5. Galaxies and Clusters - Sean Carroll - Dark Matter, Dark Energy: The Dark Side of the Universe

This lecture throws some concepts and history at you a little too fast. The course guidebook could be much more helpful if it included everything that Sean covered. But maybe there is a limit on the amount of pages per lecture. Yes, Sean does go that fast. Astronomy is my background and even I had to hit reverse several times to keep up.

So far we've assumed the universe to be smooth everywhere. So we plugged special relativity's space-time into general relativity's dynamic universe to get the Friedmann equation. That told us a great deal about the universe, but we would not be here if the universe was totally smooth. This lecture talks about the non-smooth universe and how it implies the existence of dark matter.

We see lumps in the early universe of 1 part in 100,000. To talk about how that imbalance causes formation from the atomic scale to the cosmological scale is nearly pure speculation. We really don't know much about how our own solar system formed. The solar nebulae hypothesis is so bland it makes me cringe every time I hear it. It's perfectly all right to talk about what we think happened, but it should be presented with the caveat of near ignorance on the real details.

Billions of years in gravity fields produces stars and galaxies which appear not to have enough visible ordinary matter, nor even ordinary matter to account for the dynamics of galaxies and their clusters. This work of Rubin and Zwicky is all in the course guide, so I tend to concentrate on what gets left out of the guide or what is not presented clearly.

It's funny how the famous character of Fritz Zwicky is presented differently in this course then in Alex Filippenko's course. I guess it's just expected from the nature of the lectures, but Alex gives respectful insights as he also pokes fun at some of the legendary Zwicky stories. Sean passes over without any mention of it. But Rubin agrees with Zwicky that five times the amount of ordinary matter is needed to account for the observed dynamics. Some of the techniques are not very clear or described too quickly, like the temperature gradient.

Is the dark matter ordinary? According to theories on the composition of light elements and the cosmic microwave background, the matter is not ordinary.

Since we observe the universe as flat, we call the overall energy density ρ (rho) from Einstein's equation that would make a flat universe, the critical density. But we can only account for 5% of the ordinary matter required for that critical density. Dark matter adds up to only 25% of that required for the critical density. So their total of 30% is close enough to 100% that for years it was thought that we would just discover the remaining ordinary or dark matter that makes up the missing 70%. But the critical density and overall energy density are constantly changing, even with respect to each other. The current total of 30% is a strange number because it is very close to 100%. It could be much much less than that. In the past it was actually much greater, which implies we are currently in a transition to a much smaller percentage. But that seems to be very strange and unlikely.

Only ten years ago astronomers explained it by thinking we just didn't have a good measure of the universe. The recent discovery of dark energy indeed does account for that missing 70% to make the universe flat. Yet like dark matter, its composition is in some mysterious form.

We now have under our belts, a picture of the entire universe, which is a pretty good one really. It's an impressive fact that we even have such a picture, since 100 years ago we knew almost nothing that was correct about the universe on very large scales. Yet now we have a picture that fits the data quite well.

The parts we've talked about are basically the fact that when you look into the sky, you see that it's filled with galaxies, so the universe is big. By a galaxy, we mean some collection of stars moving under their mutual gravitational attraction, maybe 100 billion stars per galaxy. There's perhaps 100 billion galaxies spread throughout the observable universe, and they're all moving away from all the other ones. So the universe is getting bigger as a function of time.

A crucial fact about this distribution of galaxies through space is that it's more or less uniform. The universe is smooth on very large scales, which is an incredibly helpful feature as far as cosmologists are concerned, because that means they can approximate the universe as being absolutely smooth. That's exactly what we did when we looked at what Einstein said about space and time and how they are joined together in one thing called spacetime in his Theory of Special Relativity. The idea that spacetime is dynamic, so that it can expand and have a geometry which can change with time, this is Einstein's Theory of General Relativity.

So if you have a universe that is perfectly smooth and looks the same everywhere, than what you can do is plug that into General Relativity and ask how such a perfectly smooth universe changes as a function of time? How does it expand, where does it come from, where will it go? We did that and derived an equation, the Friedmann equation, which relates the energy density in the universe to the way that space is curved and to the way space is expanding.

In today's lecture, we're going to go a little bit beyond that. We see a picture from the SDSS (Sloan Digital Sky Survey) of how galaxies are distributed in our local region of the universe. We said that they look pretty smooth, which was a good approximation and was very helpful. Today we will admit and take seriously the fact that they're not completely smooth. There is in fact structure there. Some of these galaxies are clumped together into groups, clusters, walls, sheets, or filaments that stretch across the observable universe.

At first glance, as cosmologists, we say that this is kind of annoying that the galaxies are not perfectly smooth, since we can really solve a lot of problems in terms of writing down equations and finding solutions, if everything really were perfectly smooth. On the other hand, it's certainly good news that there are local structures in the universe, that the density is not absolutely constant throughout space. We are much more dense than the air around us, which in turn is much more dense than the space between the stars. So there's structure in the universe, and today we're actually going to take advantage of that structure, by using it to weigh the universe, to find out how much stuff there is, in particular how much matter there is in the universe.

Of course we've already given away the punchline, so what we'll find is that there's a lot of matter in the universe, more than we seem to be able to account for with ordinary matter, with atoms, hydrogen, helium, that we're familiar with in the periodic table. Yet today we won't get quite that far. We'll just give evidence that there's a lot of mass in the universe, much more than we see directly. Yet it will take future lectures to show that we're actually pretty sure that the mass which we don't see, is some absolutely new kind of particle.

So lets think about the process by which the universe went from being very smooth, to having structures in it, to having galaxies and clusters of galaxies. We see an iconic image in cosmology, the picture of the Cosmic Microwave Background (CMB), which is the leftover radiation from the Big Bang. When the universe was less than about 400,000 years old, it was opaque, hot, dense, and giving off light. Yet that light didn't get very far, as it just kept bumping into free electrons. Yet once the universe was older than 400,000 years, it became transparent, and that light streamed across the universe unimpeded. That goes with this image we see right here.

This is a snapshot of what the universe looked like when it was about 400,000 years after the Big Bang. Now it's about 14 billion years after the Big Bang, so a lot of time has passed, and we can see the effects of this passage of time. This picture actually shows the slight differences in temperature from place to place in the CMB. The blue areas are a little colder, while the yellow and red are a little bit hotter, yet only very slightly so!

