Second-Order Tensor-Scalar Induced GWs

• The paper explains that TSIGWs arise from a mixed scalar-tensor source, producing a gravitational-wave contribution linear in both primordial scalar and tensor spectra.
• It employs radiation-era cosmological models to extract the spectral signature, showing that the scalar-tensor sector can dominate under enhanced primordial tensor conditions.
• The study highlights practical issues such as ultraviolet enhancement, gauge dependence, and observational challenges in accurately defining TSIGW energy densities.

Second-order tensor-scalar induced gravitational waves (TSIGWs) are the component of the induced stochastic gravitational-wave background generated when first-order scalar perturbations and first-order tensor perturbations mix at second order in cosmological perturbation theory. They extend the standard scalar-induced gravitational-wave framework by adding a mixed scalar-tensor source, alongside the usual scalar-scalar and tensor-tensor sectors, and they become relevant when primordial tensor fluctuations are enhanced on small scales or when both primordial scalar and tensor power spectra carry localized features (Bari et al., 2023).

1. Perturbative definition and cosmological setting

In the radiation-dominated treatments used most often in the literature, the perturbed metric is written in Newtonian gauge as

$$ds^2=a^2(\eta)\left[-(1+2\phi^{(1)})d\eta^2+\left((1-2\psi^{(1)})\delta_{ij}+h_{ij}^{(1)}+\frac12 h_{ij}^{(2)}\right)dx^i dx^j\right].$$

The first-order scalar and tensor perturbations evolve independently,

$$\psi(\eta,\mathbf{k})=\phi(\eta,\mathbf{k})=\Phi_{\mathbf{k}}\,T_\phi(k\eta),\qquad h^{\lambda,(1)}(\eta,\mathbf{k})=\mathbf{h}^{\lambda}_{\mathbf{k}}\,T_h(k\eta),$$

with

$$T_{\phi}(x)=\frac{9}{x^2}\left[\frac{\sqrt{3}}{x}\sin\!\left(\frac{x}{\sqrt{3}}\right)-\cos\!\left(\frac{x}{\sqrt{3}}\right)\right],\qquad T_h(x)=\frac{\sin x}{x}.$$

At second order, the tensor mode obeys

$$h_{ij}^{(2)''}+2\mathcal{H}h_{ij}^{(2)'}-\Delta h_{ij}^{(2)} =-4\Lambda_{ij}^{lm}\mathcal{S}^{(2)}_{lm},$$

or, in the source decomposition emphasized in later work,

$$h_{ab}^{\prime\prime (2)} + 2\mathcal{H}h_{ab}^{\prime (2)} - \nabla^2 h_{ab}^{(2)} =\Lambda_{ab}^{ij}\Big(S^{h^{(2)}_{ss}}_{ij}+S^{h^{(2)}_{st}}_{ij}+S^{h^{(2)}_{tt}}_{ij}\Big).$$

The three pieces are respectively the scalar-scalar, scalar-tensor, and tensor-tensor sources. This decomposition makes explicit that TSIGWs are not the same object as ordinary scalar-induced gravitational waves: the mixed contribution depends simultaneously on the first-order scalar and first-order tensor sectors, rather than on the scalar sector alone (Chang et al., 2022, Picard, 12 Jan 2026).

2. Source taxonomy and spectral construction

A useful organizing principle is the split of the second-order source into seven terms,

