On the asymptotic behaviour of cosmic density-fluctuation power spectra
  We study the small-scale asymptotic behaviour of the cosmic
density-fluctuation power spectrum in the Zel'dovich approximation. For doing
so, we extend Laplace's method in arbitrary dimensions and use it to prove that
this power spectrum necessarily develops an asymptotic tail proportional to
$k^{-3}$ , irrespective of the cosmological model and the power spectrum of the
initial matter distribution. The exponent $-3$ is set only by the number of
spatial dimensions. We derive the complete asymptotic series of the power
spectrum and compare the leading- and next-to-leading-order terms to derive
characteristic scales for the onset of non-linear structure formation,
independent of the cosmological model and the type of dark matter. Combined
with earlier results on the mean-field approximation for including particle
interactions, this asymptotic behaviour is likely to remain valid beyond the
Zel'dovich approximation. Due to their insensitivity to cosmological
assumptions, our results are generally applicable to particle distributions
with positions and momenta drawn from a Gaussian random field. We discuss an
analytically solvable toy model to further illustrate the formation of the
$k^{-3}$ asymptotic tail.
