Scalar-tensor theories at different scales: averaging the scalar sector

TL;DR

This paper examines how averaging matter distributions affects scalar field equations in scalar-tensor theories, highlighting differences between linear and non-linear models and implications for cosmology and experiments.

Contribution

It provides a detailed analysis of the errors introduced by averaging matter sources in scalar-tensor theories, especially in non-linear chameleon models, and discusses their cosmological and experimental implications.

Findings

Averaging commutes with the Klein-Gordon equation in linear Yukawa theories.

In non-linear chameleon theories, averaging introduces errors, especially in screened regimes.

Implications for the equation of state in cosmology and laboratory experiments are discussed.

Abstract

This article investigates the averaging of a scalar degree of freedom that couples universally to matter. It quantifies the approximation of smoothing the matter distribution before solving the Klein--Gordon equation. In the case of Yukawa theories, which enjoy a linear Klein--Gordon equation, the averaging commutes with the field equation as one might expect. While all small-scale distributions of matter lead to field distributions with the same mean, the latter can have different energy densities and pressures when the Compton wavelength of the field is smaller than the smoothing scale. In the non-linear case, such as chameleon theories, this study quantifies the error made by averaging the matter distribution before solving the Klein--Gordon equation. While field fluctuations can become arbitrarily large when the matter source is screened, the commutativity property of linear theories is recovered in the unscreened regime. Implications for cosmology -- and in particular the equation of state in extended quintessence models -- and for laboratory experiments in low density medium are discussed. This analysis, although based on a simplifying description, sheds light on the effects of the small-scale distribution of matter as well as on the care required to define their equation of state and their cosmological signature.