$$\mathcal{S}^{(2)}_{lm}=\sum_{i=1}^{7}\mathcal{S}^{(2)}_{lm,i},$$

with $\mathcal{S}^{(2)}_{lm,1}\sim \phi^{(1)}\phi^{(1)}$, $\mathcal{S}^{(2)}_{lm,2}\sim \phi^{(1)}h^{(1)}$, and $$\mathcal{S}^{(2)}_{lm,3},\dots,\mathcal{S}^{(2)}_{lm,7}\sim h^{(1)}h^{(1)}$$. The mixed term $\mathcal{S}^{(2)}_{lm,2}$ is the canonical TSIGW contribution. In kernel language, the numerical hierarchy found in radiation domination is that $$\psi(\eta,\mathbf{k})=\phi(\eta,\mathbf{k})=\Phi_{\mathbf{k}}\,T_\phi(k\eta),\qquad h^{\lambda,(1)}(\eta,\mathbf{k})=\mathbf{h}^{\lambda}_{\mathbf{k}}\,T_h(k\eta),$$0 is the largest, $$\psi(\eta,\mathbf{k})=\phi(\eta,\mathbf{k})=\Phi_{\mathbf{k}}\,T_\phi(k\eta),\qquad h^{\lambda,(1)}(\eta,\mathbf{k})=\mathbf{h}^{\lambda}_{\mathbf{k}}\,T_h(k\eta),$$1 is the second largest, and $$\psi(\eta,\mathbf{k})=\phi(\eta,\mathbf{k})=\Phi_{\mathbf{k}}\,T_\phi(k\eta),\qquad h^{\lambda,(1)}(\eta,\mathbf{k})=\mathbf{h}^{\lambda}_{\mathbf{k}}\,T_h(k\eta),$$2–$$\psi(\eta,\mathbf{k})=\phi(\eta,\mathbf{k})=\Phi_{\mathbf{k}}\,T_\phi(k\eta),\qquad h^{\lambda,(1)}(\eta,\mathbf{k})=\mathbf{h}^{\lambda}_{\mathbf{k}}\,T_h(k\eta),$$3 are smaller, so the scalar-tensor piece is typically the dominant correction from primordial tensors while the tensor-tensor sector is usually subdominant (Chang et al., 2022).

Source class	Schematic structure	Primordial-spectrum dependence
Scalar-scalar	$$\psi(\eta,\mathbf{k})=\phi(\eta,\mathbf{k})=\Phi_{\mathbf{k}}\,T_\phi(k\eta),\qquad h^{\lambda,(1)}(\eta,\mathbf{k})=\mathbf{h}^{\lambda}_{\mathbf{k}}\,T_h(k\eta),$$4	$$\psi(\eta,\mathbf{k})=\phi(\eta,\mathbf{k})=\Phi_{\mathbf{k}}\,T_\phi(k\eta),\qquad h^{\lambda,(1)}(\eta,\mathbf{k})=\mathbf{h}^{\lambda}_{\mathbf{k}}\,T_h(k\eta),$$5
Scalar-tensor	$$\psi(\eta,\mathbf{k})=\phi(\eta,\mathbf{k})=\Phi_{\mathbf{k}}\,T_\phi(k\eta),\qquad h^{\lambda,(1)}(\eta,\mathbf{k})=\mathbf{h}^{\lambda}_{\mathbf{k}}\,T_h(k\eta),$$6	$$\psi(\eta,\mathbf{k})=\phi(\eta,\mathbf{k})=\Phi_{\mathbf{k}}\,T_\phi(k\eta),\qquad h^{\lambda,(1)}(\eta,\mathbf{k})=\mathbf{h}^{\lambda}_{\mathbf{k}}\,T_h(k\eta),$$7
Tensor-tensor	$$\psi(\eta,\mathbf{k})=\phi(\eta,\mathbf{k})=\Phi_{\mathbf{k}}\,T_\phi(k\eta),\qquad h^{\lambda,(1)}(\eta,\mathbf{k})=\mathbf{h}^{\lambda}_{\mathbf{k}}\,T_h(k\eta),$$8	$$\psi(\eta,\mathbf{k})=\phi(\eta,\mathbf{k})=\Phi_{\mathbf{k}}\,T_\phi(k\eta),\qquad h^{\lambda,(1)}(\eta,\mathbf{k})=\mathbf{h}^{\lambda}_{\mathbf{k}}\,T_h(k\eta),$$9

In the simplified form displayed in the thesis treatment, the scalar-tensor source is

$$T_{\phi}(x)=\frac{9}{x^2}\left[\frac{\sqrt{3}}{x}\sin\!\left(\frac{x}{\sqrt{3}}\right)-\cos\!\left(\frac{x}{\sqrt{3}}\right)\right],\qquad T_h(x)=\frac{\sin x}{x}.$$0

while the Fourier-space solution is written as

$$T_{\phi}(x)=\frac{9}{x^2}\left[\frac{\sqrt{3}}{x}\sin\!\left(\frac{x}{\sqrt{3}}\right)-\cos\!\left(\frac{x}{\sqrt{3}}\right)\right],\qquad T_h(x)=\frac{\sin x}{x}.$$1

