Cosmological dynamics with bulk viscosity in Brans-Dicke-Rastall gravity

In this study, we investigate the role of bulk viscosity in the context of Brans–Dicke–Rastall gravity, considering a spatially flat Friedmann–Lemaître-Robertson-Walker universe (FLRW). The effective equation of state (EoS) parameter ω eff and dark energy EoS parameter ω de are introduced to incorporate the effects of bulk viscosity and modified gravity, and exact solutions are obtained under the assumption that the scalar field evolves as ϕ = ϕ 0 [ a ( t ) ] n , where ϕ 0 and n are constants. To constrain the model, we employ observational data from Hubble parameter measurements and Type Ia supernovae (Pantheon sample), performing a joint analysis to obtain best-fit values for the model parameters. We analyze the evolution of several key cosmological diagnostics, including the deceleration parameter q ( z ), EoS parameters ω eff , ω de , statefinder parameters ( r, s ), and the Om diagnostic. By utilizing parameter values constrained from the combined CC+BAO+Pantheon datasets, we explore the dynamical behavior of the universe through graphical analysis. Our findings show that both the deceleration and EoS parameters indicate an accelerated expansion phase, confirming that the proposed bulk viscous model is consistent with the observed late-time acceleration of the universe.