This lecture brings together the previous two in yet another wonderfully described presentation. The cosmological principle and general relativity combine to provide new answers and new questions. We are at mid-stage in the first cycle of learning how we got to know the universe, eventually to culminate in why we believe our current theories are true, a few lectures from now.

To characterize the universe in light of Einstein's relativity allows one to ask important fundamental questions of dynamics, composition, and evolution. The question from the last lecture on the observed expansion is one of them. The universe is changing, but changing the same everywhere throughout. Relativity explains this change by saying the space between bound systems, such as galaxies, is expanding. This expansion causes light from galaxies coming towards us to loose energy by stretching, appearing redder. So the Doppler effect does not actually cause the observed redshift because the galaxies are not really intrinsically moving. But the expansion does increase the distance between galaxies over time, so there is a scale factor involved. This is one of the most key questions ever posed, to plot this scale factor over time. But did the composition of the universe perhaps change during this time?

Einstein's field equation from the last lecture equated curvature of space-time with energy and momentum. The energy corresponds to radiation, whose amount of energy is determined by frequency. Now we just said the expanding universe stretched the light radiation of galaxies, lowering the frequency, thereby losing their energy. Add this to the fact that per unit volume, their energy was already decreasing and you have a universe today where radiation plays almost no role in the total amount of energy in the universe! Yet when the scale factor was much smaller, the universe was much younger, and the frequency of the radiation was still high, it played the larger role in total energy. The early universe was radiation dominated.

The momentum of Einstein's equation refers to slow moving particles whose energy is measured by its mass. Since mass energy is not decreasing over time like radiation energy, it's only decreasing per unit volume as space expands. It's then tempting to say the universe today is mass dominated, but that contradicts the course title of dark energy! Obviously much more to come on that.

All this decreasing density of energy adds up to the overall density energy of the universe. It actually tells us the overall curvature of space-time in Einstein's equation. But this overall curvature can be separated into one component from the expansion itself and another from the space itself. Space curvature has the three possibilities of a flat Euclidean plane (zero), a sphere (positive), or a saddle (negative). So these two sides of the equation can be made equal to one another.

Enter the Friedmann equation, derived from General Relativity, which takes the expansion curvature Ho, the space curvature K, and equates them with the overall energy density of the universe, ρ(rho):

((8πG)/3)(ρ)=H² + K

Thus the overall energy density we discussed above, drives the overall curvature of space-time, which is composed of expansion curvature and space curvature. It really ties everything together quite neatly. But even better is allowing the actual evolution of the scale factor to be determined. Extrapolating from the very early universe turns out to agree with our current one. Extrapolating from our current universe to the future is what the whole course is about.

Another basic question is answered by rho, in that we can say there is nothing more in addition to it. We're getting ahead of the story here, but we can now say that the grand total of dark energy, dark matter, and ordinary matter, makes up the whole composition of the universe. There is nothing more to it, and that is saying something very fundamental indeed.

In this lecture we get to put together the pictures we developed in the previous two. So in lecture two we talked about what the universe looks like. If you stand outside on a clear night and look with perfect vision, what do you see? Well we found that you would see a set of galaxies like our own. Every galaxy has something like 100 billion stars in it, and there's something like 100 billion galaxies scattered throughout the universe.

The good news is they're scattered uniformly, with the same density of galaxies all over the place. The more interesting news is that everything is expanding. So the universe is smooth, pretty much the same everywhere, and it's expanding, getting bigger, so galaxies are redshifted. Their light comes toward us that leaves them at a certain wavelength, which is longer when we finally receive it. It's just as if those galaxies are moving away.

The third lecture, just previous to this one, discussed Einstein's Theories of Special Relativity and General Relativity, his theories of space and time. In particular, General Relativity was the lesson that what we perceive as gravity is a manifestation of the curvature of space and time, which combine together in something called spacetime, which is one four-dimensional, dynamical spacetime that can stretch and move. The results of that is what we perceive as gravity.

So what we'll do in this lecture is take the idea of a smooth expanding universe, one the same everywhere throughout space, but growing as a function of time, and understand it in the context of General Relativity. What does Einstein's equation, the relationship between stuff in the universe and the curvature of spacetime, have to say about the idea about a smooth, expanding universe?

