Cosmological black holes: the spherical perfect fluid collapse with pressure in a FRW background

We have constructed a spherically symmetric structure model in a cosmological background filled with perfect fluid with non-vanishing pressure as an exact solution of Einstein equations using the Lema\^{i}tre solution. To study its local and quasi-local characteristics including the novel features of its central black hole, we have suggested an algorithm to integrate the equations numerically. The result shows intriguing effects of the pressure inside the structure. The evolution of the central black hole within the FRW universe, its decoupling from the expanding parts of the model, the structure of its space-like apparent horizon, the limiting case of the dynamical horizon tending to a slowly evolving horizon, and the decreasing mass in-fall to the black hole is also studied. The cosmological redshift of a light emitted from the cosmological structure to an observer in the FRW background is also calculated. This cosmological redshift includes local and cosmic part which are explicitly separated. We have also formulated a modified NFW density profile for a structure to match the exact solution conditions.