The associated power spectrum is linear in the tensor power spectrum and linear in the scalar power spectrum,

$$T_{\phi}(x)=\frac{9}{x^2}\left[\frac{\sqrt{3}}{x}\sin\!\left(\frac{x}{\sqrt{3}}\right)-\cos\!\left(\frac{x}{\sqrt{3}}\right)\right],\qquad T_h(x)=\frac{\sin x}{x}.$$2

This linear $$T_{\phi}(x)=\frac{9}{x^2}\left[\frac{\sqrt{3}}{x}\sin\!\left(\frac{x}{\sqrt{3}}\right)-\cos\!\left(\frac{x}{\sqrt{3}}\right)\right],\qquad T_h(x)=\frac{\sin x}{x}.$$3 dependence is the algebraic signature that distinguishes TSIGWs from scalar-induced and tensor-tensor induced contributions (Picard, 12 Jan 2026).

3. Distinctive spectral structure

TSIGWs differ qualitatively from ordinary scalar-induced gravitational waves. For peaked primordial spectra, they do not present resonances or a logarithmic running in the low-frequency tail, because the scalar-tensor kernel does not reach the same resonant condition that produces the familiar scalar-scalar logarithmic enhancement. In the radiation era, the would-be logarithmic divergence is regulated by the kernel prefactor, so no true resonance appears. The same mixed structure also allows TSIGWs to inherit parity violation or chirality from primordial tensor modes, which is impossible for a purely scalar-sourced background (Bari et al., 2023).

The absence of the standard scalar-scalar resonance does not imply spectral triviality. In the Dirac-delta limit with matching scalar and tensor peaks, the thesis finds

$$T_{\phi}(x)=\frac{9}{x^2}\left[\frac{\sqrt{3}}{x}\sin\!\left(\frac{x}{\sqrt{3}}\right)-\cos\!\left(\frac{x}{\sqrt{3}}\right)\right],\qquad T_h(x)=\frac{\sin x}{x}.$$4

with $$T_{\phi}(x)=\frac{9}{x^2}\left[\frac{\sqrt{3}}{x}\sin\!\left(\frac{x}{\sqrt{3}}\right)-\cos\!\left(\frac{x}{\sqrt{3}}\right)\right],\qquad T_h(x)=\frac{\sin x}{x}.$$5, and concludes that TSIGWs are comparable to or larger than SIGWs at sufficiently small scales, even though they lack the resonant peak. The scalar-tensor contribution becomes significant when the primordial tensor spectrum is enhanced; in standard inflationary settings the tensor amplitude is tiny, so TSIGWs are negligible, but if the tensor peak is near the scalar peak the TSIGW signal can become dominant over the scalar-scalar contribution (Picard, 12 Jan 2026).

A major unresolved issue is the ultraviolet enhancement problem. For the scalar-tensor sector the dangerous limits are $$T_{\phi}(x)=\frac{9}{x^2}\left[\frac{\sqrt{3}}{x}\sin\!\left(\frac{x}{\sqrt{3}}\right)-\cos\!\left(\frac{x}{\sqrt{3}}\right)\right],\qquad T_h(x)=\frac{\sin x}{x}.$$6, $$T_{\phi}(x)=\frac{9}{x^2}\left[\frac{\sqrt{3}}{x}\sin\!\left(\frac{x}{\sqrt{3}}\right)-\cos\!\left(\frac{x}{\sqrt{3}}\right)\right],\qquad T_h(x)=\frac{\sin x}{x}.$$7, where the integrand behaves like $$T_{\phi}(x)=\frac{9}{x^2}\left[\frac{\sqrt{3}}{x}\sin\!\left(\frac{x}{\sqrt{3}}\right)-\cos\!\left(\frac{x}{\sqrt{3}}\right)\right],\qquad T_h(x)=\frac{\sin x}{x}.$$8, and $$T_{\phi}(x)=\frac{9}{x^2}\left[\frac{\sqrt{3}}{x}\sin\!\left(\frac{x}{\sqrt{3}}\right)-\cos\!\left(\frac{x}{\sqrt{3}}\right)\right],\qquad T_h(x)=\frac{\sin x}{x}.$$9, $$h_{ij}^{(2)''}+2\mathcal{H}h_{ij}^{(2)'}-\Delta h_{ij}^{(2)} =-4\Lambda_{ij}^{lm}\mathcal{S}^{(2)}_{lm},$$0, where it behaves like $$h_{ij}^{(2)''}+2\mathcal{H}h_{ij}^{(2)'}-\Delta h_{ij}^{(2)} =-4\Lambda_{ij}^{lm}\mathcal{S}^{(2)}_{lm},$$1. This produces an unphysical enhancement when the primordial scalar spectrum is not sufficiently peaked. A phenomenological cure proposed in the radiation-era scalar-tensor study is to suppress long-wavelength scalar modes with

