astr-160: Frontiers and Controversies in Astrophysics
Lecture 23 - Other Constraints: The Cosmic Microwave Background [April 24, 2007]
Chapter 1. Implications of Research on Supernovae [00:00:00]
Professor Charles Bailyn: We've been talking about the Type Ia Supernovae and the data that they provide about the expansion and, as it turns out, acceleration of the Universe. And so, one way to summarize all that information is that what we have found out--we now know that the Universe is accelerating, not just expanding, but accelerating. And that acceleration comes about because of some kind of dark energy--or to turn it around, the label we give to whatever is calling that is dark energy. And this sometimes gets summarized by this quantity, which is the energy density of the dark energy relative to the critical density of the Universe. And the fact that it's accelerating means that this quantity must be greater than this other quantity, the density of matter, because the matter tends to pull things together.
And so, we know not only that it's accelerating, we know something about how much it is accelerating, and that tells us how much bigger the dark energy density is than the matter density. And we kind of know by how much. We kind of know how much bigger this is than the dark matter density.
But somebody actually asked the question that, you know, I keep saying, well, this is like ¾, and this is like ¼. If you add a bunch of matter, couldn't you just add a bunch of dark energy also, and end up with the same amount of acceleration? So, a comparable Universe. And the answer is, yes, you could. There's nothing in the supernova data that prohibits this from being .5 and this from being 1, as long as this is sufficiently greater than that by the amount necessary to give rise to the observed acceleration.
And so, if you plot these two quantities against each other--so, here's, kind of, 0, 0. Actually, this can be negative, but let's not go there.
1, 1. What you find out is that there's a kind of allowed region that sort of looks like this, allowed by the supernova data. Notice, this is--I'm assuming throughout that the dark energy really is the Cosmological Constant. I'm not thinking about Big Rip scenarios at the moment, because that would add a third dimension to this plot, and I don't want to do that just yet. So, this is all assuming that dark energy is the Cosmological Constant. But that's certainly a place to start. There's no reason not to do that.
And so, I've been kind of consistently claiming, as we've been talking about the course, that the real answer is somewhere like here. Yeah? That it's about .3 on the matter side, .25 and about ¾, &frac23, ¾ on the dark energy side. But, in fact, you could go--as far as the supernova care, you could have no matter at all and just a little dark energy, or you could have a lot of matter and a huge amount of dark energy. And you'd get the amount of acceleration, and therefore, satisfy the observational constraints.
But, in fact, it turns out, we know more than this, because there are other constraints on cosmology besides the supernovae. And, I have to tell you at this point that this course, the way I've presented it, is a little bit lopsided, because the other ways of constraining these two quantities are just as important as the supernovae, and I'm going to do them all in one lecture. And this is kind of in the spirit of the philosophy of this course, where we try and understand a few things in depth, rather than many things in breadth. But I'm going to talk about two other kinds of constraints on cosmology today, which are, in fact, just as important to modern cosmology as the supernovae are.
I like the supernovae, first of all, because they were the first evidence of dark energy. I still think they're the best evidence of dark energy. But it's also true that I think they're, in some ways, the easiest to understand. That's not quite right. What I mean is that you don't need a whole bunch of physics to understand what's going on, because it's just the expansion of the Universe, whereas some of these other things, as you will see, there's moments where I'm going to have to say, and then there's a whole bunch of complicated physics, and then, you get the result. And so, it's less satisfying to teach, and perhaps to learn, in a course like this, but equally important for modern cosmology. And what these other things do is they pick out where in the allowed region--region allowed from the supernovae, the real Universe actually is. Okay.
Chapter 2. The Cosmic Microwave Background [00:05:28]
So, with the understanding that either of the two things I'm about to talk about could have been five weeks of this course, let's move in and I'll try and explain, in brief, what these things tell us about the Universe.
The first is the Cosmic Microwave Background. This, we've encountered before. This was one of the proofs that the Big Bang really happened, as opposed to the "steady state." And, I don't know if you remember what we said about that, so, let me remind you. Recall what was going on. You're looking back in time as you look far away. So, as you look back, things are denser and hotter. And if you look back far enough, they get really hot.
