As promised in the last lecture on energy density (rho), we will now discuss the curvature (K) from the Friedmann equation. Like we were able to map the gravitational field in a cluster of galaxies by use of lensing, can we map the curvature of space in the universe by measuring the angles of some very large triangle?

Since the universe certainly appears to be flat, any curvature must be small. In fact, the Friedmann equation predicts the curvature K to decrease as the inverse of the scale factor squared. But compare this with the matter density similarly decreasing, but even more quickly at the third power. The early radiation dominated universe is similarly decreasing, but even quicker still at the fourth power. So the curvature actually grows over time in respect to the quickly fading matter and radiation. After 14 billion years of growth, we should be able to see the curvature if it exists. Otherwise the universe must be flat or the curvature is minute beyond belief. Enter the flatness problem!

Some standard sized object used as a ruler could allow measurement of curvature, just like we already found a standard candle in supernovae for measuring distance. But in addition to its standard size, we need to also know the distance of the standard object to complete the triangle. Such a triangle would tell us if the value of K is negative, zero, or positive. The 2D representations of these three 3D curvatures of space are a saddle, a plane, or a sphere. Their angles would add up to be less than 180 degrees, equal, or greater than 180. Objects themselves would also appear to be different sizes due to the light paths being pulled apart, not pulled at all, or pulled together. So they would appear smaller, the same, or larger, depending on K.

The greatest temperature variations in the CMB were caused by blobs of matter with just enough size to collapse during the 380,000 years before recombination. So we have both distance and size! These blobs would appear one degree across in a flat universe, and indeed in 2000, Boomerang observed such structures. This confirms that something is supplying the needed energy to make the universe flat. Dark energy does that for us, thus giving another reason instead of just supernovae, to support its existence.

It can be thought of as vacuum energy, negative, zero, or positive. It's smooth throughout space and persistent over time. It is inherent in spacetime itself, equal to the overall energy density. This cosmological constant was deleted by Einstein, but is now brought back as dark energy.

Yet the vacuum energy should be much larger than what we observe compared to what is expected. This is the new issue dubbed The Cosmological Constant Problem.

Another is the Coincidence Scandal, an analog to the flatness problem where K and ρ just happen to be so much alike at this instant of time. Why do the values of concordance cosmology; 70% dark energy and 30% dark matter plus ordinary matter, also just happen to be so alike at this point in the universe?

It's also curious how the universe started out being radiation dominated, then matter dominated, and will end as dark energy dominated.

The answers to this are now only speculations, and will occupy the rest of the course, along with alternate theories.

We're sort of in a good news, bad news, situation right now, with where we are in the concordance cosmology. The good news is we have a model that fits all the data. We have 5% of the universe as ordinary matter, 25% dark matter, and 70% dark energy.

Since 1998 when we first found evidence that the universe is accelerating, this idea has been tested in many different ways. It keeps coming out and passing the tests. It seems as if the universe is accelerating, just as the first supernovae observations indicated.

The bad news is we don't understand it. The dark energy part in particular is very exotic and outside our ordinary experience. In fact, the dark matter part is also very exotic. Ten years ago, we'd be giving a set of lectures just on dark matter, and that would also seem very exciting, exotic, and interesting. These days the dark matter seems almost prosaic compared to the dark energy, which is something truly different.

So clearly we want to do the best we can, to test this idea. To figure out whether or not, this hypothesis that 70% of the universe is some smoothly distributed persistent form of energy density is in fact correct. We want to use that hypothesis to make predictions, and then go out there and test those predictions. The most obvious test we can think of is the one we mentioned in the last lecture, the geometry of space.

So here is, once again, the Friedmann equation, derived from Einstein's General Theory of Relativity, that governs the relationship between energy density and the curvature of space in an expanding universe:

((8πG)/3)(ρ) = H² + K

What we're saying here is that the energy density of the universe, rho(ρ), is given by the sum of the contribution from the expansion (H) and the curvature of space itself (K). It turns out that the right amount of dark energy you need to make the universe accelerate, is also the right amount of dark energy you need to make the universe spatially flat. In other words, this satisfies the Friedmann equation, with the spatial curvature term (K) being zero.

