
The Evolution of Perturbations and Acoustic Oscillations The evolution of the initial perturbations laid down by inflation was determined entirely by the equations of general relativity. These equations are imposing nonlinear formulae. However, since the perturbations were small (CMB anisotropies are only about one part in 10^{5} ), the relevant equations can be well approximated with linear expressions. This linearity not only makes the equations much easier to understand, but also allows the perturbations to be decomposed into normal modes. Each normal mode consists of a sinusoidal (plane wave) variation in the density of the universe and the curvature of space time, and since each mode evolved independently, the evolution of the structure at each length scale (which corresponds roughly to a single angular scale in the CMB) can be studied separately. The dynamics of a particular mode depended strongly on the size of its wavelength relative to the distance a photon could have traveled since the big bang. This distance, known as the horizon scale, increased with cosmic time faster than the wavelength, so at sufficiently early times this distance was much smaller than the wavelength of any given mode. As long as the perturbation wavelength greatly exceeded the horizon scale, regions of different densities were not in causal contact, so there was no tendency for material to flow between different regions. The amplitude of the density perturbation therefore remained essentially constant and there was little, if any, bulk flow. Modes with sufficiently long wavelengths remained in this frozen state all the way until decoupling, when the universe was no longer warm enough to maintain the primordial plasma, and the photons that form the CMB were released. These frozen (acausal) modes, which correspond to the largest angular scales on the sky, produced variations in the temperature of the CMB which reflect the initial perturbations laid down during inflation. However, since there was little bulk flow on these large scales, the polarized signal should be extremely small (but see Keating et. al., 1997). Modes with short enough wavelengths reached a point before decoupling when their wavelength became comparable to the horizon scale. Regions of different densities then came into causal contact and local forces acted to redistribute material. Two forces were particularly important: gravity and photon pressure. Gravity tended to move material from underdense regions to overdense regions, while photon pressure acted in the opposite direction. The initial perturbations laid down by inflation were such that these two forces were out of balance, so material flowed back and forth between underdense and overdense regions. These acoustic oscillations continued until decoupling, when the resulting density contrasts and bulk flows left their mark on the smallscale anisotropies of the CMB. The Origin of Cosmological Polarization The Structure of Cosmological Polarization Power Spectra and the Structure of CMB Anisotropies Initial Conditions and Inflation The Evolution of Perturbations and Acoustic Oscillations Acoustic Oscillations and Apparent Temperature Variations 