So, let us imagine, we're looking at a redshift not of 1 or .8 or something like that, but at a redshift of 3,000. So, this is when the Universe was--the scale factor of the Universe was 3,000 times smaller than it is today. At that time, the average temperature of the Universe, about 10,000 degrees. Turns out, that's an interesting number, because 10,000 degrees is how hot it has to be for hydrogen to ionize. Ionization, you'll remember, is when the electrons disassociate with the protons, and then, you just have a whole bunch of charged particles moving around. And it turns out--so, before that, when the temperature is higher than this, you imagine that the entire Universe is basically filled with all this ionized hydrogen.
One of the features of ionized hydrogen, or indeed, charged particles, generally, is that they tend to be opaque. Photons don't propagate well through these things. Photons propagate well through neutral materials but not well through charged materials. So, before that, the Universe was opaque. It's like looking at the surface of a star or a big wall of opaque gas. You can't see any further. So, it can't look back any further than that. But if you look in any direction, and you're looking far enough away, you ought to see a big wall of 10,000-degree hydrogen.
And, in fact, we know what a wall of 10,000-degree hydrogen looks like, because there are stars whose surfaces are 10,000 degrees. And the Sun's a little cooler than that, but there are many stars that are 10,000 degrees. So, this should look like--so, in any direction you look, you should see something that looks like the surface of a star: hot ionized hydrogen. And this looks like the surface of a star.
And we know that that's wrong, right? You can--let's go outside and look up in some direction. If this were really true, then anywhere you look, you would see the surface of a star that's hotter and brighter than the Sun. And so, the whole of the sky would look like the surface of the Sun, only hotter. And obviously, that isn't the case. And the reason that that's not the case is because you have to remember how redshifted this is. It's redshifted by a factor of 3,000. So, the light has wavelengths that's 3,000 times longer than the light we see from stars.
So, let's work that out. You'll recall, λ that you observed is equal to λemitted, plus Δλ. Or, is equal to λemitted (1 + Z), because Z is Δλ / λemitted. And so, this is equal to the wavelength emitted, times 3,001, right? Here's the 1, here's the 3,000. We'll call that 3 x 103. And if the wavelength that's emitted is a nice optical wavelength like 5 x 10-7 meters, that looks green or yellow, times 3,000, that's 15 x 10-4. That's around 1 millimeter. Wavelengths of a millimeter. Now, wavelengths of a millimeter, you don't see with your eye. These are, more or less, what we call microwaves.
So, the prediction really is that in any direction you look, you see a whole--you see what would have been something that looks like the surface of a star, except it's redshifted so far that all you can see is microwaves instead of ordinary optical light.
This is true. This has been verified observationally, first in the 1960s. This is true. There is a Cosmic Microwave Background in any direction. Take a microwave detector, radio telescope. You point it in any direction. You see this stuff coming from the Universe. And this was taken, when it was first discovered, as evidence for the Big Bang and against the steady state. Because it was predicted by the Big Bang, because you assume that the Universe was much, much hotter and much, much denser in the past, which is what you need in order to create this stuff, and that by the time you're seeing it now, it's all redshifted.
Chapter 3. COBE and WMAP: Measuring CMB [00:11:27]
Okay, so that's fine, as far as it goes. One of the features of the Cosmic Microwave Background turns out to be that it's very smooth. And you expect that to be true, because you expect to see the exact same thing in any direction. It doesn't matter if you go back in time looking that way, or back in time looking that way. You expect the Universe to be denser and hotter in the past and about the same way. And so, sure enough, the Cosmic Microwave Background is pretty much the same everywhere you look. Same everywhere.
And, as further experiments were done, it really is very smooth. And it came--it started to-- the smoothness of this started to become a little embarrassing, because it can't be--you don't want it to be perfectly smooth. Absolutely, totally identical in all directions. Because the Universe, at the moment, is not perfectly smooth. We've got planets. We got people. We got galaxies. Then, we got big spaces where there's nothing. And so, if the Universe, at the time of the Cosmic Microwave Background, is going to evolve into our own Universe now, there must be some little perturbations that are going to grow up to be galaxies later on. If it were perfectly smooth then, it would have to be perfectly smooth now, and that's manifestly not the case. So, you suspect that there must be some irregularities, because the Universe now is clearly irregular, in the sense that it contains objects of various kinds.
