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 Incoming links: Adiabatic tensor mode

The background spatial metric is

g_{ij} = a^{2} δ_{ij}

ϵ^{0} = 0, ϵ^{i} = ω^{i}_{j} x^{j}, ω_{ij} = ω_{ji}, ω_{i}^{i} = 0,

homogeneity and isotropy imply that the purely spatial background Christoffel symbols vanish. Hence

\nabla_{i} ϵ_{j} = a^{2} ω_{ji}, Δ γ_{ij} = - 2 ω_{ij} .

The shift is constant, transverse and traceless, and is therefore the zero-momentum adiabatic tensor mode.

Δ φ (x) = - ω_{ij} x^{j} \partial_{i} φ (x) .

Fourier transformation and integration by parts in momentum space give

Δ φ (k) = ω_{ij} k_{i} \frac{\partial}{\partial k _{j}} φ (k) .

The term proportional to

ω_{i}^{i}

vanishes because the deformation is traceless.

Applying the scalar transformation from part i to both fields gives

⟨ Δ O ⟩ = ω_{ij} \sum_{a = 1}^{2} k_{ai} \frac{\partial}{\partial k _{aj}} [(2 π)^{3} δ^{(3)} (k_{1} + k_{2}) P_{φ} (k_{1})] .

δ_{ij}

and vanishes after contraction with traceless

ω_{ij}

. Removing that delta function leaves

⟨ Δ O ⟩^{'} = ω_{ij} k_{i} \frac{\partial}{\partial k _{j}} P_{φ} (k) .

Equating the two sides of the Ward-Takahashi identity and resolving the soft graviton into a polarization

s

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

For an isotropic power spectrum this is equivalently

- ϵ_{ij}^{s} k_{i} k_{j} \frac{\partial P _{φ}}{\partial k ^{2}} .

The long-wavelength adiabatic tensor mode acts on the short two-point function as an anisotropic rescaling of its momentum.