Yeah I was referencing the event horizon as the most meaningful measure of size.
And whether the density is fixed over time or not doesn't affect the question. Let's take the universe at its current average mass/energy density - whatever the "true" measure of that is.
To the best of our understanding, at large scales the density is uniform. So if we consider a suitably large spherical volume of space within our (presumably infinite) universe.. that volume will have an average mass/energy content greater than the threshold amount for a black hole of that apparent volume (again, using the external event horizon frame).
So that suggested to me that either we live in a finite universe, or we must be on the inside of an event horizon. It seems like an unavoidable conclusion.
>> Our universe is expanding. It's density is not fixed.
> None of that precludes uniform density at large scales.
According to observation, the universe is expanding. An argument that it's really static at a large scale would require contradicting observational evidence, but none exists. A theory that requires abandoning observational evidence bears a special burden, which this theory lacks.
The universe's expansion, and a black hole's increase in mass over time, are unrelated phenomena. We could have one without the other. In fact, because of Hawking radiation, in the far future we might see a larger universe accompanied by smaller black holes.
I think a point they are trying to make is that the border of a black hole is only to us outside observers, if you yourself fell into one you wouldn't notice anything specific when you crossed the boundary. The popular example of hawking radiation references a border and pairs of particles, however its actually only to help people understand the idea of what is going on
The center of a black hole is infinitely dense. That's why it even exists. The event horizon is not the black hole.
> and the fixed density of our observed universe
Our universe is expanding. It's density is not fixed.
You really want to be thinking about this in terms of entropy and not matter.