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 $Ω_m$, 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 $f(Q)=-2Λ+Q+β\sqrt{-Q}$ 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 $Λ$CDM ($j(z)=1$) and dynamically equivalent to $Λ$CDM ($w(z)=-1$). In both, GR acts as a cosmological past attractor for $f(Q)$ gravity, but not for $f(R)$.