Frontiers and Controversies in Astrophysics

ASTR 160 - Lecture 20 - Dark Matter

Chapter 1. The Scale Factor [00:00:00]

Professor Charles Bailyn: Welcome back for more cosmology on a rainy day. You’ll recall, maybe, what we were trying to do. We’re trying to determine the past and future of the Universe–which, it turns out, can be summarized by a single number, namely, the scale factor of the Universe.

It’s actually not a single number. It’s actually a function. Right? It’s a function of time. The scale factor changes with time. So, when I make this grandiose statement that I’m going to talk about the past and future of the Universe, what I really mean is, I want to determine what the scale factor is as a function of time.

And so, that’s just a standard kind of a graph. Here’s the scale factor, usually given the letter a. Function of t. Here’s time. The present is somewhere along this graph. Call that now. And so, here’s–and the Universe has some scale factor at the present time. And what we’re going to do is we’re going to call that–define the scale factor at the present time to be 1. We’re just going to–that’s called normalizing: when you set–when you multiply something by some convenient number so that it comes out to be 1 at a convenient moment. So, that’s just a definition. We define the current scale factor to be 1, to be unity. And so, there’s a point here at–right now, the scale factor is 1.

We also know what the slope of the scale factor is, because we measure that from the Hubble Constant. And so, in an appropriate set of units, the scale factor is increasing at a rate of the Hubble Constant, which is, I think, 2 x 10^-18 per second. So, if the scale factor is one right now, then one second from now, the scale factor is going to be 1 + 2 x 10^-18. So, every second, it gets 2 x 10^-18 bigger. That’s not a real large increase and, in fact, over the entire history of humanity, it won’t increase by that much. Let’s see, in a million years, there is something like–okay, in 10⁶ years, there are 3 x 10⁷x 10⁶ seconds.

That’s 3 x 10¹³ seconds. Multiply that by 2 x 10^-18 and the scale factor increases by 3 x 10¹³ x 2 x 10^-18.

That’s 6 x 10^-5. So, it still hasn’t made a whole lot difference, even a million years from now. But, keep at it for a while, and the scale factor is going to pile up and will increase. So, we know that it’s increasing.

And then, the question is, “Is this increase changing with time?” So, let’s see. If it doesn’t change–if it’s just a straight linear increase, this is the situation in which there’s no matter in the Universe. So, the Universe is not slowing down because of gravity, because there’s no matter in it. And that gets labeled with this factor Omega [Ω] being equal to 0.

Let me remind you: Ω is defined as the density of the Universe divided by the critical density. The critical density is how dense it has to be to turn around. So, another possible track for the Universe: we know it’s expanding at a particular rate now, but it could slow down in the future, turn around and fall back–could look something like this. This is the case where Ω is greater than 1. And then, the intermediate case where Ω is less than 1, but not equal to 0. It sort of does this. It falls off a little bit, continues to expand, something like that. And this Ω less than 1, but still greater than 0.

Chapter 2. Accounting for Dark Matter in Mass of Universe [00:04:41]

Okay. So, here’s the goal. We’ve got to figure out which track we’re on. At the moment, we sit here, and we measure the Hubble Constant. So, all we know is where we are and where we’re going in the short run. And what we want to do is to figure out which of these possibilities the Universe is actually following. And in so doing, decide whether we’re going to end up falling back together in this Big Crunch or expanding forever.

And we talked a little bit last time about a kind of direct way of determining this, namely, measuring Ω. This Ω factor is the density of the Universe divided by the critical density. You could, in principle, go out and measure the density of the Universe. You can compute the critical density. You divide one by another, and you know which of these possibilities turns out to be true. And that ran into trouble, as you’ll recall. So, direct measurement of Ω. This leads to trouble in the form of dark matter.

So, the Universe turns out to be full of this dark matter. If you add up all the galaxies–the mass of all the galaxies, including dark matter–so, mass in galaxies gives you an Ω that’s less than 1, that’s around ⅓ or ¼. But there might be–you know, now that we’ve already discovered that the vast majority of matter is something we know nothing about, there might be more of it between galaxies that we don’t know about. So, we want a different approach.

