The End of the Universe

John Baez

February 7, 2016

In short: the end of everything.

For a long time, the big question was whether there was enough matter in the universe to make it recollapse, or whether it would expand forever. But in the late 1990's, astronomical observations began to suggest that the expansion of the universe is actually speeding up!

Let's suppose this is true, and let's assume the most popular explanation for it: namely, that there is a nonzero cosmological constant. A cosmological constant with the right sign makes the energy density of the vacuum positive, but makes its pressure negative - and 3 times as big. This makes the universe tend to expand. Normal matter makes the universe tend to recollapse. If the effect of the cosmological constant ever beats out the effect of normal matter, the universe will keep expanding, making the density of normal matter less... so the cosmological constant will ultimately win hands down, and the universe will eventually expand at an almost exponential rate.

Let's suppose this happens. What will be the ultimate fate of the universe?

First let me set the stage. What happens in the short run, i.e. the first 1023 years or so?

First, galaxies will keep colliding. These collisions seem to destroy spiral galaxies — they fuse into bigger elliptical galaxies. We can already see this happening here and there, and our own Milky Way may collide with Andromeda in only 3 billion years or so. If this happens, a bunch of new stars will be born from the shock waves due to colliding interstellar gas, but eventually we will inhabit a large elliptical galaxy. Unfortunately, elliptical galaxies lack spiral arms, which seem to be a crucial part of the star formation process, so star formation may cease even before the raw materials run out.

Of course, even if this doesn't happen, the birth of new stars must eventually cease, since there's a limited amount of hydrogen, helium, and other stuff that can undergo fusion.

This means that all the stars will eventually burn out. The longest lived are the red dwarf stars, the smallest stars capable of supporting fusion today, with a mass about 0.08 times that of the Sun. These will run out of hydrogen about 1013 years from now, and slowly cool.

Stars become white dwarfs — and eventually black dwarfs when they cool — if they have mass less than 1.4 solar masses. In this case they can be held up by the degeneracy pressure caused by the Pauli exclusion principle, which works even at zero temperature. If they are heavier than this, they collapse: they become neutron stars if they are between 1.4 and 2 solar masses, and they become black holes if they are more massive.

In about 1014 years, all normal star formation processes will have ceased, and the universe will have a population of stars consisting of about 55% white dwarfs, 45% brown dwarfs and a small number of neutron stars and black holes. Star formation will continue at a very slow rate due to collisions between brown and/or white dwarfs.

The black holes will suck up some of the other stars they encounter. This is especially true for the big black holes at the galactic centers, which power radio galaxies if they swallow stars at a sufficiently rapid rate. But most of the stars, as well as interstellar gas and dust, will eventually be hurled into intergalactic space. This happens to a star whenever it accidentally reaches escape velocity through its random encounters with other stars. It's a slow process, but computer simulations show that about 90% of the mass of the galaxies will eventually "boil off" this way — while the rest becomes a big black hole.

(It may seem odd that first the galaxies form by gravitational attraction of matter and then fall apart again by "boiling off", but the point is, intergalactic matter is less dense now than it was when galaxies first formed, thanks to the expansion of the universe. When the galaxies first formed, there was lots of gas around. Now the galaxies are essentially isolated — intergalactic space is almost a vacuum. And you can show in the really long run, any isolated system consisting of sufficiently many point particles interacting gravitationally — even an apparently "gravitationally bound" system — will "boil off" as individual particles randomly happen to acquire enough kinetic energy to reach escape velocity. Computer calculations already suggest that the solar system will fall apart this way, barring other interventions. With the galaxies it's even more certain to happen, since there are more particles involved, so things are more chaotic.)

How long will all this take? Well, the white dwarfs will cool to black dwarfs with a temperature of at most 5 Kelvin in about 1017 years, and the galaxies will boil away by about 1019 years. Most planets will have already been knocked off their orbits by then, but any that are still orbiting stars will spiral in thanks to gravitational radiation in about 1020 years.

Then what? Well, in about 1023 years the dead stars will actually boil off from the galactic clusters, not just the galaxies, so the clusters will disintegrate. At this point the cosmic background radiation will have cooled to about 10-13 Kelvin, and most things will be at about that temperature unless proton decay or some other such process keeps them warmer.

Okay, so now we have a bunch of isolated black dwarfs, neutron stars, and black holes together with atoms and molecules of gas, dust particles, and of course planets and other crud, all very close to absolute zero.

As the universe expands these things eventually spread out to the point where each one is completely alone in the vastness of space.

So what happens next?

