Cosmological Dynamics of Hyperbolic Evolution Models in $f(Q,L_m)$ Gravity

This paper highlights cosmologically viable sine and cosine hyperbolic evolution functions in the framework of $f(Q,\mathcal{L}_m)$ gravity. The models have been tested to check the behavior of the equation of state (EoS) parameter under the variation of parametric values. The EoS parameter experiences a quintessence phase, and is approaching to $-1$ at late time. The models are showing inclined behaviour with the $Λ$CDM model at the late time. The viability of both the models is retested using the widely accepted energy conditions in both cases. The violation of the strong energy condition admits the accelerating behaviour of the models. The same has been explained through the analysis of the profile of deceleration parameter, which concretely supports the evidence that the models explain early deceleration to late time acceleration phenomena.

💡 Research Summary

This paper investigates cosmological dynamics within the framework of f(Q, Lₘ) gravity, a modified theory that extends symmetric teleparallel gravity by allowing a non‑minimal coupling between the non‑metricity scalar Q and the matter Lagrangian Lₘ. The authors adopt a specific functional form
 f(Q, Lₘ)=−Q² + α Q^μ Lₘ + β,
where α, β are constants and μ is a free exponent governing the strength of the Q–Lₘ interaction. By varying the action with respect to the metric and the affine connection, they derive the modified Friedmann equations for a spatially flat FLRW universe, together with expressions for the effective energy density and pressure.

To close the system they prescribe hyperbolic scale factors, which lead to analytically tractable Hubble‑parameter evolutions. Model I uses
 a(t)=γ cosh(γ t),
giving H(t)=γ tanh(γ t). Transforming to redshift space yields H(z)=γ √