$$h_{ij}^{(2)''}+2\mathcal{H}h_{ij}^{(2)'}-\Delta h_{ij}^{(2)} =-4\Lambda_{ij}^{lm}\mathcal{S}^{(2)}_{lm},$$2

but that prescription is explicitly presented as phenomenological rather than as a final first-principles resolution (Bari et al., 2023). The thesis later shows that third-order induced GWs and their correlation with primordial tensors suppress the overall signal but do not cancel this ultraviolet enhancement, and it identifies the full explanation or cure for the divergence as an open problem (Picard, 12 Jan 2026).

4. Gauge dependence, hypersurfaces, and physical observables

A central conceptual difficulty is that the second-order tensor perturbation extracted from the metric is gauge dependent once sourced by nonlinear perturbations. In standard perturbation theory, scalar perturbations source tensor-like pieces in the metric under gauge transformations, and the corresponding effective gravitational-wave energy density is therefore gauge dependent if defined naively. The covariant approach based on York’s transverse-traceless decomposition replaces the bare metric tensor by the extrinsic curvature $$h_{ij}^{(2)''}+2\mathcal{H}h_{ij}^{(2)'}-\Delta h_{ij}^{(2)} =-4\Lambda_{ij}^{lm}\mathcal{S}^{(2)}_{lm},$$3 of a chosen spatial hypersurface,

$$h_{ij}^{(2)''}+2\mathcal{H}h_{ij}^{(2)'}-\Delta h_{ij}^{(2)} =-4\Lambda_{ij}^{lm}\mathcal{S}^{(2)}_{lm},$$4

and defines the kinetic gravitational-wave energy density by

$$h_{ij}^{(2)''}+2\mathcal{H}h_{ij}^{(2)'}-\Delta h_{ij}^{(2)} =-4\Lambda_{ij}^{lm}\mathcal{S}^{(2)}_{lm},$$5

This $$h_{ij}^{(2)''}+2\mathcal{H}h_{ij}^{(2)'}-\Delta h_{ij}^{(2)} =-4\Lambda_{ij}^{lm}\mathcal{S}^{(2)}_{lm},$$6 is a scalar built entirely from spacetime tensors and is therefore gauge invariant by definition. In the Newtonian gauge, on the constant-time hypersurface,

$$h_{ij}^{(2)''}+2\mathcal{H}h_{ij}^{(2)'}-\Delta h_{ij}^{(2)} =-4\Lambda_{ij}^{lm}\mathcal{S}^{(2)}_{lm},$$7

so the energy density contains only the propagating tensor contribution at the relevant perturbative order. In synchronous gauge the same physical waves are recovered only after correctly identifying the transformed hypersurface, which is why the paper concludes not that standard TSIGW computations are wrong, but that their interpretation must be tied to a precise hypersurface (Ota et al., 2021).

A complementary strategy is to define induced tensor content through observables that vanish in the background and have no linear dependence on scalar perturbations. The magnetic part of the Weyl tensor and the Cotton tensor of a physical slicing satisfy this criterion. The generalized Stewart–Walker argument then implies that the scalar-induced second-order tensor contribution entering such observables is automatically gauge-invariant. In radiation domination, the magnetic-Weyl construction reproduces the Newtonian-gauge short-wavelength limit, clarifying when the usual Newtonian-gauge result is physically meaningful. However, the same paper states that the method applies only to induced tensor modes sourced by linear scalar perturbations and does not directly cover second-order tensors sourced by scalar-tensor mixing. This suggests that a comparably explicit observable-based construction for the mixed scalar-tensor sector remains incomplete (Comeau, 2023).

