qquad \mathcal{S}^{(2)}_{lm,\phi h},\qquad \mathcal{S}^{(2)}_{lm,hh},$$6 as $$\mathcal{S}^{(2)}_{lm,\phi\phi},\qquad \mathcal{S}^{(2)}_{lm,\phi h},\qquad \mathcal{S}^{(2)}_{lm,hh},$$7. A physically motivated regularization suppresses these long-wavelength scalar modes by

$$\mathcal{S}^{(2)}_{lm,\phi\phi},\qquad \mathcal{S}^{(2)}_{lm,\phi h},\qquad \mathcal{S}^{(2)}_{lm,hh},$$8

reflecting the argument that sufficiently long scalar modes should be absorbed into the background in a local inertial frame (Bari et al., 2023).

5. Concrete realizations and higher-order corrections

A concrete inflationary realization is provided by warm inflation with

$$\mathcal{S}^{(2)}_{lm,\phi\phi},\qquad \mathcal{S}^{(2)}_{lm,\phi h},\qquad \mathcal{S}^{(2)}_{lm,hh},$$9

for which the dissipation ratio grows from weak dissipation at the pivot to strong dissipation near the end of inflation (Arya et al., 2022). In the Planck-compatible window

$h_{ij}^{(2)}$0

the curvature spectrum is red-tilted on CMB scales but blue-tilted and enhanced on small scales, reaching amplitudes that in earlier work led to primordial black holes with $h_{ij}^{(2)}$1 g. The same enhancement generates scalar induced gravitational waves over

$h_{ij}^{(2)}$2

with most power coming from modes exiting near the end of inflation (Arya et al., 2022).

Standard-model radiation content also affects TIGWs. For free-streaming neutrinos, the second-order tensor equation acquires a first-order anisotropic-stress term, a genuine second-order anisotropic-stress term, and modified first-order transfer functions. Solving the Boltzmann equation to second order, Li, Lin, and Sang showed that free-streaming neutrinos “suppress the density spectrum significantly for low frequency gravitational waves and enlarge the logarithmic slope $h_{ij}^{(2)}$3 in the infrared region.” For a spectrum with $h_{ij}^{(2)}$4 Hz, “the combined effect of the first and second order could reduce the amplitude by $h_{ij}^{(2)}$5 and make $h_{ij}^{(2)}$6 jump from $h_{ij}^{(2)}$7 to $h_{ij}^{(2)}$8 at $h_{ij}^{(2)}$9 Hz” (Zhang et al., 2022).

Primordial non-Gaussianity modifies TIGWs through the full four-point statistics of the curvature perturbation. Adshead, Lozanov, and Weiner showed that both the disconnected and connected components of the primordial trispectrum contribute, and that the connected part can be comparable to, or dominate over, the disconnected contribution for local-type non-Gaussianity (Adshead et al., 2021). Their full decomposition includes Gaussian, hybrid, reducible, $$h_k''(\eta)+2\mathcal{H}h_k'(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$0, $$h_k''(\eta)+2\mathcal{H}h_k'(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$1, planar, and nonplanar pieces, and the key conclusion is that a consistent TIGW prediction in a non-Gaussian scenario cannot be reduced to the Gaussian formula with a loop-corrected scalar power spectrum (Adshead et al., 2021).

A further refinement comes from the extension of the tensor calculation to third order. Chang, Gong, Jeong, and Lee identified “missing one-loop-order contributions” from sources containing two scalar and one tensor perturbations, which generate a cross spectrum $$h_k''(\eta)+2\mathcal{H}h_k'(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$2 at the same loop order as the standard $$h_k''(\eta)+2\mathcal{H}h_k'(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$3 term (Chen et al., 2022). The new correction is “scale-invariant and negative in the superhorion region,” so it reduces the initial primordial tensor power spectrum prior to horizon re-entry. For a sharp peak with $$h_k''(\eta)+2\mathcal{H}h_k'(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$4 at $$h_k''(\eta)+2\mathcal{H}h_k'(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$5, the tensor power spectrum at the CMB scale “reduces by at most $$h_k''(\eta)+2\mathcal{H}h_k'(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$6” (Chen et al., 2022).

6. Observational status and unresolved issues

The observational window depends strongly on the source class and on the scale of the underlying primordial perturbations. In the warm-inflation example, the signal is concentrated in the high-frequency band $$h_k''(\eta)+2\mathcal{H}h_k'(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$7 Hz, largely above the operating band of current large-scale interferometers; the relevant concepts include levitated-sensor-based detectors, microwave cavities, decameter Michelson interferometers such as the Holometer, and resonant-mass detectors (Arya et al., 2022). In the tensor-scalar scenario, combining PTA, CMB+BAO, and PBH bounds shows that TSIGWs can be used to constrain both small-scale primordial curvature perturbations and primordial gravitational waves, and “TSIGWs generated by monochromatic primordial power spectra might be more likely to dominate the current PTA observations” (Wu et al., 10 Jul 2025).

At much lower frequencies, the baryon–CDM relative-velocity example shows that second-order tensor perturbations can source CMB $$h_k''(\eta)+2\mathcal{H}h_k'(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$8-modes even when they do not behave as a freely propagating stochastic background, although in that case the effect is many orders of magnitude below current and near-future sensitivity (Gurian et al., 2021). This suggests that the observable notion of TIGWs depends on the measurement channel: timing residuals, redshift observables, interferometric strain, and CMB polarization need not isolate exactly the same intermediate variables.

Three unresolved issues recur across the literature. First, gauge ambiguity is resolved for the observable strain but not fully for the gravitational-wave energy density: the Newton-gauge TT tensor matches the measured second-order strain, but the effective GW stress-energy tensor remains subtle at second order (Domènech et al., 17 Dec 2025). Second, mixed-source sectors can develop long-wavelength divergences or gauge-sensitive enhancements that require careful treatment of local inertial frames and mode separation (Bari et al., 2023). Third, a consistent one-loop description of secondary gravitational waves must include higher-order interactions and iterative solutions beyond the standard $$h_k''(\eta)+2\mathcal{H}h_k'(\eta)+k^2 h_k(\eta)=4S_k(\eta),$$9 term (Chen et al., 2022).

Taken together, these results place second-order tensor induced gravitational waves at the intersection of nonlinear relativistic perturbation theory, primordial small-scale structure, and multi-band gravitational-wave cosmology. The mature picture is no longer limited to scalar–scalar induced backgrounds: it includes gauge-invariant strain observables, mixed scalar–tensor and tensor–tensor sectors, neutrino damping, primordial non-Gaussianity, and one-loop corrections to the primordial tensor spectrum itself (Hwang et al., 2017, Chang et al., 2020, Gong, 2019).