That is a testable prediction, since you can try to imagine measuring the curvature of space itself. So in this lecture we'll actually go out and measure the curvature of space on very large scales. In other words, what we're doing is the same kind of thing we were doing when we used gravitational lensing to measure the weight of a galaxy or a cluster of galaxies.

What we were doing then was letting light rays go by a cluster or some other massive object, and we were mapping out the local geometry of space, near that gravitating object. Since Einstein tells us that space and time are warped and bent by the presence of matter, mapping out the geometry of space and time is a way to tell how much stuff there is.

So we can do that very straightforwardly with individual clusters of galaxies, yet now we're going to do that with the whole universe. We're going to look at how light propagates through space, to measure the geometry of all of space, all at once. That's how we can be sure we're not missing anything. When we're looking at individual clusters, it's always possible in principle that there's something outside. When you look at the whole universe, you're guaranteed to find everything that there is.

So here are the possibilities for what the curvature of space could be. Remember what this means. Since the universe is uniform everywhere, there is a certain single number which tells us how much space is curved. That number could be positive, it could be zero, or it could be negative. That number is the curvature of space.

It's related to Ω, the density parameter, which tells us what the total density of the universe is, divided by the critical density in the sense that when Ω=1, the universe is spatially flat, when Ω>1 it is positively curved like a sphere, and when Ω<1, space is negatively curved like a saddle. You can think of the directions of curvature of space, bending in different directions.

Now the curvature of space is something, like many of the things in cosmology, that is hard to visualize in reality. Whenever you're drawing pictures, representations of the curvature of space, the best you can do is draw a two-dimensional version. The actual space in which we live is three dimensional. As far as we know, this three-dimensional space in which we live, is not embedded in any bigger space, so it is all there is. So when we're talking about the geometry of space, we're talking about the intrinsic geometry of space, not how it looks to an outside observer, but things you can do when you are inside space, to measure it's geometry.

When we say the geometry is flat, we mean that the kind of geometry that Euclid invented thousands of years ago, is the right kind to describe space on very large scales. Euclid looked at tabletops, as a paradigm for where to do geometry. You would draw right angles and triangles, and come up with different laws on how geometric shapes would fit together.

We want the three-dimensional version of that, so we want to draw triangles, straight lines, and circles, within a big three-dimensional space, looking at their intrinsic properties. So there are different intrinsic properties that are telling you how space is possibly curved. The most obvious one is the angles inside a triangle. If you draw a triangle on a tabletop, or a triangle just on a three-dimensional space that is flat, no matter how you draw that triangle, no matter how it's oriented or how big it is, when you add up the angles of those three angles inside, you will always get 180 degrees. The way to think about the statement, "space is flat," is to translate it into the statement, "every single triangle I can draw anywhere in the universe, has angles inside that add up to 180 degrees."

If space is positively curved, a similar statement would hold true about triangles, yet it would say that the angles always add up to more than 180 degrees. We can visualize that by imagining our drawing a triangle on the surface of a two-dimensional sphere. It would be the case that the angles inside add up to greater than 180. So we have the three dimensional version of that. We're imagining that space itself in which we live, is a three-dimensional version of a sphere. Just like on a regular sphere, if you start at one point and walk around, you will eventually come back to where you left. So in a three-dimensional sphere, in a universe with positive spatial curvature, if you walk off in one direction, you will also eventually come back to where you left. It will take you billions of years, yet you would eventually get there.

Negative curvature then, means that you once again draw a big triangle, so that in a negatively curved space, every triangle you can draw, has angles inside that add up to less than 180 degrees. That's the kind of thing you can do, yet it's not the only thing you can do. A famous postulate of ordinary Euclidean geometry is the parallel postulate. It says if you start with two straight parallel lines and let them go, if the lines are initially parallel, they will always remain exactly parallel. That is to say, if they're initially having the property that the distance between the two lines is a constant, not changing as you go past the lines, that will always be true no matter how far you go.

