Qualitative behaviour of higher-curvature gravity with boundary terms i.e the f(Q) gravity models by dynamical system analysis

The higher-curvature gravity with boundary terms i.e the $f(Q)$ theories, grounded on non-metricity as a fundamental geometric quantity, exhibit remarkable efficacy in portraying late-time universe phenomena. The aim is to delineate constraints on two prevalent models within this framework, namely the Log-square-root model and the Hyperbolic tangent-power model, by employing the framework of Big Bang Nucleosynthesis (BBN). The approach involves elucidating deviations induced by higher-curvature gravity with boundary terms in the freeze-out temperature ($T_{f}$) concerning its departure from the standard $\Lambda$CDM evolution. Subsequently, constraints on pertinent model parameters are established by imposing limitations on $\vert \frac{\delta T_{f}}{T_{f}}\vert$ derived from observational bounds. This investigation employs dynamical system analysis, scrutinizing both background and perturbed equations. The study systematically explores the phase space of the models, identifying equilibrium points, evaluating their stability, and comprehending the system's trajectory around each critical point. The principal findings of this analysis reveal the presence of a matter-dominated saddle point characterized by the appropriate matter perturbation growth rate. Subsequently, this phase transitions into a stable phase of a dark-energy-dominated, accelerating universe, marked by consistent matter perturbations. Overall, the study substantiates observational confrontations, affirming the potential of higher-curvature gravity with boundary terms as a promising alternative to the $\Lambda$CDM concordance model. The methodological approach underscores the significance of dynamical systems as an independent means to validate and comprehend the cosmological implications of these theories.