5. Primordial spectra, data analysis, and detector relevance

The phenomenology of TSIGWs depends strongly on the assumed primordial power spectra. One class of analyses adopts log-normal scalar and tensor peaks at the same comoving scale,

$$h_{ij}^{(2)''}+2\mathcal{H}h_{ij}^{(2)'}-\Delta h_{ij}^{(2)} =-4\Lambda_{ij}^{lm}\mathcal{S}^{(2)}_{lm},$$8

while another class studies monochromatic limits. In these setups the present-day energy density is split into scalar-scalar, scalar-tensor, and tensor-tensor pieces,

$$h_{ij}^{(2)''}+2\mathcal{H}h_{ij}^{(2)'}-\Delta h_{ij}^{(2)} =-4\Lambda_{ij}^{lm}\mathcal{S}^{(2)}_{lm},$$9

The stated purpose of this program is to use SGWB observations across different scales to constrain small-scale primordial curvature perturbations and primordial gravitational waves. In that framework, TSIGWs generated by monochromatic primordial power spectra might be more likely to dominate the current PTA observations (Wu et al., 10 Jul 2025).

A more direct observational test compares cosmological TSIGW models with the nanohertz background measured by PTA collaborations. In the Newtonian-gauge study explicitly confronting CMB, BAO, PTA, and LISA information, the second-order tensor spectrum contains one scalar-scalar piece, one scalar-tensor piece, and five tensor-tensor pieces, and the Bayes factor analysis suggests that TSIGW+PGW might be more likely to dominate current PTA observations compared to SMBHB. The same analysis emphasizes that the presence of primordial gravitational waves introduces a degeneracy between $$h_{ab}^{\prime\prime (2)} + 2\mathcal{H}h_{ab}^{\prime (2)} - \nabla^2 h_{ab}^{(2)} =\Lambda_{ab}^{ij}\Big(S^{h^{(2)}_{ss}}_{ij}+S^{h^{(2)}_{st}}_{ij}+S^{h^{(2)}_{tt}}_{ij}\Big).$$0 and $$h_{ab}^{\prime\prime (2)} + 2\mathcal{H}h_{ab}^{\prime (2)} - \nabla^2 h_{ab}^{(2)} =\Lambda_{ab}^{ij}\Big(S^{h^{(2)}_{ss}}_{ij}+S^{h^{(2)}_{st}}_{ij}+S^{h^{(2)}_{tt}}_{ij}\Big).$$1, since both contribute to the second-order GW spectrum, and that the PTA-compatible region generally implies a small LISA signal even though tensor contributions add power at high frequency (Wu et al., 2024).

Within radiation domination, monochromatic benchmark calculations further show that the mixed scalar-tensor term $$h_{ab}^{\prime\prime (2)} + 2\mathcal{H}h_{ab}^{\prime (2)} - \nabla^2 h_{ab}^{(2)} =\Lambda_{ab}^{ij}\Big(S^{h^{(2)}_{ss}}_{ij}+S^{h^{(2)}_{st}}_{ij}+S^{h^{(2)}_{tt}}_{ij}\Big).$$2 is the dominant correction from primordial tensors, whereas the tensor-tensor pieces are negligible for $$h_{ab}^{\prime\prime (2)} + 2\mathcal{H}h_{ab}^{\prime (2)} - \nabla^2 h_{ab}^{(2)} =\Lambda_{ab}^{ij}\Big(S^{h^{(2)}_{ss}}_{ij}+S^{h^{(2)}_{st}}_{ij}+S^{h^{(2)}_{tt}}_{ij}\Big).$$3. The practical implication is that the main TSIGW handle on observables such as LISA or PTA is usually the mixed $$h_{ab}^{\prime\prime (2)} + 2\mathcal{H}h_{ab}^{\prime (2)} - \nabla^2 h_{ab}^{(2)} =\Lambda_{ab}^{ij}\Big(S^{h^{(2)}_{ss}}_{ij}+S^{h^{(2)}_{st}}_{ij}+S^{h^{(2)}_{tt}}_{ij}\Big).$$4 source rather than the purely tensor-tensor sector (Chang et al., 2022).

