What a great question! It's no coincidence.
I'm going to treat this equality only to an approximate degree (say a factor of 3 or so) , because I'm not familiar with the precise calculation, or even sure what coordinate system you're using.
First, let me help other readers catch up with you. The Schwarzschild radius of a black hole is the radius at which the escape velocity equals the speed of light, c. You can calculate it approximately using standard Newtonian gravity, getting about GM/c2
, where G is the universal gravitational constant, and M is the mass of the black hole. (Specialists will note that I've dropped a factor of 2.)
Since the universe has been expanding at an approximately constant rate since shortly after the Big Bang, 13.7 bil years ago, the most distant 'observable' parts are 13.7 bil light-years away. Saying that the Schwarzschild radius calculated from the mass in that region is the same as the actual radius is equivalent to saying that the universe is very nearly expanding at 'escape velocity', i.e. that the amount of mass in it is about what's needed to cause it to re-contract.
We should be clear that there are two kinds of mass (or, equivalently, energy) around. One is ordinary mass, whose density drops as the universe expands, and whose gravity tends to pull the universe together. This currently accounts for about 30% of the total energy. The other is a background "dark energy", whose density stays constant (or nearly so) as the universe expands, and whose gravity must therefore, according to General Relativity, push the universe apart. About 70% of the energy is this latter kind.
So now we get to your question: why would we have anything like an equality between that 13.7 bil light years and GM/c2
for either the ordinary mass or the total mass-energy? There are two stages to the answer, neither completely clear-cut.
The first is that some process in the early universe caused it to be extremely near to 'flat', which makes your equation work, at least with the right numerical prefactor, when you include both
ordinary mass and the dark energy in "M". The standard view is that the process was rapid inflation, an exponential expansion that results when the background energy density is large. The reasons for the large early energy density are the subject of active speculation. An alternative view, tentatively supported by some subtle observations, is that the universe started off flat, with the Bang being a collision between two such flat 'branes'. At any rate, there seems to be some basic reason why the universe is very nearly flat.
That brings us to stage two: why is the density of mass in the universe some appreciable fraction of the total energy density? Why isn't it 0.000000001 of the total, with .999999999 in the background? 0r 3000 times the total, with -2999 in the background? Here's where life gets interesting. If the mass density were much higher, the universe would have long since re-collapsed, and nobody would have evolved to ask such good questions. If the mass density were much less, nothing would clump up enough to make the stars and then elements needed for life, so again, nobody would be asking.
It looks as if the background dark energy were 'tuned' to just the right density so that the remaining mass density needed to make the universe flat would be in the narrow range that could support life. Why would that be? Again, we don't know, but perhaps the best guess is that some huge number of different possible background densities, allowed by string theory, all occur. In the universes where the wrong ones occurred, nobody is asking why. An excellent book on this topic is "The Cosmic Landscape", by Lenny Susskind.
(published on 03/19/09)