So, here’s something you can do that’s really quite a different approach to this problem. Let’s draw that plot up again. And so, here’s now, and then, there are these various possibilities. And it might go like this, or it might, you know, fall all the way back down like that. So, here’s something you can do. Remember that you can look back into the past. You can look into the past by looking at very distant things. And you see the light coming to you from the time in the past. So, you look into the past because of the light travel time. And so, if now is here, we can actually look at objects which were existing at some time in the past, back, say, here, I don’t know–by looking at things that are three billion light years away, or something like that.

And the question is, if you can do that–you can figure out how far in the past it is, by the way, that’s easy. Measure distance and time in the past is just distance. Miles per hour. Which way up is it going to be? Divided by the speed of light. So, if you can measure the distance to something you know how long–how far in the past it was. So, that tells you where it is on this axis.

So, then, all you need to do is measure the scale factor at that time. And you can see that if you could do this, you would be able to determine whether at some point in the past you are here, if the scale factor was low, or here, or here. And therefore, if you made accurate measurements of the scale factor in the past, you could figure out which one of these tracks you’re on. And if you could figure that out, then you can infer, if that track continues without some other form of physics kicking in, you could figure out where it will go in the future. So, you’ve got to be able to determine what the scale factor was, which doesn’t seem like it’s actually all that easy, but it turns out that this can be done.

And the way to think about this is you have to think about the redshift, the cosmological redshift, in different terms from what we’ve been thinking about it so far. So, a different view of redshift. You remember this is where we started, right? We measured the distance to things. We measured their velocity. We got Hubble’s Law; therefore, the Universe is expanding. And as we were doing that, we had the idea that the redshift is telling you something about velocity. So, one view is that the redshift tells you something about velocity, and the velocities are all moving away from us. And therefore, the Universe is expanding.

But you remember that the cosmological redshift caused by the expansion of the Universe–that velocity is a little different in kind from the way we ordinarily think of velocity. It’s not–objects are not moving through space. It’s just, the space itself is expanding and that leads to some differences. For example, in principle, the cosmological redshift can cause things that look like the velocity is potentially greater than the speed of light, and things like that. So, the cosmological redshift is not the same as an ordinary velocity.

And one way to think about what the redshift is, is the following. So, another view. The Universe is expanding. But a lot of the things in the Universe are not expanding. You’re not expanding. The galaxy’s not expanding. We talked about this a little. That’s because there are forces holding it together. The Earth isn’t expanding because its gravitational force holds it together. I’m not expanding because the chemical bonds between the atoms in my body hold it together. The galaxy is not expanding because gravity holds it together. But a wavelength of light doesn’t have such a force to hold it together. So, as the Universe expands, wavelengths of light expand along with it. So, this is another view of where the redshift comes from: that wavelengths of light expand along with the Universe. That’s an interesting thought.

So, the question is, “Why do we observe light from distant galaxies to be redshifted?” In this interpretation, it’s not because these things are moving away from us. It’s because, during the time between when that wavelength was emitted and when we see it, the Universe has expanded and the wavelength has expanded along with it. And, therefore, we see a longer wavelength than we would have from a photon that hadn’t been traveling as long, because the Universe, in that case, wouldn’t have expanded as much. Wavelengths of light expand along with the Universe. And so, when we observe distant objects, the wavelength we observe is longer than when it was emitted, because the Universe has been expanding during that time. This is–yes, go ahead.

Student: So, if the Universe is expanding as the light is traveling, by the time the light reaches this Universe, will be somewhat larger than when–than when the light was emitted?

Professor Charles Bailyn: Yeah.

Student: So, is the time–is the time that we, you know, that we even have [inaudible] in the past, is that the distance divided by–I mean, is that the original distance divided by the speed of light or the–

Professor Charles Bailyn: Okay. So, this is this complicated thing about, what do we mean by distance in an expanding Universe? And what you mean–this is a good question. What you mean by distance is how much–okay. So, the first thing to say is that cosmologists worry about this a lot, and there are different kinds of distances that can be defined. There’s the coordinate distance, which doesn’t change as the Universe expands. There are other kinds of distances. In the particular case of what we’re doing right now, the key way you measure distance, remember, is the standard candle. So, you define what’s called the luminosity distance, which is how much fainter something looks.

