Spatial curvature in coincident gauge $f(Q)$ cosmology

TL;DR

This paper explores spatial curvature in $f(Q)$ cosmology within the coincident gauge, revealing multiple solution branches and their equivalence to $f(T)$ models, while clarifying conceptual issues and invariance properties.

Contribution

It presents the first explicit construction of the coincident gauge for arbitrary spatial curvature in $f(Q)$ gravity, analyzing solution branches and their relation to $f(T)$ models.

Findings

All solution branches can be studied in the coincident gauge.

Flat and curved solutions are equivalent to $f(T)$ models.

Identifies and clarifies conceptual issues in the coincident gauge approach.

Abstract

In this work we study the Friedmann-Lema\^{i}tre-Robertson-Walker cosmologies with arbitrary spatial curvature for the symmetric teleparallel theories of gravity, giving the first presentation of their coincident gauge form. Our approach explicitly starts with the cosmological Killing vectors and constructs the coincident gauge coordinates adapted to these Killing vectors. We then obtain three distinct spatially flat branches and a single spatially curved branch. Contrary to some previous claims, we show that all branches can be studied in this gauge-fixed formalism, which offers certain conceptual advantages. We also identify common flaws that have appeared in the literature regarding the coincident gauge. Using this approach, we find that both the flat and spatially curved solutions in $f(Q)$ gravity can be seen as equivalent to the metric teleparallel $f(T)$ models, demonstrating a deeper connection between these theories. This is accomplished by studying the connection equation of motion, which can be interpreted as a consistency condition in the gauge-fixed approach. Finally, we discuss the role of diffeomorphism invariance and local Lorentz invariance in these geometric modifications of gravity.