4.3. The results of coherence
Having presented the toy model, let as look at the behavior of the real cosmological variables. Figure 8 shows (the photon fluctuations) as a function of (conformal) time measured in units of *, the time at last scattering.
Figure 8. The evolution of for two different wavelengths (upper and lower panels) as a function of time for an ensemble of initial conditions. Each wavelength shows an initial period of growth (squeezing) followed by oscillations. The initial squeezing fixes the phase of oscillation across the entire ensemble, but different wavelengths will have different phases. The two chosen wavelengths are maximally out of phase at * the time of last scattering.
Two different wavelengths are shown (top and bottom panels), and each panel shows several members of an ensemble of initial conditions. Each curve shows an early period of growth (squeezing) followed by oscillation. The onset of oscillation appears at different times for the two panels, as each mode ``enters'' RJ at a different time. As promised, because of the initial squeezing epoch all curves match onto the oscillatory behavior at the same phase of oscillation (up to a sign). Phases can be different for different wavelengths, as can be seen by comparing the two panels.
To a zeroth approximation the event of last scattering simply releases a snapshot of the photons at that moment and sends them free-streaming across nearly empty space. The left panel of Fig. 9 (solid curve) shows the mean squared photon perturbations at the time of last scattering in a standard inflationary model, vs k.
Figure 9. The left panel shows how temporal phase coherence manifests itself in the power spectrum for in Fourier space, shown here at one moment in time. Even after ensemble averaging, some wavelengths are caught at zeros of their oscillations while others are caught at maximum amplitudes. These features in the radiation density contrast are the root of the wiggles in the CMB angular power shown in the right panel.
Note how some wavenumbers have been caught at the nodes of their oscillations, while others have been caught at maxima. This feature is present despite the fact that the curve represents an ensemble average because the same phase is locked in for each member of the ensemble.
The right panel of Fig. 9 shows a typical angular power spectrum of CMB anisotropies produced in an inflationary scenario. While the right hand plot is not exactly the same as the left one, it is closely related. The CMB anisotropy power is plotted vs. angular scale instead of Fourier mode, so the x-axis is ``l'' from spherical harmonics rather than k. The transition from k to l space, and the fact that other quantities besides affect the anisotropies both serve to wash out the oscillations to some degree (there are no zeros on the right plot, for example). Still the extent to which there are oscillations in the CMB power is due to the coherence effects just discussed.
As our understanding of the inflationary predictions has developed, the defect models of cosmic structure formation have served as a useful contrast [12, 13]. In cosmic defect models there is an added matter component (the defects) that behaves in a highly nonlinear way, starting typically all the way back at the GUT epoch. This effectively adds a ``random driving term'' to the equations that is constantly driving the other perturbations. These models are called ``active'' models, in contrast to the passive models where all matter evolves in a linear way at early times. Figure 10 shows how despite the clear tendency to oscillate, the phase of oscillation is randomized by the driving force. In all known defect models the randomizing effect wins completely and there are no visible oscillations in the CMB power. This comparison will be discussed further in the next section.
Figure 10. Decoherence in the defect models: The two curves in the upper panel show two different realizations of the effectively random driving force for a particular mode of . The corresponding evolution of is shown in the lower panel. There is no phase coherence from one member of the ensemble to the next.