Large scale density perturbations from a uniform distribution by wave transport

It has long been known that a uniform distribution of matter cannot produce a Poisson distribution of density fluctuations on very large scales 1/k > ct by the motion of discrete particles over timescale t. The constraint is part of what is sometimes referred to as the Zel'dovich bound. We investigate in this paper the transport of energy by the propagation of waves emanating incoherently from a regular and infinite lattice of oscillators, each having the same finite amount of energy reserve initially. The model we employ does not involve the expansion of the Universe; indeed there is no need to do so, because although the scales of interest are all deeply sub-horizon the size of regions over which perturbations are evaluated do far exceed ct, where t is the time elapsed since a uniform array of oscillators started to emit energy by radiation (it is assumed that t greatly exceeds the duration of emission). We find that to lowest order, when only wave fields $\propto$ 1/r are included, there is exact compensation between the energy loss of the oscillators and the energy emitted into space, which means P(0)=0 for the power spectrum of density fluctuations on the largest scales. This is consistent with the Zel'dovich bound; it proves that the model employed is causal, has finite support, and energy is strictly conserved. To the next order when near fields $\propto$ r⁻² are included, however, P(0) settles at late times to a positive value that depends only on time, as t⁻² (the same applies to an excess (non-conserving) energy term). We further observe that the behavior is peculiar to near fields. Even though this effect may give the impression of superluminal energy transport, there is no violation of causality because the two-point function vanishes completely for r>t if the emission of each oscillator is sharply truncated beyond some duration. The result calls to question any need of enlisting cosmic inflation to seed large scale density perturbations in the early Universe.