Well, everybody loves to talk about how all matter eventually turns to iron thanks to quantum tunnelling, since iron is the nucleus with the least binding energy, but unlike the processes I've described so far, this one actually takes quite a while. About 101500 years, to be precise. (Well, not too precise!) So it's quite likely that proton decay or something else will happen long before this gets a chance to occur.

For example, everything except the black holes will have a tendency to "sublimate" or "ionize", gradually losing atoms or even electrons and protons, despite the low temperature. Just to be specific, let's consider the ionization of hydrogen gas — although the argument is much more general. If you take a box of hydrogen and keep making the box bigger while keeping its temperature fixed, it will eventually ionize. This happens no matter how low the temperature is, as long as it's not exactly absolute zero — which is forbidden by the 3rd law of thermodynamics, anyway.

This may seem odd, but the reason is simple: in thermal equilibrium any sort of stuff minimizes its free energy, E - TS: the energy minus the temperature times the entropy. This means there is a competition between wanting to minimize its energy and wanting to maximize its entropy. Maximizing entropy becomes more important at higher temperatures; minimizing energy becomes more important at lower temperatures — but both effects matter as long as the temperature isn't zero or infinite.

Think about what this means for our box of hydrogen. On the one hand, ionized hydrogen has more energy than hydrogen atoms or molecules. This makes hydrogen want to stick together in atoms and molecules, especially at low temperatures. But on the other hand, ionized hydrogen has more entropy, since the electrons and protons are more free to roam. And this entropy difference gets bigger and bigger as we make the box bigger. So no matter how low the temperature is, as long as it's above zero, the hydrogen will eventually ionize as we keep expanding the box.

(In fact, this is related to the "boiling off" process that I mentioned already: we can use thermodynamics to see that the stars will boil off the galaxies as they approach thermal equilibrium, as long as the density of galaxies is low enough.)

However, there's a complication: in the expanding universe, the temperature is not constant — it decreases!

So the question is, which effect wins as the universe expands: the decreasing density (which makes matter want to ionize) or the decreasing temperature (which makes it want to stick together)?

In the short run this is a fairly complicated question, but in the long run, things may simplify: if the universe is expanding exponentially thanks to a nonzero cosmological constant, the density of matter obviously goes to zero. But the temperature does not go to zero. It approaches a particular nonzero value! So all forms of matter made from protons, neutrons and electrons will eventually ionize!

Why does the temperature approach a particular nonzero value, and what is this value? Well, in a universe whose expansion keeps accelerating, each pair of freely falling observers will eventually no longer be able to see each other, because they get redshifted out of sight. This effect is very much like the horizon of a black hole - it's called a "cosmological horizon". And, like the horizon of a black hole, a cosmological horizon emits thermal radiation at a specific temperature. This radiation is called Hawking radiation. Its temperature depends on the value of the cosmological constant. If we make a rough guess at the cosmological constant, the temperature we get is about 10-30 Kelvin.

This is very cold, but given a low enough density of matter, this temperature is enough to eventually ionize all forms of matter made of protons, neutrons and electrons! Even something big like a neutron star should slowly, slowly dissipate. (The crust of a neutron star is not made of neutronium: it's mainly made of iron.)

But what about the black holes?

Well, they probably evaporate due to Hawking radiation: a solar-mass black hole should do so in 1067 years, and a really big one, comparable to the mass of a galaxy, should take about 1099 years.

Actually, a black hole only shrinks by evaporation when it's in an enviroment cooler than the temperature of its Hawking radiation — otherwise, it grows by swallowing thermal radiation. The Hawking temperature of a solar-mass black hole is about 6 × 10-8 Kelvin, and in general, it's inversely proportional to the black hole's mass. The universe should cool down below 10-8 Kelvin very soon compared to the 1067 years it takes for a solar-mass black holes to evaporate. However, before that time, such a black hole would grow by absorbing background radiation — which makes its temperature decrease and help it grow more!

If a black hole ever grew to about 1022 solar masses, its Hawking temperature would go below 10-30 Kelvin, which would allow it to keep growing even when the universe has cooled to its minimum temperature. Of course, 1022 solar masses is huge — about the mass of the currently observable universe! But it would take a nontrivial calculation to show that reasonable-sized black holes have no chance of getting this big. I think it's true, but I haven't done the calculation.

For now, let's assume it's true: all black holes will eventually shrink away and disappear — none of them grow big enough to stick around when it gets really cold.

As black holes evaporate, they will emit photons and other particles in the process, so for a while there will be a bit of radiation like this running around. That livens things up a little bit — but this process will eventually cease.

