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

Abstract

ABSTRACT 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 < ns < 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 $k^{n_\mathrm{ s}-1}$ 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 ns.