Now we can culminate our study of the laws of General Relativity and the Standard Model into theories and observations of the universe. We start by applying these laws to the early universe and compare the predictions with our current universe. This not only tells us if we are on the right track, but allows a glimpse on types of properties dark matter particles may have.
As usual, the course guide provides the raw technical details of the lecture, in this case the conservation rules of particle decay. What it lacks is the broader summaries from the actual lecture that tie everything together, though sometimes all too briefly. This is a fine example for one good use of these lecture reviews; the chance to document or discuss any missing or unclear material from the lecture or course guide. Some course guides are verbatim, so nothing is really left out. But I find this specific course guide to be woefully inadequate for really understanding the complex concepts of the course.
As the high temperature of the early universe cools over time, it will eventually equal the energy of the particles themselves and cease to create them. Those with more kinetic energy are hot and light, while those with more rest mass energy are cold and heavy. The lighter particles take less energy to create, so hot particles are more abundant. The massive particles take more energy to create, so cold particles are less abundant.
As the early universe expands, the particle/anti-particle annihilation decreases, allowing a freeze out of certain particles to occur. Neutrino/anti-neutrino pairs need to be very very close in order to annihilate each other. Over time they will be too far apart to interact. More of the weakly interacting particles survive and less of the strongly interacting particles survive.
The energy of the particles at the time of their freeze out creates two possibilities. If a particle naturally loses enough energy as the universe cools before freeze out, it will be a cold relic particle. If the freeze out occurs before it loses energy, it will be a hot relic particle.
Since we see dark matter clumping in galaxies and clusters, it must be a heavy, slow moving, cold relic particle. It also must interact weakly to have enough particles survive. Thus the acronym WIMP for weakly interacting massive particle. The next lecture discusses more specifics on WIMPs and alternate scenarios.
So by now, we know the laws of physics. When we say that, we mean basically all the fundamental laws from which all the rest of physics is derived We know General Relativity, Einstein's theory of curved spacetime, which manifests itself as gravity, and we know the standard model of particle physics. We didn't dig down into the quantum mechanical underpinning of the standard model, yet the standard model plus General Relativity is enough to cover every experiment we've ever done, and get the right answer.
Nobody believes these are the final answers to the laws of physics, yet as far as all the experiments we can do here on earth are concerned, they're good enough to match the data. The best indication we have that there's physics beyond the standard model and General Relativity, comes from the cosmos, from dark matter and dark energy, which we're trying to understand in these lectures.
So today, we'll take the laws of physics and put them to work in the universe. We're going to take the universe we see, the large universe that is smoothly distributed with galaxies, dark matter, and dark energy, and wind the movie back to the very early parts of the universe, plug in the laws of physics, and make predictions for what we're supposed to see now at the end of the entire process. Using the laws of physics and what we know about the early universe, what should we be seeing today?
In other words, we're going to be doing paleontology, essentially. We'll be looking for fossils, yet not the 100 million year old fossils in the ground, but the 14 billion year old fossils in the sky, these leftover particles from the earliest moments of the universe's existence. Partly this is just because we want to make sure we're on the right track. We think we have a picture that fits the data, and we want to use that picture to make predictions and then go test them.
Yet also partly we want to learn about what can come next. We think we have excellent evidence that there is early dark matter in the universe, for example. So the dark matter particles are going to be relics from the very early universe, and we need to understand how those particles are created, how the laws of physics can be used to predict how many kinds of particles of each kind there are. So we need to understand the different features of particles in different circumstances. We need to understand what particles do when they're by themselves, and what they do in a box with a whole bunch of other particles. Ultimately that box will be the entire universe.
So what we mean when we say "what particles do," is to either decay or be stable. It can sit there forever, not moving, or can turn into other particles. It can also bump into other particles, which would be relevant for the early universe when the densities are very high. So lets just think about particles decaying versus being stable.
