## To Build A Universe

So I kept thinking about what the distribution of elements might be in an infinitely old universe. It’s a tough problem to consider, if you want to do it exactly right, since you have to consider how stars turn lighter atoms in a blistering array of possibilities. Besides the nearly hundred different elements — which represents the count of how many protons are in the nucleus — each element has multiple isotopes — representing how many neutrons are in the nucleus — and I don’t know how many there are to consider but it’s certainly at least several hundred to deal with. There’s probably a major work in the astrophysics literature describing all the ways atoms and their isotopes can get changed over the course of a star’s lifetime, either actually existing or waiting for an indefatigable writer to make it her life’s work.

But I can make a toy model, because I want to do mathematics, and I can see what I might learn from that. This is basically a test vehicle: I want to see whether building a more accurate model is likely to be worthwhile.

For my toy model of the universe I will pretend there are only three kinds of atoms in the universe: hydrogen, iron, and uranium. These represent the lighter elements — which can fuse together to release energy — and Iron-56 — which can’t release energy by fusing into heavier or by fissioning into lighter elements — and the heavier elements — which can fission apart to release energy and lighter elements. I can describe the entire state of the universe with three numbers, saying what fraction of the universe is hydrogen, what fraction is iron, and what fraction is uranium. So these are pretty powerful toys.

Over time the stars in this universe will change some of their hydrogen into iron, and some of their iron into uranium. The uranium will change some of itself into hydrogen and into iron. How much? I’m going to make up some nice simple numbers and say that over the course of a billion years, one-quarter of all the hydrogen in the universe will be changed into iron; three-quarters of the hydrogen will remain hydrogen. Over that same time, let’s say two-fifths of all the iron in the universe will be changed to uranium, while the remaining three-fifths will remain iron. And the uranium? Well, that decays; let’s say that two-fifths of the uranium will become hydrogen, two-fifths will become iron, and the remaining one-fifth will stay uranium. If I had more elements in the universe I could make a more detailed, subtle model, and if I didn’t feel quite so lazy I might look up more researched figures for this, but, again: toy model.

I’m by the way assuming this change of elements is constant for all time and that it doesn’t depend on the current state of the universe. There are sound logical reasons behind this: to have the rate of nucleosynthesis vary in time would require me to do more work. As above: toy model.

So what happens? This depends on what we start with, sure. Let’s imagine the universe starts out made of nothing but hydrogen, so that the composition of the universe is 100% hydrogen, 0% iron, 0% uranium. After the first billion years, some of the hydrogen will be changed to iron, but there wasn’t any iron so there’s no uranium now. The universe’s composition would be 75% hydrogen, 25% iron, 0% uranium. After the next billion years three-quarters of the hydrogen becomes iron and two-fifths of the iron becomes uranium, so we’ll be at 56.25% hydrogen, 33.75% iron, 10% uranium. Another billion years passes, and once again three-quarters of the hydrogen becomes iron, two-fifths of the iron becomes uranium, and two-fifths of the uranium becomes hydrogen and another two-fifths becomes iron. This is a lot of arithmetic but the results are easy enough to find: 46.188% hydrogen, 38.313% iron, 15.5% uranium. After some more time we have 40.841% hydrogen, 40.734% iron, 18.425% uranium. It’s maybe a fair question whether the universe is going to run itself all the way down to have nothing but iron, but, the next couple billions of years show things settling down. Let me put all this in a neat little table.

Composition of the Toy Universe
Age
(billion years)
Hydrogen Iron Uranium
0 100% 0% 0%
1 75% 25% 0%
2 56.25% 33.75% 10%
3 46.188% 38.313% 15.5%
4 40.841% 40.734% 18.425%
5 38% 42.021% 19.979%
6 36.492% 42.704% 20.804%
7 35.691% 43.067% 21.242%
8 35.265% 43.260% 21.475%
9 35.039% 43.362% 21.599%
10 34.919% 43.417% 21.665%
11 34.855% 43.446% 21.700%
12 34.821% 43.461% 21.718%
13 34.803% 43.469% 21.728%
14 34.793% 43.473% 21.733%
15 34.788% 43.476% 21.736%
16 34.786% 43.477% 21.737%
17 34.784% 43.478% 21.738%
18 34.783% 43.478% 21.739%
19 34.783% 43.478% 21.739%
20 34.783% 43.478% 21.739%

We could carry on but there’s really no point: the numbers aren’t going to change again. Well, probably they’re changing a little bit, four or more places past the decimal point, but this universe has settled down to a point where just as much hydrogen is being lost to fusion as is being created by fission, and just as much uranium is created by fusion as is lost by fission, and just as much iron is being made as is being turned into uranium. There’s a balance in the universe.

At least, that’s the balance if we start out with a universe made of nothing but hydrogen. What if it started out with a different breakdown, for example, a universe that started as one-third hydrogen, one-third iron, and one-third uranium? In that case, as the universe ages, the distribution of elements goes like this:

Composition of the Toy Universe
Age
(billion years)
Hydrogen Iron Uranium
0 33.333% 33.333% 33.333%
1 38.333% 41.667% 20%
2 36.75% 42.583% 20.667%
3 35.829% 43.004% 21.167%
4 35.339% 43.226% 21.435%
5 35.078% 43.345% 21.578%
10 34.795% 43.473% 21.732%
15 34.783% 43.478% 21.739%

We’ve gotten to the same distribution, only a tiny bit faster. (It doesn’t quite get there after fourteen billion years.) I hope it won’t shock you if I say that we’d see the same thing if we started with a universe made of nothing but iron, or of nothing but uranium, or of other distributions. Some take longer to settle down than others, but, they all seem to converge on the same long-term fate for the universe.

Obviously there’s something special about this toy universe, with three kinds of atoms changing into one another at these rates, which causes it to end up at the same distribution of atoms.