Primordial gravitational waves could be non-Gaussian, just like primordial scalar perturbations. Although the tensor two-point function has thus-far remained elusive, the three-point function could, in principle, be large enough to be detected in cosmic microwave background temperature and polarization anisotropies. We perform a detailed analysis of tensor and mixed tensor-scalar non-Gaussianity through the Planck Public Release 4 bispectrum, placing constraints on 11 primordial templates, spanning various phenomenological and physical regimes including modifications to gravity, additional fields in inflation, and primordial magnetic fields. All analysis is performed using modern quasioptimal binned estimators and yields no evidence for tensor non-Gaussianity, with a maximum detection significance of $2\sigma$. Our constraints are derived primarily from large scales (except for tensor-scalar-scalar models) and benefit greatly from the inclusion of $B$-modes. Although we find some loss of information from binning, mask effects, and residual foreground contamination, our $f_{\mathrm{NL}}$ bounds improve over those of previous analyses by (20–700)%, with 6 of the 11 models being analyzed for the first time. Unlike for scalar non-Gaussianity, future low-noise experiments such as LiteBIRD, the Simons Observatory, and CMB-S4, will yield considerable improvement in tensor non-Gaussianity constraints.

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• Cosmic microwave background
• Cosmological parameters
• Inflation
• Primordial magnetic fields
• Relativistic aspects of cosmology