And so, in the 1990s, it turns out, you can see microwaves from the ground, but it's better to do it from space. And so, in the 1990s, they launched a satellite called COBE. This is the Cosmic Background Explorer. This is a satellite launched in the early 1990s. And COBE finally found the variations in the Cosmic Microwave Background. And they were very small, ten--1 part in 100,000. So, 10-5 to 1.
Now, that contrasts with the variations we see in the current Universe. In the current Universe, the variations in density can be 30 orders of magnitude, 1030. Remember, the average density of the Universe is something like 10-27 kilograms per cubic meter. In here, we've got 103 kilograms per cubic meter. That's the density of water, for example. And so, the density of the Universe now varies by 1030, but then, it only varied by 10-5. So, the sixth decimal place in the density was found to be variable. The first five were always the same.
Now, 1030. And so, that seems like this Universe is quite different from this Universe. But there are ways that structure grows and that, I'll come back to in a minute. That's the second thing I want to talk about, is the growth of structure.
But, at least, there was some variation. People were very excited about this, because this, again, is a situation where something was predicted, and then, you build a complicated piece of apparatus, and you see what you predict. And that's always a good thing to have happen. That gives you confidence that you know what you're talking about.
And in their excitement, a number of scientists became, sort of, overexcited. There's a famous quote from Stephen Hawking. At the time, he said--about the COBE results, he said it's like seeing the face of God. This is a very dangerous thing to say, because, you know, most people in the world don't understand how similes work. So, he said, it's like seeing the face of God. And this was reported: Scientist Sees God. So, you have to watch what you say. And, you know, the Weekly World News got a hold of this and then it was trouble. All right. So, they saw God by seeing the variations in the Cosmic Microwave Background. Who knew? All right, now, then.
So, that was nice, but COBE was just the first crack at this. And, while they saw variations, they didn't see it very precisely. It was a pretty blurry image. So, then, there was another satellite, still operating, actually, called WMAP. WMAP is the Wilkinson Microwave Anisotropy Probe. So, anisotropy means irregularity. Wilkinson--Dave Wilkinson was one of the pioneers of this kind of radio astronomy, and he was the leader of this project until he died shortly before launch, and so, then, they named it after him.
And then, in 2003, WMAP announced its results, and they got a much more precise map of the Cosmic Microwave Background; so precise that they could measure the size of these irregularities. Not just how irregular they were, but how big they were. And that, it turns out, gives you cosmological information. And I'll come back to that in a second. But I want to start by showing you some pictures, here. Let's see.
All right. This is the COBE results. So, let me explain what you're looking at. What you're looking at is a map of the entire sky. They've projected it onto the sky. And the way it works is this. Down the middle here, that's the plane of the galaxy. That's where the Milky Way is going to be. And then, any position on the sky maps onto this oval. And so, this is a common way of expressing the whole sky. Now, this is actually kind of the raw data, where red--I can't remember which way it goes. I think red is slightly denser, or actually, slighter hotter, is the way they measure it, and blue is slightly cooler.
So, there's this kind of yin-yang pattern here. And that's because we're moving in some direction through the Universe. And so, if you look at the Microwave Background in the direction we're moving, then it's a little bit blueshifted, which makes it look a little bit hotter. Whereas, if you look back where we're coming from, that's a little bit redshifted, so it looks a little cooler. And so, we're moving in this direction, away from here, and that effect--it's a very small effect of our motion through the Universe--turns out to dominate the irregularities in the Cosmic Microwave Background.
But, of course, that's not what we care about. We want to see what the background itself is doing. It's also true that that's quite easy to remove, because we know how fast we're moving. You can tell from this what direction we're moving in, and also how fast. And you can just take that out of the whole map, and adjust the map to be how we would see it if we weren't moving anywhere. That looks like this.
And now, there's a clear signal. This is the Milky Way, here. There are sources of microwaves that aren't the Cosmic Microwave Background. They're objects in our own galaxy that emit a lot of microwaves. But the thing is that the spectrum of the microwaves they emit is very different from the spectrum of the Cosmic Microwave Background.
And so, if you take not just an image of this, but you take an actual spectrum, which they did, you can tell the difference between things coming--emission coming from the background, and emission coming from actual objects in our own galaxy. And so, you can take out the ones--you can artificially, digitally, remove the objects that have the spectrum associated with galactic objects. And so, you can do that. You can take this out. And that's what you end up.