And so, basically, distance is defined–so let me–this is kind of a parenthesis. What do I mean by distance? Distance is defined by this equation. So, what you mean by distance, in this case, is how much fainter are the objects you’re looking at than otherwise. And so, something that’s twice as far away–what does that mean for something to be twice as far away? It means the light from a standard candle is four times fainter. And so, you define distance as the dimming of light, because that’s the way you measure the thing. And then, you have to be careful when you’re doing this out in strict mathematical terms. You have to be careful that that kind–that the way you’ve defined, you don’t get confused about which kind of distance you’re working on. So, that defines distance.

Then, time, the lookback time, turns out to be, more or less, certainly for short distances, correctly defined as Distance / c. But you do have to worry about this when you get into large cosmological distances and I don’t want to go into that, but the different kinds of distances are related to each other in complicated ways.

Okay. So, this is basically back to the question of how you get the x-axis on that graph. I’m now talking about the y-axis, the scale factor. So, then, how do you measure the scale factor? The scale factor–so, the scale factor now, of the Universe now, divided by the scale factor then, when you emitted that wavelength, is going to be equal to the observed wavelength, divided by the wavelength of the light when it was emitted. Because that’s how much larger the Universe has gotten, and the wavelengths have gotten big along with it.

So that’s equal to λ_emit plus Δλ, as we’ve defined it before, over λ_emit, which is equal to 1 + Z. Because the redshift, Z, remember, is the change in the wavelength divided by the wavelength when it was emitted. So, by measuring redshifts, you can tell what the scale of the Universe was relative to the scale of the Universe now, when that light was emitted.

Now, remember, I defined the scale factor now to be 1. So, that means 1 / a is equal to 1 + Z, or the scale factor of the Universe in units where it’s currently equal to 1, is equal to 1 / (1 + Z). And Z, you measure. And so, in principle, you can measure both axes of that graph.

And so, if I now go back to my plot, you could measure some points on this plot. Now, all of them are in the past, not the future, because you can’t look into the future by this light travel time trick. But you could go out there and you could discover, if you made, you know, really excellent measurements, you could discover that there’s a point right here. That a particular galaxy, or whatever it is you’re looking at, is far away such that the time back is this. And then, you measure the redshift. You get a scale factor. And you could measure that at that time the Universe had such and such a scale factor. And if you measured a whole bunch of these things and they sort of lined up in this nice way, then you might conclude, well, the Universe is doing this, and you might then extrapolate to determine what the Universe is going to do in the future.

And so, this is just the Hubble Diagram interpreted in a slightly different way and being looked at at a distance sufficiently far away that the rate of expansion would have been different back then, than it is now. So, I’m going to draw it all again. Here’s a. Here’s t. There’s this. Here’s one of these. And here’s this. But, you know, that’s not usually how we plot the Hubble Diagram. What we’re usually plotting, as you’ll recall, is distance versus velocity, because those are the things we actually measure. Velocity is redshift, Z. And remember that 1 / (1 + Z) = a, or–let’s see.

1 / a = 1 + Z

a / a

1 / a

1 ‒ a / a, yeah.

1 ‒ a / a = Z.

So, this represents the scale factor, this axis, whereas this–what we actually, usually, plot is m - M, which, you’ll recall, is equal to 5 log (D / 10 parsecs). And so, this scale represents time, because distance can be converted to time, because we know the speed of light. So, this is a time axis in really weird units. And this is a scale factor axis in really weird units.

But, you can, therefore, transform each of these lines, which is scale factor versus time, into an equivalent line on this plot. So, the green line is the plot where the Universe just expands and there’s no mass. Nothing slows down. So, it kind of looks like this. All right.

Now, what are the points? This is now in the time factor, and up here is then. Okay, so, you’re going from now to then. So, you’re kind of sitting on this point, looking back this way. And here is scale factor a = 1, and here’s scale factor a = small. Okay? Because the Z is getting bigger, therefore a is getting smaller.

So, what would the red curve look like on this plot of the things we actually measure? Well, when it’s close, the red curve sits right on top of the green curve. So, it kind of starts out like this, with no distinction. Then, at a certain point in the past, let’s see. How do I want to do this? Let’s pick a particular time. So, that’s a particular distance. At that time, the scale factor is smaller on the red curve than it is on the green curve. If the scale factor is smaller, the redshift has to be bigger. So, what’s going to be happening, here, is the red line is going to gradually diverge from the green line in just the same way it does here, in such a way that at a given distance–that is to say, a given time in the past, the redshift is bigger, which means the scale factor is smaller. And then, the brown curve will kind of do the same thing, only more so.

