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

TL;DR

The paper develops a unified dynamical-systems framework for spatially flat FLRW cosmology in $f(Q)$ gravity that applies identically to all three affine-connection branches without fixing the model a priori. By using Hubble-normalised variables and the auxiliary functions $m(Q)$ and $r(Q)$, it closes the autonomous system and unveils generic, model-independent features such as de Sitter attractors in the nontrivial Γ2/Γ3 branches and ΛCDM-like backgrounds realized on an invariant submanifold $ ext{S}_{x_3}$, even when the underlying theory differs from GR. The work demonstrates that, for a broad class of $f(Q)$ models, late-time acceleration can arise without fine-tuning, and it provides a route to extend dynamical analyses to broader theories via the $m_i$-hierarchy. It also explores observational implications through the varying effective gravitational coupling $G_{ m eff}=1/f_Q$ and shows that background degeneracies between branches require perturbation-level studies to distinguish between geometries. Overall, the framework offers a robust, cross-branch methodology to assess the viability and phenomenology of $f(Q)$ cosmologies and sets the stage for future perturbative and data-driven investigations.

Abstract

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.

Figures (7)

• Figure 1: Upper panel: Physically viable regions in the $n$-$\Omega$ plane for $\Gamma_1$ with $f(Q) = \alpha Q + \beta (-Q)^n$, satisfying $\Omega \geq 0$, $Q < 0$, $f_Q > 0$ and $f-2Qf_Q\geq0$ for $\alpha = 1$ and $\beta>0$ in (a) and $\beta<0$ in (b). Lower panel: Evolution of the deceleration parameter $q$ (black curve), matter density parameter $\Omega$ (green curve), $\frac{\alpha (1 - n)}{\Omega n - 2n + 1}$ (red curve) and $\frac{(\Omega-1)}{\beta (\Omega n-2n+1)}$ (blue curve) for $n = 0.4$, $\alpha = 1$, $\beta = 1$.
• Figure 2: In this figure, we depict the left- and right-hand sides of Eq. (\ref{['finalvariation']})—which constrains the allowed relative variation of the effective coupling—using solid and dashed curves, respectively. We fix $\alpha = 1$, $\beta H_0^{2n - 2} = -1$ and take $\frac{D}{H_0} = 0.62, 0.01, 0.0015$ and $0.0005$ in panels (a), (b), (c) and (d) respectively; hence these panels scrutinize the bounds inferred from large-scale cosmic and local solar system dynamics, the former being looser. Figs. (e) and (f) are the enlarged versions of some regions in Figs. (c) and (d) respectively. In each plot, red curves correspond to $n = 0.4$ and black curves correspond to $n = 1.5$. Our analysis shows that this observational constraint is more easily satisfied towards the present epoch when $n < 1$, consistently with the preliminary assessment based on the structure of the Lagrangian density, which suggests that this branch of the model is more suitable for late-time cosmology Anagnostopoulos:2022gej.
• Figure 3: Projection of phase trajectories \ref{['eq:ds_2_example_redefined']} on the intersection of pairs of invariant submanifolds, shown in the following panels: (a), We intersect $\mathcal{S}_{\Omega}$ with $\mathcal{S}_{x_2}$ obtaining a point $\left(0,\frac{1}{3}\right)$ in red, which belongs to the $\mathcal{Q}_1$ in table \ref{['tab:C2_1']} as an attractor node, while a curve $\mathcal{Q}_3$ (in red) is a saddle; (b) We intersect $\mathcal{S}_{\Omega}$ with $\mathcal{S}_{x_3}$ and display also the line $x_2=3 x_4-1$ (in red) further confirming the attractive nature, as a node, of $\mathcal{Q}_1$; this numerical analysis in both panels corroborates the analytical discussion on the stability of the set of points $\mathcal{Q}_1$; (c) Finally, we intersect $\mathcal{S}_{\Omega}$ with $\mathcal{S}_{x_1+x_2}$ and choose $n=-2,$ and confirm that $\mathcal{S}$ (in red) is a saddle. The qualitative behaviors of the trajectories depicted here are independent of the parameter $n$.
• Figure 4: The projection of phase trajectories \ref{['eq:ds_2_example_redefined']} on the $(x_1,x_2,x_3,x_4)=\left(x_1,x_2,0,\frac{2}{3}\right)$-plane which shows that $(0,0)$ is saddle and hence a point $\mathcal{Q}_2$ is saddle. Here we have taken $n=0.5$. The diagonal red line corresponds to the invariant submanifold $\mathcal{S}_{x_1+x_2}$. One must be mindful that the above picture is just a projection of the complete higher-dimensional phase space. Note that although no physical trajectory actually crosses the invariant submanifold $\mathcal{S}_{x_1+x_2}$, it may seem to do so in the projected phase portrait due to dimensional reduction.
• Figure 5: (a) Evolution of the dynamical variables $x_1+ x_2, x_3$ and the quantity $\frac{x_2(1-n)}{nx_1+x_2}$, whose positive value corresponds to $f_Q>0$, (b) Evolution of energy density parameter $\Omega$ and the deceleration parameter $q$ along a typical trajectory passing through the vicinity of $\mathcal{Q}_2$. Here, $f(Q) = \alpha Q + \beta (-Q)^n$, $n = 0.5$. The initial conditions are taken to be close to the matter-dominated fixed point $\mathcal{Q}_2$ at $z=6$: $x_1(z=6)=0.001, x_2(z=6)=0.001, x_3(z=6)=-0.001, x_4(z=6)=0.5$. The plots are made within a range of $z$ high enough so that one can see the transition between the matter-dominated saddle $\mathcal{Q}_2$ and the de-Sitter attractor $\mathcal{Q}_1$. In figure (a), notice that $x_3=0$ throughout the evolution, implying that the entire transition takes place within $\mathcal{S}_{x_3}$. The positivity of the quantity $\frac{x_2(1-n)}{nx_1+x_2}$ implies preservation of the condition $f_Q>0$ throughout the evolution, while its constancy is an artefact of $\dot{Q}=0$ on $\mathcal{S}_{x_3}$. However, also note that $(x_1+x_2)$ transitions from $0$ to $1$, which means that even if $Q$, and thus $f_Q$, remains constant, the theory of gravity does not remain GR. This difference is because the Hubble-normalised contribution of the nonlinear term in the action, $\frac{(n-1)\beta(-Q)^n}{H^2}$, gains prominence along the course of the evolution.