That is a postulate of Euclidean geometry, yet not a necessarily true fact about the world. If we lived in a positively curved space, two lines that were initially parallel, would eventually come together. That makes perfect sense. Thinking about a sphere, you can draw two lines as parallel as you want, yet if you trace them down, they will eventually hit each other somewhere. In a negatively curved space meanwhile, you start with two parallel lines, you follow them down, and guess what? They're going to peel off, becoming more and more far apart.

That is a thing you can empirically imagine doing in space, that would reveal whether space is positively curved, negatively curved, or flat. So we've briefly touched upon something called the flatness problem in actual cosmology. The flatness problem has a sort of informal version and a formal version. The informal version is that the universe seems to be close to spatially flat. When you calculate the actual density of matter in the universe, putting aside dark energy for a bit, if you knew only about the matter, you would say that we've reached 30% of the critical density necessary to make the universe flat. Now 30%, as these things go, is awfully close to 100%. It makes you think that we're almost there. So there's this feeling that you're just missing something, and should be exactly flat if you knew what everything was. That's the sort of informal version of the flatness problem.

Yet there is a more formal version of the flatness problem, a statement that is a little more quantitative that drives home, exactly how surprising it is, that the universe is close to being spatially flat, yet without exactly being there. Basically it is a statement that the universe doesn't want to be flat! If the universe is a little bit non-flat, if there's a little bit of curvature to space, the amount of curvature becomes more and more important as the universe grows. So if you live in an old universe, it should be very curved, if there were any curvature at all.

You can see this roughly from the Friedmann equation once again. It relates the energy density of the universe, rho(ρ), to the sum of the contribution from the expansion (H) and the curvature of space itself (K). So you have three terms in that equation. Yet remember that we have very well-defined set of rules about how ρ changes as the universe expands.

For ordinary matter, for dark matter, or for any particles that are slowly moving compared to the speed of light, as the universe grows, the density goes down and the volume goes up, so that ρ goes down exactly like the number density goes down, since particles just become more dilute.

For radiation, ρ goes down even more quickly, because the number density goes down. Space becomes bigger, but also, every single particle of radiation is losing energy. So we have very well-defined rules about how ρ changes as the universe expands.

There's also a very well-defined rule about how K changes as the universe expands. You can imagine blowing up a balloon, which looks very curved when small, yet as it gets larger, every little cubic inch on the balloon looks flatter and flatter. In the same way, curvature dilutes away as the universe expands. However, when you plug in the numbers, K dilutes away more slowly than ρ in matter or radiation. To be technical:

K goes like 1/(scale factor)²

ρ of matter goes like 1/(scale factor)³

ρ of radiation goes like 1/(scale factor)^4

So as the universe gets larger, as the scale factor grows, ρ in matter and radiation, fall off more rapidly than the contribution to the spatial curvature. So imagine that there is some non-zero spatial curvature in the very early universe. That the term K in the Friedmann equation is not exactly 0 at early times. Even if it's fairly close to zero, the relative importance of that term, compared to the importance in ρ of matter or radiation, grows. We live today in a very old universe of 14 billion years. It's had that long for the spatial curvature to overtake and overwhelm the ρ of matter and radiation.

Yet it hasn't! That is the technical statement of the flatness problem. In order to get a 30% of the critical density universe today, in just matter and radiation, the difference between being absolutely flat and being curved in the early universe, had to be incredibly infinitesimal and finely tuned to just the right, tiny, tiny, tiny amount, so that today it would be comparable to the value of ρ in matter and radiation. This seems like an unlikely situation, and that is the flatness problem.

So 30% of the critical density, Ω=0.3, the amount that we've found in ordinary matter, is a weird number to have. That's why before we found the dark energy, theoretical physicists who were very convinced by this, were thinking that Ω matter, the contribution to ρ just from ordinary matter plus dark matter, had to be 1.0. They were believing that the universe had to have the critical density, Ω=1, since it was so close that it would make no sense to not quite go all the way. They believed in a flat universe.

