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Hello the number of black holes is much bigger than the number of black holes in the past right?Let $b$ be the number of black holes in the universe.Obviously $\frac{db}{dt}>0$ but what about $\frac{d^{2}b}{dt^{2}}$?

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  • $\begingroup$ There could be as many as 100 million stellar black holes in the Milky Way, a new one every century. $\endgroup$ Commented Nov 12, 2023 at 4:13

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Black holes are produced from the core collapse of massive stars. As such stars are short-lived, the rate of production of black holes follows the rate of production of such massive stars. This rate is decreasing exponentially for 10 billion years, as the universe slowly runs out of hydrogen gas. (see this paper)

It's not completely obvious that the number of black holes will increase, as black hole mergers will cause the number of black holes to decrease. However this is currently going to be a small number in comparison to the the production in supernovae.

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  • $\begingroup$ The simplest model would be that b' = SFR(t)*k1 - b*k2, where SFR is the star formation rate, k1 is the fraction of stars turning into black holes, and k2 would be the decay rate due to mergers. One can complicate this with different mass ranges, merger rates separate from evaporation etc. but for a crude model this might be enough and produces "peak black hole" at some point (plausibly before 10^20 years). $\endgroup$ Commented Nov 13, 2023 at 15:48

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