Kerr-anti-de Sitter/de Sitter black hole in perfect fluid dark matter background

We obtain the Kerr-anti-de-sitter (Kerr-AdS) and Kerr-de-sitter (Kerr-dS) black hole (BH) solutions to the Einstein field equation in the perfect fluid dark matter background using the Newman-Janis method and Mathematica package. We discuss in detail the black hole properties and obtain the following main results: (i) From the horizon equation g(rr) = 0, we derive the relation between the perfect fluid dark matter parameter alpha and the cosmological constant Lambda when the cosmological horizon r(Lambda) exists. For Lambda = 0, we find that alpha is in the range 0 < alpha < 2M for alpha > 0 and -7.18M < alpha < 0 for alpha < 0. For positive cosmological constant Lambda (Kerr-AdS BH), alpha(max) decreases if alpha > 0, and alpha(min) increases if alpha < 0. For negetive cosmological constant -Lambda (Kerr-dS BH), alpha(max) increases if alpha > 0 and alpha(min) decreases if alpha < 0; (ii) An ergosphere exists between the event horizon and the outer static limit surface. The size of the ergosphere evolves oppositely for alpha > 0 and alpha < 0, while decreasing with the increasing | alpha |. When there is sufficient dark matter around the black hole, the black hole spacetime changes remarkably; (iii) The singularity of these black holes is the same as that of rotational black holes. In addition, we study the geodesic motion using the Hamilton-Jacobi formalism and find that when alpha is in the above ranges for Lambda = 0, stable orbits exist. Furthermore, the rotational velocity of the black hole in the equatorial plane has different behaviour for different a and the black hole spin a. It is asymptotically flat and independent of alpha if alpha > 0 while is asymptotically flat only when alpha is close to zero if alpha < 0. We anticipate that Kerr-Ads/dS black holes could exist in the universe and our future work will focus on the observational effects of the perfect fluid dark matter on these black holes.