A Googol of Particles?

A few weeks ago, some friends asked me whether or not there are a googol particles in the universe. I copped out at the time, pleading hunger, but it's an interesting question, so I found myself trying to answer it yesterday.

First off, a googol is defined as 10¹⁰⁰, or a 1 followed by 100 zeros. It's a big number, which is why the Google people wanted to name their company after it; unfortunately one of their first buyers wasn't very good at spelling... I'm also going to define "in the universe" to mean "in the known, observable universe." Obviously, if the actual universe is infinitely large, then there will be more than a googol particles in it. Second is the question of what we mean by "particle." I'll start by assuming that particle means "atom," specifically hydrogen or helium atoms, since there are relatively few other atoms around. I'm also going to take "known universe" to mean "observable universe," and for both of these terms to refer to the entire observable universe at an age of 13.7 billion years old. This isn't actually how the universe is observed; because it takes time for light to travel to us, the farthest away we can see in the universe corresponds to when the universe was very young. Taking this into account would needlessly complicate the question and the calculations.

To get a feel for numbers of particles in big, massive things: how many particles are there in the Sun? The mass of the Sun is roughly 2 x 10⁵⁷ times that of the mass of the proton, and since about one out of every four atoms in the Sun is a helium atom (which has the same mass as four protons), there are roughly 10⁵⁷ particles in the Sun.

So how many particles are there in our galaxy, the Milky Way? The mass of the Milky Way is difficult to define, and we have to be careful to only talk about the baryonic mass for now—that is, the mass that is in the atoms we are trying to count. Let's say it is 10¹¹ solar masses; the number of particles in the Milky Way is therefore 10¹¹ x 10⁵⁷ = 10⁶⁸.

From here, we can estimate how many particles are in the known universe if we have a good estimate for how many galaxies are in the known universe. This is a tricky number to estimate, because we can't actually see all of these other galaxies. We are also assuming that the Milky Way is a typical galaxy of typical mass. The internet gives a lot of different numbers for the "total number of galaxies in the universe." NASA's "ask an astronomer" page claims 125 billion, which is the same as 1.25 x 10¹¹; other sources give similar answers, so I'll use 10¹¹. Under these assumptions, we calculate that there are on the order of 10⁷⁹ "particles" (specifically, atoms) in the known universe.

Another way to do this calculation is to first figure out the number density of atoms for the universe, and then multiply by the total volume of the known universe to obtain a total number of atoms. I don't really feel like going into cosmology right now, but essentially, the density of the universe is inexorably linked with its geometry. We know that that universe is pretty damn flat, so we know what the density is fairly well. ("Flat" here means just what it sounds like; you can think of it as meaning that the three angles of a triangle have to add up to 180°.) This "critical" density is approximately 10^-29 g/cm³. This is really really tiny: for comparison, the density of water is 1 g/cm³, the density of air at sea level is roughly 10^-3 g/cm³, and the density of air in the best vacuum that can be made on earth is 10^-20 g/cm³. Cosmologists tell us that only 4% of the universe is made up of atoms, so the density of atoms is more like 4 x 10^-31 kg/m³. The mass of a typical atom is still about the mass of two protons, so this corresponds to about 1 x 10^-7 atoms per cubic centimeter. (If you don't like fractional atoms, you can think of this as about 1 atom in every ten cubic meters instead.) The volume of the observable universe is determined by its radius. Even though the universe is 13.7 billion years old, its radius is not 13.7 billion lightyears; it's actually more like 93 billion lightyears, or about 9 x 10²⁶ m. This gives us a volume of 3 x 10⁸¹ m³, and a total number of atoms in the observable universe of about 10⁷⁷. I put a lot of assumptions and simplifications into these calculations, so it isn't too surprising that they give slightly different results. When I was doing the calculations yesterday, I was getting more like 10⁸⁰ for both methods.

So we have determined that there are fewer than a googol atoms in the observable universe. This number won't increase by much if we expand the definition of "particle" to mean all electrons and quarks, the most fundamental particles of matter. But what if we include photons and neutrinos? The ground-up way to do this calculation is to start with a number density of photons (approximately 400 per cubic centimeter) and a number density of neutrinos (approximately 200 per cubic centimeter). Multiplying by the same volume as above, we now get a total of 2 x 10⁸⁹ particles. There are therefore fewer than a googol known particles in the known universe. An easy check this number—as well as an alternate way of doing the calculation—involves using the known baryon-to-photon ratio of roughly 10^-10. ("Baryon" being the word used in cosmology for "stuff that turns into atoms.") This is in fact approximately the ratio between the two total number of atoms and the total number of photons (and neutrinos) we calculated.

I said above that there this means that there are fewer than a googol known particles in the known universe. I have already mentioned that once we no longer restrict ourselves to the observable universe then there can clearly be more particles, but the distinction of known particles is important as well. I said earlier that about 4% of the universe by mass is made up of atomic-like particles; about 20% of the universe is made up of some other kind of mass that we don't really know what is, known as dark matter. A popular assumption is that the dark matter is some kind of as-yet unknown, un-detected particle—with some currently unkown mass. Using the calculation above, if there are going to be a googol dark matter particles in the observable universe, then the dark matter particle would need to be about 10^-20 as massive as a proton—that's tiny—so stupidly small that it is in fact ruled out by the fact that we observe the universe to have structure. Such a small mass for the dark matter particle would lead to what is known as the "Hot Dark Matter" scenario; essentially, if the dark matter particle has very little mass, then regular matter won't get cold enough to condense and form nice things like stars and galaxies.

In conclusion, a googol is in fact a very very large number.