So, this is what we expect to see. This is Ω greater than 1.

Ω less than 1, but still greater than 0.

Ω = 0.

So, all the Hubble Diagrams we’ve looked at so far have been down at this end, here, where they’re all lying on top of each other. But you can imagine that if you get out to big redshifts, redshift of 1. At a redshift of 1, the scale factor is half of what it is today. You’re halfway back to the Big Bang in terms of scale factor. You can imagine that these things would really look quite different, as indeed they do in this little plot I’ve drawn, when the scale factor is half of what it is today. If the scale factor is 1% less than it is today, you’ll have a tough time telling these things apart.

So, what do you need to do? What you want to do is measure redshifts and distances. Plot them on this plot, and see which of these lines turns out to be true. What do you need to do that? What you need is a really bright standard candle. Why does it need to be bright? Because you need to be able to see it at large distances.

And the thing that happened in the 1990s, and that has transformed cosmology, is that we are now able to make these kinds of measurements all the way out to a redshift of 1, and in some cases, further. And the particular standard candle we’re looking at are the so-called Type 1a Supernovae. I’ll discuss what these are later on, but for the moment let’s just take the point that they’re standard candles, therefore we know how bright they are. And they’re really, really bright–absolute magnitude of something like -19. So, they’re as bright as whole galaxies. In fact, they often outshine the galaxy they’re in. These are exploding stars and, as I say, we’ll talk about the details of exactly how these things work later on. So, Type 1a Supernovae turn out to be these really bright standard candles, and you can make this kind of measurement on them.

I have the data here, but it’s–I have to talk about one more technical thing about the way the data’s presented. Let’s see. Ordinarily, the way you tend to plot these things, as I said, is you plot redshift versus apparent minus absolute magnitude. This is a quantity known as the distance modulus, and it’s related to distance by the equation we had before.

And so, then, what you get is, you get this, and you get this, and you get this, as we’ve written it down before. And the problem is that it’s hard to tell the differences between these lines because this is now being measured in magnitude. The separation out here at a redshift of 1. This is about–this interval here, in magnitude–let me write it that way–is about–oh, I don’t know, a third of a magnitude or something like that. But this whole plot is, you know, if you go all the way back to time 0, this whole plot is, I don’t know, can be 10 to 20 magnitudes.

And so, it’s just kind of hard to see these little differences between the things. So, they plot it differently so that you can bring out these slight differences. And what they do is, they do this. They plot a quantity called Δ (m ‒ M). What’s that? That’s the difference between the measured distance modulus and the distance modulus that you would expect at that particular redshift in an empty Universe. So, m - M at a given red shift in an empty Universe.

So, what does the green curve look like? The green curve is Ω = 0. That is to say, an empty Universe.

Student: [Inaudible]

Professor Charles Bailyn: Yeah, yeah, yeah. It’s just a straight line at 0, because it’s always true that the empty Universe, minus what you expect for the empty Universe, is 0. So, what you’ve done, basically, is you’ve taken this plot and tipped it 45º so that this green line is straight at 0. And then the brown line will look like this–sorry, the red line will look like this. The brown line will look like this. This, again, Ω greater than 1, Ω less than 1. Ω, 0.

So, that’s just convenient, because then, you can have your scale here go from, I don’t know, -0.3 to +0.3, or something like this. And so, you can see the whole thing. Actually, I’ve got the signs wrong, haven’t I? Because bright things are a minus. So, let’s just put it as bright and faint.

Chapter 3. Discovery of Dark Energy [00:27:58]

So, we’ve done our very best to confuse you, right? First of all, we started with a magnitude system that’s upside down and logarithmic. Second of all, we took the intuitively simple plot of scale factor versus time, which actually tells you what you need to know about the Universe: is it getting bigger? Is it getting smaller? Is it going to have a Big Crunch? And we transformed it into this rather stranger thing, where the things we are plotting are the things we measure–but it’s not at all obvious how this translates into what the Universe is doing. Then, not content with that, we take that whole thing and subtract something you don’t understand from it, so as to flatten out one of the lines. Okay? So, it’s a three-part exercise in confusion.

