This lecture gets specific as promised, with a dense description of how our parameters of the early universe work to make the change into our current one. Sean admits this is a difficult lecture but the payoffs will be worth the effort.
Primordial nucleosynthesis is most confusing when presented as specific events at specific times. I prefer more general statements of trends, especially when just being introduced to the topic. Sean gives a good mix of both, but I will concentrate on the trends.
The freeze-out concept from last lecture is useful to understand how protons and neutrons can begin to survive as the universe expands, instead of being annihilated by anti-protons and anti-neutrons. The lighter protons are more abundant than the heavier neutrons, so their ratio is 7 to 1 (or 14 to 2). When 2 deuterium nuclei (1 proton, 1 neutron) fuse into one helium nucleus (2 neutrons and 2 protons), there are 12 protons remaining as free hydrogen nuclei. That ratio of 4/16 or 25% He to 12/16 or 75% H, is one of the great triumphs of cosmology. We can observe quasar light from the early universe and confirm this ratio.
This abundance of H, He, and other trace elements like Li, changes with the overall density of the universe. Our observations of these abundance ratios implies a density of 5% for ordinary matter. Thus another sign that dark matter is a new and different type of particle, since otherwise the abundances would have correlated so well.
The expansion rate and radiation are the key parameters of the early universe. As previously mentioned, the radiation of the early universe had not yet lost its tremendous energy that it quickly will, when the universe expands. If this radiation parameter is any different, such as another family of neutrino, it would greatly affect the Big Bang nucleosynthesis model. A different expansion rate would affect the production of helium, or imply a different gravitational constant, again affecting the model. It all fits together into a working unit.
The analogy from Sean is one of sinking a cross-court hook shot while blind folded. Yes, it is improbable that we have come to knowing these fundamentals about the universe. But I'm wary of these types of comparisons. The voyager slingshot assist from Jupiter is like sinking a pool shot from thousands of miles, using the cushion. The Hubble image is like seeing a dime in LA from NY. No real perspective is gained.
One can just as easily point out how much there is that we don't know. Ten years ago we didn't consider that the universe was accelerating or that 95% of it was of complete unknown composition. Ten years from now there could be another totally different focus in cosmology, the field is changing that fast. Maybe a lecture on Kuhn's paradigm shifts is in order? It will be interesting to see how the latter half of the course treats these issues.
In the last lecture we learned about how to go from the soup in the early universe, of particles and radiation bouncing into each other, to figure out what would be left over at late times. How to go from very high temperatures to very cold temperatures, like we have in our universe today.
In this lecture we'll give an example of this process, perhaps the most important example of which we actually have data about in the universe right now. That's the example of the light elements. We have the universe when it was one minute old, acting like a nuclear reactor, turning individual protons and neutrons into helium, lithium, deuterium, and other light elements. Today we can observe the amounts of helium, lithium, and deuterium, from the early universe. We can make predictions on the basis of our understanding of cosmology and particle physics, and test those predictions today, to see if we correctly understand what was going on in the very earliest moments of the universe's history.
So this is a fairly specific lecture. We're not going to talk about generalities anymore, but will dig into some very precise quantities. Yet it's worth the effort to follow what is going on, since the payoff is enormous. Compared to talking about dark matter and dark energy, it can seem kind of pedestrian to talk about nuclear physics, something that perhaps some 50 years ago was the sexy, hot topic, yet now we understand pretty well.
The point is that Big Bang nucleosynthesis, as it is called, gives us our best empirical handle on what was happening at the first moments of the universe's history. From that handle, we can infer things, not only about what was going on back then, but what is true right now. In particular, Big Bang nucleosynthesis provides us with a very believable and absolutely comprehensive measurement of the total amount of ordinary matter in our current universe.
This ordinary matter, from the standard model of particle physics, the actual mass density of that stuff, comes from protons and neutrons. In an atom, in a molecule of this table or in us, most of the mass and energy is in the atomic nuclei, in the protons and neutrons. Nucleosynthesis is intimately involved with how many protons and neutrons are laying around. For a different number of them, you would get a different prediction for helium and lithium and so forth. So the results of Big Bang nucleosynthesis tell us that the amount of ordinary matter in the universe is about 5% of the critical density, which is the density of stuff needed to make the universe spatially flat.
