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

The matter power spectrum, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>P</a:mi><a:mo stretchy="false">(</a:mo><a:mi>k</a:mi><a:mo stretchy="false">)</a:mo></a:math>, 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>d</e:mi></e:math> spatial dimensions, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>P</g:mi><g:mo stretchy="false">(</g:mo><g:mi>k</g:mi><g:mo stretchy="false">)</g:mo></g:math> has a universal <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>k</k:mi><k:mrow><k:mo>−</k:mo><k:mi>d</k:mi></k:mrow></k:msup></k:math> asymptotic scaling with the wave number <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>k</m:mi></m:math>, for <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>k</o:mi><o:mo>≫</o:mo><o:msub><o:mi>k</o:mi><o:mrow><o:mi>nl</o:mi></o:mrow></o:msub></o:math>, where <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msubsup><q:mi>k</q:mi><q:mrow><q:mi>nl</q:mi></q:mrow><q:mrow><q:mo>−</q:mo><q:mn>1</q:mn></q:mrow></q:msubsup></q:math> 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 <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>k</s:mi></s:math> dependence of the power spectrum. A parallel is drawn to Batchelor turbulence in hydrodynamics, where large scales mix smaller ones via tidal interactions. The <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msup><u:mi>k</u:mi><u:mrow><u:mo>−</u:mo><u:mi>d</u:mi></u:mrow></u:msup></u:math> scaling is also derived by expressing <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>P</w:mi><w:mo stretchy="false">(</w:mo><w:mi>k</w:mi><w:mo stretchy="false">)</w:mo></w:math> 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 <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>d</ab:mi></ab:math>-dimensional submanifold of phase space—underpins both approaches. The theory is accompanied by 1D Vlasov-Poisson simulations, which confirm it.