And I would say that the number one conceptually difficult thing in this entire course is to understand how this plot relates to this plot, and more difficultly, how this plot relates back to the scale of the Universe plot. And if you can understand that, you can understand twenty-first century cosmology. And so, we will work on this a little bit, so, if you’re not quite getting it yet, don’t freak out. You’ll have plenty more opportunities to discuss it, I assure you. In any case, even if you’re a little puzzled at the moment, let’s look at this plot because this is how the data show up.

So, there’s an Ω = 0 line, which is straight. And then, lines below it are increasingly large values of Ω. And so, if you go out and measure a bunch of points, you expect to see them kind of lying in a straight line here. And then, you expect to see them falling below that straight line, and depending on how far below the straight line they fall, you can determine whether the Universe is going to re-collapse or not. So, if they end up, you know, down here, it’s going to re-collapse. If they end up here, it’s going to expand forever.

Okay? How are we doing? Ready for some data? Okay. So, in the 1990s, as I mentioned, they actually made this measurement. In fact, two different groups made the measurement, and they were rivals. They did things slightly differently. They’re measuring the same kinds of objects. Here’s what they found. Okay. So, let’s see.

This is this Δ (m ‒ M). So, 0 on this plot is the empty Universe. And then, this dotted line is Ω. So, let me color code this the way I had color-coded everything earlier. So, I’m going to just take this and draw the relevant lines. So, here’s the green line at exactly Ω = 0. And then, this thing is the brown line–sorry, the red line. Got to get my colors straight. Red line. This is Ω less than 1. In fact, this particular line is for Ω = 1 / 4. And then, the brown line is this thing here, which is Ω = 1. So, if you fall below the brown line, the Universe is going to re-collapse. Right. What else do I want to say about that? Nothing, because I’ve now put some points on there. Yes?

Student: Why are they lines [inaudible] whereas the ones that you used before they were curved?

Professor Charles Bailyn: Oh, these are actually curved. It’s just, I exaggerated the curvature. The reason that it comes out less curved on this is it’s only going up to a redshift of 1. If you continued this out much further, the curvature would become more obvious. But these are not, in fact, straight lines. The green one’s a straight line because it’s just at everything equals 0. But there is curvature on these lines.

Okay. So, the problem’s obvious. Here are the observed points, and they don’t go through any of these lines. And you can first of–okay. So, there are many things to say about this.

First, the error bars. Let’s talk about the error bars. Redshifts are relatively easy to observe. You can do that to high accuracy. This distance modulus is more complicated. There are two kinds of errors incorporated into these error bars. One is measurement errors–that is to say, how accurately can you actually measure the brightness of these things? That’s an error in m, you know, how bright–how accurately can you measure these things? They’re pretty faint by the time they’re out at a redshift of .6.

But the other is how good a standard candle is. How well do you know exactly what the absolute magnitude is supposed to be? So, accuracy of the standard candle. And that’s an error in big M, because you don’t know–these things turn out to have a little bit of a spread in brightness, and you don’t know that. And so, that’s why the nearby ones, these ones down on this end, which you can measure, really, quite precisely, still have substantial errors, because you don’t know how bright the standard candle is supposed to be to perfect precision.

And so, you can see that down below a redshift of .2, down in this region here, you’re going to have a really tough time telling the difference between these lines. Because the lines are separated by a tiny fraction of a magnitude and the errors on these things, on any individual object, is several tenths of a magnitude. So, you know, pretty much all of these points go through all of the lines, and so, that isn’t going to tell you a great deal.

By the time you’re out in here, you’re starting to get several tenths of a magnitude difference in the prediction. Now, the errors are still several tenths of a magnitude. So, observing one such object isn’t going to tell you anything because, supposing I observe any one of these things, I don’t care. Here’s an error bar. And while it doesn’t go through this line, that’s only, you know, two times the error. Two Sigma error for those of you have studied statistics, isn’t such a crazy thing. And so, one of these points would not tell you all that much.

But on this plot, which was published in 2003 and contains data up until about 2001, there are a couple dozen points. Now, this line here, the red line with Ω = 1 / 4. We already know that Ω has to be equal to at least 1 / 4, because we’ve measured all that dark matter out there. We added it up, and it comes out to about Ω = 1 / 4 or 1 / 3, or something like that.

And so, if there’s nothing else in the Universe beyond what we already know about, you expect it to be on this line. If there is matter in the Universe that we don’t know about, you expect it to be below that line.