Why do some particles last forever, and others go away? Well there are rules that govern when one particle can decay into some set of other particles. For our purposes today, there are basically two rules. One is that heavier particles decay into lighter particles. The second rule is that there are conserved quantities. There are numbers that don't change in any process of particle physics, including the decay of one heavy particle into a set of lighter particles. So lets put both of those rules to work in the particle physics we already know about.
The first rule says that heavier particles decay into lighter ones. This is basically a consequence of the conservation of energy. We know that mass is one form that energy can take. To a very good understanding, the energy of a particle is the energy in its mass, even if just sitting there, plus the kinetic energy of its motion through space. Those two forms of energy together tell you the total energy of the particle.
So if we have a heavy particle and ask what it can decay into, it can certainly not decay into an even heavier particle. There's no way for the energy of the final state to match that of the initial state. A light particle decaying into a heavier one, would violate the conservation of energy. So when a heavier particle decays into lighter particles, it can match conservation of energy, even if the masses do not precisely align. Since there's only a finite number of particles, it would be very surprising in fact that the masses of the first particle were exactly the sum of the masses of the second particles. The extra energy is taken up into kinetic energy of the outgoing particles flying away.
A classic example of an allowed decay of a heavy particle into lighter ones, is the decay of the neutron into a proton, electron, and antineutrino. We see that happen in nature, since the lifetime of the neutron is something like 10 minutes. So it must be the case that the mass of the neutron, the energy it has from E=mc², is greater than the sums of the masses of the proton, electron, and neutrino.
The neutrino is very light, so we don't have to worry, since it almost plays no role in this. Yet when we look at the numbers, the neutron and proton are both much heavier than the electron, with the neutron being a little heavier than the proton. In units of masses of the electron, the neutron is 1879 times as heavy as the electron. The proton is 1877 times heavier than the electron. So the neutron, 1879 electron masses, decays into the proton with 1877 electron masses, plus the electron, which is 1 electron mass, and get a sum which is 1878. This is still less than the 1879 that is the neutron, and that is why it's allowed to happen.
The fact that these numbers are so close, that the mass of the proton plus the electron plus neutrino is almost equal to the mass of the neutron, is one of the reasons why the neutron is so long lived. There are not a lot of possible arrangements for the outgoing particles.
Now consider the possibility of the proton decaying. What could it decay into? When you think about it, you might want the proton to decay into a neutron, a positron, and a neutrino. That would conserve electric charge, since the neutron and neutrino are neutral, and the positron has a positive charge just like the proton. Yet this event never happens since it doesn't match energy conservation. The proton is lighter than the neutron and therefor will never decay.
It's interesting to speculate that the universe would be a very different place if the masses of the elementary particles were a little bit different. What if the mass of the proton were a little bit larger than the mass of the neutron? Then it would be the other way around, and the neutron would be stable, with nothing for it to decay into. Yet the proton could decay into a neutron, spitting off a positron. What do all those positrons do, but go around and annihilate with all the electrons. In other words, we'd be left in a world with nothing but neutrons and neutrinos if the proton were heavier than the neutron. That would make for a very different world than the one in which we now live. There would be no atoms, no chemistry, no molecules, no anything like that. So we're very lucky that the mass of the proton is a tiny bit less than that of the neutron.
The second rule relevant to what can happen in particle physics, is that conserved quantities don't change. That's what the word conserved means. The most obvious example of a conserved quantity is of course, electric charge. It's sort of intuitively obvious to us that the electric charge shouldn't change in particle physics reactions. So for example, the electron for historical reasons has a charge of -1, the proton has a charge of +1, since it's made of three quarks , two up quarks with +2/3 each, and one down quark with -1/3 charge. The neutron is electrically neutral, just as the neutrino is, and comes from two down quarks at -1/3 each, and one up quark at +2/3 charge.
So in any process that can happen, the electrical charge of all the particles at the beginning, must equal the sum of the electrical charges of all the particles at the end. So for example, a neutron cannot decay into a proton, electron, and a positron. That would be allowed by energy conservation, but you would have a total charge of +1 at the end, yet a total charge of 0 at the start. So that is not allowed, and no one would have even guessed it to be, since we sort of understand immediately that electric charge is not created or destroyed.