I should say that the scale between red and blue changes by about a factor of 10 between each one of these plots. And so, the difference between red and blue, here, is much, much smaller than the difference between red and blue up here.
Anyway, this was the face of God, according to Hawking, because it wasn't smooth, and because you could, after some exhaustive error analysis, convince yourself that you really believe that these pieces of the Cosmic Microwave Background are actually hotter, and therefore denser, than these pieces of the Cosmic Microwave Background.
So, that was a big triumph. And these irregularities, then, grow up to be galaxies, or groups of galaxies, or groups of groups of galaxies, because, you know, they're pretty big. They extend, you know, some significant fraction across the whole Universe. This is, you know, a map of the whole Universe, because it's mapping the entire sky. So, they're not going to grow up to be individual galaxies, but they are going to grow up to be irregularities in the Universe.
Okay, so now, this is COBE in the 1990s. To compare that to the WMAP results. Here is, on a different visualization color scheme. WMAP changed colors on us so that you wouldn't get confused between their results and the COBE results. So, this is the COBE map again and this is what WMAP saw.
And you can see that it's seeing the same things. This over-density here, that's this stuff here. This over here, that's these, only it's seeing it much, much, much more precisely. And, you know, it's kind of the difference between a Hubble Space Telescope image and a ground-based image. It's seeing the same thing, but the resolution here is much, much higher. And you can see that you might have a chance of measuring the size of some of these clumps in this image in a way that you wouldn't be able to do in this image here. And so, that was what WMAP was about, was actually measuring the sizes of these little bumps in the Cosmic Microwave Background.
So, why is that a good thing to do? And here's where, if I had five weeks, I'd explain that to you, but I don't, so I won't. So, let me just say that from the sizes, you can get cosmological information. And I'm not going to tell you how that works. Basically, it comes about because if you--what you find out from this is that there are certain sizes-- the clumps like to be certain sizes, and there are very few clumps that are other sizes, so they're particularly favored sizes. And the favored sizes turn out to relate to how old the Universe is at that moment and what has been contained in the Universe up until then.
But I won't go into that in any more detail except to say that the outcome of this is that what you constrain, what you find out from this is what they refer to Ωtot, which is the sum of Ωλ and Ωmatter. And this turns out to be equal to 1, to as close as they've been able to measure it. It's about 10% by now. One, probably plus or minus, I don't know, 0.1. It's maybe a little better than that by now, a few percent.
But now, that's a very good thing to know. Because, remember what this looked like from the supernovae alone. The allowed region of the--so, here's 1 and 1. The allowed region of the supernovae looks something like this. The Cosmic Microwave Background has these two things equaling 1, and so, you force yourself to be on a straight line that looks like that.
And so, the two of these experiments put together gives you a much better constraint on what's going on. So, this is from the Microwave Background. It's actually some kind of region that looks like this. And so, now, you've got an allowed region that's much smaller than either of these two experiments individually. Okay.
Chapter 4. Growth of Universe Structure: Large-Scale Clustering [00:24:40]
As I said, there's a third constraint and this comes about by the growth of structure. You have to get from these tiny perturbations that, after great effort, have been seen in the Cosmic Microwave Background, those had better grow up to be galaxies or groups of galaxies or something. So, at Z = 3,000 you have 10-5 perturbations. At Z = 0, you have 1030plus--even greater than that, you know. Basically a black hole is infinitely dense, so I guess you could have infinite perturbations, if you think about it from that point of view.
And you've got to get from one to the other. So, how does this work? Supposing you have a slight over-density. Well, what happens? The region that is over-dense, a little denser than the stuff around it, has more gravity than the surrounding regions. And so, it then pulls in material from the neighboring areas, because the gravitational force on any particular piece of mass will put it toward the place where the matter is densest, because that's where the gravity is coming from. Pulls in nearby material.
But, of course, that action makes it denser, because it's just pulled in more stuff from the nearby region, and it makes the other parts less dense. And so, this is a runaway process. Then, the denser regions are denser still. They have yet more gravity. They pull stuff in even faster. And so, you can start with tiny perturbations, and they grow and grow and grow and grow, until it becomes the difference between a galaxy and not a galaxy, or a group of galaxies. Turns out that the--even on the WMAP plot, the kinds of perturbations we're looking at are what's called large--this leads to the Large-Scale Structure, which is distributions of galaxies and galaxy clusters.
