Greybody factor and perturbation of a Schwarzschild black hole with string clouds and quintessence

In this paper, we investigated the Dirac and Klein-Gordan equations, as well as the greybody factor for a Schwarzschild black hole (SBH) immersed in quintessence and associated with a cloud of strings. Primarily, we study the Dirac equation using a null tetrad in the Newman-Penrose formalism. Next, we separate the Dirac equation into radial and angular sets. Using the radial equations, we study the profile of effective potential by transforming the radial equation of motion into standard Schr odinger wave equations form through tortoise coordinate. Similarly, we study the Klein-Gordan equation in this spacetime. The Miller-Good transformation method is employed to compute the greybody factor of bosons. To compute the Gerybody factor for fermions, we use the general method of semi-analytical bounds. We also investigate the effect of string clouds and quintessence parameters on Hawking radiation. According to the results, the greybody factor is strongly influenced by the shape of the potential, which is determined by the model parameters. This is consistent with the ideas of quantum mechanics; as the potential rises, it becomes harder for the wave to penetrate.