Evolution of the Power Spectrum and the Self-Similarity in the Expanding One-Dimensional Universe

We have investigated time evolutions of power spectra of density fluctuations for long time after the first appearance of caustics in the expanding one-dimensional universe. It is found that when an initial power spectrum is sale-free with a power index $n$, a self-similarity of the time evolution of the power spectrum is achieved. We find that the power spectrum can be separated roughly into three regimes according to the shape of the power spectrum: the linear regime ($k < k_{nl}$ : the regime {\cal 1}),the single-caustic regime($k_{nl} < k < k_{snl}$ : the regime 2), and the multi-caustics regime($k > k_{snl}$ : the regime 3). The power index of the power spectrum in each regime has the values of $n,-1$, and $\mu$ which depends on $n$, respectively. Even in the case of an initial power-law spectrum with a cutoff scale, there might be the possibility of the self-similar evolution of the power spectrum after the appearance of the caustics. It is found, however, the self-similarity is not achieved in this case. The shape of the power spectrum on scales smaller than the cutoff scale can be separated roughly in two regimes: the virialized regime ($k_{cut}< k < k_{cs}$ : the regime 4), and the smallest-single-caustic regime ($ k > k_{cs}$ : the regime 5). The power index of the power spectrum is $\nu$ which may be determined by the distribution of singular points in the regime 4. In the regime 5, the value of the power index is -1. Moreover we show the general property about the shape of a power spectrum with a general initial condition.