So one thing that we mentioned, and is worth emphasizing again, is that the fact that the light from distant galaxies is redshifted when it gets to us, is not strictly speaking a Doppler shift. That is what happens when you have two objects with a relative velocity. One object is moving toward the other one, and when it emits something, whether a sound wave or light wave, that something has its wavelength squeezed, and we observe it as a shorter wavelength thing, whether a higher frequency sound, or a bluer ray of light. If it's moving away from us, the wavelength would be stretched. That's the Doppler effect.

A very similar thing happens according to General Relativity, but it's not the Doppler effect. It's not that this thing is moving toward or away from you, but it's that the space itself is expanding in between. That is the origin of the cosmological redshift. It's that spacetime itself is dynamical. So the amount of space is getting bigger, and we're going to try and understand that in a slightly more quantitative way.

Talking about quantitative ways, we'll now ask, "How big is the universe?" Is that even a question that makes sense? The answer is that we don't know, both how big the universe is, or whether the question even makes sense. Well we know that we don't see any boundary to the universe, because in Einstein's universe, space and time are dynamical, and we can imagine different shapes that space could take.

In Newton's universe, it was kind of obvious that if you went out in space, you'd just go on forever and ever because space just extended in infinite directions. On the other hand, in Einstein's universe, things can be curved. If things can be curved, they can be curved in on themselves. So we can certainly imagine a finite universe, one with the shape of a sphere for example.

What we see when we look out into the universe, is that we do not see the back of our heads, or any direct evidence that the universe we observe is finite in size. That doesn't mean that it's not finite in size, but just that we don't see any wrapping around or any edge to the universe.

Yet that's not a complete surprise, since we only see a finite part of whatever universe there is, for two very simple reasons. First, the universe has a finite age, at least since the Big Bang. For reasons we'll talk about later, the time between now and the Big Bang is fairly well determined to be about 14 billion years. Our best guess right now is something like 13.7 billion years. Yet it's not a guess, since it's something the data are telling us is true.

So the universe is 13.7 billion years old, yet remember that light travels at a finite speed, one light year per year is the best way of thinking about it. So if light travels at a finite speed, and the universe has a finite age, light has only been able to travel a finite distance between now and the Big Bang.

So even if the universe were infinite. Even if it did go forever, we couldn't see all of it, but only a certain patch from which light can get to us in less than 14 billion years. So we don't know the size of the universe. It could be infinite or finite, yet we nevertheless talk about the universe expanding! So how does that make sense? If you don't know how big something is, how do you know it's expanding?

Well what we know is the relative size of the universe. By this, we really mean the relative distance between different things in the universe. So just take two galaxies, and we know they are moving away from each other. So we can ask quantitatively, how many years it will be, before the distance between these galaxies is 10% more? How many years will it be before that distance is twice what it was?

The answer to those questions is the same for any two galaxies! In other words, because the distance to different galaxies is proportional to their velocities we see, it doesn't matter which galaxy you pick, as long as you ask the question, how many years will it take for the distance to grow by a certain number? You will always get the same answer. So we can talk about the relative size of the universe, and say it was half its current size. That means the distance between any two galaxies was then half of what it is today. That's a perfectly sensible concept.

So cosmologists invent something called the scale factor of the universe. This is a number we set by convention equal to 1 today. It tells us the relative size of the universe, so when it was ½, that means all the galaxies in the universe were half their current distance. When it is 2, they will be twice as far apart as it is now. So one of the major projects that you want to complete if you consider yourself to be a cosmologist, is to understand what the scale factor was doing as a function of time.

So we see a plot showing the basic idea of what the scale factor does in an expanding universe. Right now its getting bigger, as galaxies are going apart from each other and we say that the universe is expanding. We trace it backwards and it is, first, very much smaller. We can trace it all the way back to zero. We don't necessarily think we know what happens at zero, so we call it the Big Bang!

