Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 2 / c / Solution

Neglecting baryon loading, the Sachs-Wolfe combination obeys a harmonic-oscillator equation with sound speed $1/\sqrt{3}$. Adiabatic initial conditions give a nonzero initial displacement and negligible initial velocity, henceProjection maps $k$ approximately to $\ell /{\chi }_{\ast }$, and the angular power is quadratic in the transfer function, producing the observed approximate ${\cos }^2(k{r}_s)$ sequence of acoustic peaks. A ${\sin }^2$ phase would instead indicate vanishing initial displacement and nonzero initial velocity, characteristic of an isocurvature rather than adiabatic primordial mode.