Extend the unified dynamical-system framework to cosmological perturbations

Develop a complete perturbation-level cosmological analysis for specific f(R) and coincident-gauge f(Q) gravity models, complementing the background dynamical-system analysis with the matter-growth dynamics relevant to the observational quantity fσ8.

Background

The paper formulates a unified background dynamical-system framework for f(R) gravity and the coincident-gauge branch of f(Q) gravity using cosmographic variables, dark-energy equation-of-state variables, and theory-space parameters. Near the end of the paper, it derives common sub-horizon, quasi-static equations for the matter-density perturbation δ_m and the growth rate f, thereby connecting the background phase-space variables to fσ8.

The authors explicitly state that a complete cosmological study should include both background and perturbation analyses, but leave the perturbation analysis unresolved and for future work. The open task is therefore to implement and analyze the perturbation equations within the proposed framework for concrete models, rather than merely to write down the formal growth equations.

References

The full linear analysis of such equations, without assumptions on the QSA or SHA, is left for future work.

— Matter density perturbations in Quadratic Gravity  (2609.29046 - Bessa, 24 Sep 2026) in Section 4, Discussion and Conclusion

A complete cosmological study of any given model should, of course, involve both a background and a perturbation analysis; the latter is beyond the scope of the present article and is reserved for future work.

— Formalizing a unified dynamical system framework for $f(R)$ and $f(Q)$ gravity  (2609.24733 - Chakraborty, 21 Sep 2026) in Section Summary and Outlook