A unified dynamical systems framework for cosmology in f(Q) gravity: generic features across the connection branches

Abstract We develop a unified dynamical systems framework for spatially flat FLRW cosmology in f ( Q ) gravity, covering all three connection branches using a single set of Hubble-normalised variables without fixing the function f ( Q ) a priori. This model-independent and connection-agnostic approach enables direct comparison across connection choices and uncovers structural features of the cosmological dynamics not visible in connection-specific formulations. While the existing works narrowly focus only on fixed point analysis, in our work we put special efforts to identify the invariant submanifolds, model-independent trajectories and physically viable regions of phase space across connections. For a broad class of viable f ( Q ) models, we establish the generic existence of de Sitter attractors and matter-dominated fixed points in the two branches of connection, offering a robust route to late-time acceleration without fine-tuning. We further identify an invariant submanifold that yields $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> CDM-like background evolution despite underlying dynamics distinct from General Relativity, providing a geometric origin for cosmic acceleration distinguishable only at the perturbation level. We also derive a first integral on this submanifold, allowing analytic reconstruction of the dynamical connection and uncovering hidden conservation laws. Another key feature we found is that while trivial connections exhibit strong parameter dependence, the nontrivial branches often feature parameter-independent behaviours. We also study the variation of the effective gravitational coupling, $$k_{\text {eff}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>k</mml:mi> <mml:mtext>eff</mml:mtext> </mml:msub> </mml:math> , across branches and show how this can be constrained using astrophysical observations, which bridges theoretical viability with observational consistency in a novel way. Applying our framework to the illustrative model $$f(Q)=\alpha Q + \beta (-Q)^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>α</mml:mi> <mml:mi>Q</mml:mi> <mml:mo>+</mml:mo> <mml:mi>β</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> , we find late-time acceleration and $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> 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 $$m_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> -hierarchy method. This framework enables closure for models previously inaccessible to standard approaches and offers a diagnostic tool for identifying structurally viable cosmologies within modified gravity theories.