Now we don't know ahead of time that the actual, total density of the universe, is the critical density, but we can measure densities that we observe in terms of the critical density, and Big Bang nucleosynthesis says that protons and neutrons are about 5%. Yet we've already said that the total amount of matter in the universe, the amount of stuff that has accumulated in galaxies and clusters of galaxies, is something like 30%. That's why we know that not only is there dark matter, but that it's something new and different, not ordinary stuff, even as hidden as it maybe. So the goal of this lecture is really to present a convincing understanding of this chain of logic.
So we're going back to about one minute after the Big Bang. Again, we don't know what happens at the Big Bang itself, but using the Friedmann equation of cosmology, using our understanding of General Relativity, and how the curvature of space and the expansion of the universe responds to the energy density of the universe, we know what the temperature was about one minute after the Big Bang, or even one second afterward.
Back at that time, we had free protons and neutrons which couldn't join together to make atomic nuclei because the temperatures were too high. Yet as the universe cooled, the temperature lowerd enough so that a neutron and proton can now join and stick together.
Now if you had a bunch of neutrons and protons at a low temperature and you let them go to their lowest energy state, you let all the neutrons and protons transform into the state that they really wanted to be in, they would all be in the form of iron. This is the configuration of protons and neutrons that is the most stable. If you break apart an iron nucleus, the resulting particles will always have greater energy, no matter how much smaller than iron they are. Likewise if you take a heavier element than iron and break it down into iron, you will release energy, which is how nuclear reactors work. So you can either take much heavier elements and have them undergo fission, coming closer to iron, like uranium or plutonium, breaking into smaller nuclei, or you can have fusion where you have tiny little nuclei joining and thus going towards iron.
So if the universe lasted an infinite amount of time, and cooled off very gradually, it's easy to predict what the outcome of Big Bang nucleosynthesis would be. All of those protons and neutrons would turn into iron. The reason why that doesn't happen is because there's not enough time, since the universe is expanding quite rapidly. So one the one hand, the temperature is going down which allows nucleons to fuse into heavier nuclei, yet on the other hand the density is also going down which in turn lowers the frequency of fusion. So because they don't interact with each other, they can't fuse, and we don't go all the way to iron.
We're stuck with helium and all the other light elements. So lets run the movie backwards and start with our universe today, asking what would happen when the universe was smaller and things were closer together? It's extremely analogous to taking a bunch of stuff in a piston and just compressing it. That compression is very much like the shrinkage of the universe as you go backwards in time. So what happens is of course, things get pushed together and the density goes up. Yet as you push a piston in, the temperature of the stuff inside the piston goes up because things are banging together more frequently.
So the temperature and density of the universe goes up. That temperature is basically the average amount of kinetic energy of every particle, as we've said before. So the temperature of our current universe, which is the temperature of the relic photons of the CMB, is about 3 Kelvin. One degree Kelvin is equal to one degree Celsius, but the zero point of the scale has been moved, so that zero degrees Kelvin is the lowest possible temperature that exists. It's the temperature at which everything is absolutely zero, so it's called absolute zero, which corresponds to about -270 degrees Celsius.
So today, the universe is at about 3 degrees Kelvin and the earth is at about 300 degrees Kelvin, or something close to it. In other words, the universe is not in thermal equilibrium, and the temperature is different from place to place. So if we take this universe with the photons today being three degrees Kelvin, and we squeeze them as we go back in time, the temperature goes up, inversely to the squeezing. So when the universe was 1/1000th its current size, the temperature was about 3000 degrees Kelvin, which was when the CMB was formed.
When the size of the universe was a billionth of the size it is today, its temperature was about 3 billion degrees Kelvin, which is the era of Big Bang nucleosynthesis. We can also tell what times these correspond to, by using the Friedmann equation. It tells us that recombination, the moment when the CMB was formed, is about 380,000 years after the Big Bang. The time of nucleosynthesis is only at about one minute after the Big Bang, when the temperature was a billion times higher than today.
