Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 75 / 1 / b / Solution

For adiabatic initial conditions, the coupled photon-baryon fluid begins with a coherent displacement and approximately zero initial acoustic velocity. Neglecting slowly varying gravitational driving, its effective temperature satisfies an oscillator equation with phase $k{r}_s$, whereThus the Sachs-Wolfe combination has a dominant cosine phase, while the velocity has a sine phase. At last scattering, the first three temperature Cosmic microwave background acoustic peaks correspond approximately to $k{r}_s=\pi ,2\pi ,3\pi$: first compression, first rarefaction, and second compression of the initially displaced plasma in a potential well. Baryon inertia shifts the oscillator's equilibrium and enhances compressions relative to rarefactions, producing the familiar odd/even peak contrast. Potential decay and radiation driving alter the amplitudes, especially of the earliest peaks.

Adjacent displacement extrema differ by $\pi$, soHere $D_M$ is the transverse comoving distance to last scattering; in a flat model it is ${\tau }_0-{\tau }_{\ast }$. If $c_s$ is treated as constant, this reduces to $\Delta \ell \simeq \pi ({\tau }_0-{\tau }_{\ast })/(c_s{\tau }_{\ast })$. No numerical spacing is determined without a cosmological distance and sound horizon. Gravitational phase shifts, baryon loading and projection displace the exact peak locations from integer multiples of this spacing. These are primarily coherent density-plus-potential acoustic peaks; the label “Doppler peaks” does not mean the velocity term alone creates their sequence.