THE POWER SPECTRUM OF DENSITY-FLUCTUATIONS IN THE ZELDOVICH APPROXIMATION

The Zel'dovich approximation, combined with an initial spectrum, appea rs to yield a surprisingly good prescription of the large-scale matter distribution for the evolution of structure in the Universe; in parti cular, it describes the evolution of structure fairly accurately well into the non-linear regime, and is thus superior to the standard Euler ian linear perturbation theory. We calculate the evolution of the powe r spectrum P(k, a) of the density field in the Zel'dovich approximatio n, which can be reduced to a single one-dimensional integral. The resu lting expression reproduces the result from linear perturbation theory for small values of the cosmic scale factor a. On the other hand, the power spectrum as obtained from the Zel'dovich approximation predicts the generation of power on small scales, mainly as a result of the fo rmation of compact structures and caustics. In fact, it is shown that, for k-->infinity, P(k, a) behaves like k(-3) on scales for which diss ipative processes are negligible; this asymptotic behaviour is not an artefact of the Zel'dovich approximation, but is due to the occurrence of pancakes. We evaluate the power spectrum for standard hot dark mat ter (HDM) and cold dark matter (CDM) spectra; in the latter case, we e mploy the truncated Zel'dovich approximation which has been shown prev iously to yield much better agreement with the results from N-body sim ulations in cases where the primordial power spectrum contains large a mounts of power on small scales. We obtain a simple fitting formula fo r the smoothing scale used in the truncated Zel'dovich approximation.