Theoretical upper bounds for the mass of supermassive black holes at the centers of galaxies or galaxy clusters are not well-defined and can vary depending on the assumptions and models used. However, there are some constraints and estimates based on current scientific understanding.
The largest known supermassive black hole is located at the center of the galaxy Holmberg 15A, with an estimated mass of around 40 billion times that of our Sun. This provides an empirical upper bound for the mass of supermassive black holes, at least within the observable universe.
Theoretical studies have proposed upper limits based on the concept of the Eddington luminosity, which is the maximum luminosity that a black hole can achieve before the radiation pressure of the emitted light overcomes the gravitational attraction. By assuming that a black hole accretes mass at the Eddington-limited rate over its lifetime, an upper limit for the mass can be estimated. However, these limits are based on simplifying assumptions and do not take into account complex astrophysical processes.
Another approach to estimating upper bounds is based on the growth of supermassive black holes through accretion of surrounding matter. By considering the available mass in the vicinity of the black hole and the efficiency of accretion processes, theoretical models can provide estimates for the maximum achievable mass. These estimates often consider the maximum amount of matter that could have been accreted since the formation of the black hole.
It's important to note that these theoretical upper bounds are subject to considerable uncertainties and are still areas of active research. The study of supermassive black holes and their growth mechanisms is a complex field, and our understanding continues to evolve as new observations and theoretical advancements are made.