Posted by Okay. Yesterday, I explained to you that the only thing that determines how the Universe expands is the amount of energy density in it. But many of you wanted more details. So, by popular demand — including one insistence that there is no equation that tells us how the Universe expands — here is the simplest explanation with math that I can come up with for the expanding Universe. If you hate or don’t care about math, do not read this. This article is not for you. But if you’re curious, read on.
Image credit: Wayne Hu.
First, let me tell you the underlying assumption behind all of it. You assume that the Universe is — on average — identical in all directions (what we call isotropic) and the same everywhere in space (homogeneous). This means that it doesn’t matter where you are or what direction you look in; the Universe is the same. On small scales, like the size of stars or galaxies, this isn’t true. But on the largest scales, like scales larger than galaxy clusters, our observations support this. Take a look at the observations:
as well as at the theory:
These match up pretty well! So, with this assumption that the Universe is isotropic and homogeneous, there are two ways to proceed. You can — as one option — choose General Relativity as your theory of gravity, write down the spacetime that describes an isotropic, homogeneous Universe (the Robertson-Walker metric), and solve for the equations that govern the evolution of the size and scale of the Universe as a function of time, known as the Friedmann Equations. The end result that you get out of that is this relationship between the Hubble expansion rate, H, and the energy density of the Universe, ρ:
Now, the curvature of our Universe, k, turns out to be zero, and the cosmological constant, Λ, is just another type of energy density. So if we like, we can rewrite this equation in a more simple form:
0 = KE + PE, 0 = ½ m v2 – G m M / r.
Or, I could do a little algebra to cancel out the little m, and make everything positive, in which case I get:
G M / r = ½ v2.
That looks okay, but it isn’t quite what I want. I don’t really know what this big mass, M, is, do I? Well, I know that mass is equal to density times volume. And — if I remember Gauss’ law for gravity — I will recall that the volume I’m interested in is the volume of a sphere. So I need to replace that mass, M, in my equation with this:
M = ρ V
where the volume, V, is:
V = 4/3 π r3,
and so my equation that I got from kinetic and potential energy becomes this:
G (ρ 4/3 π r3) / r = ½ v2.
Now, this looks pretty good! I’m going to cancel one of the rs from the numerator and denominator, and I’m going to multiply both sides of the equation by 2. So our equation looks like this now:
8/3 π r2 G ρ = v2.
Hmm. Looks like we’re getting really close! Let’s divide both sides by r2, and see what we get:
8/3 π G ρ = v2/r2.
Nice! The only thing is, instead of having H2, we’ve got v2/r2. But hang on. Remember what I said about what the Hubble expansion rate is?
That thing that the Hubble expansion rate equals, å/a, is just the velocity that an arbitrary point in the Universe moves away from us (å) divided by the distance to that point (a).
Well, that’s what we have, a velocity divided by a distance! So all we have to do is subsitute in H for v / r, and look at what we’ve got:
H2 = 8/3 π G ρ,
the equation that governs the expansion of the Universe. And all I need was a little bit of high school physics, geometry, and algebra. And now you know how to do it, too!
August 14, 2009
August 14, 2009
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