Acoustic oscillations receive a boost at horizon crossing $a_H$ due driving from gravitational potential decay. The perturbations then settle into a pure acoustic mode and are subsequently damped by photon diffusion. Together the potential driving and diffusion damping effects form the acoustic envelope. After diffusion damping has destroyed the acoustic oscillations, the underlying baryon drag effect becomes apparent. Since $\Psi$ is here constant for $a \gg a_{eq}$, $|R\Psi|$ increases with time. This contribution itself is damped at last scattering $a_*$ by cancellation. Well after last scattering $a \gg a_*$, isotropic perturbations collisionlessly damp creating angular fluctuations in the CMB. The model here is adiabatic CDM, $\Omega_0=1$, $h=0.5$, $\Omega_b=0.05$.

Visibility functions for the photons and baryons. In the undamped $k\rightarrow 0$ limit, the photon acoustic visibility and the Compton visibility are equivalent $\hat \Vg = \Vg$ and the baryon acoustic visibility equals the drag visibility, $\hat\Vb = \Vb$. If $\Omega_b h^2 \simlt 0.03$, $\Vb$ peaks at later times than $\Vg$, \ie\ $\eta_d > \eta_*$. For small scales, the acoustic visibilities, which weight the acoustic contributions from the tight coupling regime, are increasingly suppressed at later times by damping and the width and amplitude of the acoustic visibilities decrease as $k$ increases.

Photon transfer function. Plotted is the present rms photon temperature fluctuation relative to the initial curvature fluctuation $\Phi(0,k)$ for adiabatic fluctuations and the initial entropy fluctuation $S(0,k)$ for isocurvature fluctuations assuming standard recombination. The intrinsic acoustic amplitude is approximated by the analysis in Appendix B for scales well inside the horizon at equality $k \gg k_{eq}$. The damping is well approximated by the tight coupling expansion for high $\Omega_b h^2$ and is slightly overestimated for low values. Below the damping tail, the baryon drag offset clearly appears in (b) and (c) where the gravitational potential is not dominated by acoustic density fluctuations. An analytic treatment of this effect is given in Appendix A and C.