Emergence of Dark Phases in Scalar Particles within the Schwarzschild-Kiselev-Letelier Spacetime

This work focuses on the emergence of dark phases (dark energy imprints) in the radial wave function of scalar particles. We achieve this by presenting novel solutions of the Klein-Gordon equation in a spherically symmetric spacetime, encompassing a black hole, a quintessential fluid, and a cloud of strings. We determine the exact solution for the spacetime metric, analyze the admissible ranges for its physical parameters, and discuss the event horizon formation. Subsequently, we detail the solution of the Klein-Gordon equation and explore three distinct cases of dark phases, corresponding to the quintessence state parameter $\alpha_{Q}$ taking the values $0$, $1/2$, and $1$. Notably, the case where $\alpha_{Q} = 1$ holds particular significance due to current observational constraints on dark energy.