Investigating shadow of a rotating charged black hole with a cosmological constant immersed in the perfect fluid dark matter

Abstract: In this paper, we mainly investigate the shadow of a rotating charged black hole with a cosmological constant immersed in perfect fluid dark matter. We first obtain the charged spherically symmetric black hole with a cosmological constant solution immersed in perfect fluid dark matter by using the gravitational decoupling method. Based on the mass function seed source of the spherically symmetric solution, we construct a rotating charged black hole with a cosmological constant immersed in perfect fluid dark matter, and study the effects of the perfect fluid dark matter parameter $\alpha$, cosmological constant $\Lambda$, electric charge $Q$, and rotating parameter $a$ on the event horizons of the rotating black hole. We then derive the geodesic equation of photon of the rotating charged black hole with a cosmological constant immersed in perfect fluid dark matter. In addition, we analyze the influences of the four black hole parameters on the effective potential functions of photon, the boundaries, and the deformations of the rotating black hole shadows. Particularly, we find that the effects of the $\alpha$ on the event horizons, the effective potential functions, the boundaries, and the deformations of the black hole shadows are more obvious than the other three parameters ($\Lambda, Q$, and $a$). We expect our results will be useful in the future to relate the theoretical models of perfect fluid dark matter and dark energy with observations of celestial bodies immersed in dark matter and dark energy.