So, there's individual galaxies. Galaxies like to live in clusters. Turns out, the clusters are grouped together. That's the Large-Scale Structure. And you can plot this by using--by doing galaxy surveys. And the problem with the galaxy survey is, how do you know where it is in the third dimension? How do you know how far away it is?
So, you have position in the sky. That tells you two of the coordinates of a galaxy. And then, you measure its redshift. And you use the redshift in order to make a guess at how far away it is, just from using the Hubble Law. So, instead of trying to determine the distance in order to measure the value of the Hubble Constant, you assume a value of the Hubble Constant that you take from somewhere. And then, if you measure--then, you can--and, once you've assumed the value of the Hubble Constant, you can use redshift to tell you what the distance is supposed to be.
And so, then, you get a 3D map of the sky, and that tells us what the structure is now. You can then compare that to--you can then simulate the growth of structure by computer. You start with the Cosmic Microwave Background, with what you know about the Cosmic Microwave Background. You apply the laws of gravity. And you can also put it in a little dark energy if you feel like, to change how the--and that actually does change how the growth works, as you might imagine. But you can stick in any cosmological parameters you want, and just apply them as the Universe grows from a redshift of 3,000 to now. And then, you determine what the simulation looks like now, where now is defined to be, you know, 13.7 billion years after you start. And you ask, does it look alike? Look the same as the observations?
Now, along the way, when you're applying gravity in all this stuff, you have to assume cosmological parameters--how much dark energy there is, but especially, how much dark matter there is. Because, it's the matter that's pulling these density perturbations together--how much dark matter and some of its properties. And so, then, what you do is you just run your computer program a whole bunch of times, over and over again, with different amounts of dark energy, dark matter, and stuff like that. And you see which one of these simulations produces a Universe that actually looks like the one we see. So, which input parameters give you the observations?
So, this is, in general, how computational science works. This is how, you know, the big global warming models, all these kinds of things of work this way. You put in as much physics, as much chemistry, whatever, as you can understand. Make a big computer simulation. You pick a set of starting conditions. In this case, that's easy, because you observe the Cosmic Microwave Background. You run the thing for a while, and you ask, does it look like real life? And to the extent that it looks like real life, you believe that you've put in all the important physics and all the important science into your simulation.
Then you ask, all right, given the results of this simulation that seems to represent real life, what does it predict? So, you can do things, for example, in the global warming models, you add a whole bunch of carbon to the atmosphere, and then you ask, now what's going to happen? And with various results.
And so, this is a kind of example of a general kind of science that is now more and more frequently being done--this kind of computational simulation of reality, which you then check in some way against reality. And then, to the extent that it works, you use it to predict reality in other sets of circumstances.
So, let me show you some of that. All right. Here's a map of the Universe. These are observations. This is a famous redshift survey. So, what you've got is redshift going up this way, or actually, they plotted in velocity and position on the sky in this direction. This is kind of a slice of the Universe, so it's got no third dimension coming out of the blackboard, because they just haven't used it. And, you can see, each dot is a galaxy. They are not distributed randomly in this plot. This little guy, here, is obviously hugely over-dense. And there are also places--they think they're complete out to the edge here. There are also places where they're almost no galaxies. These are called voids. There are walls and voids.
And then, as they took more and more slices to build up the third dimension, they ended up having things that look like this. And there are clear over-densities at certain--this is named the Great Wall. There are clear over-densities in some places and clear under-densities in others. And it turns out that the Large-Scale Structure of the Universe is sort of like a bubble, some soap bubbles.
And so, they've got walls, and then, there are voids in the middle of the walls. And where those walls connect are particularly dense clumps. Another way you can think about it is that there are these kinds of these filaments, things that go through space where there are unusually large numbers of galaxies.
So, the most recent redshift survey, which goes out much, much further than the one I just showed you, that was the first. That one sort of goes out to here. But now they've the 2DF, Two Degree Field. This is a telescope that can look at--measure the redshifts of galaxies within two degrees of each other all at the same time. So, you get many, many redshifts. I think they have close to--maybe 100,000 redshifts. Here. Each blue dot is a galaxy.
