I'd like to live in a Universe that has evolved subject to only two simple constraints: 1: That it be
big enough to accommodate an indefinite amount of interesting latently complicated stuff and
happenings and 2: That all this would fit and work together in a strictly logical way. No fairy
doings allowed!
I’d prefer this universe to extend to infinity straightforwardly with Euclidean geometry and, since
it may to begin with have been too compact for stuff to explore latent prospects, it should be
expanding. For stuff to fit and work together logically, cause and effect should be able to link
freely everywhere. I’d not allow barriers inside my universe that inhibit cause and effect! So an
infinite, expanding, Euclidean, causal universe would suit me just fine.
Let’s model a region of such an Ideal (for me) universe with simple physics (this is Physics forums, so it's allowed?). Choose this region to be a sphere, radius R, mass/energy M, average mass/energy density d, so that M = 4/3 pi R^3d in a Euclidean way.
Then, if geometry is to be Euclidean while the model sphere expands against the pull of its own
gravity, say at a fractional linear rate of H per second, General Relativity (our best theory of
gravity) proscribes a critical density of d = 3H^2/8 pi G, where G is Newton’s constant.
But sadly the region’s mass generates an unwanted (by me) spherical cause-and effect-inhibiting
barrier, called a Schwarzschild Horizon, at a radius of GM/c^2, where c is the (always locally
measured) speed of light. Happily, by restricting the ratio (Horizon radius/R) to unity, this
unwanted barrier can be banished to the sphere’s surface and then be disposed of entirely by
letting R run off to infinity, so that the model region covers all of an infinite (albeit spherical)
universe with the Schwarzschild Horizon banished to ‘just outside’, wherever that is. When we combine the
expressions for M, d and horizon radius for this Ideal expanding model, we find simply that
(RH)^2 = 2 c^2.
Let’s now compare this Ideal universe with the ‘real’ Universe we live in. Observations show that
light would take T = 13.7 billion years (about 4.3 x 10^17 sec) to reach us from the Universe’s
origin, so this light could have covered a distance of c T metres (always locally measured) since the
Big Bang. Suppose we use R = c T as a lower-bound estimate of R. Since the presently
measured fractional linear expansion rate, measured from redshift observations, is accepted to be
about H = 2.17 x 10^(-18) per second, a rough estimate for our Universe is (RH)^2 = 0.45 c^2,.
The fact that both 0.45 and 2 are of the same order of magnitude as unity may be coincidental. Or, as I
hope, it may lend some credibility to the constraints I proposed earlier, to help answer questions like "why are things as they are?
Anyway I’m happy to be in a Universe so quantitatively close to my Ideal!