6. Extensions, environmental effects, and unresolved issues

Several extensions modify the basic TSIGW picture without eliminating its defining mixed source. Free-streaming neutrinos provide a notable example. In the detailed second-order treatment including scalar-scalar, scalar-tensor, and tensor-tensor couplings, the tensor equation acquires neutrino-anisotropic-stress terms and nonlocal damping kernels. The reported effect is that free-streaming neutrinos suppress the density spectrum significantly for low frequency gravitational waves and enlarge the logarithmic slope $$h_{ab}^{\prime\prime (2)} + 2\mathcal{H}h_{ab}^{\prime (2)} - \nabla^2 h_{ab}^{(2)} =\Lambda_{ab}^{ij}\Big(S^{h^{(2)}_{ss}}_{ij}+S^{h^{(2)}_{st}}_{ij}+S^{h^{(2)}_{tt}}_{ij}\Big).$$5 in the infrared region. For the spectrum of $$h_{ab}^{\prime\prime (2)} + 2\mathcal{H}h_{ab}^{\prime (2)} - \nabla^2 h_{ab}^{(2)} =\Lambda_{ab}^{ij}\Big(S^{h^{(2)}_{ss}}_{ij}+S^{h^{(2)}_{st}}_{ij}+S^{h^{(2)}_{tt}}_{ij}\Big).$$6, the combined effect of the first and second order could reduce the amplitude by $$h_{ab}^{\prime\prime (2)} + 2\mathcal{H}h_{ab}^{\prime (2)} - \nabla^2 h_{ab}^{(2)} =\Lambda_{ab}^{ij}\Big(S^{h^{(2)}_{ss}}_{ij}+S^{h^{(2)}_{st}}_{ij}+S^{h^{(2)}_{tt}}_{ij}\Big).$$7 and make $$h_{ab}^{\prime\prime (2)} + 2\mathcal{H}h_{ab}^{\prime (2)} - \nabla^2 h_{ab}^{(2)} =\Lambda_{ab}^{ij}\Big(S^{h^{(2)}_{ss}}_{ij}+S^{h^{(2)}_{st}}_{ij}+S^{h^{(2)}_{tt}}_{ij}\Big).$$8 jump from $$h_{ab}^{\prime\prime (2)} + 2\mathcal{H}h_{ab}^{\prime (2)} - \nabla^2 h_{ab}^{(2)} =\Lambda_{ab}^{ij}\Big(S^{h^{(2)}_{ss}}_{ij}+S^{h^{(2)}_{st}}_{ij}+S^{h^{(2)}_{tt}}_{ij}\Big).$$9 to $$\mathcal{S}^{(2)}_{lm}=\sum_{i=1}^{7}\mathcal{S}^{(2)}_{lm,i},$$0 at $$\mathcal{S}^{(2)}_{lm}=\sum_{i=1}^{7}\mathcal{S}^{(2)}_{lm,i},$$1, which may be probed by PTA in the future (Zhang et al., 2022).

Higher-order terms do not simply decouple from the TSIGW sector. At the level of the correlator,

$$\mathcal{S}^{(2)}_{lm}=\sum_{i=1}^{7}\mathcal{S}^{(2)}_{lm,i},$$2

so third-order cross-correlations $$\mathcal{S}^{(2)}_{lm}=\sum_{i=1}^{7}\mathcal{S}^{(2)}_{lm,i},$$3 contribute at the same perturbative order as the ordinary second-order induced background. The explicit third-order analysis finds that these terms generally suppress the SGWB around the peak scales considered, but some of them are enhanced when the primordial scalar spectrum is not sufficiently peaked, and no cancellation of the known scalar-tensor ultraviolet divergence is found (Picard et al., 9 Sep 2025).

Parity-violating scalar-tensor theory provides a different extension. In the “Qi-Xiu” framework, $$\mathcal{S}^{(2)}_{lm}=\sum_{i=1}^{7}\mathcal{S}^{(2)}_{lm,i},$$4 modify linear tensor propagation, whereas $$\mathcal{S}^{(2)}_{lm}=\sum_{i=1}^{7}\mathcal{S}^{(2)}_{lm,i},$$5 and $$\mathcal{S}^{(2)}_{lm}=\sum_{i=1}^{7}\mathcal{S}^{(2)}_{lm,i},$$6 enter exclusively through the second-order source term for induced gravitational waves. During radiation domination, the resulting SIGWs exhibit characteristic deviations from GR around the peak frequency and acquire a nonzero degree of circular polarization. For the broader TSIGW program, this is significant because it shows that a mixed or induced background can carry helicity information even when the linear tensor sector is tuned to look effectively GR-like (Feng et al., 7 Feb 2026).

The open problems identified repeatedly across the literature are therefore threefold: a full explanation or cure for the ultraviolet enhancement in the scalar-tensor sector, a more complete gauge-invariant treatment of the second-order tensor observable, and systematic exploration of models with enhanced primordial tensors, parity violation, or nontrivial small-scale spectra (Picard, 12 Jan 2026).