In that era, the matter was not dominated by matter and dark energy like it is today. Dark energy, as we'll talk about later, has an approximately constant density as the universe expands. Matter dilutes away in energy as the universe expands, because the density of particles goes down. Yet radiation has a density that goes down even faster, not only because the number of particles goes down, but also because the energy per particle goes down as particles get redshifted. So of the three kinds of energy, radiation goes away the fastest as the universe expands. Contrary wise, as the universe contracts into the past, the radiation goes up the fastest.
So even though today, radiation is a small part of the energy budget of the universe, at early times it was the dominant part. The early universe was radiation dominated, so that the things making up most of the energy budget of the universe were photons and neutrinos, light particles moving close to the speed of light.
So when the scale factor was about a billionth, the universe was very hot. The reason why that's an especially important time is that the universe was hot enough that individual atomic nuclei could not survive. The temperature of about 3000 Kelvin is the scale factor of about 1/1000th, when the temperature was too high for atoms to survive, so that electrons are torn from their atomic nuclei. Yet it's at 3 billion degrees K that individual nuclei cannot survive. So even if you had some atomic nucleus of oxygen, carbon, iron, or whatever, back then the temperature was so high that the protons and neutrons would just be torn apart.
So at times less than a minute after the Big Bang, at temperatures more than 30 billion K, you didn't have any atomic nuclei, just individual protons and neutrons. Before even then, you didn't have individual protons and neutrons, since the temperature was high enough to resolve them into their constituent quarks and gluons. Yet by one second or one minute after the Big Bang, you certainly did have protons and neutrons, and they kept bumping into each other.
We see a schematic picture of thermal history of the universe, as you go from the very earliest times when the temperatures were very high, a series of transitions occurs as the temperature goes down. The one we're talking about today, is the one where you make the light elements.
So lets start at about one second after the Big Bang. You have protons and neutrons going around, and one second is chosen because not only do you not have atomic nuclei, but one second is before the protons and neutrons have frozen out. Remember that freezing out is the moment when particles stop interacting with each other, the rate at which things happen dips down because the densities become too low.
At times earlier than one second, protons and neutrons were interacting with the surrounding plasma, which means they were being created and destroyed all the time. They were not in a total quark number 0 state, so there was more protons and neutrons at that time, than there were anti-protons and anti-neutrons, otherwise they would have all annihilated away.
There's some net baryon number laying around for reasons that we don't completely understand due to some unknown process of baryogenesis. Yet they're there and the forms that they want to take are protons and neutrons. So when we say they've not frozen out, we mean that protons and neutrons are converting into each other very rapidly.
So we talked about the process of a proton bumping into an electron and making a neutron and a neutrino. This can happen if the energies are high enough. So we're saying that when the universe was less than one second old, the energies were indeed high enough. Neutrons would interact with neutrinos to make protons and electrons. Then vice versa, protons and electrons would react with each other to make neutrons and neutrinos.
So everything was in equilibrium, which means you have more light particles than heavy, since they're harder to make. So the fact we know that the proton for example, is a little bit lighter than the neutron, when you can convert protons and neutrons back and forth rapidly, you'll end up with more protons than neutrons because the proton is lighter. It's easier to have it around.
So in fact, if you run the numbers, you plug in the equation, and find that at the moment the protons and neutrons do freeze out, when these weak interactions which are converting from protons to neutrons and back again, slow down so much that they don't happen anymore, that's a moment when you have about one neutron for every six protons. We can't derive that in real time, yet that's what comes out of the equations. You have about six protons and one neutron for every seven baryons lying around.
So then what happens? This is the time when you've frozen out, so protons and neutrons don't convert anymore. They just sit there! Yet of course the proton really does just sit there, and nothing happens to it. Yet the neutron isn't stable, since we know their decay rate is a half life (a 50/50 chance of decay) once every ten minutes.
Now we're talking about a timescale when the universe is only a few seconds old, so not many neutrons decay away. Yet between one second and one minute, some of the neutrons will decay away. Their lifetime is ten minutes so most will be around, but some will go away. So by the time we've reached a moment of about a minute after the Big Bang, we've gone from having one neutron for every six protons, to having one neutron for every seven protons. This is a number worth remembering since we'll put it to good use.