The amount of variation in temperature of the CMB is only about 1 part in 100,000. So what that means, is that there were slight variations in the density of stuff back at that time. The density of the universe from place to place, some 400,000 years after the Big Bang, was nearly perfectly smooth, yet not exactly so. We may have had 100,000 particles in one region of space, and 100,001 in this other region, and 99,999 in that region, tiny variations over an ultimately very smooth distribution.

So in lecture 11 we'll be talking in great detail about what kinds of patterns we see in the CMB, how they got there, and what we learned from them. Today we'll just mention that the universe was a lot more smooth back then than it is today, where we see galaxies in some places, yet no galaxies in others. There's a huge amount of contrast between the density in a galaxy and the density outside.

So what happened from the microwave background era to today, is gravity. It worked on these tiny perturbations, these tiny ripples, in the density of space, to increase their magnitude. Gravity works as a contrast knob where you turn up the contrast in the universe. You take dense regions and they become even more dense, and less dense regions in the universe become even less dense.

What happens in that region where there's 100,001 particles is that the force due to gravity is a little bit more than in the region with only 100,000. So the region with one extra particle, pulls things toward it. Then it has two extra particles, so it has a little bit more mass, so it can pull even more things to it. This is a very slight effect and takes literally billions of years to happen. Yet due to this effect, we go from an almost perfectly smooth distribution, to one that, though still very smooth on large scales, on small scales has stars, planets, and galaxies.

So that happened over the last 14 billion years. Gravity turned up the contrast knob on the universe, and it did so in a hierarchical fashion. What that means is, on smaller scales, things can happen more quickly. It takes less time to build up a density region in some very tiny region of the universe, than it does over megaparsecs. Such dense structures that are millions or billions of light years across, will literally take millions or billions of year to create, respectively.

So the universe forms in a bottom-up way. You first form planet and star sized things, than galaxy sized things, clusters of galaxies, and beyond. So it's no surprise that we see more structures on small scales than we do on large scales in the universe.

We mentioned before that when you actually go and look out into the sky, what you see are a collection of galaxies. So here again is the Hubble Deep Field, an image of a randomly selected part of the sky. There's not anything special or especially interesting in that region, but the universe is more or less the same everywhere, so it doesn't matter.

We take a picture in some region of the sky and we see that it's filled with galaxies. One thing we notice about them, is that they're very pretty, beautiful, and aesthetically pleasing. Personally Sean has always wondered just why that is the case? He's really sure why we find galaxies beautiful, and it's hard to imagine some explanation of this from evolutionary psychology! A million years ago when we were hunters and gatherers, we didn't know what galaxies were looking like. So it's hard to imagine some kind of pressure on us, so that we would look at the galaxies sometime in the future and find them to be beautiful. Yet beautiful they certainly are!

The other thing is that the galaxies look different. Some are red, blue, smooth, or irregular. Partly this is because we are looking at galaxies from different moments of time. When you look at the sky, because the speed of light is finite, one light year per year, you're looking backwards in time. If you see a galaxy that is a billion light years away, you're seeing it as it looked a billion years ago. So it's not surprising in some sense, that the galaxies look different. We're taking snapshots of them at different stages in their evolution.

Yet on the other hand, even today the nearby galaxies that are all the same age, still do look different. Galaxies have a slightly individualistic character, which is going to be important once we get to talking about what we can learn by studying individual galaxies.

So what we're caring about in this course, dark matter and dark energy, is how to weigh the galaxies. How do we know how much stuff there is in a galaxy? Well the simplest thing to do is just look and count. To say, well you know there's 100 billion stars and that the average mass of the star is something like the mass of the sun, you're going to guess there's about 100 billion solar masses worth of stuff in a typical galaxy.

Yet it's always possible that, since even 100 years ago, people recognized that there was stuff in galaxies that we couldn't see directly. So you want a better way to get a handle on the total mass of the galaxy. How do you do that? Of course, the answer is given to us again by gravity. Einstein showed that gravity is universal. Everything that exists in nature, everything that has energy, will give rise to a gravitational field. So even though you don't see it directly, you can tell how much stuff there is in a galaxy, by measuring its gravitational field. The more mass there is in the galaxy, the stronger the gravitational field will be.

Now in fact, we don't really even need Einstein's laws to tell us this, since Newton's laws could have done it. Einstein invented a theory of gravity that is better than Newton's theory, that can describe things which Newton's theory can't describe. The expansion of the universe is not something that makes sense for Newtonian gravity, for Isaac Newton's space and time were fixed in absolute structures. He couldn't have talked about the Big Bang as some actual expansion of the universe itself.

However, Newton could have talked very easily about a galaxy, since it's in a regime of a part of the universe where Newton's theory of gravity is an excellent approximation for what is really going on, according to Einstein. So we can think about Newtonian gravity. We can think about the fundamental law of Newtonian gravity which is the inverse square law. According to this, if you have two objects, they exert a force on each other, proportional to their mass. So the heavier they are, the larger the force is. It's also inversely proportional to the distance between them, squared. So as things get further apart, the gravitational force between them gets less and less.

That means that as one object is orbiting around another one, when the orbiting object goes further away, it needs to orbit more slowly in order to follow an ellipse around the object it orbits. So in the solar system, this is perfectly clear. Mercury is moving around the sun very quickly, since it's the closest planet. The earth is moving at some medium speed, and Pluto is moving very slowly. It's not just that it takes Pluto longer, because it has longer to go, but it is actually moving more slowly around the sun, since the force of gravity is much weaker out there where Pluto is.

This is, in fact, how we measure the mass of the sun! We can't put the sun on a scale and weigh it. Yet what we can do is to look at the velocity with which planets are orbiting around the sun, and measure its gravitational field. Knowing the latter will tell us the mass of the sun.

So lets do the same thing with a galaxy. Let's measure the velocity of things moving around a galaxy. We see a picture of Andromeda, and we'd want to look at stuff that is on the outskirts of the galaxy. The difference between a galaxy and the sun, is that the galaxy is not all concentrated in one small region. It's clear where the sun is, and outside the sun, there's not a lot of mass.