Yet this is a good paradigm, a good example of something that happens and something that doesn't go away, and we have more subtle examples before us that play a big role in particle physics. The two other example we'll mention of conserved quantities of numbers that don't change in particle physics interactions, are quark number and lepton number. Basically the quark number is the total number of quarks. The lepton number is the total number of leptons.
The only subtlety there is that when we say the total number, anti-particles count as negative. So the difference between a particle and an anti-particle, is that all of the conserved quantities for the particle, are the opposite for the value of those conserved quantities for the anti-particle. So you know the electron has charge -1, and the positron has charge +1, since they are opposites.
Similarly, the lepton number of the electron is +1, so that of the positron is -1. The total amount of conserved charge for a particle and anti-particle will always add to zero and always be opposites. Quarks have quark number of +1, and lepton number of 0. There's no lepton number associated with quarks, just as there is no quark number associated with leptons. So then you add them up, inside a proton or neutron, and get a total quark number of three. There are three quarks inside a proton, therefor the quark number is three. Likewise for a neutron.
Remember that we defined different ways that quarks can get together to form heavier particles. We called a baryon any collection of three quarks. All the three different colors combined to give you one white baryon. So sometimes people talk about baryon number, which is exactly equal to quark number divided by three. So the baryon number of the proton is 1, since it has a quark number of 3.
Now consider the pion for example, which is a meson that's made of one quark and one anti-quark. So if you think about it, the quark number for a pion is 0, and likewise the baryon number also 0. So even though it's made of quarks, the total quark number is still 0. You can make as many pions as you want, you're not changing the quark number.
So the way to think about this is almost as if there's some stuff, some substance that changes its form, rearranges how it appears, yet is never created or destroyed. It's some nugget of quarkiness that can change from being an up quark to a down quark, or vice versa, as neutrons and protons intercommute, yet it never disappears. It never goes from being a quark to an electron, or something like that. So all the reactions we can possibly have in the universe will keep the quark number the same, as well as the lepton number.
Here is an example. You might think, if you were only thinking about electric charge, that the proton could decay into a positron, an anti-electron, and a neutrino. That would keep the electric charge constant, with +1 for the proton, a +1 for the positron, and 0 for the neutral neutrino. The reason why protons don't decay in the real world, is because quark number is conserved. So again this is a happy feature of the world in which we live, since if quark number were not conserved, the protons would all dissolve in to positrons which would annihilate. We'd be left in the universe with nothing but neutrinos and positrons, again not a very exciting place.
Likewise, lepton number tells you that certain things can't happen. Imagine a neutron trying to decay into a proton plus an electron. This is what people used to think that neutrons actually did. They saw the neutron sit there, and that the photon and electron would come out. The total electric charge is 0, before and after. The total baryon number is 1, before and after. The one neutron with three quarks turns into one proton with three quarks, plus an electron.
However we now know that this can't happen due to violation of lepton number conservation. There were no leptons before, and then there's an electron lying there afterward. That's why when a neutron decays, there must be an anti-neutrino involved. That anti-neutrino carries lepton number -1, so that the total lepton number at the end of the day, with the proton, electron, and anti-neutrino, adds to 0. So the neutron is allowed to decay, if there's that anti-neutrino there. It wouldn't have been able to decay if there weren't.
Now when we talk about historically, how do you invent these conservation laws? How do you figure out what is conserved and what is not? You don't do it by saying, "Aha, I think some number should be conserved, let me go look at the interactions!"
Usually you do particle physics, you measure what interactions actually do happen, which ones happen quickly or slowly, then try to figure out why certain things don't happen. That's how people come across things like the conservation of quark number or baryon number. People knew that baryon number was conserved, long before they knew what a quark was. They could see that neutrons didn't decay into neutrinos or something like that, so there must be some conserved quantity there. So that's how the game is usually played out.