So we have one neutron lying around and about seven protons. The time is about one minute after the Big Bang and the temperature about 3 billion degrees Kelvin. What's so special about this time? The temperature is then low enough to start to form nuclei. Protons and neutrons can stick together without being continuously dissociated by bumping into other particles.
So it's worth pausing at this important moment in the universe's history, to emphasize that we're assuming a lot here. We weren't there at the Big Bang, or one minute after. We weren't looking at what was going on. So how do we know all these things we're saying? How do we know there was one neutron per seven protons, the temperature at one minute, etc?
The way we know is, we never know for sure, but we do have a theory. We have the Theory of General Relativity, we have the Friedmann equation for the expansion of the universe which was derived from General Relativity, and we use that theory, plus particle physics, plus the standard model, to make a prediction. We go from what we see today, to what the universe must have been like.
Yet we're not satisfied with making a prediction, so we're now going to test it. We'll say, "If the universe were doing that, certain things should be true today." So we're going to check it. Since those things will turn out in fact, to be true today, we're pretty confident that we understand what the universe was doing one second or one minute after the Big Bang.
So what happens? The temperatures now dip down low enough, that we can make nuclei out of protons and neutrons. So of course, you start with individual protons and neutrons, and you make the nuclei step by step. It's not going to happen that a whole bunch of protons and neutrons come together all at once, and make one heavy nucleus. Rather, particles will interact with each other one by one.
So the first thing that can happen is a neutron and a proton can come together to make a nucleus of what is called deuterium. This is heavy hydrogen, one neutron and one proton in an atomic nucleus. Now that's easy to describe, yet in fact a deuterium nucleus is very fragile. So this slows down what we might have thought would be the process of nucleosynthesis, since you can make a deuterium nucleus, but it's so fragile that they keep getting disrupted at this early time of high temperature in the universe's history.
Yet the temperature is dropping bit by bit, continually going down, so eventually two deuterium nuclei can last long enough to bump into each other, and they make a nucleus of helium. This has two protons and two neutrons, which is sticky and robust. It is not very fragile so once it's made, it stays there.
So basically we start with one neutron, and seven protons. Or if you like, we double the number, two neutrons and 14 protons. Basically every single neutron that you have lying around, goes into a nucleus of helium. So these two neutrons and 14 protons become one helium nucleus, and 12 protons.
The helium nucleus has four particles in it, two neutrons and two protons, and you have 12 protons lying around. So 25% of the mass that you started with, has now become helium. This 25% of the baryons, of the total number of protons and neutrons, are now stuck in a helium nucleus.
What you'd like to have happen next, if you could just stay at that temperature forever, is for those helium nuclei to then stick together. You would start making carbon and oxygen and heavier elements. Yet what actually happens is that the universe expands, so it doesn't happen. The helium just gets stuck there.
So realize that this statement which we make with such confidence, of course depends on details of what was going on, just three minutes after the Big Bang. It depends on the expansion rate of the universe, since if it were much slower, then you would have time to go from helium to heavier elements. In the model we're working with, in the Theory of General Relativity and the Friedmann equation, you end up with almost all helium, about 25% by mass.
You can do better than that. Beside the fact you get 25% helium, you get trace elements that contribute just a little bit. Not all of the deuterium is sucked up into helium, and there's some little bit left over. Not all of the helium nuclei stick around by themselves. Sometimes the helium nucleus will take on other little particles, and perhaps become a nucleus of lithium. Lithium-7 has three protons and four neutrons in it, another light element. You can also have helium-3, with two protons and one neutron.
So in other words, primordial nucleosynthesis takes individual protons and neutrons, and from that makes a whole bunch of helium, all the helium it can make, plus trace abundances of deuterium, helium-3, and lithium. In particular, it does not make anything heavier, like carbon, oxygen, iron, all the stuff that is very important for our lives. So when this theory was first being invented, it was in the late 1940s actually. The first paper on primordial nucleosynthesis was written in 1948 by George Gamow and Ralph Alpher. They were trying their best to make all the elements that we know, in the Big Bang. They were trying to make all the heavy elements, yet they eventually realized it doesn't quite work.
