On the asymptotic behaviour of cosmic density-fluctuation power spectra

TL;DR

This paper analyzes the small-scale asymptotic behavior of cosmic density-fluctuation power spectra, proving a universal $k^{-3}$ tail in the Zel'dovich approximation that is independent of cosmological models.

Contribution

It extends Laplace's method to arbitrary dimensions and proves the universal $k^{-3}$ asymptotic tail of the power spectrum, applicable across various cosmological scenarios.

Findings

Power spectrum develops a $k^{-3}$ tail at small scales.

The asymptotic behavior is independent of cosmological models.

Characteristic scales for non-linear structure formation are derived.

Abstract

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.