Yet a galaxy is more spread out. It's much more diffuse. So you want to look at objects that are as far away as possible, since they would be testing the accumulated gravitational field from everything that they are orbiting. If you want to make sure you're getting all the mass, you want to go as far away as possible. The prediction from Isaac Newton would be that the further you go away, the slower things are revolving around the galaxy.

If you measure the speed at which they're rotating, you can figure out the mass. So this project was undertaken in the 1970s, by an astronomer at the Carnegie Institute in Washington, named Vera Rubin. She measured what are called rotation curves of galaxies. So she observed gas and dust that orbits around a galaxy. Her optical images in visible light, only show part of the galaxy, the biggest, most massive part.

Yet if you also use radiotelescopes, there is stuff you can also see, little wisps of gas and dust, further outside. So she could use those as gravitational tests for what the gravitational field is. What Rubin found is that even though she looked at gas further and further outside the visible galaxy, the velocity of the orbiting gas, did not decrease. As she went further out, the stuff kept orbiting at the same velocity, more or less.

The simple interpretation of this, is that as you go further out, you're feeling the effects of more and more stuff. You have not succeeded in getting outside the entire galaxy. There is still stuff outside. However, we've looked at gas that is far outside anything we can visibly observe inside the galaxy. So the conclusions from Vera Rubin's observations of the rotation curves of galaxies, is that there's stuff creating gravity in galaxies that we don't directly see. This extra stuff is distributed in a much bigger, puffier halo around the galaxy.

So if Rubin is right, and current cosmologists think she certainly is, the galaxy that we see is a decoration. It's a lit up little thing, inside a much bigger structure. A halo made of dark matter. If you plug in the numbers, what you find, according to Rubin's calculations, is that most of the stuff in the galaxy, most of the mass, most of the grams or ergs, is actually in dark matter, and not in the visible matter.

There's certainly also the possibility that we don't understand gravity. Perhaps Newton's theory does not apply in this regime. Yet that would be a big surprise, and we'll talk about limits on that later. So for right now, the simplest explanation is that there really is dark matter there.

Yet that's a surprising result, the first evidence that we have for dark matter being an important part of galactic dynamics. So we'd like to check that, since whenever you get a result that is this important, which says there's a whole new kind of stuff out there in the universe, you don't just accept it and move on. You want to find more evidence that there's something like that going on, or that contradicts it.

So what you can do is look at rotation curves of clusters of galaxies. Rubin measured the rotation curves around individual galaxies, which actually have good and bad features for such a purpose. The good features are the galaxies are very organized. They're very regular, and it's clear there is a disk, that you're looking at stuff which is outside the visible matter, and you can see what you're doing, basically.

Yet clusters are a little bit messier. They're not in disks, but are more disorganized into somewhat spherical shapes, so that you might not be able to measure as cleanly what is actually going on. You need to be a little bit more clever about the techniques you use to weigh the total amount of stuff in the cluster. We see a picture of a cluster of galaxies, where we see at least dozens of different galaxies. In fact a typical cluster will often have hundreds of galaxies in it.

The good part about clusters, even though they're a little bit messy, is that they have a good chance to be a fair sample of the universe. The worry when you look at an individual galaxy, is that they are all different. Some are small and ratty, while some are big and very smooth. Clusters are all big enough, that over the lifetime of the universe, there isn't time for different kinds of stuff to be segregated. Within a cluster, you basically are getting a fair sample of every kind of thing in the universe.

The analogy that astronomers sometimes use is that you're trying to measure the density of people in the US, in different places. Yet the data you have, is not the total density of people, but what you have is the membership directory for the American Astronomical Society, broken down by city. You can see for example, that there's a giant metropolis called Pasadena, California, and there's a tiny suburb called Los Angeles, California! There's a lot more astronomers in Pasadena, since Caltech and the Jet Propulsion Laboratory are there, than there are in Los Angeles.

Yet the problem is, the reason you reach this untrue conclusion, is because you didn't use a fair sample of the universe. Astronomers do not completely, accurately trace the population at large. They are a biased sample. So the good thing about clusters of galaxies is that they are not a biased sample. You get a big enough chunk of the universe, and what you see, should be representative of what is actually going on.

So the next picture we see is a computer simulation, not a photograph using data. It's the result of a computer trying to figure out how we went from an early universe with an almost smooth cosmology, to the late universe where we see these structures lighting up. The good news is that these simulations do a very good job of matching the data, so that we can understand in our computers, where different types of matter might be.

An interesting thing about these galaxies, is that they're helping us to answer the question of where the ordinary matter is located in galaxies. It might be that there's dark matter, and we'll say that there is. Yet even without dark matter, you can still say that just because you've seen a galaxy that's lit up, shining with stars, it doesn't mean that this is where the ordinary stuff is. What if there's just gas in between the galaxies? How do you know?

Again, clusters provide the way to answer that question, because they are a fair sample of the universe. What happens in a cluster is that the gas in between the galaxies is pulled into the center of the cluster, because of the force of gravity. So even if that gas is dark, and is not shining and forming stars, even if in the desolate cold of intergalactic space, it would be completely invisible, in the cluster where it clumps together, it heats up. The gas bumps into other gas and goes up to a very high temperature, so starts to emit x-rays.

We see an image from the Rosat x-ray satellite, of a cluster of galaxies which have a much smoother distribution than if we just saw it in visible light. We see that in between the galaxies which we thought we were observing, there's a lot of stuff there. This is the hot x-ray gas in between the galaxies.

We can do the calculation and ask how many hydrogen atoms for instance, are in the hot gas in between the galaxies, versus the hydrogen atoms actually in the galaxies, shining in the form of stars. The answer is that there's more mass in between the galaxies in ordinary matter, just in hydrogen atoms, than there is in the stars in the galaxies themselves! About 2/3 of the total amount of ordinary matter in a cluster, is in between the galaxies themselves!

So because we believe that clusters are a fair sample of the universe, that tells us that when we were weighing individual galaxies, we were missing a lot of ordinary matter. Forget about the dark matter, we didn't even get a fair sample of the ordinary matter, since most galaxies are not even in clusters. They're out by themselves and we can measure it, yet we had no way of knowing what the total amount of invisible hydrogen was. In a cluster of galaxies, we can find that. It's lit up, because it is heated up, and now emits x-rays.

