We have hitherto also been neglecting the energy density of the radiation in comparison to the matter. The matter-to-radiation ratio scales as and so is also of order unity at recombination for reasonable parameters. Moreover fluctuations corresponding to the higher peaks entered the sound horizon at an earlier time, during radiation domination.
Including the radiation changes the expansion rate of the Universe and hence the physical scale of the sound horizon at recombination. It introduces yet another potential ambiguity in the interpretation of the location of the peaks. Fortunately, the matter-radiation ratio has another effect in the power spectrum by which it can be distinguished. Radiation drives the acoustic oscillations by making the gravitational force evolve with time [Hu & Sugiyama, 1995]. Matter does not.
The exact evolution of the potentials is determined by the relativistic Poisson equation. But qualitatively, we know that the background density is decreasing with time, so unless the density fluctuations in the dominant component grow unimpeded by pressure, potentials will decay. In particular, in the radiation dominated era once pressure begins to fight gravity at the first compressional maxima of the wave, the Newtonian gravitational potential and spatial curvature must decay (see Figure 3).
Figure: Radiation driving and diffusion damping. The decay of the potential drives the oscillator in the radiation dominated epoch. Diffusion generates viscosity , i.e. a quadrupole moment in the temperature, which damps oscillations and generates polarization. Plotted here is the numerical solution to Equation (18) and Equation (19) for a mode with wavelength much smaller than the sound horizon at decoupling, .
This decay actually drives the oscillations: it is timed to leave the fluid maximally compressed with no gravitational potential to fight as it turns around. The net effect is doubled since the redshifting from the spatial metric fluctuation also goes away at the same time. When the Universe becomes matter dominated the gravitational potential is no longer determined by photon-baryon density perturbations but by the pressureless cold dark matter. Therefore, the amplitudes of the acoustic peaks increase as the cold dark matter-to-radiation ratio decreases [Seljak, 1994,Hu & Sugiyama, 1995]. Density perturbations in any form of radiation will stop growing around horizon crossing and lead to this effect. The net result is that across the horizon scale at matter radiation equality the acoustic amplitude increases by a factor of 4-5 [Hu & Sugiyama, 1996]. By eliminating gravitational potentials, photon-baryon acoustic oscillations eliminate the alternating peak heights from baryon loading. The observed high third peak [Halverson et al, 2001] is a good indication that cold dark matter both exists and dominates the energy density at recombination.