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

Erik Jensko

TL;DR

The paper analyzes general FLRW cosmologies with arbitrary spatial curvature in symmetric teleparallel $f(Q)$ gravity using a coincident-gauge (vanishing connection) formulation. It constructs four coincident-gauge coordinate branches (three flat, one curved) adapted to the cosmological Killing vectors and derives the gauge-fixed FLRW metrics and gauge-invariant non-metricity scalars $Q$. By treating the connection equation as an off-shell diffeomorphism-consistency constraint, it shows that flat branches 1 and 2 force $Q$ to be constant (reducing to GR with a cosmological constant), while flat branch 3 and the curved branch lead to dynamics equivalent to $f(T)$ gravity in certain regimes; notably, the curved, negatively curved case yields an exact $f(Q)$–$f(T)$ correspondence for background cosmology. The results illuminate the deep connections between the two teleparallel formalisms and clarify the role of diffeomorphism and local Lorentz invariances in these gauge-fixed modifications, providing a robust framework for exploring curved spacetimes in $f(Q)$ gravity and guiding future perturbative studies.

Abstract

In this work we study the Friedmann-Lemaître-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.

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

TL;DR

The paper analyzes general FLRW cosmologies with arbitrary spatial curvature in symmetric teleparallel gravity using a coincident-gauge (vanishing connection) formulation. It constructs four coincident-gauge coordinate branches (three flat, one curved) adapted to the cosmological Killing vectors and derives the gauge-fixed FLRW metrics and gauge-invariant non-metricity scalars . By treating the connection equation as an off-shell diffeomorphism-consistency constraint, it shows that flat branches 1 and 2 force to be constant (reducing to GR with a cosmological constant), while flat branch 3 and the curved branch lead to dynamics equivalent to gravity in certain regimes; notably, the curved, negatively curved case yields an exact – correspondence for background cosmology. The results illuminate the deep connections between the two teleparallel formalisms and clarify the role of diffeomorphism and local Lorentz invariances in these gauge-fixed modifications, providing a robust framework for exploring curved spacetimes in gravity and guiding future perturbative studies.

Abstract

In this work we study the Friedmann-Lemaître-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 gravity can be seen as equivalent to the metric teleparallel 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.
Paper Structure (20 sections, 140 equations, 1 figure)

This paper contains 20 sections, 140 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Teleparallel gravity
  3. Symmetric teleparallel gravity
  4. Gauge-fixed formalism
  5. Spatially curved FLRW cosmology
  6. Symmetries
  7. Gauge-fixed implementation
  8. Transformation to the coincident gauge
  9. Spatially flat $k=0$ solutions
  10. Spatially curved $k\neq0$ solutions
  11. FLRW metric in coincident gauge coordinates
  12. Field equations and solutions
  13. Constraint equations
  14. Zero spatial curvature $k=0$
  15. Non-zero spatial curvature $k \neq 0$
  16. ...and 5 more sections

Figures (1)

  • Figure 1: Representation of coordinate transformation (\ref{['A_trans']}) from cosmological coordinates $(t,r)$ to coincident gauge coordinates $(\tilde{t},x)$ for the choice $R(t)=t$. The dashed grey lines are lines of constant $t$, while the dashed blue lines are lines of constant $r$. The first panel \ref{['fig:A']} shows $k=0$, the second \ref{['fig:B']} shows $k=+1$, and the final panel \ref{['fig:C']} shows $k=-1$, where the coordinates are bounded by $\tilde{t} > |x|$.