So we can find out where the stuff is, in a cluster of galaxies. There is a complementary way of doing it, so not only can we look at x-rays being emitted by the hot x-ray gas, but we can also look at the shadow that the cluster casts on the CMB. This is coming from way back when the universe was only 400,000 years old. If a microwave from the CMB passes through a cluster of galaxies, it will scatter off of that hot gas. So when we look at the cluster, we see a dark spot instead of the CMB, because it's been scattered by the hot gas. That is another way, besides using the x-rays, to measure the amount of stuff, and we get a consistent answer. We find that 2/3 of the ordinary matter in a cluster of galaxies, is not in the galaxies themselves.

OK, so that tells us where the ordinary matter is located. Yet what about this dark matter stuff? Should we weigh the clusters of galaxies and get the same answer? Well it turns out there are two different ways to weigh the clusters of galaxies, one is to do the sort of analogous thing to what Rubin did with rotation curves. Unfortunately, clusters are not aligned in a nice little disk, so you can't get a nice little rotation curve. Yet what you can do, is measure the individual velocities of every single galaxy. There's a theorem in classical Newtonian physics, that the average velocity of every galaxy moving around, is related to the total mass inside. The more mass you have, the faster they will be moving. That is the analogous statement to the fact that the more mass you have in the galaxy, the faster something will be orbiting around it.

So this was first looked at, actually back in the 1930s, very long ago, by Fritz Zwicky, who was a Caltech astronomer at the time. He looked at the Coma cluster of galaxies, measuring their individual velocities. He plugged in the numbers and said, "Look, there's no way to explain these numbers on the basis of the matter we see. There must be matter there which we don't see."

He wasn't absolutely taken seriously at the time. It was sort of too early, and he was on the spot, before people were ready for it. People figured, "Well there was stuff there that we just don't see. There's hydrogen or whatever, ordinary stars and planets that we haven't taken into account. Someday we will find them" Today of course, we can do better. We can take account of the stars and planets, and we confirm that don't find enough.

The other thing to do is look at the temperature profiles of the hot gas, to find out the temperature of the gas, starting from the center of the galaxy, and moving our way outward. Not only does this tell us how much gas there is, but also the total gravitational field of the cluster of galaxies. That total gravitational field, depends on the total amount of mass.

The answer once again is completely consistent, namely that even though there's a lot more matter in hot x-ray gas in between the galaxies, than in the galaxies themselves, it is still not nearly enough to account for the total amount of matter in the cluster of galaxies, either by the dynamics of the galaxies zooming around, or by measuring the total amount of gravitational field, using the x-ray profile of the cluster. We get the same answer, which is that the total amount of mass is something like five times what we actually have in ordinary matter. This seems to be very good evidence that there needs to be something new in there, something that we call dark matter.

This kind of argument doesn't completely rule out that it's not some very sneakily hidden ordinary form of matter, but that it's pointing in that direction. So what can we learn from that? Is it possible that we're missing something? Is it possible that there is some dark form of ordinary matter that we haven't seen? It is certainly possible, just from what we've said so far.

In later lectures, we'll put an absolute upper limit on the total amount of ordinary matter in the universe. We'll do that by going back to the very early universe when it was one minute old, or a few hundred thousand years old, look at the relics we get from the early universe in the form of helium, lithium, and other light elements, and also in the form of the CMB. It turns out that properties of these relics from the early universe, depend on the total amount of ordinary matter in the universe, in a very specific way. Back then, ordinary matter was not separated into stars, gas, dust, and planets. It was all absolutely smoothly distributed.

So once and for all, we can get a measurement of the total amount of ordinary matter. We'll find that it's very consistent with what we think is there, from what we see from hot gas in clusters of galaxies and elsewhere. In other words, we have more or less, a rigorous argument that the matter which we now think must be there, from the dynamics of clusters of galaxies, cannot be ordinary matter hidden in some clever way. It has to be something new.

So lets' explore one way of thinking about what this answer is. We keep talking about the fact that the total amount of dark matter is about five times as much as the total amount of ordinary matter. Yet what units is that in? How should we think about this? Well remember the Friedmann equation is used to govern the expansion of the universe. It relates the total energy density on average in the universe, rho(ρ), to the expansion rate (H) and to the curvature of space (K):

((8πG)/3)(ρ)=H² + K

In other words, if you know (ρ) and (H), then you can tell what (K) is, right from the equation. In yet other words, there is one particular value of the (ρ), which for any given value of (H), will mean that the universe must be flat. In other words, there is a critical value of (ρ) that says if we measure (H), and know that the actual (ρ) is equal to that critical value, we know that the spatial value of the geometry of the universe must be Euclidean, flat, with no curvature.

This is not to necessarily say that we have that (ρ), but that is "a" density that we can define using the Friedmannn equation. So this is very convenient for astronomers, to measure the (ρ) that we see, in units of how much density you would need to make the universe flat, the so-called critical density.

It turns out that the ordinary matter we see in the universe, which we'll eventually show is limited very strongly by Big Bang nucleosynthesis and the CMB, and what right now we're inferring from the matter we actually see, is about 5% of that critical density. Ordinary hydrogen atoms and other forms of stuff in the periodic table, makes up about 5% of what you would need to make the universe spatially flat.

So the other stuff in the universe, the dark matter that we see evidence for in clusters of galaxies, then makes up about 25% of the critical density, which is about five times as much as the ordinary matter. So that's a very interesting result, because 5% + 25% = 30%. We see in matter, both ordinary and dark, about 30% of what you need to make the universe spatially flat, about 30% of the critical density.

The reason why that's an interesting number, is that 30% is close to 100%, yet it's not equal to it. So this number could have been anything, like one billionth of 1%, or several trillion percent! It could have been any number, yet it's 30%, which is pretty close to being 100%.

That makes us suspicious. So 10-20 years ago, talking to most cosmologists, they would have said, I bet that we just haven't found all the matter yet. Once we go and do better survey's of the universe, we'll bring that 30% all the way up to 100% and be much more aesthetically pleasing to have exactly the critical density, than if we had only 30% of it."

In the back of their minds was one other fact, which is that this 30% is a number which is not a constant. It changes with time as the universe expands. So the energy density changes in time in a very specific way, in a matter dominated universe, but the critical density also changes, and they change with respect to each other.