It's important to emphasize that the Big Bang is not a crucial part of our understanding of cosmology. It's a label we give to the place where we don't know what's going on. We know very well what's going on one second after the Big Bang, so we talk about the Big Bang as if we understand it, yet really it's just a marker for our ignorance.

Nevertheless, 14 billion years ago there was something going on that we call the Big Bang. We understand that everything there was squeezed on top of everything else, and it was expanding. So we believe, as we'll talk about real soon, that the rate of expansion in the past was bigger than the rate is now. So when we draw this plot, it has a sharper slope in the past than it does today.

One of the things we'd like to understand about this plot is its exact form, from the Big Bang, to today, and into the future. One of the most surprising results of cosmology in the last ten years, is that today the expansion rate of the universe is going up! The universe is not only expanding, but is accelerating, which is why we believe in dark energy.

So if you want to understand the phenomenon of the expansion of the universe in the context of General Relativity, we remember that Einstein told us it was stuff in the universe that leads to the curvature of spacetime. So the first question we should ask is, "What kind of stuff is there in the universe?" Even before we answer that question, we should know what kind of stuff there could possibly be. What are the different forms of matter or stuff that could possibly fill the universe, before we actually even go out and look at the universe, asking what we actually see.

So lets first think about the things with which we are most familiar, the earth, sun, galaxies. These are all collections of particles. The earth is made of particles. We talked earlier about atoms. The earth is made of atoms. The sun is made of individual atomic nuclei and electrons moving around, yet they also are made of particles. These particles are bound together so that the earth is also bound together under the gravitational field of all the stuff in the earth. Likewise for the sun, and likewise for the galaxy. The Milky Way galaxy is bound together by the mutual gravitational force of all the 100 billion stars in it.

So the first crucial thing about the expansion of the universe is that bound systems do not expand. So if you have two galaxies in the universe, each one is orbiting under it mutual gravitational force. The distance between the two galaxies is getting bigger, yet the galaxy itself is not getting bigger. The earth is not getting bigger, we are not getting bigger, or if you are, it's not due to the expansion of the universe but because of those donuts!

So the bound systems do not get stretched along with the universe. This is sort of a subtle thing that people are initially reluctant to believe, yet it really is true. In fact it's the only way that you could even make sense of the claim that the universe was expanding. If everything in the universe, including humans and atoms, expanded along with space, that would be exactly the same as nothing expanding. We only measure the distances between two objects in terms of the size of something else.

If the size of everything increases, your rulers get stretched, and the number of meters in between you and some distant galaxy wouldn't be seen to change. When we say that space is expanding, what we mean is the number of atoms it would take to stretch from us to a distant galaxy is getting bigger. That's because the atoms are fixed, and the galaxies are moving away.

Absolutely strictly speaking, there's an equally good way of thinking about this, in which the distance to galaxies is constant and all the atoms are shrinking! That's just a really silly way of thinking about it, so we don't do that. It's just more convenient to think of atoms and all of us, our sun and galaxy, as remaining fixed in size, with the space in between the galaxies getting bigger.

Yet while the universe is expanding, even though atoms or particles do not stretch along with it, they become more dilute. More and more space is coming into existence. So if we draw a really big box, one that contains many hundreds of galaxies, we can ask how many galaxies per volume of space are there? That number will be going down. Space is expanding, the number of galaxies is basically not changing, the number of atoms is certainly not changing, so the number of atoms per cubic megaparsec, or the number of atoms per cubic centimeter, is going down. So we say that things are diluting away as the universe expands.

Now matter, to a cosmologist, is any set of particles that are moving slowly compared to the speed of light. Matter are things that get their energy from their mass. So Einstein, among his many other accomplishments, said E=mc². The interpretation of that equation is not really energy equals mass times the speed of light squared. We need to attach some extra words to that equation to make sense of it.

E=mc² is telling us that the rest energy of an object, the energy it has when it's not moving, is its mass times the speed of light squared. That's the minimum amount of energy something can possibly have, and it's a constant. The mass is not changing, the speed of light is not changing. So the reason we're going through this is to emphasize that the energy per particle is not changing, if the particle is moving slowly compared to the speed of light. That, to a cosmologist, is what we call matter.