So, if you take the points at redshifts greater than .2, the interesting fact is, all of them lie above this line, and none of them lie below that line. Now, each of them has a different error. But nevertheless, the points are all above the line.

So, if the Universe truly were following this red line here, then what would you say? You would say that half of them should lie above and half of them–and points should lie above–below just by random chance. So, this is throwing two dozen coins and having them all come up heads because they’ve all, by random chance, if the Universe is following this line, landed above the line. So, you can figure out what the probability of that is. It’s–you know, if there are 20 points and they’re all heads, that’s 2²⁰. That’s about–a probability of one in a million. There was another team doing the same experiment. They measured about the same number of points, got the same result. So, that’s two incidences of a one in a million chance. That’s one in a trillion. So, random errors are not going to get you back down onto the redline, much less there.

Now, look at the green line. This is an empty Universe. Nothing in it at all. Maybe we’re just totally wrong and we’ve screwed up all the dark matter, and we’ve got only two points below that. That’s almost equally improbable. And so, what we think is going on is that this line actually goes above the green line. And one current hypothesis for what such a line should be is this solid line up here, which nicely goes through the points in such a way that, kind of, half of them are above and half of them are below–the way points ought to be.

So, this is quite strong evidence that, in fact, what the Universe is doing is not going below that line, but going above it. So, what does that tell us? All right. Let’s do this whole exercise backwards.

Δ (m ‒ M) versus Z. Here’s the empty Universe, and here’s what we see. So, how does that go onto the next plot? This is now m - M versus redshift again. Here’s the empty Universe. And so, what appears to be happening is this. How does that, then, map onto the scale factor versus the redshift? Scale factor versus time, sorry. Here’s now. Here’s 1. Here’s this. This is now still the empty Universe. It’s a straight line in this case, and so, what is happening here?

If you go back and you look at a particular distance–that is to say, a particular time ago–so, you’re looking here. The purple line here has a redshift that is smaller and therefore the scale factor is larger. So, you ought to be here, right? So, that’s what we’re observing–the implication of which is that the future is going to look like this.

So, the Universe in this is–Universe is expanding. We already knew that. And the expansion is accelerating. Something is pushing the Universe outwards. So, we had a thought all along. What’s going to happen? Gravity is going to pull it back together again. It’s going to be slowing down. No, not true. The Universe is being pushed outward by some kind of force, by some kind of energy, which has the characteristics of a repulsive gravity. So, it’s anti-gravity, if you like. This is what the science fiction writers would certainly call it. It’s something that pushes things away from each other. And this pervades the Universe in such a way that it actually dominates the behavior of the Universe as a whole. We give this the name dark energy, and we have absolutely no idea what it is.

But, of course, as my colleague, Charlie Baltay, likes to point out, we never actually admit to people that we don’t know what we’re talking about. So, you observe this kind of thing. You don’t say: we have no idea what’s going on in the Universe. No. You say: we discovered dark energy. Big triumph.

And so, we have discovered dark energy, and furthermore, we know something about how much of it there is–namely, because we can actually measure this curve. And so, you can ask: what is the density of the dark energy compared to matter?

Now, you know how to compare energy to matter, right? You have to use E = mc². So, if you have energy–some amount of energy per meter cubed, you can transform that into energy / c² per meter cubed, and that now is in the same units as a matter density.

And so, you can have Ω of the matter, which we think is about 1 / 3, or 1 / 4, or something like that. And you can ask: how much energy is there. You can ask: how much Ω is there in energy, in this dark energy? Not photons. Not ordinary kinds of energy, but this repulsive–this mysterious, repulsive dark energy. It’s about three times the amount of matter–where Ω, remember, is equal to density over the critical density, and, you know, it doesn’t matter whether it’s an energy density divided by c², or some amount of matter.

Chapter 4. Understanding Dark Energy and Einstein’s Biggest Mistake [00:42:06]

Okay. That’s what we think the Universe consists of. So, here is the famous pie chart of the Universe, which tells you what’s going on. So, you know how pie charts work? You start with a circle. That’s everything, and here’s us. So, this is all ordinary matter, and all ordinary kinds of energy, photons, what have you, that we know about. And it’s something like 4% of the Universe.