So if it's the critical density that we need, and if we only have 30% today of it, then in the past we had a much closer value to the critical density. Things were very close to being spatially flat. So if we really lived in a universe that had 30% of the critical density and nothing else, we would be right in the middle of a transition regime from a spatially flat universe, where the curvature was negligible and you couldn't measure it, to a universe that was highly spatially curved, and the density would be going from almost the critical density and swooping down to almost zero percent of it.

That's very strange to find us living today, in some important part of the universe's history. For a long time, it was just thought that we don't have evidence that we don't exactly have a good measure of the total amount of matter in the universe. Less than 10 years ago of course, in 1998, the true answer was revealed. The answer is that there is stuff out there, that is not matter. It's dark energy, that makes up almost precisely the 70% that we need to make the universe spatially flat. We do have the critical density in the universe, yet most of it is not matter, but something even more mysterious.

4. Cosmology in Einstein's Universe - Sean Carroll - Dark Matter, Dark Energy: The Dark Side of the Universe

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.

3. Space, Time, and Gravity - Sean Carroll - Dark Matter, Dark Energy: The Dark Side of the Universe

Sean says at the beginning that this is actually his favorite lecture. Now I know exactly what he means. I've never had the Einstein/Newton comparison made quite so clearly to me. The topic I always got tripped up on, is now made into something understandable. Relativity is now also my favorite subject, instead of the usual stumbling block. Sean is as much a physicist as cosmologist. If he can describe relativity so well, I can't wait for the actual astronomy aspects.

OK, enough of the hype. The lecture does proceed at a fast pace, not that he talks fast, but he goes through the concepts pretty quickly. I often wonder if any given lecturer could grasp all the information presented by someone else in a different field. Unfortunately the course guide does a rather poor job of conveying this lecture, so a second viewing wouldn't hurt, or reading this review might also work!

The concepts of Newton's "absolute" changing to "relative" in special relativity, and Newton's "fixed" changing to "dynamic" in general relativity is hammered home quite effectively. Special relativity is further clarified by contrasting Newton's concept of time being distinct from space, into Einstein's time being like space. Also Newton's limit of simply moving forward in time changes into Einstein's speed limit of light. General Relativity is clarified from Newton's concept of gravity as a force into Einstein's concept of space-time. The traditional representation of curved space-time as a marble on a sheet is used, but I've heard others critique this as flawed in the same way as Sean dissed the balloon and raisin bread analogies!

In any case, until now I have never seen a lecture actually spell out Einstein's field equation,

Rμν - ½Rgμν = (8πG)Tμν

The curved space-time equals energy and momentum, implying not only Newton's mass creates gravity but everything does. This infers we can use gravity to detect mass and energy, or the pull of dark matter and the acceleration of dark energy. This dynamic space-time allows one to rephrase the distance velocity relation as more space coming into existence between galaxies. New questions of why, how and where could be asked of space-time, although still not answered.

Sean Carroll conveys difficult scientific concepts at an appropriate pace. Compared to the recent lectures of Steven Pollock on Classical Physics and even Alex Filippenko's massive Intro to Astronomy, this course really sets a new level of standard in the TTC library of courses.

This is perhaps Sean's favorite lecture. We shouldn't hold him to this, since he may say that other lectures in the future may be his favorite! This is the lecture in which we take the expanding universe that we talked about last time, and we ask what it really means. We won't answer that question until the next lecture, but this question is going to take us into trying to understand how space and time work.

This is the real thing that made Einstein quite a famous international celebrity. He is the person who figured out how space and time work. He figured out how they're related to gravity. This is something in which Einstein did better than Isaac Newton, which is a hard thing to do!

So we'll go from what Newton did, up through Einstein, explaining along the way how we get there. It's really is a set of deep ideas about the structure of space and time that are counter-intuitive. So we'll be thinking pretty hard about why things are like that, and why they're one way than another.

So lets start just by thinking about what the ideas of space and time are supposed to be. Space and time are what help you locate things. Space tells you where things are, and time tells you when things are. So lets think about space to begin. Scientists like to say that space is three-dimensional. What does that mean? It means if you want to locate the position of an object, in principle you need to give three numbers. The way we can do that can be different, dependent on the circumstances. So for example, the three numbers might be the height of something, the length of something, and the breadth of something.

Once you get those three numbers, you've specified its dimensions entirely. If you only get two numbers, it wouldn't be enough. So we say that space is three-dimensional. Saying the same thing from a different way, if you want to meet someone for coffee, you need three numbers to say where to meet them. One way of thinking about that, would be latitude, longitude, and height above ground. Ordinarily we're lucky enough to know the height above ground for the people we're going to meet, yet you might tell them which of the two cross-streets you'll meet at!

So it takes three pieces of information to locate something in the universe, and that's what it means to say the universe is three-dimensional. Space is the arena in which things play out. Space is where you locate things, how two things come together, and how things are distributed all throughout the universe.

Time, on the other hand, tells us when things happen. So say we'll be meeting that friend of ours for coffee and give them the three numbers that say where we're going to meet them. That is not enough. We'll not successfully meet them if we give them only these three numbers, since we need a fourth number, which is when we're going to meet them.

So it takes four numbers to specify a unique point in the universe, the three numbers to tell us where we are in space, and the one to tell us what time it is. One could think about all these four numbers as being one, four-dimensional thing, called spacetime. For along time, people didn't think that way, because there was no point! Space is something very clearly different than time. Part of the revolution of relativity as Einstein understood the universe, is to realize that space and time are in fact two different aspects of the same thing, a single four-dimensional thing that we call spacetime.

So what we'll do in this lecture is start with Isaac Newton, who was probably without a doubt the greatest physicist who ever lived. One of the things he did was to put together a very sensible picture of space and time. In Newton's picture of space and time, they are both absolute, fixed structures of the universe. They are the stage in which the drama of physics plays itself out.

Now Einstein comes along with his collaborators also, since he wasn't just alone. By 1905 he put the finishing touches on something called Special Relativity. This was a replacement for Newton's notions of space and time. Special Relativity first said that space and time are different aspects of the same four-dimensional thing called spacetime. So in Special Relativity there was still a fixed structure, yet the thing that was fixed was one, four-dimensional spacetime, not separately the three dimensions of space, and the one dimension of time.

