Evolution of the Power Spectrum and the Self- Similarity in the Expanding One-Dimensional Universe
We have investigated the time evolution of the power spectrum in two cases of initial conditions, the scale-free case and the cut-off case. In the scale-free case, we can observe a self-similar evolution of the power spectrum. The scaled power spectra P*(k *) coincide each time. We can roughly divide the power spectrum into three regimes. One is the linear regime (k < k nl : regime 1). The value of the power index in this regime remains the initial power index, n. The second regime is the single-caustic regime (k nl < k < k snl : regime 2). In this regime, the power index becomes — 1 and is independent of the initial conditions. This is caused by the appearance of caustics on this scale, which determine the power index of the power spectrum in this regime. The third regime is the multi-caustics regime (k > k snl : regime 3). In this case, the distribution of the “whirlpool” in phase space determines the value of the power index. Therefore, on this scale, the power index μ has a value which depends on the initial conditions.