What about the neutron stars? Well, if they don't ionize first, ultimately they should quantum-tunnel into becoming black holes, which then Hawking-radiate away.

Similarly, if the black dwarfs and planets and the like don't evaporate and their protons don't decay, they may quantum-tunnel into becoming solid iron — as I already mentioned, this takes about 101500 years. And then, if this iron doesn't evaporate and nothing else happens, these balls of iron will eventually quantum-tunnel into becoming black holes, which then Hawking-radiate away. This would take about 10100000000000000000000000000 years — that's 26 zeros.

This is a much longer time than any I've mentioned so far, so I wouldn't be surprised if some other effect we haven't thought about happens first. Indeed, this whole discussion should be taken with a grain of salt: future discoveries in physics could drastically change the end of this story. It's also possible that the intervention of intelligent life could change things — I've avoided discussing that here. Cosmology has been full of surprises lately, and there will probably be more to come.

But the overall picture seems to lean heavily towards a far future where everything consists of isolated stable particles: electrons, neutrinos, and protons (unless protons decay). If the scenario I'm describing is correct, the density of these particles will go to zero, and eventually each one will be cut off from all the rest by a cosmological horizon, making them unable to interact. Of course there will be photons as well, but these will eventually come into thermal equilibrium forming blackbody radiation at the temperature of the cosmological horizon — perhaps about 10-30 Kelvin or so.

This is why it's really a bad idea to keep putting things off for tomorrow.

However, Leonard Susskind has recently pointed out that in thermal equilibrium at any nonzero temperature, any system exhibits random fluctuations. The lower the temperature they smaller these are, but they are always there. These fluctuations randomly explore the space of all possible states of your system. So eventually, if you wait long enough, these random fluctuations will carry the system to whatever state you like. Well, that's a bit of an exaggeration: these fluctuations can't violate conservation laws. But conservation of energy doesn't count here, since at a nonzero temperature, a system is really in a state of all possible energies. So it's possible, for example, that a ice cube at the freezing point of water will melt or even boil due to random fluctuations. The reason we never see this happen is that such big fluctuations are incredibly rare.

Carrying this thought to a ridiculous extreme, what this means is that even if the universe consists of more or less empty space at a temperature of 10-30 kelvin, random fluctuations will occaisionally create atoms, molecules... and even solar systems and galaxies! The bigger the fluctuation, the more rarely it happens - but eternity is a long time. So eventually there will arise, sheerly by chance, a person just like you, with memories just like yours, reading a webpage just like this.

In short: maybe the universe has already ended!

However, the time it takes for really big fluctuations like this to occur is truly huge. It dwarfs all the time scales I've mentioned so far. So, it's probably not worth worrying about this issue too much: we don't know enough physics to make reliable predictions on such long time scales.

References

Many of the numbers in the above article were taken from table 10.2 in this book:

John D. Barrow and Frank J. Tipler, The Anthropic Cosmological Principle, Oxford U. Press, Oxford, 1988.

Tipler subsequently wrote a book of speculations about the fate of intelligent in the far future. He assumed there would be a big crunch, which no longer seems to be true, so I do not use his ideas here.

Freeman Dyson has discussed the fate of intelligent life in the far future assuming a perpetually expanding universe, but assuming the cosmological constant is zero. In this situation the temperature of the universe decreases ever closer to absolute zero, and Dyson figured out that in principle, intelligent life could last forever and think an infinite number of thoughts, although slower and slower:

Freeman J. Dyson, Time without end: physics and biology in an open universe, Rev. Mod. Phys. 51 (1979), 447–460.

For more, try:

The notion of a nonzero lower bound for the temperature of the universe are discussed in Scientific American articles appearing in the April 1999 and November 1999 issues, the latter written by Krauss and Starkman.

The rough estimates of 10-30 Kelvin for the limiting temperature of the universe and 1022 solar masses for the smallest black hole that would never evaporate are derived here:

Neal Dalal and Kim Griest, Black holes must die.

Leonard Susskind, The anthropic landscape of string theory.

I don't think any of these sources mention the "ionization" effect I discuss here. I would like to know the rate of this process, but I'm busy, so if you can figure it out it, go ahead and let me know the answer.

This article arose from a discussion among the contributors to sci.physics.research, and I thank all these people for their help in putting this together, especially Ted Bunn and Keith Ramsay.

The sun's not eternal. That's why there's the blues. — Allen Ginsberg

© 2012 John Baez

baez@math.removethis.ucr.andthis.edu