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

For two neighboring comoving points separated along the zeroth-order direction $\hat{\mathbf{n}}$ by coordinate distance $\delta \chi$, the synchronous gauge spatial metric gives the cosmological proper distanceDifferentiate with respect to cosmic time at fixed coordinate separation. To first order,A photon crosses this neighboring interval in $\delta t=\delta r$ when $c=1$, so its local relative recession velocity isAt each infinitesimal step, the local Doppler effect gives $d\log \nu =-\Delta v$. ThereforeThe first term integrates to the homogeneous cosmological redshift, $\bar{\nu }\propto a^{-1}$. Subtracting it isolates the metric contribution to the fractional frequency shift. A shifted blackbody radiation spectrum has its temperature shifted by the same fraction as its photon frequencies, givingThe dot denotes a partial time derivative at fixed comoving position, evaluated along the zeroth-order null geodesic. It is not the total derivative of $h_{ij}$ along the ray, so the integral cannot in general be replaced by a difference of $h_{ij}$ at the endpoints. Deflection of the ray or its direction multiplies an already first-order metric perturbation and is second order here. This is the metric part of the Sachs-Wolfe effect in the negative-spatial-perturbation convention; an intrinsic emission temperature fluctuation or an emitter velocity would add separate boundary terms. The time derivative is present in the original PDF, although missing from the converted TeX's displayed integral.

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.