Then Einstein still thinks he's not done yet, and tries to incorporate gravity into his theory of Special Relativity, and realizes they are incompatible. He eventually throws away Special Relativity and replaces it with something better, which we call General Relativity. It's just as good as before, but now it also includes gravity.

Yet the most profound difference is that now, nothing is fixed. Nothing is given to you ahead of time in General Relativity. The very structure of space and time themselves are dynamical, so they can change. Spacetime has a geometry and a curvature which we actually interpret as gravity.

That's the short version of the story, so lets go slowly through the longer version. Here we see a picture of Sir Isaac Newton, who as we said, is arguably the greatest physicist and/or mathematician of all time. He invented calculus among other things, and among the pieces of physics he invented was the fact that when you pass white light through a prism, it separates into colors. He invented the law of gravity that we know and love, the inverse square law. The gravitational force between two objects goes down as they get further away. If their separation is doubled, the force of gravity decreases by a factor of four, which is the square of the factor of two for which we doubled their distance.

So this inverse square law of gravity will play a very crucial role in what is to come, but it's not what we'll be talking about in this lecture right now. What we care about today, right now, is what Newton said about space and time. So we think that space and time are so immediately obvious, that there's nothing more to say. That's where things happen, since space tells us how to locate things and time tells us when they happen. What more could one need to know?

One of the great features of the Newtonian revolution in physics, was being quantitative, attaching equations to things. It wasn't good enough to just say that space is where things happen. Now you needed to have some equations and mathematical structure. So Newton did that by telling us what it meant to say that space was an absolute, three-dimensional set of points, and that time is a different absolute one-dimensional set of points.

The basic picture of the Newtonian universe is shown here in a diagram. It's taking the four dimensions we need to slice up the universe, and slicing them up int moments of constant time. Sadly on a two-dimensional picture we can't draw all four dimensions of spacetime, so the two-dimensional planes that we've compressed diagonally in the picture, are supposed to represent all three dimensions of space.

So this picture should not be intimidating. It's a fairly straightforward representation of the universe, the universe according to Isaac Newton. We know better now, and we'll get to just why that is, but this is how spacetime worked according to Newton. There was space and there was time. So what we see here are little clocks, where each clock tells a different time located in a different moment of space.

So the Newtonian universe is one where space happens over and over again, at different moments in time. At every moment objects are going to be located in slightly different places. So we call that motion through space. Things change as a function of time. So that's how Newton's universe worked.

So the point we're emphasizing about Newtonian spacetime is that it is fixed and absolute. So what does that mean? Those are two different words. Fixed means it never changes. There is this structure that space and time are staying in, and don't change into something else. The rate at which time flows does not depend on where you are. There's one fixed thing called the flow of time, and it's the same all throughout the universe. That's what fixed means.

Absolute means it's the same for everybody. The rate at which time flows, or the distance between two objects, doesn't depend on who you are, where you are, or how you're moving through space. It's just a notion that for example, it took three seconds between this event and that event. Everyone agrees on the fact that it takes three seconds between these events, no matter where they are, or how they're looking at it, as long as they take accurate observations.

These notions will change, once we get to relativity. Absolute in the Newtonian sense, as opposed to relative. So the absolute fact that a certain object has a certain length, is going to turn into a relative fact in relativity. That means that the length of an object will depend on who is measuring it, possibly depends on where they are, and certainly depends on how fast they're moving. "Fixed" is what we'll get rid of when we go from Special Relativity to General Relativity, where spacetime is dynamical and can change, taking on different forms, no longer being absolutely fixed.

So to understand the change from Isaac Newton's absolute space and time, to Albert Einstein's relative notion of space and time, we have to understand how we'd go about operationally making sense of Newtonian spacetime? In other words it's one thing for Sean to say that there are moments of time that are the whole three-dimensional universe of space, and those three-dimensional pieces of space repeat themselves over and over again. Yet what does that mean and how does one make sense of it?

The way you do that is in something you see in spy movies all the time. You synchronize your watches. So when we say that there's a moment of time that extends throughout the universe, what that means is that you have a clock right here that says a certain time, and we can set up clocks all throughout space that agree with this clock. So when this clock says that it's 3:00, then it's 3:00 here and everywhere else! These are just a set of words. How do we make them real?

What we do is, first to set clocks throughout all the universe. That the easy part! We're doing thought experiments here, and these are not real experiments. We have a clock sitting at every point in space. Yet we have to synchronize our clocks, so we take the clock that we have here. We take for granted that we have good clocks, so these are clocks that run at the same rate. They are reliable clocks.

So we take the clock we have here and leave it. Yet then we take another clock and set it to say the same thing as our first clock, and then this second clock that agrees with the first one, we move throughout the universe. We move this clock as shown in the diagram from before, and move it to all the other clocks. When we get to another clock, we align them so they are reading the same time. By that procedure, we synchronize our clocks all throughout the universe. It doesn't matter what the clocks are doing, how they're moving, or where they are. When we do this, we take our one clock and make sure that every other clock, all throughout the universe, agrees with it, and then we are done. We've synchronized our watches as it were, all throughout the universe.

That's what Isaac Newton says happens. There's only one rule in Newton's universe, which says you have to move forward in time. In other words, if it's 3:00 here, then when Sean's watch goes one hour to the future, he will be at 4:00 no matter where he is in the universe, no matter what he's done in the meantime.

Then Einstein and his friends came along in the early 20th century and said, "Actually it's not like that!" We need to replace the absolute notions of space and time that Newton had, with relative notions of space and time. In particular the fundamental insight of Einstein can be summed up in the phrase, "Time is like space."

So what is that supposed to mean? Suppose you measured the distance between two points. You have a path that goes from one place to another. Well you have to actually go down that path, and measure the distance along it. That distance could be different, depending upon the path you take.

Einstein says that time is like that. The rate at which time flows, depends on what you do. This is really hard to get through your brain, because it's very non-Newtonian. In some sense, even though Newtonian mechanics is a great triumph of the human imagination, it's still quite intuitive and makes sense to us, once we understand it.

Yet Relativity is very counter-intuitive. Einstein says that the amount of time you feel elapsing is personal, so it's not a fact of the universe, it's a fact of what you do. It can be different for two different observers, even if they start and end at the same point.

