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Gravitational turbulence: The small-scale limit of the cold-dark-matter power spectrum

Yonadav Barry Ginat1,2,*, Michael L. Nastac1,3, Robert J. Ewart1,4,5, Sara Konrad6, Matthias Bartelmann6, and Alexander A. Schekochihin1,7

  • 1Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Parks Road, Oxford, OX1 3PU, United Kingdom
  • 2New College, Holywell Street, Oxford, OX1 3BN, United Kingdom
  • 3St. John’s College, St. Giles, Oxford, OX1 3JP, United Kingdom
  • 4Balliol College, Broad Street, Oxford, OX1 3BJ, United Kingdom
  • 5Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, New Jersey 08544, USA
  • 6Institute for Theoretical Physics, Heidelberg University, Germany
  • 7Merton College, Merton Street, Oxford, OX1 4JD, United Kingdom

  • *Contact author: yb.ginat@physics.ox.ac.uk
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Phys. Rev. D 112, 063501 – Published 3 September, 2025

DOI: https://doi.org/10.1103/ychs-2d5p

Abstract

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 knl−1 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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Phys. Rev. D 112, 063501– Published 3 September, 2025
  • Received 1 January 2025
  • Accepted 8 August 2025
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