Having said that, we should also admit that some rules are more solid than some other rules. Electric charge really is conserved. It is true that quark number and lepton number have never been seen to be violated in any experiments we've ever done, yet we wouldn't be all that surprised if someday we found that they were violated. In fact there are theories in which they are violated, it's just very slow that it happens so we haven't seen it yet.
Yet electric charge is something we really don't think is ever violated. If someone claimed to do an experiment and found electric charge created or destroyed, basically no one would believe them, unless they had some cache for some other reasons. It's something that no one expects on any theoretical basis. You can notice for example, that when you look at the whole universe and add up its electric charge, you get a number that is 0 as far as we can tell. There are an equal number of electrons and protons in the universe, so it's electrically neutral.
That is not true for quark number, and may not even be true for lepton number. The truth is we don't know, since we don't know how many neutrinos are out there, neutrinos versus anti-neutrinos contribute to the lepton number, and we can't count them. Yet we can count the quarks, which show up in protons and neutrons. There are many more quarks in the universe than there are anti-quarks. It could be that this is just how the universe started, so quark number is truly conserved and there has been this imbalance all along. Yet honestly nobody believes that. All believe that in the very early universe the quark number was 0, and there was some dynamical process that really did violate quark number, creating that imbalance.
That whole game that we try to play, creating quark number where there wasn't any before, is called baryogenesis, creating baryon number out of zero baryon number. We don't understand baryogenesis very well, so it's an open research project we'd like to understand better. Part of the ways we try to do this, is actually doing experiments here on earth that would witness baryon number being violated. There are proton decay experiments which conceptually are the simplest things in the world.
You get a big vat of stuff that is not radioactive, stuff that doesn't emit any form of high energy particles. You put it deep underground where no one will bother it, where no cosmic rays can come in and bump into it, and you wait for something to happen. You're waiting for something like a proton to turn into a positron plus a neutrino. That keeps electric charge and lepton number conserved, yet violates quark number. We've never seen it happen, but we're looking. As soon as we do see it happen, that will be a great advance in our understanding of the laws of physics.
So those rules are telling us what happens to a particle sitting all by itself, what particles can decay into others, and which ones are stable. Many of the particles we've talked about so far are stable. The neutrino, the electron, and proton, all are very stable. Why, if you think of it, can't we give any set of particles that those guys can decay into, while both being less massive and keeping all the conserved quantities conserved? The proton is the lightest particle that carries baryon number, that carries quark number, therefor it can't go into anything else! The neutrino is the lightest particle that carries lepton number, and the electron is the lightest particle that carries electric charge. So that's telling us why these particles can't decay. They have nowhere to go.
Dark matter is another particle which seems to be stable. One nice way to make new theories of dark matter, is to invent new conserved quantities that explain why the dark matter particle has lasted for the 14 billion years that the universe has been around.
Now lets take some of that understanding and put it together in the early universe where you don't just have one particle sitting there and decaying, but you have a bunch of particles. So the early universe is compact, things are squeezed together, the density is very high, and this means two things. One thing is that interactions happen very frequently, so particles are right next to each other and keep bumping into each other.
The second thing is that since the temperature is high, the energy per particle is very high. That's basically the definition of temperature, the average energy per particle. So if the energy of a particle is its rest energy, its mass, plus its kinetic energy, then when the energy is very high, the particles are moving all very quickly. This means that new things can happen.
Consider an interaction where a proton bumps into an electron. Now that interaction already happens all the time in a hydrogen atom. That's what a hydrogen atom is, an electron spinning around a single proton. You can ask what happens and the answer is nothing. The hydrogen atom can sit there forever.
Yet if we go to the early universe where the temperature was very high, instead of the electron just sitting there on top of the proton, we have an electron and proton running into each other with a large amount of energy. Then they can collide and turn into a neutron plus an electron neutrino. The total charge of that interaction goes from 0 to 0, quark number goes from 3 to 3, lepton number goes from 1 to 1, so everything is conserved. The only thing you're trying to do is add energy to the system so you can turn lighter particles, a proton and an electron, into a heavier combination. You can do that and conserve energy, if you're at high temperature.
