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A sufficiently long adiabatic tensor mode acts locally as an anisotropic change of spatial coordinates. Write the spatial metric as $a^2(e^{\gamma }{)}_{ij}$, with ${\gamma }_{ij}={ϵ}_{ij}^s{\gamma }^s$ transverse and trace-free. Removing a constant long mode maps the short wavevector to ${\tilde{k}}_i=k_i-{\gamma }_{ij}{k}_j/2$ at first order. The trace-free deformation has no first-order volume Jacobian, so its effect on an isotropic scalar power spectrum isCorrelating this response with the long graviton givesThe prime removes the momentum-conserving $(2\pi {)}^3{\delta }^3(\sum \mathbf{k})$, and $P_{\gamma }$ is the power of the same polarization amplitude used in ${\gamma }_{ij}={\sum }_s{ϵ}_{ij}^s{\gamma }^s$. 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.

The background spatial metric is $g_{ij}=a^2{\delta }_{ij}$. For the large spatial diffeomorphismhomogeneity and isotropy imply that the purely spatial background Christoffel symbols vanish. HenceThe shift is constant, transverse and traceless, and is therefore the zero-momentum adiabatic tensor mode.

A scalar transforms by its Lie derivative:Fourier transformation and integration by parts in momentum space giveThe term proportional to ${\omega }_i{}^i$ vanishes because the deformation is traceless.

Applying the scalar transformation from part i to both fields givesThe derivative of the Dirac delta distribution is proportional to ${\delta }_{ij}$ and vanishes after contraction with traceless ${\omega }_{ij}$. Removing that delta function leavesEquating the two sides of the Ward-Takahashi identity and resolving the soft graviton into a polarization $s$ yields the cosmological soft-graviton consistency relationFor an isotropic power spectrum this is equivalentlyThe long-wavelength adiabatic tensor mode acts on the short two-point function as an anisotropic rescaling of its momentum.