Viscous Accretion Disks around Regular Black Holes Embedded in a Quintessence Dark Energy Field: Beyond the Novikov--Thorne Approximation

Abstract

We develop a comprehensive relativistic framework for geometrically thin, optically thick Shakura--Sunyaev $\alpha$-viscous accretion disks around rotating Hayward regular black holes embedded in a quintessence dark energy (DE) field. The static, spherically symmetric building blocks of our spacetime are each exact solutions of the Einstein field equations, sourced respectively by the Hayward non-linear electromagnetic field and a quintessence fluid with equation-of-state parameter $\omega<-1/3$; we combine and rotate them following standard practice for this class of models, and we are explicit throughout about the resulting metric's phenomenological status. Abandoning the stress-free inner boundary of the Novikov--Thorne--Page (NTP) model, we analytically incorporate a non-zero viscous torque $\Tcal_{\rm in}$ at the innermost stable circular orbit (ISCO) via the relativistic vertical epicyclic frequency $\Obot(r)$. We prove that the bolometric efficiency $\eta=[1-E(\rISCO)]\times100\,\%$ is strictly independent of the viscosity parameter $\alpha$ but sensitive to both the Hayward length scale $l$ and the DE density $\rho_0$, establishing a rigorous two-observable degeneracy-breaking strategy. At benchmark parameters ($j=0.4$, $l=0.5M_{\BH}$, $\rho_0=2\times10^{-4}$), the combined geometry yields $\eta=8.71\,\%$, substantially above the vacuum Kerr value $7.51\,\%$ at the same spin. The viscosity correction $\delta\Fcal/\Fcal_{\rm NTP}$ diverges at $r\to\rISCO^+$, amplified by the geometric odification of the boundary pressure: the viscosity amplification ratio rises from $0.80$--$4.52\,\%$ (vacuum Kerr) to $0.87$--$6.76\,\%$ (Hayward+DE) at $\alpha=0.1$, providing a clean, monotonic observational discriminator accessible to \textit{NICER} and \textit{NuSTAR}, independent of the bulk spectral normalisation.