EVOLUTION OF THE POWER SPECTRUM AND SELF-SIMILARITY IN THE EXPANDING ONE-DIMENSIONAL UNIVERSE

We have investigated time evolutions of power spectra of density fluct uations for a long time after the first appearance of caustics in the expanding one-dimensional universe. It is found that when an initial p ower spectrum is scale-free with a power index n, self-similarity of t he time evolution of the power spectrum is achieved. We find that the power spectrum can be separated roughly into three regimes according t o the shape of the power spectrum: the linear regime (k < k(nl): regim e 1), the single-caustic regime (k(nl) < k < k(snl): regime 2), and th e multicaustics regime (k > k(snl): regime 3). The power index of the power spectrum in each regime has values of n, - 1, and mu, respective ly, which depend on n. Even in the case of an initial power-law spectr um with a cutoff scale, there might be the possibility of self-similar evolution of the power spectrum after the appearance of the caustics. It is found, however, that selfsimilarity is not achieved in this cas e. The shape of the power spectrum on scales smaller than the cutoff s cale can be separated roughly into two regimes: the virialized regime (k(cut) < k < k(cs): regime 4), and the smallest single-caustic regime (k > k(cs): regime 5). The power index of the power spectrum is nu, w hich may be determined by the distribution of singular points in regim e 4. In regime 5, the value of the power index is - 1. Moreover, we sh ow the general property of the shape of a power spectrum with a genera l initial condition.