So that's the kind of thing that becomes possible in the early universe when all the particles are zooming around very rapidly and bumping into each other. So this gives us a way of asking what can happen when you go from the very early universe when things are hot, dense, and bumping into each other, to the very late universe that we see today where things are very dilute, not bumping into each other all the time, and things are very cold. The universe right now, technically speaking, is not in what we call thermal equilibrium. We are not at the same temperature as the sun or the CMB.
Yet roughly speaking, when we talk of the temperature of the universe, the thing we refer to is the temperature of that CMB. That is the relic, leftover stuff, from the plasma that used to be in thermal equilibrium. In the early universe, those photons that we now see as the CMB, were rapidly bumping into everything else, and their evolution in temperature is what set the scale for how hot the universe is.
The way it works is very simple. As the scale factor goes down in the early universe, as things squeeze together, the temperature goes up. It goes exactly inversely, so when the temperature was twice as high, the scale factor was only half its current size. In particular, when the scale factor was 1/1000th its current size, the temperature was 1000 times as high, and that's when the CMB itself was formed.
If we keep going back, until the universe was very hot, then we can create new particles that aren't laying around today. Remember that the temperature is the rest energy plus the kinetic energy, the average total energy of the particles. So by increasing that temperature, by increasing the total energy, we have enough energy to create heavier particles than we find in the universe today. Whether or not they're stable, they will come into existence.
For example, an electron and a positron can come together and annihilate. We ask what would be the result, which would turn out to be any pair of particles and anti-particles, since the total of all conserved quantities for an electron plus a positron is zero. The total amount of conserved quantity in any "particle, antiparticle" pair, always adds to 0. So any "particle, anti-particle" pair, can always convert into any other "particle anti-particle" pair!
So lets imagine we invent a new kind of particle, called particle X. It's a fermion and comes with its anti-particle, the anti-X particle. The X particle is heavy, so we don't make it in the lab today, yet in the early universe when the energies were very high, electrons and positrons kept bumping into each other, and creating X particles and anti-X particles.
What we want to know is, how does the abundance, the density of those X particles and anti-X particles, change as the universe expands? So there are two events in the thermal history, in the lifespan of this X particle and anti-X particle. One is that it becomes cold. So if you think about it, the temperature tells you of the average energy of a particle, the energy is the rest mass plus the kinetic energy, and there's two different regimes you can think about. One in which the rest energy is much larger than the kinetic, and the other one in which the rest energy is much smaller.
If the rest energy is larger than the kinetic energy, that translates directly into the statement that the particles are moving slowly compared to the speed of light. They're more or less at rest, relativistically speaking. We call such particles cold. They're moving slowly and the temperature of the surrounding gas is small compared to the mass of that particle. In the other limit, when the temperature is very high, the particle's energy is dominated by its kinetic energy. This means it's moving close to the speed of light, and we call such particles hot.
So of course, as the universe cools down, the temperature of every particle goes down, and there will be a moment in its life when it goes from being hot, to being cold. That's the moment when the temperature, the average energy, is approximately equal to the mass of the particles. This is one important event.
Another important event is when particles freeze out. This is when particles stop interacting with the particles around them. Now in the early universe, not only is everything high temperature and high energy, it's also densely packed, close together. So things keep bumping into each other, and things can for example, annihilate. A positron will bump into an electron and annihilate, for example.
Today however, particles are more spread apart. Imagine that you have a positron here in this room, and you're in a strange situation where the nearest anti-positron (an electron) is a mile away. It's still true that if an electron and positron come together, they will annihilate, but if they're a mile away, they don't even know that each other exists. They can't find each other to annihilate.
So when things freeze out, and become so dilute that they don't find each other, they can exist even though there are both particles and anti-particles laying around. They could in principle annihilate, yet they can't find each other. They are left over as a relic abundance. It sounds unlikely for positrons and electrons, since they find each other very easily. Yet then when you talk about neutrinos, or other particles that interact very weakly, they need to be very close together to interact a lot. Neutrinos can freeze out very easily, since they interact so weakly that they stop finding anti-neutrinos and stop annihilating. That's the general picture.
