Monday, December 13, 2004
Our simple little universe
Plane trips are great. You have time to think about all sorts of random things that you would never get to if you had a working internet connection. I wanted to write about cosmology and the argument from design, but I got sidetracked into thinking about how complicated our current universe is, at least those aspects that are relevant to human existence. The mass of the top quark, for example, doesn't affect your life very much, unless you are a high-energy physicist (and we can ignore them for the moment).
So let's imagine we are given the basic set-up of the standard models in particle physics and cosmology -- the gauge groups and representations of the particles, a cold dark matter candidate, the dimensionality and signature of spacetime, a flat homogeneous and isotropic universe with a scale-free spectrum of adiabatic fluctuations. These are the fundamental discrete facts that describe the framework of our universe. But within that framework, there are a number of parameters, which we could imagine taking on all sorts of values. How much information is required to specify those values?
We can easily list the continuous parameters that matter to human life: the masses of the electron and up and down quarks, the QCD scale, the Fermi constant, Newton's constant of gravitation, the fine-structure constant, the amplitude of density perturbations, and the densities of baryons, dark matter, and vacuum energy. So I count eleven parameters -- really only ten, since only dimensionless ratios of mass scales matter -- each of which needs to be specified to three significant digits or fewer. A number expressed to three significant digits in decimal notation requires about ten bits (210 = 1024), so ten such numbers requires about one hundred bits of information. And there you have it, our whole universe. Or at least, the statistical properties of our branch of the wavefunction of the universe. It wouldn't be enough information to predict, for example, how long it will take the Cubs to win the World Series.
Have I missed anything important? Some of these numbers (in particular, the quark masses) aren't even known to three significant figures. But other numbers that could in principle be derived from them (like the masses of the neutron and proton) are known quite well, and are pretty important to the way the universe looks.
There are a lot of other parameters, of course, both in particle physics and in cosmology: masses of neutrinos and the heavier fermions, CP-violating angles, amount of isocurvature perturbations, and so on. But all these numbers have values that are safely removed from our everyday lives; we could change them by quite a lot and you'd never know. For a look at what might happen if you really messed around with different parameters, see Robert Cahn's The eighteen arbitrary parameters of the standard model in your everyday life (postscript). You wouldn't want to live in a world where the muon was the lightest charged lepton.
Of course it would be nice to have something even more economical -- maybe even just one number! But still, it's interesting that it takes so little information to specify the workings of our universe. As a homework problem, look up the actual values and turn them into a hundred-digit binary number. It would make a nice T-shirt.
Update: I realized I was a little too optimistic about expressing each number with just ten bits. Even if the number of significant figures is only three, you still need to keep track of the overall size of the number; i.e., if you want to express 2.85x10-16, you need to encode the -16 as well as the 2.85. So a few more bits will be necessary, but not very many; perhaps 15 bits per parameter.