And you can see the structure. You can see the structure really cleanly with these void areas surrounded by walls. And then, when the walls run into each other, there are particularly dense clumps of galaxies. So, that's the current structure of the Universe that we want our simulation to end up looking like. Yes?
Student: Why does [Inaudible]
Professor Charles Bailyn: Oh, because after around here, they stop being complete. They just don't see all the galaxies, because there are faint ones that they can't measure. And so, beyond about here, they're just not measuring all the galaxies that are there.
Okay, so now, let me show you a simulation. Let's see. Come on. Here we go. All right. I'll come back and I'll tell you what you're looking at in just a second. But here is the evolution of the Universe as done by computer. All right. So now, let me--where are we starting from? The redshift is up here. And so, we're going to go from a redshift of about thirty, when the Universe was thirty times smaller than it is, to a redshift of zero, which is today. And we're going to look at this box. This is a little chunk of the Universe. Each one of these dots is supposed to be a galaxy. Now, at a redshift of thirty, they're not galaxies yet. They're just over-densities of gas.
Now, two things are going to happen, since a redshift has started by now. One is that the box is going to get thirty times bigger. The whole Universe is going to expand. And that is included in the simulation, but we're not going to include it when we make the movie, because it makes it hard to see what's going on.
And so, if you'll remember the coordinate system, the way it works is there's this scale factor times x, y, z position. We're taking the scale factor out. So, this box is a single part of the Universe. It's going to expand by a factor of 30, but you're not going to see the expansion here. All you're going to see is the motion in x, y and z, not the scale factor, as these things moved toward each other.
Now, as you start out, it looks like it's pretty evenly distributed, but, in fact, there are little perturbations in the density. And what will happen is the galaxies will stream toward the perturbations and the perturbations will get bigger. And the--keep track of redshift up here. The rate at which the redshift changes, varies with time, because what we're actually counting, here, is not redshift, but time steps. And the relationship between time and redshift is not obvious.
So, let us take another look at the growth of structure, here. See them all moving towards each other, collecting in these filaments, abandoning whole regions into voids. And then, you end up--well, you end up back where you started, the way this film loop works. But let me show you a different visualization of exactly the same data.
You know, this is all on the computer. You can do anything you want. They make these great things where you sort of fly through the Universe while it's evolving and stuff. I don't think--That's amusing, but I don't think it's very educational. One thing that is useful: they took that box and they just kept rotating it as this film looped right around it. And that's useful, because you get to see these structures from different angles, and I think it makes it easier to visualize. So, let's try that.
See, it's the same simulation, except the box is rotating, and so, you get a better sense, I think, of what these structures actually look like. I'll do it once more. As they collect into these filaments and the filaments collect together. And so, what you can do with the output of something like this, with what it is at the very end--whoops, I missed. Try it one more time. Let me see if I can pause before it gets--before it regroups.
All right. So, here is a small redshift. When you get to this point, there are mathematical ways of saying, you know, how clumpy is it? How stringy is it? And you can compare the results of simulations of this kind with actual observations from the redshift surveys. And so, in this particular case, it worked. So, let me go back to here--yes--oh, it's down here--whoops. There we go. There, that's where I want to be for the moment.
And the consequence of these kinds of things is, it turns out that most, as I said before, mostly what you constrain, and so, you compare the simulations to redshift surveys. And the consequence of that is that you constrain the amount, and also, to a lesser extent, the type, the behavior, of dark matter. Because it's the matter that's pulling things together. And in a way, you can think about these little clumps and filaments. That's places where the Universe has already stopped and contracted. So, it had a little extra density. So, it hit--so, it was beyond the critical density. And so, that little piece of the Universe stopped and turned around, even though the Universe as a whole continues to expand.
Chapter 5. The Degree of Clustering and the Potential Fates of the Universe [00:39:30]
All right, so, now. What do we have? Let's go back to the plot. There's 1. Here's 1. And so, we have the supernovae forcing you to be in this box, here. We have the Cosmic Microwave Background forcing you to be in this box, here. And now, we're going to have a straight vertical line in this plot, because we know how much dark matter there is from the structure simulations.
