Cosmic microwave background acoustic peak (CMB acoustic peak)

Cosmic microwave background acoustic peaks arise from photon-baryon oscillations caught at successive extrema at recombination. Their approximate phase is set by $k{r}_s=n\pi$ and their angular positions measure the sound horizon relative to the distance to last scattering.

For an outward sightline, Fourier velocity $v_b^i=i{k}^i{\theta }_b$, and sudden last scattering, intrinsic photon temperature projects with a Spherical Bessel function, while the Doppler term projects with its derivative. Density and velocity acoustic sources are approximately a quarter cycle out of phase. Their amplitude is suppressed by photon diffusion, giving a power damping envelope $e^{-2k^2/k_D^2}$. Projection maps the sound scale to peak spacing approximately $\pi D_{\ast }/r_s$ and the damping scale to angular multipoles of order $k_D{D}_{\ast }$.

Adiabatic initial conditions select coherent cosine phases of the photon-baryon acoustic oscillator. Its extrema at last scattering satisfy $k{r}_s\simeq n\pi$ and project to $\ell \simeq k{\chi }_{\ast }$. Thus ${\ell }_n\simeq n\pi {\chi }_{\ast }/r_s({\eta }_{\ast })$, with comoving sound horizon $r_s=\int d\eta /\sqrt{3(1+R)}$. Projection, driving and finite last-scattering width modify the exact peak locations.