Yet they did realize they could make helium very efficiently, and that the observations in 1948 weren't very good. Yet the predictions that they made, fit very well that the observations that helium actually existed. This was the first quantitative thought that we could extrapolate our knowledge from the current universe, all the way back to moments after creation. So Alpher was actually a graduate student of Gamow's at the time, and this work was his Ph.D. thesis. It's defense was attended by 300 people from the Washington DC area, and the Washington Post had a front page story the next day, "Universe Created in Five Minutes."
Seans' Ph.D. thesis did not appear on any front page of the daily newspapers, but he imagines it to be a nice thing when that would happen! Yet Alpher was not always happy with how things turned out, since Gamow was quite a practical joker. So when they submitted their paper, by Alpher and Gamow, he added the name of Hans Bete into the middle, so that the author list read Alpher, Bete, and Gamow! It was a joke, supposed to sound like alpha, beta, and gamma, from the Greek alphabet. He didn't even tell Hans Bete, who is a famous nuclear physicist. He didn't even tell Ralph Alpher, was was the lead author on the paper. Alpher was upset since now he was going to lose some credit, with two very famous collaborators, not just one.
Nevertheless, we now have a theory of where the heavier elements came from, and know that they were not born in the Big Bang, but were born in stars. They are both created while stars burn, like our sun right now creating carbon and oxygen, and then they'll be scattered when the stars explode in supernovae, which will at the same time make even heavier elements. The very heavy elements of gold, plutonium, and uranium that we see in the universe, are created not in the centers of stars, but in the actual process of the explosion of a supernovae.
So to test Big Bang nucleosynthesis, we have a set of predictions. We have 25% helium roughly, trace amounts of deuterium, lithium, and helium-3. The problem is that the late universe is spoiled by all these supernovae explosions. There's more and more elements processed by stars, so the abundances we see today are not the primordial abundances. What we want to do then, is to take observations of pristine parts of the universe, where there aren't any stars. It turns out that we have now learned how to do that. We can look at light from stars and quasars, shining through otherwise empty regions of the universe, which are thinly dispersed with gas and dust.
So you have a back light that is shining through some primordial material, which has never been processed by going through a star or supernovae. So we can try to measure the abundances of light elements, in those primordial clouds, and try to compare them with the predictions of Big Bang nucleosynthesis.
The good news is, it works! The abundances we actually see in unspoiled parts of the universe, match the predictions of Big Bang nucleosynthesis. So here is an actual plot of some data versus some theoretical curves for what the predictions of Big Bang nucleosynthesis are, and what we observe.
We see predictions for helium-4, which we just call helium, deuterium, helium-3, and for lithium-7. No we'll notice that in this plot it's not just a single number prediction for each element, but a curve. There's some function, some parameter that is being varied to get different predictions. That parameter is the density of ordinary matter. So this plot, this set of predictions, assumes that we don't know ahead of time, the number of baryons in the universe, the number of protons and neutrons.
The plot is saying that as you change the number of protons or neutrons in the universe, we'll make different predictions for the amount of helium, deuterium, etc. There is a value for the number of protons and neutrons in the universe, for which we get the right answer. This is how Big Bang nucleosynthesis fixes the number of ordinary matter particles in the early universe. If there were more baryons in the universe, more protons and neutrons, you would make more helium. The production of helium would be more efficient. You would make correspondingly less helium, deuterium, and lithium-3, because you'd be getting rid of it into the helium.
So there's a prediction that if the helium abundance is higher, it means there are more baryons in the universe, but the deuterium abundance would be lower and so forth. There is a concordance, a match, between all the observations we have, and all the theoretical predictions, at one specific value for the abundance or ordinary matter in the universe. That abundance is 5% of the critical density, the density you would need to explain the spatially flat universe. So 5% of that, is existing in ordinary matter, baryons, protons, and neutrons.
So this is worth emphasizing, since it's one of the very crucial building blocks, on our road to understanding dark matter. We can measure the total amount of matter in the universe pretty efficiently, by using things like the rotation curves of galaxies and the dynamics of clusters of galaxies. We can find that there is matter there, and extrapolate it to the rest of the universe. In fact, we could imagine that we have missed some matter, something in between galaxies and clusters, yet we can't imagine that we've overestimated the amount of matter. The matter we see is certainly there.
