Interaction of a black hole with scalar field in cosmology background

Recent data from elliptical galaxies indicate that the growth in the masses of black holes exceeds what is expected from the accretion of surrounding matter, and this growth appears to be dependent on the expansion of the universe. This phenomenon can be explained by considering the accretion of dark energy, which is responsible for cosmological expansion, into these black holes. In this paper, we investigate the perturbative interaction of a black hole with a real scalar field $\phi$ (which can represent dark energy) in a cosmological background using an appropriate metric. We derive solutions for the field $\phi(t, r)$, the black hole mass $M(t)$, and the expansion rate $H(t)$, and discuss the behaviour of the scalar field $\phi$ in the vicinity of the black hole, with respect to exterior and interior observers. We obtain the energy density of $\phi$ inside of the black hole and find that energy density converges and tends to take a non-vanishing fixed values (stable) at late time of cosmological expansion. We find that on horizon surfaces, it is possible to make the field $\phi$ continuous and bounded (maybe differentiable) function.