Primordial standing waves

TL;DR

This paper explores the concept of primordial fluctuations as standing waves, analyzing their statistical properties, how they evolve through squeezing, and implications for their detectability depending on phase and theory modifications.

Contribution

It introduces the conditions for primordial standing waves, emphasizing the importance of specific correlators and their evolution under squeezing, including effects of modified dispersion relations.

Findings

Only sine-phase standing waves survive squeezing in translationally invariant systems.

Primordial standing waves may be undetectable at late times depending on their phase.

Modified dispersion relations alter which standing wave phases survive at late times.

Abstract

We consider the possibility that the primordial fluctuations (scalar and tensor) might have been standing waves at their moment of creation, whether or not they had a quantum origin. We lay down the general conditions for spatial translational invariance, and isolate the pieces of the most general such theory that comply with, or break translational symmetry. We find that, in order to characterize statistically translationally invariant standing waves, it is essential to consider the correlator $\langle c_0({\mathbf k}) c_0({\mathbf k}')\rangle$ in addition to the better known $\langle c_0({\mathbf k}) c_0^\dagger({\mathbf k}')\rangle$ (where $c_0({\mathbf k})$ are the complex amplitudes of travelling waves). We then examine how the standard process of "squeezing" (responsible for converting travelling waves into standing waves while the fluctuations are outside the horizon) reacts to being fed primordial standing waves. For translationally invariant systems only one type of standing wave, with the correct temporal phase (the "sine wave"), survives squeezing. Primordial standing waves might therefore be invisible at late times -- or not -- depending on their phase. Theories with modified dispersion relations behave differently in this respect, since only standing waves with the opposite temporal phase survive at late times.