Large Scale White Noise and Cosmology

TL;DR

This work demonstrates that large-scale white noise (LSWN) is an inevitable consequence of non-linear mode coupling in local non-linear PDEs describing cosmological inhomogeneities, even when initial perturbations are small and sub-Poissonian. By developing a Leading-Order Born Approximation (LOBA) and applying it to cosmology-like systems with and without sound, the authors derive general expressions for the LSWN contribution in terms of linear mode functions, Green functions, and initial-condition correlators, revealing relic versus active LSWN regimes and the No-No-Scale constraint. They show that LSWN can dominate on the largest scales if small-scale power is present and the large-scale spectrum is suppressed, placing novel constraints on the small-scale cutoff and allowing running of the spectral index to mitigate LSWN. The results imply that cosmological observables on the largest scales carry a mixed imprint of primordial fluctuations and non-linear small-scale physics, with significant implications for inflationary model-building and for interpreting CMB and large-scale structure data. The framework generalizes beyond cosmology, linking LSWN to universal phenomena in turbulence, wave dynamics, and nonlinear optics, and offering a robust, quantitative tool for probing non-linearities in large systems.

Abstract

The generation of white noise on large scales is a generic property of the dynamics of physical systems described by local non-linear partial differential equations. Non-linearities prevent the small scale dynamics from being erased by smoothing. Unresolved small scale dynamics act as an uncorrelated (white or Poissonian) noise (seemingly stochastic but actually deterministic) contribution to large scale dynamics. This white noise exists even when the dynamics is very nearly linear. In cases where the power spectrum is sub-Poissonian on large scales, this noise will dominate on the largest scale power no matter the amplitude of the inhomogeneities. Such is the case in the standard model of cosmology, where the primordial density power spectrum is expected to have an almost Harrison-Zel'dovich, $P[k]\sim k$, spectrum on a much broader range of scales than can be observed. Even though linear gravitational evolution dominates non-linear corrections by a factor $\sim10^5$, the non-observation of white noise on the Hubble scale precludes the extrapolation of this power law below the comoving $1\,$pc scale. More generally, observation or non-observation of large scale white noise provides a powerful probe of the universe on very small scales in the early early universe. Gravitational radiation, phase transitions, vorticity, and running of the spectral index are all phenomena that can be probed with large scale white noise. Large scale white noise is a non-optional feature of all cosmological models but one which has not heretofore been appreciated.

Figures (2)

• Figure 1: Evolution of dust inhomogeneities in Lagrangian coordinates using the analytic solutions of Eq. (\ref{['eq:OverdensitySolutions']}) with periodic boundary conditions. Initial growing mode amplitudes, $\mathsf{g}$, drawn from a Gaussian distribution with power spectrum in Eq. (\ref{['eq:PowerLawGaussianCutoff']}) with $n_\mathrm{s}=4$ and $\sigma=0.0003\,L$, where $L$ is the size of the box. Green, blue, and red points denote the power spectra of the linearized solution $\mathfrak{d}=\delta\lambda_\mathrm{linear}$, full analytic solution $\delta\lambda$, and leading order Born approximation solution $\delta\lambda_{\rm LOBA}$, respectively. The green band gives the expected power spectrum for linear evolution, which should track the green points. The pink band gives the expected power spectrum for LOBA evolution, which should track both red and blue points (provided LOBA is accurate).
• Figure 2: For a range of times parameterized by the sound horizon, $h$ as defined in Eq. (\ref{['eq:RadiationSoundHorizon']}) we plot the estimated power spectrum from Eq. (\ref{['eq:RelicRadiation']}) for $\delta$ in blue and the linear theory prediction in green. Initial conditions are of the form of Eq. (\ref{['eq:PowerLawSpectrumGaussianTruncation']}) with $n_\mathrm{s}=5$ and normalized such that $\delta_\mathrm{rms}^\mathrm{linear}$ peaks at $10^{-5}$. All quantities are normalized to the cutoff scale, $\sigma$. Plotted in green is the linear theory prediction. Evident are the acoustic oscillations on small scales and as well as large scale white noise (LSWN). Linear theory evolution dominates initially while non-linearities generate additional large scale power which eventually also grows like linear theory, i.e. growth outside the sound horizon and damping inside.