Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 67 / 5 / ii

Let $q=\mathbf{k}\cdot \hat{\mathbf{n}}$ and retain the negative-spatial-perturbation convention of the preceding part. Since $h=h_S$, the isotropic pieces of the supplied Fourier mode contraction cancel:For $h=A(\mathbf{k}){\tau }^2{k}^2$, the factor $1/2$ in the line-of-sight integral cancels the derivative's factor two. ThusAn antiderivative convenient even at $q=0$ isIt follows directly, or by two integrations by parts, thatThe source sign and both endpoint coefficients are fixed by the metric convention; no division by $q$ is needed.

To interpret the large-angle Cosmic microwave background anisotropy, restore the observer-centered propagation phase $e^{-iq{\tau }_0}$, which can equivalently be absorbed into the definition of each Fourier mode amplitude. Put $D={\tau }_0-{\tau }_{\mathrm{dec}}$. One mode contributesThe first term is a local angular monopole plus a dipole, so it is removed when discussing observed angular multipoles $\ell \ge 2$. For modes outside the horizon at decoupling, $k{\tau }_{\mathrm{dec}}\ll 1$, the higher-multipole emission signal is dominated byIts transfer amplitude has no leading positive power of $k$: the two explicit powers in $h$ have canceled in the integral. Projection onto the sky supplies Spherical Bessel functions $j_{\ell }(kD)$, with modes of order $kD\sim \ell$ contributing to each angular scale. This is the large-angle Sachs-Wolfe effect; the emission velocity-like correction is relatively of order $k{\tau }_{\mathrm{dec}}$. For wavelengths much larger than the entire observed region, $kD\ll 1$, removing the monopole and dipole leaves a signal starting at order $A(kD{)}^2$, rather than a finite observable anisotropy from a spatially uniform mode.

The stochastic scale dependence of $A(\mathbf{k})$ is not specified by the deterministic metric solution. If one additionally assumes a scale-invariant primordial spectrum, $k^3{P}_A(k)=\text{constant}$, thenfor $\ell \ge 1$, giving the familiar large-angle Sachs-Wolfe plateau, $\ell (\ell +1)C_{\ell }\simeq \text{constant}$, for the measured $\ell \ge 2$ multipoles. A tilted primordial spectrum changes that trend. The full numerical temperature transfer also includes intrinsic last-scattering and Doppler terms, omitted from the stipulated metric-only integral.