A Unified Dynamical Systems Framework for Cosmology in $f(Q)$ Gravity: Generic Features Beyond the Coincident Gauge

We present a unified dynamical systems framework for spatially flat FLRW cosmology in $f(Q)$ gravity, covering all three connection branches via a single set of Hubble-normalised variables without fixing $f(Q)$ \textit{a priori}. This connection-agnostic, model-independent approach enables direct comparison across branches and reveals generic structural features that are not apparent in model or connection-specific analyses. Beyond fixed points, we identify invariant submanifolds, model-independent trajectories, and viable phase-space regions common to multiple branches. For a broad class of viable $f(Q)$ models, we find generic de Sitter attractors and matter-dominated points in non-coincident branches, ensuring late-time acceleration without fine-tuning. An invariant submanifold is shown to reproduce $Λ$CDM-like backgrounds despite dynamics distinct from GR, offering a geometric origin for cosmic acceleration detectable only at the perturbation level. On this submanifold, a first integral enables analytic reconstruction of the dynamical connection and uncovers hidden conservation laws. While trivial connections display strong parameter dependence, nontrivial branches often exhibit parameter-independent behaviour. We also analyse the variation of the effective gravitational coupling $κ_{\text{eff}}=\frac{1}{f_Q}$ across branches, providing observational constraints that bridge theory and data. Applying the framework to $f(Q)=αQ+β(-Q)^n$, we recover late-time acceleration and $Λ$CDM-like behaviour without vacuum energy. Finally, we propose a general route for extending dynamical systems analysis to broader classes of $f(Q)$ models using the $m_i$-hierarchy method, which enables closure of the autonomous system for models previously inaccessible to standard approaches.

💡 Research Summary

The paper develops a unified dynamical‑systems framework for spatially flat FLRW cosmology in the metric‑affine theory known as f(Q) gravity, where Q is the non‑metricity scalar. In symmetric teleparallel gravity the metric and an independent affine connection are both flat and torsion‑free, but three inequivalent connection branches (Γ₁, Γ₂, Γ₃) exist, each characterized by a time‑dependent function γ(t). The authors construct a single set of Hubble‑normalised dimensionless variables that works for all three branches without fixing the functional form of f(Q) in advance. By introducing the auxiliary “m‑hierarchy’’ variables (e.g. m₁ ≡ −Q f_{QQ}/f_Q, m₂ ≡ Q² f_{QQQ}/f_Q, …) they close the autonomous system for generic f(Q) models, a step that is usually obstructed in higher‑order theories.

For the trivial branch Γ₁, the connection does not affect the dynamics, and the phase space collapses to a one‑dimensional system, reproducing earlier results. In the non‑trivial branches Γ₂ and Γ₃ the phase space is four‑dimensional. The analysis goes far beyond the usual fixed‑point study: the authors identify invariant sub‑manifolds, model‑independent trajectories, and physically viable regions of the phase space that are common to several branches.

Key generic findings are:

1. Matter‑dominated saddle points (Ω_m≈1, w_eff≈0) exist in all branches, guaranteeing a standard early‑time matter era.
2. De Sitter attractors (H=const, w_eff=−1) appear in Γ₂ and Γ₃ independently of the specific f(Q) form, providing a robust route to late‑time acceleration without fine‑tuning.
3. A ΛCDM‑like invariant sub‑manifold where the background expansion mimics ΛCDM even though the underlying dynamics differ from General Relativity. On this sub‑manifold a first integral exists, allowing an analytic reconstruction of the dynamical connection γ(t) and revealing hidden conservation laws.
4. Effective gravitational coupling κ_eff = 1/f_Q becomes a dynamical quantity in the non‑trivial branches. Its evolution can be constrained by current astrophysical observations (SNe Ia, BAO, CMB, growth data), linking theoretical parameters to measurable quantities.
5. Parameter‑independent behaviour: while the trivial branch shows strong sensitivity to model parameters, the non‑trivial branches often display dynamics that are largely independent of the specific values of α, β, n in the illustrative model f(Q)=αQ+β(−Q)^n.

The authors illustrate the framework with the power‑law model f(Q)=αQ+β(−Q)^n (α,β≠0, n≠0,1). They show that for a broad range of n the system possesses the generic matter‑dominated saddle and the de Sitter attractor, and that the transition redshift can be tuned by the ratio β/α and the exponent n. This reproduces late‑time acceleration and ΛCDM‑like background evolution without invoking a cosmological constant.

Finally, the paper proposes a systematic extension of the method using the m_i‑hierarchy. By treating the m_i as independent variables and employing recursive relations among higher‑order derivatives of f(Q), one can close the autonomous system for models that were previously inaccessible (e.g., exponential, logarithmic, or mixed forms). This opens the door to a comprehensive dynamical‑systems classification of a wide class of f(Q) theories.

In summary, the work provides a powerful, connection‑agnostic tool to explore the cosmological viability of f(Q) gravity. It uncovers universal dynamical features, clarifies the role of the affine connection as a source of effective dark energy, and establishes concrete pathways to confront the theory with observations, thereby advancing the prospects of modified gravity as a solution to current cosmological tensions.