Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 312 / 4

Project the photon Boltzmann equation onto Legendre polynomials. The angular average uses $⟨\Theta ⟩={\Theta }_0$ and $⟨\mu ⟩=0$, so the collision term has no monopole andFor the dipole, the recurrence relation makes free streaming couple ${\Theta }_1$ to ${\Theta }_0$ and ${\Theta }_2$. The gravitational term $-ik\mu \Psi$ contributes $k\Psi /3$, while projection of $i\mu v_b$ gives the baryon-velocity source. ThusNeglecting the photon quadrupole gives

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In the tight-coupling approximation, finiteness of the photon Euler equation as $\Gamma \to \infty$ requiresLet $S=3{\Theta }_1+v_b$. Rearranging the baryon Euler equation and substituting the leading relation only on its right-hand side givesInsert this slip into the photon Euler equation:The photon continuity equation gives ${\Theta }_1=-({\Theta }_0^{\prime }-{\Phi }^{\prime })/k$. Eliminating the dipole yieldswhere the photon-baryon sound speed is

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For $R=0$, $c_s=1/\sqrt{3}$. If the Newtonian potentials are constant and equal, the equation in part (b) becomesThe Sachs-Wolfe combination $D={\Theta }_0+\Psi$ therefore obeysAdiabatic initial conditions have vanishing initial fluid velocity, hence $D^{\prime }(0)=0$. With the sound horizon $r_s(\eta )={\int }_0^{\eta }{c}_s,d{\eta }^{\prime }$, the solution is

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The $\ell =2$ projection of the Boltzmann hierarchy isIn tight coupling, ${\Theta }_2=O(k/\Gamma )$ and ${\Theta }_3$ is higher order. Neglecting ${\Theta }_2^{\prime }$ relative to $\Gamma {\Theta }_2$ at leading order givesso $C=2/5$.

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The continuity equation and the cosine solution givewhen the potentials are constant. Part (d) then impliesSince Cosmic microwave background polarization is sourced by the local quadrupole at last scattering,The monopole contribution to the CMB temperature oscillates as $\cos (k{r}_{s\ast })$, so its acoustic extrema occur near $k{r}_{s\ast }=m\pi$, whereas polarization extrema occur near $k{r}_{s\ast }=(m+\frac{1}{2})\pi$. The principal polarization peaks are therefore interleaved with the principal temperature peaks. Projection, baryon loading, time-varying potentials, and the temperature Doppler contribution shift the observed angular peaks from this idealized phase relation.

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