Primordial tensor power spectrum ( $Δ_{t}^{2}$ )

For the two graviton polarizations in canonical single-field inflation,

Δ_{t}^{2} (k) = \frac{8}{M _{Pl}^{2}} (\frac{H}{2 π})^{2}

at horizon exit.

Graviton power spectrum ( $P_{γ} (k)$ )

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.

A sufficiently long adiabatic tensor mode acts locally as an anisotropic change of spatial coordinates. Write the spatial metric as

a^{2} (e^{γ})_{ij}

, with

γ_{ij} = ϵ_{ij}^{s} γ^{s}

transverse and trace-free. Removing a constant long mode maps the short wavevector to

k_{i} = k_{i} - γ_{ij} k_{j} /2

at first order. The trace-free deformation has no first-order volume Jacobian, so its effect on an isotropic scalar power spectrum is

δ P_{φ} (k) = - \frac{1}{2} γ_{ij} k_{i} \partial_{k_{j}} P_{φ} (k) .

Correlating this response with the long graviton gives

lim_{q \to 0} \frac{⟨ γ _{q}^{s} φ _{k} φ _{- k - q} ⟩ ^{'}}{P _{γ} ( q )} = - \frac{1}{2} ϵ_{ij}^{s} k_{i} \partial_{k_{j}} P_{φ} (k) .

The prime removes the momentum-conserving

(2 π)^{3} δ^{3} (\sum k)

, and

P_{γ}

is the power of the same polarization amplitude used in

γ_{ij} = \sum_{s} ϵ_{ij}^{s} γ^{s}

. Thus polarization normalization must be consistent on both sides.^[ref]

This consistency relation concerns a squeezed bispectrum configuration: it relates a three-point correlation function to a coordinate response of a two-point function. It requires the long tensor to be adiabatic and the state/dynamics to respect that response. Additional physical long-wavelength effects or incompatible initial-state assumptions cannot simply be replaced by this identity.