We started this quest with a set of simple questions. What is the biggest number you can visualize? What is the biggest number you know? What is the biggest number you can attribute to a physical phenomenon or quantity? We decided to try and come up with a way to answer the third question with the answer to our second question—the googol, or 10^100.
The quest began with an attempt to count physical quantities, like individual particles, or total distances, such as the volume of the universe, but in each case we came up short with only 10^80. We need a new approach, but one that we can still argue has a physicality to it.
The best approach (in my opinion) will come from measurements of time. This may seem counterintuitive—the universe is only 13.8 billion years old (1.38 x 10^9 years)—but we can make predictions of how long things can last. In Astronomy, we’re not just interested in the past, or in how the Universe was or is interacting; we also care a great deal about how long certain processes/phenomena will last and how the Universe will ultimately end up. So let’s start small, with star formation: stars form from interstellar gas and dust, but eventually that pool of material will cease and star formation will end–how long will that take?
The answer…not that long. If you look at galaxies today, they come in a few types. Spiral galaxies, like our own Milky Way, are very blue and contain lots of nebulous material from which stars can, and are, forming. But even now there are large elliptical galaxies, which have or soon will exhaust all of their fuel for star formation. The thing is, when galaxies collide there can often be lots of associated star formation, so we really have to wait until all the galaxies that will collide, do, and then see what’s left. Current estimates say that star formation will be done in about roughly 10^14 (100 trillion) years.
Okay, that failed. But one can assume that even if the Universe can have an end, it should take a long time, so let’s follow this exercise of hypothetical fate to its conclusion and see where we end up. When star formation in the Universe ends, what’s left? All the gas will be completely used up, so that leaves all stars as either brown dwarfs, white dwarfs, neutron stars, or black holes. Other than that, you’ll have some planets and asteroids lying around too. All of this normal matter (non-black holes) is called degenerate matter, a state in which all of the electrons in atoms are scrunched down to their lowest bound states, and the shells are filled. Think of it as a bottle of marbles. Even if you try pushing another marble into the bottle, it just has nowhere to go—in degenerate matter, this is true for either the electrons, or in the case of a neutron star, for the neutrons. During this time, the Universe is said to have therefore entered the Degenerate Era.
So how long does the Degenerate Era last? Well, it’s hard to say, because it has to do with what’s known as Proton Decay. It’s quite an interesting story! You see, it’s known that a neutron, one of the fundamental particles in atoms, will decay given enough time into a proton and an electron (in fact, a free neutron, or one not bound in a nucleus, actually only lasts about 10 minutes before it decays). But what about protons? Can they too decay?
In physics research today, a lot of attention is given to theories in which they try to combine all fundamental forces in the universe except gravity (i.e. electromagnetism, strong nuclear force, and weak nuclear force) into one, known as Grand Unified Theories (GUTs). These make some interesting, testable predictions, one of which is the timescale of proton decay. There are currently experiments to try and measure proton decay (if it exists), and to date no decays have been detected. For more in depth info, you can go here or here. These experiments have at least led us to believe the decay half-life of a proton must be at least 10^33 years!
This looks promising, so let’s pull it back into our discussion of the Degenerate Era of the Universe. What happens in the Universe now? Well, the remaining objects will either get flung out into the vastness of empty space, or will fall into black holes. So we’ll have a bunch of black holes, and a bunch of lone, degenerate masses. If protons decay, then eventually the degenerate matter will just decay away into the basest of subatomic particles. However, if you’ve learned about decay mechanisms in a science/math class, you may remember that it’s exponential. This means that even if the half-life is really long, it only takes a small number of half-lives to be left with next to nothing left. So even though it might take 10^33 years for protons to decay half way, it’ll only take about 10^40 years (and that’s being a little generous) for all the protons (and neutrons) to decay away, ending the degenerate era.
Darn. Well, we can keep going. We’ve now entered what’s called the Black Hole Era, because that’s all the Universe has left—black holes and a soup of elementary particles. But if Stephen Hawking is correct about black holes, then even they will evaporate over time into electrons and positrons. So how does that take? If you calculate the radiation timescale for a supermassive black hole, you get roughly…wait for it…a googol years! That’s right—the evaporation time scale for a supermassive black hole is roughly 10^100 years!
There you have it! We’ve succeeded in our quest to ground the googol with a physical phenomenon: it is the time it takes (in years) for a supermassive black hole to completely evaporate. It’s also the rough time it takes for a universe with proton decay to reach the so-called Dark Era, in which the Universe is nothing but a soup of cold particles. Though we’ve already done a lot of speculation, it’s even worse when trying to determine the Universe’s fate in the Dark Era. It’s likely that it will just keep expanding forever and cooling until we reach something known as the Heat Death of the Universe. Thermodynamics dictates that in order for work to be done, and things to interact to make structure, entropy (disorder) in the Universe must always increase. But once we reach a maximum entropy, no work can ever be done, and the Universe is effectively dead. Despite it’s name, the Universe would be a cold, dead place…a somewhat bleak end for the beautiful place in which we exist.
If you’re interested in learning more about potential timelines of the Universe, you can check out this University of Oregon lecture material, as well as look up a fun timeline. Hopefully you enjoyed our fun quest to ground the googol, and this is just one way in which I found an answer. Perhaps you came to another way to tie the googol to a physical quantity or phenomenon, or maybe tie down the googolplex? Leave it in the comments!