So lets see how that works. Imagine that you do in Einstein's universe, that is to say the universe in which we actually live, what you tried to do in Newton's universe. You try to synchronize your clocks all throughout space. So you take the clock that you leave here, that's reading a certain time, you take your movable clock and make them agree, then send out your movable clock to synchronize all the other clocks all throughout space.

You think you've done it, but then you bring your clock back to where it started, and you notice that the two clocks don't say the same time anymore. It's not true that this clock, even though it's a perfectly good clock, clicked at the same rate as it traveled throughout space, synchronizing all the other ones. In fact, it turns out that in Special Relativity, there is no way to make that happen. You didn't make a mistake, since it's just a feature of space and time, that when a clock moves around, it measures a different amount of time than a clock that stays still.

This seems weird to us, because to make the effect obvious, you need to move close to the speed of light, which is the magical velocity in the theory of relativity, at which all these weird things become obvious. Since we're always in our every day lives moving much slower than the speed of light, we don't notice them. We think its very strange for someone to say that the amount of time that clicks off on our wristwatch, depends on how we walk, depends on the journey we take through space and time.

Yet once Einstein tells us that time is kind of like space, there is a way of thinking about it that makes perfect sense. So in a new diagram, we draw an analogy between distances and times, because Einstein says the two things work the same way. You draw two different curves that connect two points in space. One curve is a straight line and the other is a sort of curvy thing that goes back and forth.

Nobody in the world is surprised with the fact that the length along those two curves is different, even though they begin and end at the same point. Why should they be the same? They move in different ways! So Einstein says that time works in the same way.

Take two points in spacetime. Take two events, two points where you're both located in space and time. Two people begin at the same point, and they end at the same point, but they take different journeys. One just sits there not moving, and one zips around. In space, it was the case that the shortest distance between two points was a straight line. Any curvy path you went along, gave you a longer distance.

Einstein says that a very similar thing happens, with one interesting twist. The longest time that elapses between two moments is if you just sat there and waited for time to go by you. Yet if you zip off and come back, you will have experienced less time than your friend who stayed behind. The shorter time is the path through spacetime that zips around!

There had to be some difference, because we know time is not exactly like space, but Einstein is telling us that deep down, space and time work in similar ways. Just like the the distance around a path depends on how you move, the time that elapses on your wristwatch, depends on how you move through spacetime. So that's the fundamental difference between spacetime in Einstein's universe and spacetime in Newton's universe.

Yet remember there was a rule in Newton's universe, where you had to move forward in time. That rule gets replaced in Special Relativity, and in fact is more stringent than the one in Newton's universe. Newton's universe allowed you to move as fast as you wanted, as long as you kept moving forward in time. Yet in Einstein's universe, the speed of light is special. So if there's an event in spacetime, and you want to get somewhere else in spacetime, you can only plausibly get there if you move slower than the speed of light.

So if you imagine drawing a picture of spacetime, and you have an event where you imagine all the light rays that are leaving that event, those rays represent the ultimate speed limit. They represent the velocity faster than which you cannot travel. So there is no allowed trajectory through space and time, that moves faster than the speed of light. Everything you're allowed to do, moves up in that diagram, and eats up more time than space, if you like.

So that is the fundamental rule in Special Relativity. Newton's rule is one has to move forward in time, Einstein's rule is one has to move slower than the speed of light. That set of things, faster than which you cannot go, is called a light cone. The set of all points connected to your original event, by things moving at the speed of light. So the only thing you need to remember is that whatever the speed of light is, where you are in the universe, you need to move more slowly than that. That is the fundamental rule.

So this was a replacement as we mentioned, for Newton's concept of space and time. Newton was the greatest physicist who had ever lived. He invented not only this notion of space and time, but also the theory of gravity and many other things. So Einstein knew that if he's going to replace Newtonian space and time, he would have to reproduce the other successes that Newton had.

You had a very good theory of how gravity worked, which was able to predict how planets moved around the sun, or the moon around the earth. This theory of Newton's was verified to high accuracy over and over again by experiments. Yet here comes Einstein saying that it was not right. So clearly you have to reconcile the new theory of spacetime, Special Relativity, with what Newton thought gravity was. Gravity was the inverse square law, where the force was proportional to the mass of the thing that is pulling you, and inversely proportional to the mass between the two objects.

So after inventing Special Relativity in 1905, Einstein devoted himself to this task of trying to figure out how to reconcile gravity with his new notion of spacetime. It took him ten years to do it, and the way he ultimately did it was he realized you needed to throw away what Newton had said about gravity. The way that gravity was going to work in relativity, is just very different than how it works in Newtonian mechanics.

So here we see a picture of Einstein, which we've all seen before. Yet usually these are when he was in his older ages, and when he had let himself go a little bit! He's got the hair going up and he's wearing a sweatshirt for days and days! This is a picture from 1912 when Einstein was young, a sharp dressed man, and somebody was combing his hair. He was thinking about the nature of space and time and how to make it compatible with gravity. Ultimately he hit on a thought experiment that provided the key to understanding how gravity would work in the context of relativity.

The thought experiment actually traces itself back to Galileo, the famous Italian physicist who lived just before Newton, who really invented a lot of the framework that Newton made sharp and quantitative and attached equations to. Galileo was one of the best physicists of all time. One of the many thing he did, was to understand that gravity was universal. The famous experiment that he may or may not have done, he probably didn't but serves as a nice visual image, was to drop objects off of the leaning tower of Pisa, and notice that no matter what the objects are made of, no matter what the mass is, they always fall at the same rate. In other words, gravity as a force has a very strange characteristic where everything responds to it in the same way.

So that's different from, say, the force of electricity where the electromagnetic force has things called positive charges and negative charges. They move in different ways and in opposite directions in an electric field. Yet gravity has everything moving the same way. It has this feature of universality where everything responds to it identically.

So Einstein, because he's smarter than us, realized that this was the key to understanding how gravity works, the fact that it works the same on everything. What that meant was that gravity was undetectable! That's a bit of a surprise to say this, since it seems pretty obvious that if we let something fall, we see the force of gravity at work.

Yet what Einstein says was that imagine you're in a box (or elevator) so that you can't see outside. Someone asks you the question if there was gravity in that box, or in that room? Well you can drop things and see them falling, so you say yes there is gravity.