Yet at the end of the day, you can believe whatever you want, it's not going to make you any money, you still have to go out there and look. So how do you actually measure the spatial geometry of the universe? Ideally you want to do what Karl Gauss did, when inventing the concepts of non-Euclidean geometry. He actually made a big triangle and measured the angles inside.

So we want to make a big triangle in the universe. Of course we can't travel to distant galaxies, stringing lasers or apiece of wire from one place to another, we have to take celestial objects as they're given to us, and use them to construct a big triangle, adding up the angles inside.

So one way to do this is if we had a standard ruler. A standard candle is something whose brightness we know. So the further away it is, the dimmer it will look. A standard ruler is something whose size we know, so the further away it is, the smaller it looks. So a standard ruler is just as good a way to measure distances to objects as a standard candle is. The reason you don't hear as much about standard rulers is because there just aren't as many of them. There aren't that many objects in cosmology or astrophysics whose size is miles across that you actually know ahead of time. Galaxies, stars, different astrophysical objects, appear in different sizes.

However imagine that you not only had a standard ruler, but imagine that you actually knew how far away it was. Imagine you had some object whose size you knew and whose distance you knew. Then you would think that you must be able to figure out exactly how big it will look. That would be true, assuming that you knew the geometry of space. The angular size that the ruler will take up, if you know how big it is, and it's distance, is telling you the geometry of space.

If it has a certain angular size, one degree across in a flat universe, then in a positively curved universe, it will look bigger than one degree across. That's because the angles your subtending from the light rays that come from you to the object or vice versa, are pulled together by the positive curvatures. So they can start out parallel and they will end up at the different angles of your standard ruler. Similarly if you have a negatively curved universe, that ruler which would have looked one degree across, is now going to look smaller than one degree across,

However, this is of course, asking a lot. You're asking to have some object, not only whose size we know, yet we also want to put that object in some place and know exactly how far away it is. How lucky do we have to be to have an object that is a fixed size and a known distance? Well, we got lucky! The universe provides us with exactly that, in the form of the temperature fluctuations in the CMB.

We already talked about the CMB, the leftover radiation from the Big Bang. It's a snapshot of what the universe looked like about 400,000 years after the Big Bang. At times earlier than that, the universe was so hot, that individual atoms were ionized and the universe was opaque, so light could not travel very far, before bumping into an electron. After 400,000 years, the universe had cooled down enough that electrons and nuclei had gotten together, the universe became transparent, and light traveled unimpeded through the universe. So we see what the universe was looking like about 400,000 years after the Big Bang.

The universe at that early time was much smoother than it is today, where it is smooth on large scales, yet on small scales we see individual planets, stars, galaxies, and clusters. At very early times, the universe was smooth on essentially all scales. It's the tiny deviations from perfect smoothness of that early time, that grew under the force of gravity, into stars, galaxies, and clusters.

So if you look at the CMB, observing not only that it exists, but also delicately measuring the temperature of the CMB at different points in the sky, you're measuring the imprint of those primordial fluctuations in density. We see the classic picture of the CMB, from the WMAP satellite. It's an all-sky picture, the whole sphere as we look out onto the sky, then projected onto an ellipse in this image. What we're seeing are just the tiny fluctuations in temperature, at only one part in 100,000. The blue spots are a little bit cooler, the red a little hotter. This is telling us where the density fluctuations were located, on a sphere 400,000 years after the Big Bang.

So we can learn a lot from these density fluctuations. We don't know any theory that predicts a precise place for where they should be, and don't ever expect to even have such a theory, yet we do know the statistical properties that they should have. Already in an earlier lecture, we talked about the fact that the properties of these splotches on the CMB, the hot spots and cold spots, provide evidence for dark matter. That's because what you have in the primordial universe, is a hot plasma that is experiencing acoustic oscillations.