Then, there’s a big chunk that’s dark matter. And that, we discussed last time. The dark matter is this stuff we can’t see. It might be WIMPs. It might be MACHOs, we don’t know. We can’t see it, but we know it’s there. Why do we know it’s there? Because it exerts gravitational force on galaxies and stars in galaxies, and we can measure how much of them there is by looking at stars–galaxies and clusters of galaxies, and inferring the mass of them by observing the orbits of things around them, just as we did with black holes; just as we did with planets. This is a fairly straightforward astronomical technique.

The rest of it–this region, here, is dark energy, which not only can we not detect directly, but it has a physical effect unlike anything else we know anything about at all–namely, it pushes out. It’s a purely repulsive force of some kind and that purely repulsive force provides ¾ of the mass energy of the Universe. Well, that’s bad, because we have even less idea–oh, and I should say–So, we’ve been counting up frontiers and controversies with time. This is Frontiers and Controversies in 2007. What the heck is going on with this pie chart? What’s this? What’s this and why? That’s what we need to find out.

Now, oddly enough, it turns out, Einstein predicted this. Einstein was a genius. Only, he then retracted it and decided it was the greatest mistake he had ever made. And then, because he said it was a great mistake, everybody ignored it for seventy years, until we discovered the dark energy, and then, somebody said: oh, you know, Einstein told us about this already. And here’s how–why he did it.

Remember, Einstein wanted a static Universe. This is before Hubble discovered that the Universe was expanding. And so, Einstein was looking at the implications of his equations of general relativity, and he thought it would be nice if the Universe was static. And the problem is, there’s all this matter in the Universe, so it attracts itself. So, it can’t be static, because if you try and hold it in one place, it’ll fall down. It has to either be expanding up or expanding down, just like the pen is either going up or going down. But that’s bad, thought Einstein, because it ought to be static.

So, he invented an additional term for his equation, which he wrote down as a capital lambda [Λ] and called the cosmological constant. And this had generated a repulsive force that would balance gravity and lead to a static Universe. And it was allowed in the equations. It fit fine into the equations. There was no problem there. Those of you who have done calculus, this shows up as a constant of integration, so, it can be any value you like. If you don’t know calculus, ignore that last sentence. Okay.

And so, he invented this repulsive force, this cosmological constant, to balance gravity, so that you could have a static Universe. Then, Hubble discovers that the Universe isn’t static–discovers expansion. And Einstein says: damn, I could have predicted that, and then, I would have been famous. So, Einstein says: Λ was my biggest mistake. Because if he hadn’t been so insistent that the Universe needed to turn out to be static, he could have said, you know, my equations predict that the Universe either has to be expanding or contracting. It has to be moving around. And then, Hubble would have verified Einstein’s prediction and Einstein might have won the Nobel Prize, or something. So, Λ was my biggest mistake.

Except, it turns out that it might be true. So, this is one of those fables, right? The fable of Einstein’s biggest mistake. And the moral is: interesting ideas can turn up in a different context and turn out to be even more important than you thought they were–can turn up in other contexts.

So, for seventy years, you know, the idea of this Λ term, this cosmological constant, was kind of buried in the textbooks. It would be, you know, Problem 53 at the end of chapter 7. “Supposing there is a cosmological constant, then, what would happen?” And generations of grad students thought about this for fifteen minutes and then forgot. And then, all of a sudden, it turns out to be an explanation for this very mysterious force.

Now, just because you stick it in an equation doesn’t tell you what it is. So, what might it be? The particle physicists, it turns out, have a little bit of an explanation for this, called the vacuum energy. I’ll talk more about that later. Never mind what it is. I just wanted to point out that if you calculate what the vacuum energy is, what you discover is that Ω associated with this Λ is equal to 10¹²⁰. That’s a particle physics calculation.

The observed Ω is equal to about ¾, okay? So, this is the wrongest calculation in the history of science. It’s off by 120 orders of magnitude. And in fact, it would be very bad if this was true, because if there was that much repulsive energy, no galaxies would have formed. No planets would have formed. We wouldn’t be here. All our stuff would have been sprayed out into the far reaches of the Universe.

So, we knew already that that couldn’t really be true. And so, what the particle physicists said was, you know, obviously, this can’t be true, so it must somehow cancel out. So, it must be 0, because otherwise we have this embarrassing problem, which we know can’t be true. The fact that it’s small, but not 0, is completely mysterious, and that’s where we’ll start next time. Yeah, hand in problem sets if you’ve got them.

[end of transcript]

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