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

Solution

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.

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