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4.2. A toy model

During the ``outside RJ'' regime, there is one growing and one decaying solution. The simplest system which has this qualitative behavior is the upside-down harmonic oscillator which obeys:

Equation 11 (11)

The phase space trajectories are show on the left panel in Fig. 7. The system is unstable against the runaway (or growing) solution where |q| and |p| get arbitrarily large (and p and q have the same sign). This behavior ``squeezes'' any initial region in phase space toward the diagonal line with unit slope. The squeezing effect is illustrated by the circle which evolves, after a period of time, into the ellipse in Fig. 7.

Figure 7

Figure 7. The phase space trajectories for an upside-down harmonic oscillator are depicted in the left panel. Any region of phase space will be squeezed along the diagonal line as the system evolves (i.e. the circle gets squeezed into the ellipse) . For a right-side-up harmonic oscillator paths in the phase space are circles, and angular position on the circle gives the phase of oscillation. Perturbations in the early universe exhibit first squeezing and then oscillatory behavior, and any initial phase space region will emerge into the oscillatory epoch in a form something like the dotted ``cigar'' due to the earlier squeezing. In this way the early period of squeezing fixes the phase of oscillation.

The simplest system showing oscillatory behavior is the normal harmonic oscillator obeying

Equation 12 (12)

This phase space trajectories for this system are circles, as shown in the right panel of Fig. 7. The angular position around the circle corresponds to the phase of the oscillation. The effect of having first squeezing and then oscillation is to have just about any phase space region evolve into something like the dotted ``cigar'' in the right panel. The cigar then undergoes rotation in phase space, but the entire distribution has a fixed phase of oscillation (up to a sign). The degree of phase coherence (or inverse ``cigar thickness'') is extremely high in the real cosmological case because the relevant modes spend a long time in the squeezing regime.

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