As opposed to matter we have radiation, which could be photons or other particles moving at the speed of light. If you're moving at the speed of light, your mass is zero. Your energy is not coming from E=mc² if you're moving at the speed of light. It comes from the frequency at which your vibrating. Yet that frequency is changing as the universe expands, because of this redshift. The cosmological redshift takes short wavelength, high-energy photons, and stretches them into long-wavelength, low-energy photons.

So for particles that are moving slowly, E=mc² and E is a constant. The mass per particle is a constant, so the energy is a constant. Yet for particles moving at the speed of light, the energy per particle decreases as the universe expands. Photons and other particles that cosmologists call radiation, lose energy as they age, in the history of the universe. So that's the important difference to cosmologists between matter and radiation, how the energy density changes.

The number density of particles is just how many particles there are per cubic centimeter. So the energy density is how much energy per cubic centimeter. In a region that grows along with the universe, the number density is constant, you're not creating new particles. Yet in a fixed, cubic centimeter of space, the energy density will go down.

The number density will go down as particles become more dilute, and therefor the energy density will go down. Yet the important difference between matter and radiation, is that the energy density in radiation goes down more quickly. In radiation, not only do particles dilute away, becoming less and less per cubic centimeter, but every particle looses energy.

So we see a picture of basically what is happening as the universe expands. This is not a fixed amount of space, as our box is getting bigger as the universe expands. So the number of particles in the box is a constant. The number of photons in the box is a constant. Yet the density per cubic centimeter of both particles and photons is going down. The energy per cubic centimeter is going down. Yet the energy in photons and radiation is going down more quickly, since every photon is losing energy.

So why belabor this point? The point is that the total energy in all the universe in radiation, goes down more quickly than the total energy in matter. So you expect that as the universe gets older and older, matter will eventually win. There will eventually be more energy density in matter than in radiation, because the radiation has an energy that is going way more quickly.

Contrary wise, in the past the energy density of matter is less important than that of radiation. In the past when all the photons were squeezed together, with shorter wavelengths and higher energies, it was the photons that were winning. The photons were dominating. So a cosmologist will say that in the past, the universe was radiation dominated. It was the radiation that was important for the expansion of the universe. After that, the universe became matter dominated.

So if you didn't know anything about the universe. If you didn't look out at the universe and discover the dark energy, if you pretended you were a cosmologist or astronomer from 20 years ago, you would say that the things that could possibly be in the universe were either matter or radiation. It seems like those are the only possibilities, particles moving slowly compared to the speed of light, and particles that are moving fast compared to the speed of light. That is to say, at the speed of light, or only slightly less.

Then what we want to do now, is to put that idea to work, in order to understand in the context of General Relativity, how matter and radiation create the curvature of spacetime. So in general, if you weren't in a simple context like the expanding universe, the concept of the curvature of spacetime can be arbitrarily complicated. Space can be curved in all sorts of different ways. Just imagine a simple two-dimensional sheet of paper and all the different ways you can crumple it, bend it, and twist it.

A four-dimensional spacetime can be crumpled, bent, and twisted in correspondingly many more ways. So spacetime curvature can be very complicated. Yet luckily for us, the universe is a simple place. So for the next few lectures, we'll be ignoring the fact that there are lumps in the universe, well for this lecture anyway! We'll be approximating the universe as perfectly smooth, imagining that all over the place, you have exactly the same density of stuff.

Then in that very nice circumstance, the curvature of spacetime becomes very simple. There are only two different forms that the curvature of spacetime can take. One is the expansion of space itself. The very fact that space is getting bigger, adds a component to the curvature of spacetime. So one way in which spacetime can be curved, is that space gets bigger as a function of time.

The other way is that even without expansion, space can be curved all by itself. So forgetting about the fact that the universe is getting bigger, just looking at the universe at one instant of time, you have a three-dimensional space and you can ask, "Is that three-dimensional space, flat or curved? What kind of curvature does it have?" There are different possibilities for this curvature as well.

So lets start with that possibility. Lets start with the curvature of space, before adding in the expansion of the universe. So now we get to forget about time, we get to think about three-dimensional space, and ask if it's flat or curved? So what does that mean?

