Probing Bardeen-Kiselev black hole with cosmological constant caused by Einstein equations coupled with nonlinear electrodynamics using quasinormal modes and greybody bounds

In this work, we investigate a static and spherically symmetric Bardeen-Kiselev black hole with cosmological constant which is a solution of the Einstein-non-linear Maxwell field equations along with a quintessential field. We compute the quasinormal frequencies for Bardeen-Kiselev black hole(BH) with cosmological constant due to electromagnetic and gravitational perturbations. By varying the BH parameters, we discuss the behaviour of both real and imaginary parts of the BH quasinormal frequencies and compare frequencies with Reissner-Nordstr\"om-de Sitter BH surrounded by quintessence (RN-dSQ). Interestingly, it shows that the response of Bardeen-Kiselev BH with cosmological constant and RN-dSQ under electromagnetic perturbations are different when the charge parameter $q$, the state parameter $w$ and the normalization factor $c$ are varied, but for the gravitational perturbations, the response of Bardeen-Kiselev BH with cosmological constant and RN-dSQ are different only when the charge parameter $q$ is varied. Therefore, compared with the gravitational perturbations, the electromagnetic perturbations can be used to understand nonlinear and linear electromagnetic fields in curved spacetime separately. Another interesting observation is that due to the presence of quintessence, the electromagnetic perturbations around the Bardeen-Kiselev BH with cosmological constant damps faster and oscillates slowly, and for the gravitational perturbations, the quasinormal mode decays slowly and oscillates slowly. We also study the reflection and transmission coefficients along with absorption cross section for the Bardeen-Kiselev BH with cosmological constant, it shows that the transmission coefficients will increase due to the presence of quintessence.