You have an ionized plasma, yet in some regions is a little bit more dense, while in others a little bit less. Under the force of gravity, a region is going to collapse under its own gravitational field and heat up. That will increase the temperature fluctuation in that region, so it's now hotter than it used to be. The dark matter of course, continues to fall in, yet the ordinary matter bounces due to pressure, and it becomes less hot than it used to be. That's also a fluctuation, yet now it's a lower temperature rather than higher. It goes back and forth exactly like that.

Let's ask a question. What size would you expect the largest amount of fluctuation to be? The point is that if a region is going to collapse that is very large, it has no time to do so in any appreciable way, so its temperature fluctuation is going to be whatever it was stuck with in the very early universe. It it's too small, the region collapses, and expands, and collapses, and bounces back and forth, until it gets damped. So eventually it settles down into a configuration that is not very fluctuated or different from the surrounding plasma.

However, there is one particular length for which a region has time to collapse and heat up, becoming fairly substantial in terms of the difference it has in temperature, yet doesn't have time to bounce back yet. That will correspond to a physical size in light years, of the age of the universe in years at the moment when the CMB was formed.

In other words, since the CMB is a snapshot of 400,000 years after the Big Bang, regions that are 400,000 light years across, will have the greatest amount of fluctuation in their temperature. So we have a prediction for at that moment in the CMB, we know hot large the most noticeable hot-spots and cold-spots should be, which is 400,000 light years across.

Furthermore, we know how far away the CMB is. Using the Friedmann equation, using some theory of what the universe is made of, which we have, we can predict the distance from here today, to there, 400,000 years after the Big Bang. In other words, we have all the ingredients for making a big triangle! We have a standard ruler on the microwave sky. It's the hot and cold spots with the largest amplitude, and they are predicted to be one degree across, if the universe is spatially flat.

In other words, if the universe is spatially curved, positively or negatively, that's like taking the map you would predict in a flat universe, and either magnifying it, or shrinking it. So these are the data that we have from WMAP, and we can compare them to the prediction. What is the answer? The answer is that the universe is spatially flat!

This was in fact, not first found by WMAP, as there were other experiments that found this. The Boomerang experiment from Antarctica was actually the first to make a precision measurement of the size of the most dominant fluctuations in the CMB. They all knew ahead of time, what they were looking for, so if it was one degree across, that would be evidence that the universe was spatially flat. This was in the year 2000, soon after dark energy had been discovered, when people were still very excited, though skeptical. They wanted to know if this picture was hanging together?

So the Boomerang experiment and other experiments following up on it, looked for fluctuations in the CMB of one degree across, and that's exactly what they found. In other words, the CMB is telling us that the universe is spatially flat, and that the total density of stuff in the universe is the critical density. It fits perfectly with the concordance cosmology of 5% ordinary matter, 70% dark energy, 25% dark matter.

So exciting this was, that by 2003 when WMAP came along, Science magazine declared it to be the breakthrough of the year, even though what they really did was just say that, "We didn't make a mistake in 1998!" Back in 1998, the acceleration of the universe was the breakthrough of the year, which suggested the concordance cosmology. Then in 2003, WMAP came along and said, "Yes, that's right! This crazy universe with 5%, 25%, 70%, really is the universe we live in." This was sufficiently surprising, that again, it was the breakthrough of the year!

One thing to take very seriously is the fact that the CMB tells us the value of ρ. So if you just take evidence from the CMB, that ρ is the critical density, so that Ω=1, and you add that to the evidence from clusters of galaxies, ordinary galaxies, gravitational lensing, that matter only adds up to 30% of the critical density, then you don't need to mention the word supernova to be convinced that there's something called dark energy, that is 70% of the universe. It's just 100% - 30% = 70%.

In other words, whether or not we have a correct theoretical understanding of the physics of type Ia supernovae, whether or not the two supernovae groups did a good job in explaining their error bars and collecting data that is reliable, you're still forced to the conclusion that dark energy exists. The constraints that we have right now, over-determine the kind of universe in which we live. You can't wriggle out of the conclusion that there's a lot of dark energy, just by being skeptical about the supernovae results.

