Adiabatic acoustic-peak spacing (${\ell }_n\simeq n\pi {\chi }_{\ast }/r_s({\eta }_{\ast })$)

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.