# 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 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 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 at 1:31
• – Mr Anderson Jul 1 at 11:29
• I've removed a number of comments that were attempting to answer the question and/or responses to them. Please keep in mind that comments should be used for suggesting improvements and requesting clarification on the question, not for answering. Even partial or summary answers should be posted as answers or not at all. – David Z Jul 26 at 1:28