So the place we are, is that we're stuck with a universe in which 70% of ρ, is dark energy. We now must face up to the same kind of problem that we had with dark matter. Given that there is this stuff, what could it be? Let's just foreshadow a little bit, by saying what the simplest possible candidate for dark energy would be, and that is something called vacuum energy.

Remember that the two important properties that we know dark energy has, are that it is spatially uniform, more or less the same in every location in space, and it's persistent in time. The density ρ, the amount of energy per cubic cm, isn't changing very much for the dark energy as the universe expands. So the simplest idea for something that is more or less smooth in space, and more or less constant in time, is something that is exactly constant throughout both space and time. In other words a form of ρ that is inherent in spacetime itself. It's just the statement that every little cm³ of space, contains energy, whether there's any stuff in that cm³ or not.

So we see an image that is the closest Sean could come to an artist's representation of what this idea of vacuum energy, really looks like. You imagine taking a little cube of completely empty space, and ask yourself the question of how much energy is there in this cm³ of empty space?

According to General Relativity, there's no reason why the answer has to be zero. The energy density of empty space, is a constant of nature. It could be negative, positive, or zero. That number, whatever it is, is the vacuum energy. So the hypothesis that this energy is the correct value to explain 70% of the critical density, fits with the hypothesis that this is the dark energy. The dark energy is in fact, the energy density of empty space, the vacuum energy.

This is the same thing that Einstein was talking about when he first invented what he called the cosmological constant. Einstein added this term to his equation as we'll talk about later. Because he was not able to explain a static universe within his theory. We now know the universe is not static, so Einstein said this was a great mistake of his. Yet now we're bringing back the cosmological constant, that's exactly equivalent to this idea of vacuum energy.

So that's a very good idea. We don't need to complicate the idea, the idea that the dark energy is vacuum energy inherent in empty space, plus our ideas about dark matter and ordinary matter, are enough to fit the data. Why then do we even contemplate other possibilities? Well there's one point, which is that we were just surprised with the existence of dark energy itself. We had a preference, a prejudice ahead of time, that matter made up the critical density, and we were wrong! So our prejudices about what ideas are simple and make sense, shouldn't be taken too seriously. We're keeping an open mind about this possibility.

In fact, just like the flatness problem, there are sort of naturalness and fine tuning problems associated with the concept of a vacuum energy. One is called the cosmological constant problem, which is just the statement that once you admit that there can be vacuum energy, you can start asking how big should it be? The answer is, as we'll see, that it should be much larger that what we would observe! Once you admit that there can be any energy density in empty space, the surprise is that's it's so tiny.

The other problem is called the coincidence scandal. This is an exact analog to the flatness problem. Remember the flatness problem, said look, ρ in stuff that decays away very quickly, the equivalent of ρ in K decays away slowly, so why would it be the case that we are born right at the right moment, when these two things are comparable to each other? These two things evolve with respect to each other, K and ρ. Wouldn't it be a coincidence if there we were, at the right time to observe both of them?

This kind of argument convinced people that K must be zero, that we must live in a flat universe, which turned out to be right. Yet exactly the same set of words applies to the vacuum energy. It doesn't evolve at all. As a function of its ρ, it doesn't change as the universe expands. It doesn't go up or down. Yet ρ in matter or radiation, does go down as the universe expands. These two numbers change with respect to each other by quite a bit.

However, we're inventing a universe now, claiming that it fits the data in which 30% of ρ is matter, and 70% is vacuum energy. Those numbers are close to each other. In the past, it was almost all matter, in the far past it was almost all radiation, in the future it will be almost all vacuum energy. Why are we lucky enough to be born at the right time when the vacuum energy is comparable to the matter in the universe? That's the coincidence scandal. Sean actually has no good ideas for why that might be the case!

So we have a theory that fits the data, and we keep getting more data that the theory keeps fitting. Yet we recognize that the theory we have, has holes in it, in the sense that we don't understand why certain parameters take on the values that they do. That will encourage us to keep looking at more theories with different possibilities. So in the rest of these lectures we'll take some of these very seriously.

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