# Mathematical coincidence of the Schwarzschild radius of the Universe?

I read here that a black hole with a mass of the observable universe, $$M=8.8\times10^{52}kg$$, would have a Schwarzschild radius of $$r_s=13.7$$ billion lightyears.

I immediately noticed that at the speed of light, to travel a distance of $$r_s$$, it would take nearly exactly the age of the universe. Is that a coincidence, or is there a connection?

Normally, I'd brush this off as a coincidence and go on with my day. However, the top answer to this post details that the previous values are related to the Hubble constant $$H_0$$; the reciprocal of which is known as Hubble time, only varying in value from the age of the universe (due to the non-linear expansion of the universe), by a dimensionless factor of about $$0.96$$

Thus, considering that $$r_s$$ is related to $$H_0$$, which is related to the age of the universe, is that value of $$13.7$$ billion (light)years a coincidence, or is there a direct mathematical relationship?

As an undergraduate in math, I'm not extremely familiar with these concepts, and may have glossed over something obvious to the more atrophysically atuned. Hence, this is mere curiosity.

• So... I was searching this up, and I saw very many conflicting sources on what the mass of the observable universe is. Wikipedia "observable universe" page says that the 10^53 figure is for ordinary matter only, but then I've seen other stuff that say that that figure is for all matter. And then there's the fundamental problem that there are multiple competing definitions for mass-energy in GR because of various technicalities about energy being dependent on reference frame, gravitational potential energy being weird, etc. – qwyxivi Jun 30 '20 at 7:36
• I guess being within a few orders of magnitude of the Schwarzchild radius would still be an interesting coincidence, though. – qwyxivi Jun 30 '20 at 7:38
• @qwyxivi I forgot how complicated it must be to get a figure for the mass for the of the observable universe. I wonder how the zero-energy hypothesis muddles the matter more. – Graviton Jul 1 '20 at 1:31
• – Mr Anderson Jul 1 '20 at 11:29
• This is answered here. astronomy.stackexchange.com/questions/32443/…. Also, the relationship the OP mentions (as Pela explains in the link) is the universe is flat. – Mr Anderson Sep 1 '20 at 11:48