Evolution of the Power Spectrum and Self‐Similarity in the Expanding One‐dimensional Universe

We have investigated time evolutions of power spectra of density fluctuationsfor long time after the first appearance of caustics in the expandingone-dimensional universe. It is found that when an initial power spectrum issale-free with a power index $n$, a self-similarity of the time evolution ofthe power spectrum is achieved. We find that the power spectrum can beseparated roughly into three regimes according to the shape of the powerspectrum: the linear regime ($k < k_{nl}$ : the regime {\cal 1}),thesingle-caustic regime($k_{nl} < k < k_{snl}$ : the regime 2), and themulti-caustics regime($k > k_{snl}$ : the regime 3). The power index of thepower spectrum in each regime has the values of $n,-1$, and $\mu$ which dependson $n$, respectively. Even in the case of an initial power-law spectrum with acutoff scale, there might be the possibility of the self-similar evolution ofthe 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 powerspectrum on scales smaller than the cutoff scale can be separated roughly intwo regimes: the virialized regime ($k_{cut}< k < k_{cs}$ : the regime 4), andthe smallest-single-caustic regime ($ k > k_{cs}$ : the regime 5). The power index of the power spectrum is $\nu$ which may bedetermined by the distribution of singular points in the regime 4. In theregime 5, the value of the power index is -1. Moreover we show the generalproperty about the shape of a power spectrum with a general initial condition.Comment: Accepted for pubrication in the ApJS, vol.118, October, 1998; 15pages, uses aaspp4.sty, 12figure