Formalizing a unified dynamical system framework for f(R) and f(Q) gravity
Abstract
Going beyond the traditional Hubble-normalized framework, we formalize a unified dynamical system formulation for f(R) and coincident gauge f(Q) gravity using three distinct sets of dimensionless variables: (1) kinematical cosmographic parameters, (2) the dark energy equation of state parameter w (with its N=ln a derivatives) and matter density abundance Ωm, and (3) theory space parameters {mi} characterizing the shape of the theory function. Our formulation is free of auxiliary variables, closing via three equivalent ways: (1) specifying a theory, (2) requiring a specific cosmological evolution, or (3) requiring a specific dark energy equation of state evolution. Each closure strategy is illustrated with explicit examples. The theory closure approach studies a case of Hu-Sawicki f(R) and f(Q)=−2Λ+Q+β√(−Q) models, where traditional formalisms fail. A heteroclinic trajectory from a GR-matter-dominated epoch to a late-time de-Sitter epoch is general for f(Q), but sensitive to initial conditions for f(R). The cosmographic and equation-of-state closure approaches compare f(R) and f(Q) gravities kinematically equivalent to ΛCDM (j(z)=1) and dynamically equivalent to ΛCDM (w(z)=−1). In both, GR acts as a cosmological past attractor for f(Q) gravity, but not for f(R).
Related
- Stability analysis of de Sitter solution in the Starobinsky-Grisaru-Zanon gravity using the dynamical system method
- Unimodular Gravity with Arbitrary Diffusion Function: A Dynamical System Reconstruction Approach
- Is DESI Seeing Dynamical Dark Energy, or a Cosmic Glitch in Gravity?
- A novel phenomenological parametrization of dynamical dark energy via matter-gravity coupling
- Unified Spectral, Dynamical, and Correlation Signatures of an Exceptional Point in Cavity Optomechanics
BibTeX
@misc{chakraborty2026formalizing,
title = {{Formalizing a unified dynamical system framework for $f(R)$ and $f(Q)$ gravity}},
author = {Saikat Chakraborty},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24733},
eprint = {2609.24733},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24733},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Formalizing a unified dynamical system framework for f(R) and f(Q) gravity
AU - Saikat Chakraborty
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.24733
UR - https://arxiv.org/abs/2609.24733
AB - Going beyond the traditional Hubble-normalized framework, we formalize a unified dynamical system formulation for f(R) and coincident gauge f(Q) gravity using three distinct sets of dimensionless variables: (1) kinematical cosmographic parameters, (2) the dark energy equation of state parameter w (with its N=ln a derivatives) and matter density abundance Ω_m, and (3) theory space parameters {m_i} characterizing the shape of the theory function. Our formulation is free of auxiliary variables, closing via three equivalent ways: (1) specifying a theory, (2) requiring a specific cosmological evolution, or (3) requiring a specific dark energy equation of state evolution. Each closure strategy is illustrated with explicit examples. The theory closure approach studies a case of Hu-Sawicki f(R) and f(Q)=−2Λ+Q+β√(−Q) models, where traditional formalisms fail. A heteroclinic trajectory from a GR-matter-dominated epoch to a late-time de-Sitter epoch is general for f(Q), but sensitive to initial conditions for f(R). The cosmographic and equation-of-state closure approaches compare f(R) and f(Q) gravities kinematically equivalent to ΛCDM (j(z)=1) and dynamically equivalent to ΛCDM (w(z)=−1). In both, GR acts as a cosmological past attractor for f(Q) gravity, but not for f(R).
ER -