Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 312 / 2 / iv / Solution

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})] .

The derivative of the Dirac delta distribution is proportional to

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

yields the cosmological soft-graviton consistency relation

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.

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