So the total amount of matter we see is something like 30% of this critical density. Yet Big Bang nucleosynthesis provides a measurement of the total amount of ordinary matter in the universe, and it's 5%. It doesn't matter what happens to those baryons, those protons and neutrons, after Big Bang nucleosynthesis. They could get absorbed into black holes, they could collapse into brown dwarfs and other compact objects, they could be spread out in gas across the universe, they would all count in what Big Bang nucleosynthesis is sensitive to.
So it doesn't matter what form the baryons take, the fact that the predictions of Big Bang nucleosynthesis agree with observations of the abundances of the light elements, implies that the total amount of ordinary matter in the universe is not enough to be all of the matter in the universe. Not only is there dark matter, not only is there matter we don't see, but dark matter is not some hidden form of ordinary matter. It's some different kind of particle. That is the important implication of the success of primordial nucleosynthesis, for the story of dark matter and dark energy.
There are other implications we can get from the success of nucleosynthesis. For one thing, we mentioned that the process of nucleosynthesis was first, strictly dependent, very precisely, on the expansion rate of the universe at the time when it was just one minute old. Another is that the universe at that time was radiation dominated. The relevant amounts of radiation come from photons and neutrinos. Those are the light particles that were zipping around at very early times, dominating the energy density of the universe.
So if we changed the expansion rate of the universe by just a little bit, you would predict different things from Big Bang nucleosynthesis, and we would not get the right answer. How could you possibly change the expansion rate of the universe at different times? Well one way is you could change the value of Newton's concept of gravity. We have a constant that appears in Einstein's equation, so it reappears in the Friedmann equation. This constant tells you how strong gravity is. How much of a gravitational field, or how much curvature of spacetime is created by some amount of mass.
So this is a constant that has been around since Isaac Newton's time, and certainly describes how gravity works here in the solar system. The success of Big Bang nucleosynthesis is telling us that it is also successfully describing gravity, one minute after the Big Bang. Nothing dramatic has happened to change the strength of gravity. There's no reason to suspect that anything did, but now we know, based on data, that it didn't happen.
Another thing we know is how much radiation there is in the universe. If there was more of it, then the Friedmann equation would tell us that back then, during Big Bang nucleosynthesis, the universe would have been expanding more quickly. That would have altered the predictions of Big Bang nucleosynthesis. So for example, if there were not just three different varieties of neutrinos, and if there were four instead, with four different families of fermions in the standard model, that would be an extra contribution to radiation in the early universe. If there was such a fourth, light neutrino, then the universe would have expanded faster, things would have cooled off, and you'd get less helium, in contradiction to what you observe in the data.
So one of the reasons we're confident that only three families or generations appear in our standard model of particle physics, that we're not going to keep finding more and more families, is that Big Bang nucleosynthesis is inconsistent with extra kinds of neutrinos, as long as they are light, in the early universe.
So it's a little bit mundane when we think about nuclear physics, compared to dark matter and dark energy, but you should be impressed with the success of Big Bang nucleosynthesis. We take a bunch of laws of physics, all of which we've figured out in the course of the last 100 years. We're in a 14 billion year old universe, and we used these laws of physics to extrapolate back to what the universe was like one second after the Big Bang. We made a prediction, and it became right. That prediction turned out to be true! We know what the universe was doing a tiny fraction of a minute after the Big Bang.
It's like you're on a basketball court, standing at one end, and you try a hook shot with a blindfold on, at the other basket on the other end, and you hit nothing but net! It's that impressive, except that it's not an accident. You do it 100 times and you hit nothing but net every time. We're able to go from our knowledge of the current universe, all the way back then and get the right answer.
The reason why that's important for dark matter and dark energy, is because they are things we don't understand. Since there are so many things we don't understand in the universe, it's crucially important to have backup to the claims that we do understand something! The success of Big Bang nucleosynthesis tells us that we do understand the basic story of the Big Bang, and we do understand the basic equations that govern the expansion of the universe.
Our inference that the universe today is governed by dark matter and dark energy is buttressed by these facts. So now our job is to think about what those other things we don't understand, the dark matter and dark energy, could possibly be.