Spatial correlations of dark energy from quantum fluctuations in inflation

TL;DR

This study investigates how quantum fluctuations of a scalar field during inflation could explain dark energy's spatial correlations, refining previous models and comparing stochastic and field theoretical approaches.

Contribution

It provides a comprehensive derivation of dark energy correlations from quantum fluctuations, compares stochastic and field theory methods, and introduces a reduced sound speed to reconcile discrepancies.

Findings

Stochastic and field theory predictions differ for sub-Hubble correlators.

Introducing a reduced sound speed aligns stochastic results with field theory.

Both methods agree on 2-point functions but diverge on 4-point functions.

Abstract

This paper contains a detailed study of the properties of a simple model attempting to explain dark energy as originated from quantum fluctuations of a light spectator scalar field in inflation. In [1] we recently outlined how Starobinsky's stochastic formalism can be used to study the spatial correlations imprinted on dark energy by its quantum origin in this model and we studied their possible role in relieving the Hubble tension. Here we provide a more comprehensive derivation of the results in [1] and we refine some of our estimates, comparing to the approximate results obtained previously. Among the main results, we analyze the non-coincident correlators predicted by a full field theoretical treatment and their relation with those computed within the stochastic formalism. We find that in the region where stochastic theory predicts significant sub-Hubble correlators it is in disagreement with field theoretical predictions. However, agreement can be restored by introducing a reduced speed of sound for the scalar field. We also discuss an alternative approach to the problem of studying correlators within the stochastic formalism based directly on the evolution of probability distributions. We find that the two approaches give the same answer for 2-point functions of the field, but not for 4-point functions relevant to density correlators and we discuss the behaviour of the two methods with respect to Wick's theorem.