Loop Quantum Gravity Black Holes Surrounded by Quintessence

Quantum gravity corrections act on the near-horizon geometry of a black hole (BH), while dark energy acts on its asymptotic region, and how the two imprints combine is still poorly understood. We study a static, spherically symmetric BH of Loop Quantum Gravity (LQG) surrounded by a quintessence field. The metric function carries the LQG parameters [Formula: see text] and the quintessence parameters [Formula: see text], and for suitable values it has three horizons, a structure found neither in pure LQG BHs nor in classical quintessence BHs. We analyze null and timelike geodesics, photon spheres and the shadow seen by a static observer placed between the event and the cosmological horizons. Sixth-order WKB quasinormal modes (QNMs) of scalar and electromagnetic fields confirm stability over the ranges examined. The quantum term behaves as a short-range repulsive correction: it shrinks the photon sphere and the shadow, raises the QNM oscillation frequency and lowers the light deflection. Quintessence works the other way on the photon sphere, the ringing frequency and the bending. With the Gauss-Bonnet method we obtain a weak deflection angle valid for [Formula: see text], which we check against direct numerical integration, and in which quintessence dominates at large impact parameters and the LQG terms at small ones.