A flat geometry to space, means the Euclid was right. Our good-old, high-school geometry that you get by drawing things on a tabletop which is perfectly flat, has certain rules. For example, if you draw a triangle in space, and add up the angles inside the triangle, there's a rule that says you'll 180 degrees every time. That is actually not a hard and fast law of nature, but a feature of a particular kind of geometry which we call flat.

It turns out to be true on this table top, yet if you're a professional geometer, you imagine other possibilities. For a long time, people tried to prove that it needed to be the case that the angles inside a triangle would have to add up to 180. Eventually they gave up because they realized it wasn't true! There were other possibilities, so we can imagine that space itself it curved. The good news is that space is the same everywhere, so this curvature of space is just a number, one that changes by the function of time. Yet right now, we're looking at the universe at one moment in time, and we say space is either positively curved, negatively curved, or zero curvature, which is to say, flat. There's only three possibilities.

We see a picture showing these three possibilities, which demonstrate the operational difference between space being flat, positively curved, and negatively curved. If you're in a positively curved space and draw a big triangle, it's like drawing a triangle on the surface of a sphere. We can do that at home, and add up those angles inside. Every triangle we draw on the surface of a sphere, will have the property that the angles inside add up to greater than 180 degrees. The precise number of degrees they add up to, will depend on the size of the triangle you draw. In a limit, when you go to a very tiny triangle, you'll come close to 180. Yet as the triangles become larger and larger, they will have larger and larger angles inside.

So you could do that to the universe and ask what it's curvature was. You could even get that the angles inside added up to a number less than 180 degrees. In that case, we'd call it negative curvature. Instead of living on the surface of a sphere, it would be like living on the surface of a saddle, something that bent in different directions in different regions. So this is something you can go out there and test.

The first person to actually think about these possibilities in any systematic way, was Carl Gauss, a famous 19th century mathematician. In fact, he went out in the field and tried to test this. He drew a big triangle in the mountains near Hanover and measured the angles inside. He got 180 degrees up to his experimental error. It's an interesting thought experiment to ask that if he had better precision instruments, would he had invented General Relativity? Probably not is the answer, but he was a smart cookie, so you never know.

So these are the possible scenarios for the possible curvatures of space. It could be flat, and Euclid would be right. Yet space could also be positively curved, like that of a sphere, or negatively curved like the surface of a saddle. The curvature of space is one of the two contributions to the curvature of spacetime in a smooth expanding universe. The other contribution is just the expansion of the universe.

So Einstein says that the curvature of spacetime is being driven by the stuff that is in the universe. In other words, there should be some equation that relates the stuff that is in the universe, and the expansion rate of space. This equation was in fact derived from Einstein's equation by Alexander Friedmann, who was a Russian cosmologist. He derived this equation in 1922, which turns out to be the most important equation in cosmology. Remember that General Relativity had only been invented by 1915, and there was something else going on at the time called World War I when it was put together!

Yet soon thereafter, Friedmann, who served in WWI, looked at what General Relativity had to say about the expansion of the universe, and he came up with an equation. Sadly he died by 1925 of typhoid, so it wasn't until 1929 that Edwin Hubble found that the universe is in fact expanding.

So here is Friedmann's equation, the most important in cosmology. It's what tells us how the expansion of the universe responds to the stuff inside the universe.

((8πG)/3) ρ = H² + K

On the left hand side, we have again some constants,some numbers, 8, pi, G, and 3. G is Newton's constant of gravity, familiar from Isaac Newton. The Greek letter rho (ρ) is the energy density, the amount of energy in every cubic centimeter of space. It's going to change as the universe expands, but in its approximation where everything is very smooth, it's the same number from place to place, all over the universe. So rho, the number of ergs per cubic centimeter, the amount of energy in every little part of the universe, is what drives the curvature of space and time, in this way of thinking about things.

