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Formation of Gyrs old black holes in the center of galaxies within the Lema\^ıtre--Tolman model

In this article we present a model of formation of a galaxy with a black hole in the center. It is based on the Lema\^ıtre--Tolman solution and is a refinement of an earlier model. The most important improvement is the choice of the interior geometry of the black hole allowing for the formation of Gyrs old black holes. Other refinements are the use of an arbitrary Friedmann model as the background (unperturbed) initial state and the adaptation of the model to an arbitrary density profile of the galaxy. Our main interest was the M87 galaxy (NGC 4486), which hosts a supermassive black hole of mass $3.2\cdot 10^{9}M_{\odot}$. It is shown that for this particular galaxy, within the framework of our model and for the initial state being a perturbation of the $Λ$CDM model, the age of the black hole can be up to 12.7 Gyrs. The dependence of the model on the chosen parameters at the time of last scattering was also studied. The maximal age of the black hole as a function of the $Ω_m$ and $Ω_Λ$ parameters for the M87 galaxy can be 3.717 or 12.708 Gyr.