Gravitationally driven fragmentation and classical turbulence: a quantitative assessment of the spectral analogy

The postinflationary inflaton condensate fragments gravitationally, a process well described by the Schrödinger–Poisson (SP) equations. Under the Madelung transformation the SP system is exactly equivalent to a compressible, inviscid, irrotational self-gravitating fluid in which a quantum potential replaces the thermodynamic pressure. That exact correspondence shows that vortex stretching, viscous dissipation and scale-separated forcing are all absent, so Kolmogorov scaling is not predicted. We then analyse the density-fluctuation power spectrum published by Musoke, Hotchkiss and Easther for the snapshot at which the Universe has grown by a factor of 200 since inflation ended, extracted from their figure with an uncertainty budget, bootstrapped confidence intervals, a scan of the fitting interval, and information-criterion model comparison. Over the decade of wavenumbers immediately above the horizon scale the spectrum follows a power law of slope − 3.73 ± 0.18 , which for that choice of interval is consistent with the − 11 / 3 obtained by transcribing the Kolmogorov spectrum into a three-dimensional spectral density and inconsistent with − 5 / 3 . The agreement is interval-dependent and is not a robust feature of the data: the slope varies between − 3.1 and − 7.4 as the fitting interval is changed, a curved spectrum is mildly preferred, and the scatter is dominated by realization noise rather than by figure extraction. The exponential high-wavenumber tail extends past the Nyquist frequency of the source simulation, so we attach no dissipation-range interpretation to it. The correspondence therefore holds over about 1 decade; we set out the measurements that would make the test decisive.