Cuscuton gravity as a classically stable limiting curvature theory

TL;DR

This paper investigates cuscuton gravity, a minimal modification of general relativity that naturally incorporates limiting curvature, demonstrating its stability and non-singular cosmological solutions through detailed perturbation analysis.

Contribution

It provides a comprehensive analysis of cosmological perturbations in cuscuton gravity, confirming stability even at H=0 crossings and extending results to models within Horndeski theory.

Findings

Cuscuton models possess non-singular, stable cosmological solutions.

Perturbations remain stable at H=0 crossings with appropriate gauge choices.

Sound speed remains close to unity in the ultraviolet, and curvature perturbations stay constant during bounces.

Abstract

Finding effective theories of modified gravity that can resolve cosmological singularities and avoid other physical pathologies such as ghost and gradient instabilities has turned out to be a rather difficult task. The concept of limiting curvature, where one bounds a finite number of curvature-invariant functions thanks to constraint equations, is a promising avenue in that direction, but its implementation has only led to mixed results. Cuscuton gravity, which can be defined as a special subclass of $k$-essence theory for instance, is a minimal modification of gravity since it does not introduce any new degrees of freedom on a cosmological background. Importantly, it naturally incorporates the idea of limiting curvature. Accordingly, models of cuscuton gravity are shown to possess non-singular cosmological solutions and those appear stable at first sight. Yet, various subtleties arise in the perturbations such as apparent divergences, e.g., when the Hubble parameter crosses zero. We revisit the cosmological perturbations in various gauges and demonstrate that the stability results are robust even at those crossing points, although certain gauges are better suited to analyze the perturbations. In particular, the spatially-flat gauge is found to be ill defined when $H=0$. Otherwise, the sound speed is confirmed to be generally close to unity in the ultraviolet, and curvature perturbations are shown to remain essentially constant in the infrared throughout a bounce phase. Perturbations for a model of extended cuscuton (as a subclass of Horndeski theory) are also studied and similar conclusions are recovered.