The two different regimes, hot and cold, can happen at different times with respect to the event of freeze out. In other words, particles freeze out when they stop interacting, that's one event. The other event is particles going from being hot to being cold, and there's two different possibilities. You could first go from being hot to cold, and then freeze out. So you freeze out while you are cold, and are then called a cold relic particle. Or you freeze out while you're hot.
The only difference in these two different cases is how many particles you are left with. If you're hot and moving very close to the speed of light, your mass is very tiny compared to the energy that is going around. Then it's easy to make you, it's easy to make light, relativistic, particles. They don't weigh that much, so will be very abundant. When you're cold however, that means that you're heavy, you're massive, you're lumbering. It's hard to make X particles and anti-X particles if they're cold. Therefor if a relic is cold when it freezes out, there will be many fewer of them left around.
Now the obvious application to this kind of analysis is dark matter. You're going to say that you think there's a lot of dark matter in the universe. It's not a particle you see very easily, so could it be a leftover from some freeze out process in the early universe? This is where you start examining in detail the possibility that neutrinos could be the dark matter. They are the only particle in the standard model that could stand any chance of being the dark matter. They are neutral and weakly interacting.
So what happens to them in the early universe? When do they freeze out? The answer is they freeze out when they are hot. They are very light particles, so are almost always hot and moving close to the speed of light. So when they freeze out, they're hot and there's a lot of them. So we can calculate very accurately the number of neutrinos per cubic centimeter in the early universe, and extrapolate it with great confidence to the number of neutrinos in the current universe.
They could be the dark matter if they have a certain mass. We know very well what the mass of a neutrino would have to be, to account for the dark matter. Yet we don't think that this is right. We don't think neutrinos really are the dark matter, because while they're freezing out, they're hot, moving close to the speed of light, and therefor they don't clump together.
The characteristic feature that helps us find dark matter in the current universe, is that dark matter falls into galaxies and clusters of galaxies. We can measure the rotation curves of galaxies, and we find that most of the mass of the galaxy is dark matter. We can measure the speeds of galaxies within clusters, or use gravitational lensing, to find that most of the mass in the clusters of galaxies is dark matter.
Yet if the dark matter is hot, and moving rapidly when created in the early universe, instead of falling together under its mutual gravitational field, it would zip around and smooth everything out. The feature that you get when the dark matter is hot, is that structure doesn't grow in the universe. The dark matter particles are moving very quickly, and instead of falling together and becoming lumpy, increasing the contrast knob of the universe, the neutrinos or other relic hot dark matter particles would simply smooth everything out.
This is not a close call either. The possibility that the dark matter is hot, has been absolutely ruled out by what we know about the existence of galaxies and so forth in the universe. That's why, when we talk about the dark matter particle, we talk about cold dark matter. We talk about a particle that was massive when it froze out, that was heavy and moving slowly compared to the speed of light.
So what you want, is a particle that is heavy, and a particle that has enough relic leftover abundance to be the dark matter. If the particle interacted very strongly, there wouldn't be enough of it left over. It would just go away. If the particle interacts weakly, then you can make enough of it to be the dark matter. So that is why the leading candidate for dark matter is called the WIMP (Weakly Interacting Massive Particle). If it's weakly interacting, even though there are new particles X and anti-X particles, they don't annihilate away because they just don't find each other.
That is the leading scenario for what the dark matter could be. It's a cold dark matter particle, moving slowly because it's heavy, weakly interacting and therefor called a WIMP. That's a very general kind of scheme for what could be going on. In the upcoming lectures we'll look at specific examples for such things that can be leftover from the early universe. We'll look at specific examples of WIMPs to be sure for the dark matter, yet also more mundane examples like helium, something that we've actually detected. Even better than that, the CMB, all of which are going to give us very great clues as to what the universe was like in those early times.