So on the right-hand side of the equation, you get two contributions to the curvature of spacetime. The first contribution comes from the expansion, the second comes from the curvature of space. So the expansion of the universe, remember, is measured by the Hubble parameter. Hubble is the one who told us, that the number H is the constant or proportionality between the velocity of a distant galaxy and the distance to it. So the bigger H is, the faster that the universe is expanding. So the particular way that H enters the curvature of spacetime, is in the form H². Sean has no simple, fun explanation for why it's H squared, rather than H cubed, but it turns out to be the Hubble parameter squared that is a feature of the curvature of spacetime.

To that we have added K, the curvature of space itself. So the Friedmann equation is telling us that if you know the energy density, the number of ergs per cubic centimeter in the universe, and you know the Hubble constant, the rate at which the universe is expanding, then you can predict the curvature of space.

Or you can do it in any other combination. If you knew the energy density (ρ) and the curvature of space (K), you could predict the Hubble constant (H). Or best of all, if you could separately measure the Hubble constant (H) and the curvature of space (K), then you would know what the energy density (ρ) is going to have to be.

This is going to be a crucial ingredient later on, when we're going to say that dark matter and dark energy are really 95% of everything that there is. You can say that if ordinary matter is 5% of the universe, and by looking at gravitational fields we find new stuff, dark matter that is 25%, and dark energy is 70%, how do you know that there's not something else that is even more, even much much more, than there is either dark matter or dark energy?

This equation provides the answer to that. If you can separately measure the Hubble constant and the curvature of space, you know the total energy density of the universe. You're not leaving anything out. The manifestation of the Theory of General Relativity is that everything gives rise to a gravitational field, so everything makes spacetime curved. Then if you can measure the curvature of space and time, you've really found everything!

So what we're going to be claiming later on in the course, is that we have measured the Hubble constant (H), and we have measured the curvature of space (K), so that we do know the energy density (ρ). We know how much stuff there is in the universe. When we add up ordinary matter, dark matter, and dark energy, we've reached that number. There is no room for anymore stuff in the universe that we haven't yet found.

There could be one percent more, here or there. There could be trace things that we don't know, yet most of the universe, we claim to now understand in these three terms. That's an important accomplishment.

More specifically, what we'd like to do as a working cosmologist, since it's nice to think we have everything, is to take this equation and solve it. So remember that what we want to do is to plot out the scale factor as a function of time. In other words, what was the size of the universe at different epochs in cosmological history? This is the equation that tells us that!

If we not only know the value of the energy density rho (ρ), the number of ergs per cubic centimeter of space, not only do we know the value of the Hubble parameter (H), and the value of the curvature of space (K), but if we know how the energy density changes as the universe expands, then we can solve this equation uniquely for the scale factor, throughout the history of the universe.

If we know what the energy density rho (ρ) is, and how it changes, the Friedmann equation can tell us a unique history for how close together things were in the universe as a function of time. So at different points later on in the lectures, we'll be saying things like, "When the universe was 1/1000th its current size, which is 400,000 years after the Big bang." How do we know when the universe was a certain size, what time it was in the age of the universe? This is how we know, by plugging into this equation and it tells us how big the universe is as a function of how old it is, the crucial question for all of cosmology.

So this is the game we're going to play for the rest of these sets of lectures. We're going to determine what is in the universe, what it's made of. Ordinary matter, dark matter, and dark energy, will be the answers. Remember in the past, the universe was radiation dominated, yet radiation today is very unimportant, so we even have to keep careful track of the radiation in today's universe, because it used to be important in the past.

Then we will use this equation to extrapolate from our current conditions, back to the earliest times in the history of the universe. With that extrapolation we will make predictions and then test them against stuff we'll observe today. It is never the case in this whole procedure that we say things like, "Well, it was Einstein who came up with this equation, so it must be right!" No, we are always trying to test these ideas.

So we have an equation and it seems to work pretty well. How well do we test it? How well do we have empirical data that says this equation must be right? So as we will see in a few lectures, we can trace this Friedmann equation back to when the universe was only a few seconds old after the Big Bang. We can use that extrapolation to make predictions about what the universe looks like right now, which will turn out to be true.

The Friedmann equation, and Einstein's equation on which it is based, are at least to a very good approximation, telling us something true about how the universe behaves. In lectures to come, we'll put that truth to use, to see how the universe has evolved.

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