Now, there are two possibilities. Supposing it comes in here‒this is a good answer, because then, everything is consistent. Alternatively, however, it could come in here. This would be awkward, because if it came in there, there would be no answer that would simultaneously satisfy all of these constraints. And so, then, something is wrong either with one of these determinations of the cosmic parameters, or with the theory as a whole. Could be either observational badness or theoretical badness.
And remember one ‒ you know, we did assume ‒ in even making this plot, you're assuming that you're dealing with a Cosmological Constant as the form of the dark energy. And, as we discussed last time, it's possible that it isn't constant, that it's something else. And so, if you ended up--if these things didn't work, you'd have to ask yourself, do I have any clue how the dark energy works? And the answer would be, no, and you would kind of have to abandon the Cosmological Constant.
I think that's the easiest way out if things don't work, because dark matte--we don't know what it is, but we do know what it does. It exerts gravity and pulls things together. So, no matter what the answer to what it is, its gravity will still be the same. Dark energy--not only do we not know what it is, we don't really know what it does, except that at the moment, it's causing a certain amount of acceleration.
And so, if you were satisfied that all the observations were right, you might also think that your simulations have left out something important. That could easily be that you don't fully understand how structure evolves from a Z of 3,000 to now. That's certainly true at some level, and so, there are various ways you could get out of it.
But the nice thing about it is it's a test of cosmology. Basically, if you have one measurement of two variables, the variables being the amount of dark energy and the amount of dark matter, that doesn't tell you--all that does is it establishes a relationship between the two variables. It doesn't actually measure either one of them. And so, you get some allowed region like this or this, but you don't actually make a measurement.
If you have two measurements of two variable--if you think of it in a mathematical sense, this is two equations and two unknowns. This you can solve. So, then, you measure values for--well, in this case, Ωλ and Ωmatter. And that's nice. Then, you can, you know, quote what the values are, once you've done this, to however--whatever the precision of your measurement turns out to be.
If you have three measurements of two variables, that's a test of the theory, because it is not guaranteed that you'll get a consistent result. In general, if you write down three lines in a two-dimensional space, you don't pick out a single point, because they don't all cross at the same place. And so, if they do, then you've got a consistent result. And then, you have tested the theory because your prediction is that all these three lines cross at the same place. They don't have to do that. So, if you measure something and it works out that way, then you've tested the theory and, kind of, come out on top.
So, let me end today's class by showing you the answer. This is what we know about the Universe. Let's see, what have I done with my--oh, there it is. This is the allowed region of the supernovae. And what they've done is, in each of the cases, there's a sort of--I think it's a one sigma region. So, there's a &frac23 chance it'll live inside here, and a--;well, over 90% chance that it lives inside the broader area.
So, the supernovae force you to be here. The Cosmic Microwave Background is the green result and it forces you to be here. And then, the clusters, Large-Scale Clustering, force you to be here, and they cross. So, this is the big triumph. You've tested the theory by measuring it in three different ways for two parameters--namely, the dark energy density and the dark matter density.
Now, there are a couple other interesting things to note on this plot. One is that, in some cases, the Universe expands forever, and in some cases, it re-collapses. And we're well into the expansion region. There's also the question of the geometry of the Universe. The more stuff there is, either dark energy or dark matter, the more curved the Universe becomes.
And so, you can ask yourself, is the Universe curved in such a way that it curves back on itself? Is it basically flat, giving you Euclidian geometry, or does it curve backwards and extend infinitely? And that's these three situations, here. If the sum is equal to 1, then the Universe is flat and Euclidian. And so, that separates these two.
Now, as somebody pointed out a few classes ago, if you have too much dark energy, then you never cross 0. And you never have a Big Bang, because the acceleration has been so great that you must have started out coming down and been bent up toward where we are now. That's this region up here. No Big Bang. And so, we're well away from that as well.
And so, this basically sums up what we currently know about the expansion of the Universe. That we have a constraint going this way, a constraint going this way, a constraint going this way. And you can see why I say that in a kind of philosophical sense, all three of these measurements are equally important, because it's the fact that you've got three of them that allows a test of the theory.
And we've talked a lot about the supernovae. You could talk just as much about the Cosmic Microwave Background or about Large-Scale Clustering, and the comparison of simulations with redshift surveys. We obviously aren't going to have time to do it, but this plot summarizes what we know about the Universe. Okay, thanks.
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