Until now we have implicitly assumed that the evolution of the perturbations plays a small role as is generally true for scales larger than the horizon at last scattering. Evolution plays an important role for small-scale scalar perturbations where there is enough time for sound to cross the perturbation before last scattering. The infall of the photon fluid into troughs compresses the fluid, increasing its density and temperature. For adiabatic fluctuations, this compression reverses the sign of the effective temperature perturbation when the sound horizon s grows to be (see Fig. 13a). This reverses the sign of the correlation with the quadrupole moment. Infall continues until the compression is so great that photon pressure reverses the flow when . Again the correlation reverses sign. This pattern of correlations and anticorrelations continues at twice the frequency of the acoustic oscillations themselves (see Fig. 13a). Of course the polarization is only generated at last scattering so the correlations and anticorrelations are a function of scale with sign changes at multiples of , where is the sound horizon at last scattering. As discussed in §3.2, these fluctuations project onto anisotropies as .
Fig. 13: Time evolution of acoustic oscillations. The polarization is related to the flows v (red) which form quadrupole anisotropies such that its product (green) with the effective temperature (blue) reflects the temperature-polarization cross correlation. As described in the text the adiabatic (a) and isocurvature (b) modes differ in the phase of the oscillation in all three quantities. Temperature and polarization are anticorrelated in both cases at early times or large scales ks much less than unity.
Any scalar fluctuation will obey a similar pattern that reflects the acoustic motions of the photon fluid. In particular, at the largest scales the , the polarization must be anticorrelated with the temperature because the fluid will always flow with the temperature gradient initially from hot to cold. However, where the sign reversals occur depend on the acoustic dynamics and so is a useful probe of the nature of the scalar perturbations, e.g. whether they are adiabatic or isocurvature ([Hu & White] 1996). In typical isocurvature models, the lack of initial temperature perturbations delays the acoustic oscillation by in phase so that correlations reverse at (see Fig. 13b).
For the vector and tensor modes, strong evolution can introduce a small correlation with temperature fluctuations generated after last scattering. The effect is generally weak and model dependent and so we shall not consider it further here.
Next: Model Reconstruction