Generalized fractional deceleration as a tool to decode the universe’s expansion

In this work, we investigate a cosmological model within the context of [Formula: see text] gravity by considering a bulk viscous fluid as the cosmic source and adopting a Bianchi type [Formula: see text] anisotropic spacetime. To derive an exact solution of the field equations, we assume a specific time-dependent deceleration parameter expressed in terms of the Hubble parameter. This choice facilitates the derivation of analytical expressions for the scale factor and Hubble parameter as functions of cosmic time t and redshift z. Additionally, we conduct a cosmographic analysis by computing higher-order kinematic parameters — jerk, snap, and lerk — which are found to asymptotically approach unity, thereby indicating consistency with the standard [Formula: see text]CDM cosmological model in the late-time limit. To further constrain the model and solve the field equations, we impose a relation between the directional scale factors as [Formula: see text]. This condition plays a crucial role in reducing the complexity of the anisotropic field equations and enables the derivation of exact expressions for key physical quantities. As a result, we obtain the temporal and redshift evolution of the energy density [Formula: see text], effective pressure [Formula: see text], and the equation of state (EoS) parameter w. The dynamical behavior of these quantities suggests a transition from a decelerated to an accelerated phase of cosmic expansion, compatible with current observational evidence.