How big would a singularity have to be... [cont]

[WestAir]

How big would a supermassive black hole have to be to stop the Galaxy clusters in our visible Universe from escaping our visible Universe due to the metric expansion of space?

In other words, how much [new] mass would I need to shove into a singularity to stop the visible universe from growing? Is it impossible due to the fact that gravity only propagates at C?

[Z-Man]

That depends a lot at where you mark the end of the visible universe and which cosmological model you want to consider and how you want to make the mass appear. But the principle answer is easy:

Figure out how the current density of the universe needs to be changed right now to completely stop the expansion. How much mass would you need to add to the visible universe to achieve that? That's your answer.

The ballpark number to expect would be the total mass of the visible universe including dark matter and dark energy. Dark and regular matter apparently was dense enough to significantly slow down the expansion once, and you need to compensate for the extra dark energy that has appeared since them. So that would be about 10⁵³ - 10⁵⁶ kg. Err, and that mass has a Schwarzschild radius of 10¹⁰-10¹³ light years... where do you plan to put it? Of course, with that size, it no longer is an ordinary black hole, it interacts in interesting ways with the cosmological curvature and dark energy. The actual radius of the horizon would likely be much smaller.

And that's the other way to get a bearing on your answer: The outer reaches of the visible universe appear to be moving away almost with the speed of light. What gravity force is required to reverse that? Right, an event horizon almost right behind them. So you can expect to need a mass big enough that it would usually generate a black hole with the radius of the visible universe.

Is it impossible due to the fact that gravity only propagates at C?

Well, if you magically let so much mass appear, you inevitably break the known laws of GR everywhere in the universe at least at one time. So you may as well do it in a way that everyone is "informed" of the new mass any time you like. Even in the "past".