Gravitational turbulence: the small-scale limit of the cold-dark-matter power spectrum - Ginat, Yonadav Barry; Nastac, Michael L.; Ewart, Robert J.; Konrad, Sara; Bartelmann, Matthias; Schekochihin, Alexander A. (Physical review, 2025)
The matter power spectrum, 𝑃⁡(𝑘), is one of the fundamental quantities in the study of large-scale structure in cosmology. Here, we study its small-scale asymptotic limit, and show that for cold dark matter in 𝑑 spatial dimensions, 𝑃⁡(𝑘) has a universal 𝑘−𝑑 asymptotic scaling with the wave number 𝑘, for 𝑘 ≫𝑘nl, where 𝑘−1nl denotes the length scale at which nonlinearities in gravitational interactions become important. We propose a theoretical explanation for this scaling, based on a nonperturbative analysis of the system’s phase-space structure. Gravitational collapse is shown to drive a turbulent phase-space flow of the quadratic Casimir invariant, where the linear and nonlinear time scales are balanced, and this balance dictates the 𝑘 dependence of the power spectrum. A parallel is drawn to Batchelor turbulence in hydrodynamics, where large scales mix smaller ones via tidal interactions. The 𝑘−𝑑 scaling is also derived by expressing 𝑃⁡(𝑘) as a phase-space integral in the framework of kinetic field theory, which is analyzed by the saddle-point method; the dominant critical points of this integral are precisely those where the time scales are balanced. The coldness of the dark-matter distribution function—its nonvanishing only on a 𝑑-dimensional submanifold of phase space—underpins both approaches. The theory is accompanied by 1D Vlasov-Poisson simulations, which confirm it.
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