Why primordial sound leaves a series of peaks

Photons push and baryons add inertia

Frequent scattering couples photons to electrons, and electromagnetic interactions couple electrons to nuclei. In tight coupling the photon-baryon fluid moves almost together. Photon pressure supplies the restoring force; baryon mass resists acceleration. Define the loading $R=3\rho_b/(4\rho_\gamma)$. Both densities are energy per physical volume.

The sound speed is $c_s=c/\sqrt{3(1+R)}$. More baryons lower it. Baryons and photons dilute at different rates, so $R$ grows as the universe expands. Sound therefore travels a different comoving distance in each interval.

Reading: Hu & Sugiyama

Solve one oscillator inside a fixed well

Let $\Theta_0=\delta T/T$ be the photon temperature monopole and use the Newtonian-gauge time potential $\Psi$. Set light speed to one. With constant potential and baryon loading, and omitting fluid-drag terms, the compression mode obeys the forced-oscillator equation below. Primes denote conformal-time derivatives.

Gravitational redshift contributes to the observed temperature, so define $X=\Theta_0+\Psi$. Moving the constant force into a shifted equilibrium gives an ordinary cosine when the initial velocity vanishes.

Reading: Hu & Sugiyama

Calculate the two turning points

Choose a fixed well with $\Psi=-10^{-5}$, $R=1/2$, initial effective temperature $X(0)=\Psi/3$ and zero initial velocity. The potential in the well is negative, so the baryon-induced equilibrium effective temperature is positive. Substitution into the cosine solution gives the dimensionless expression below.

After half an oscillation, $kr_s=\pi$ and $X=(4/3)\times10^{-5}$. After a full oscillation, $X=-(1/3)\times10^{-5}$. Their absolute amplitudes differ by a factor of four. This shows the effect of the shifted equilibrium for the specified initial conditions. A temperature spectrum also combines velocity, driving and projection, as the next section explains.

Baryons make compression and rarefaction unequal

Without baryon loading, $X$ oscillates around zero. Adding baryons shifts the equilibrium and deepens compression inside potential wells. For the usual adiabatic initial conditions, compression peaks rise relative to rarefaction peaks, producing the familiar odd-even temperature-peak pattern.

The complete temperature spectrum also contains Doppler velocities, changing potentials and photon diffusion. Increasing baryon density changes recombination, sound speed and the sound horizon together. A realistic comparison therefore needs the coupled calculation. The frozen CAMB grid retains these simultaneous effects.

Reading: CAMB source and documentation · Hu & Sugiyama

Estimate the separation between peaks

Modes with $k r_s\simeq n\pi$ reach oscillator extrema at last scattering. Projection places them roughly at $\ell\simeq kD_M$. Combining these relations gives the main acoustic spacing.

For an illustrative $r_s=144\;\mathrm{Mpc}$ and $D_M=14000\;\mathrm{Mpc}$, the angle is about $0.0103\;\mathrm{rad}$ and the acoustic multipole scale is about $\ell_A=305$. This estimates a spacing scale. Driving, phase shifts and projection move the first temperature peak away from that simple value.