Primordial tensor power spectrum (${\Delta }_t^2$)

For the two graviton polarizations in canonical single-field inflation,at horizon exit.

The graviton power spectrum is the two-point spectrum of a tensor metric perturbation. A convention may quote it per polarization or summed over both polarizations, so that choice must accompany its normalization.

An adiabatic tensor mode is a long-wavelength transverse-traceless metric perturbation whose leading local effect can be produced by a large spatial diffeomorphism. Its effect on short modes is therefore constrained by a Ward identity.

For a single adiabatic long tensor mode, the squeezed scalar-scalar-graviton bispectrum is the anisotropic momentum response of the scalar power spectrum:

The tensor spectral index is the logarithmic slope $n_T=d\log {\Delta }_t^2/d\log k$. Canonical slow-roll inflation gives $n_T=-2ϵ$ to first order.

The tensor-to-scalar ratio is $r={\Delta }_t^2/{\Delta }_{\mathcal{R}}^2$. Canonical single-field slow roll gives $r=16ϵ$.

Combining $r=16ϵ$ with $n_T=-2ϵ$ gives the leading-order consistency relation