On the asymptotic behaviour of cosmic density-fluctuation power spectra
  of cold dark matter
  We study the small-scale asymptotic behaviour of the cold dark matter density
fluctuation power spectrum in the Zel'dovich approximation, without introducing
an ultraviolet cut-off. Assuming an initially correlated Gaussian random field
and spectral index $0 < n_s < 1$, we derive the small-scale asymptotic
behaviour of the initial momentum-momentum correlations. This result is then
used to derive the asymptotics of the power spectrum in the Zel'dovich
approximation. Our main result is an asymptotic series, dominated by a $k^{-3}$
tail at large wave-numbers, containing higher-order terms that differ by
integer powers of $k^{n_s-1}$ and logarithms of $k$. Furthermore, we show that
dark matter power spectra with an ultraviolet cut-off develop an intermediate
range of scales, where the power spectrum is accurately described by the
asymptotics of dark matter without a cut-off. These results reveal information
about the mathematical structure that underlies the perturbative terms in
kinetic field theory and thus the non-linear power spectrum. We also discuss
the sensitivity of the small-scale asymptotics to the spectral index $n_s$.