Yet Einstein says that you might be very far away from any gravitational field. It might be that you're in a rocket, being accelerated at a tremendous rate through interstellar space where there's no gravity around. In that rocket, if you drop things, they will fall in exactly the same way that they would fall if there were gravity! This is not true for electricity, since we can detect an electric field by dropping a positive particle and a negative particle, then seeing them move in different directions. Yet since everything moves the same way under the influence of gravity, you can never be sure in a small region of spacetime inside a tiny little box, if there is actually any gravity at all!

So again, to us we might just say, "Well that's interesting." Yet to Einstein, he says, "That's the key. If everything responds to gravity in the same way, then gravity is not a force at all, but a feature of spacetime itself." In fact, that very feature of gravity, is the curvature of spacetime itself. Einstein's most brilliant insight was to say that spacetime has a geometry.

By 1915 he came up with General Relativity which described how the geometry of spacetime, how the curvature of space and time itself, manifests itself in what we observe as gravity. So according to Einstein we have a picture of something like the following, although it's a somewhat colorful conception of what is going on.

We have the sun that is warping the space around it. The usual analogy used here, which is pretty good, is that you put a bowling ball in a rubber sheet, that distorts the shape of the rubber sheet around it. A marble that you rolled by the bowling ball would then be deflected by the curvature of the rubber sheet.

Einstein is saying is that this is what's happening in spacetime. That is what gravity really is. The gravitational force that we attach to the sun, is really the warping of the geometry of space near the sun. Earth is just doing its best to move in a straight line. Yet there are no straight lines because the geometry itself is curved. So what we see the earth do, is to orbit the sun. This orbiting of the earth around the sun, is just the earth's response to the geometry of space and time. This was Einstein's brilliant idea.

Now again it's not enough to just say words like that, since you need to make it quantitative. So Einstein has an equation known as Einstein's equation. This is not E=mc², although it's a very good and famous equation that is Einstein's, which we'll talk more about later, but what is known to physicists as Einstein's equation is the equation for gravity. We see it here, again not because we're going to understand it in any great detail, but it's nice to see just what it looks like, to get an appreciation for the art and poetry that are these physicist's equations.

Rμν - ½Rgμν = (8πG)Tμν

We see a left-hand side and a right hand side, so that part is easy. The left-hand side explains the way in which spacetime is curved. It's a set of numbers, in fact it's a 4x4 matrix, a little array of 16 numbers whose values are telling us by how much spacetime is curved. If spacetime is flat, just like a tabletop with no geometry, then all those numbers will be zero.

On the right-hand side, we have stuff. We have (8πG) which are just constants of nature, just fixed numbers. Then we have Tμν which is the energy and momentum of the stuff in the universe. Anything that is in the universe, comes along with energy, some substance, some heat, some pressure and things like that. Einstein is telling us in very specific equations that all of those things contribute to the curvature of spacetime.

Newton was telling us was that what makes gravity is mass, yet Einstein is telling us that what makes gravity is everything, every form of energy which includes mass but also includes heat, pressure, temperature, and stress. All of these contribute to gravity in some way.

So this equation, Einstein's equation, is the one that should be famous. It's a little bit more intimidating than E=mc², a little bit less immediately relevant to our lives, but it's a tremendous accomplishment to conceptualize gravity as a feature of spacetime itself, as a feature of the curvature of space and time.

So we're all allowed to ask, "Who cares?" Why should if matter that we use the words "gravity is the curvature of space and time" rather than "gravity is a force stretching through space and time?" Is there any difference between these two sets of ideas? Well there are differences, and they will make a big difference to us in our quest to understand cosmology, dark matter, and dark energy.

The first implication of the claim that gravity is the curvature of spacetime, is that universality, the fact that everything responds to gravity in the same way, implies as its converse that everything creates gravity in the same way. There's a universal coupling between the curvature of space and time, and the stuff in the universe. In other words, you can't hide from gravity. If there is stuff in the universe, if it has mass and energy, if it exists in any substantial or physical way, it will give rise to a gravitational field.

So Einstein is giving us a surefire technique for detecting absolutely everything in the universe! There is nothing that can hide from us, as long as we can detect gravitational fields throughout the cosmos. You can't hide from gravity, so you can go the other way by detecting gravity, and can infer that there must be stuff. If you see a gravitational field pointing in some direction, there must be some stuff there, causing that gravitational field, even if you don't see the stuff.

This will be the technique that we use to infer the existence of dark matter and dark energy. Dark matter we will say, must be there because we will see stuff being pulled in some direction where there's not enough ordinary stuff to explain it. Dark energy has a more subtle effect, which is that it makes the universe accelerate. We'll talk about how this acceleration of the universe is a manifestation of the curvature of spacetime in the way that would be caused by dark energy.

The second implication of the motto that "gravity is the curvature of spacetime," is that spacetime is dynamical. Spacetime can change as a function of time. It need not be the same in the past as in the future. So it gives us a changed viewpoint on how we would think about the fact that galaxies, for example, are moving away from us.

Hubble said that we have a velocity for galaxies that is larger if they are further away. The same phenomenon makes sense according to Einstein, yet we attach slightly different words to it. This is why, when we're being careful, we refer to the apparent velocity of distant galaxies, not the actual velocity. In the real, careful way of understanding things, a la Einstein, the galaxies aren't moving. They are sitting there, located in space. What's happening is that space is expanding. Spacetime itself is dynamical and can change, so what's happening is not that the galaxies have a velocity through space, but that the space in between galaxies is growing, stretching, getting bigger. More and more space is coming into existence.

This is a way of thinking about things that Einstein gives us, that allows us to ask questions which just wouldn't have made sense to Isaac Newton. We can ask for example, where did space and time come from? What happened at the beginning? Did space and time get created, or is there something before what we know think of as the Big Bang?

To Newton, with an absolute, fixed, spacetime, these kinds of questions really don't make sense. You don't even ask where spacetime came from, because it was always there! General Relativity is saying that space and time are dynamical and can change. You're at least allowed to ask the question of why they're here, and why did they come into existence? We won't be surprised to hear that we don't yet know the answers to these questions, yet we're hopeful that our understanding of dark matter and dark energy would be part of the clues that help us answer them eventually.