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

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 $\omega <-1\mathrm{/}3$; 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 ${\text{Tcal}}_{\mathrm{i}\mathrm{n}}$ at the innermost stable circular orbit (ISCO) via the relativistic vertical epicyclic frequency $\text{Obot}(r)$. We prove that the bolometric efficiency $\eta =[1-E(\text{rISCO})]\times 100\mathrm{\%}$ 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 ($j=0.4$, $l=0.5M_{\text{BH}}$, ${\rho }_0=2\times {10}^{-4}$), the combined geometry yields $\eta =8.71\mathrm{\%}$, substantially above the vacuum Kerr value $7.51\mathrm{\%}$ at the same spin. The viscosity correction $\delta \text{Fcal}\mathrm{/}{\text{Fcal}}_{\mathrm{N}\mathrm{T}\mathrm{P}}$ diverges at $r\to {\text{rISCO}}^{+}$, amplified by the geometric odification of the boundary pressure: the viscosity amplification ratio rises from $0.80$–$4.52\mathrm{\%}$ (vacuum Kerr) to $0.87$–$6.76\mathrm{\%}$ (Hayward+DE) at $\alpha =0.1$, providing a clean, monotonic observational discriminator accessible to $\mathit{\text{NICER}}$ and $\mathit{\text{NuSTAR}}$, independent of the bulk spectral normalisation.