Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 58 / 4 / d / Solution

Restore the Fourier transform integral and the last-scattering distance:Insert the Rayleigh plane-wave expansion and use orthogonality of spherical harmonics with the complex conjugate on the projection harmonic. The angular integral selects the matching indices, givingThe complex conjugation is present in the original PDF. If $A_{\mathbf{k}}=A{k}^{n/2}$ denotes a mode coefficient with its random amplitude and phase included in $A$, substitution of ${\Phi }_{\mathbf{k}}=-6A_{\mathbf{k}}/k^2$ gives the equivalent factor $-8\pi i^{\ell }{A}_{\mathbf{k}}/k^2$ inside the Fourier integral. The monopole and local observer dipole removed in part (b) do not change this formula for $\ell \ge 2$.