On the asymptotic behaviour of cosmic density-fluctuation power spectra of cold dark matter

Understand

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 $k$.