 Now, it wasn't so much that the astronomers were completely wrong. What happened is that a new ingredient got added about 10 years ago, and the new ingredient is what's called dark energy. And it comes from the fact that in 1998, astronomers discovered that the expansion of the universe is not flowing down under the influence of gravity, as everyone had expected, or almost everyone had expected, up until that point. But in fact, the universe today is actually accelerating, and in fact it's been accelerating for about the last five billion years or so. And that in fact means that there's a kind of repulsive gravity going on in the universe today. We don't know of any other explanation for this other than some form of repulsive gravity. And the stuff, which again has to have a negative pressure that drives this repulsive gravity, we don't know exactly what it is, so we call it dark energy. So dark energy is the energy that, by definition, the energy that's causing this expansion. And even though we don't really know what the dark energy is, we can, in fact, figure out how much of it there has to be to produce the right acceleration, and it's by adding in that extra energy from the dark energy that brings omega up to 1 from the previous value of 0.2 and 0.3. So the astronomers were, in fact, right about the contributions to omega from the things they were looking at, but it turned out that there's, in fact, the larger contribution to the total energy of the universe in the form of this dark energy, which was not discovered until 10 years ago. Finally, and perhaps most impressively, inflation makes a prediction for the small-scale non-uniformity of the universe. Remember, 0.1 was the large-scale uniformity, and the largest scale of the universe looks unbelievably uniform. But of course, on smaller scales, like this room, there are lots of non-uniformities, also on the scales of galaxies and things like that. And the cleanest way to measure these non-uniformities is in the cosmic background radiation, because it's essentially unprocessed in the cosmic background radiation. It has that time to undergo nonlinear and complicated evolution. And even though I told you that the intensity of the cosmic background radiation is unbelievably uniform, uniform to one part in 100,000, that's not a bound. The astronomers actually see non-uniformities at the level of one part in 100,000. And now they can even measure those non-uniformities with rather fantastic precision. And one actually wants to ask, where did these non-uniformities come from? Now, in conventional cosmology, that is, without inflation, there was really nothing that could be said about it. Without inflation, the initial conditions were really whatever you made them up to be. So there were initial ripples if you put in initial ripples. If you didn't put in initial ripples, there were no initial ripples. There was no theory that controlled how those initial ripples might be created. In inflationary cosmology, there's a different answer. Inflation can actually attribute these ripples to quantum fluctuations. At the classical level, inflation simply smooths the universe out by this enormous expansion. An enormous expansion is to smooth everything out. But the world is not classical. The world is quantum mechanical. And when one puts in the quantum fluctuations to describe inflation, then you would expect that the mass density would on average be uniform. But because of quantum fluctuations at any given place, it could be a little bit above the norm or a little bit below the norm. And you can actually calculate the properties of these density fluctuations and compare with observations. So inflation allows us to understand these ripples in terms of quantum fluctuations. Now I should add that usually we think of quantum fluctuations as being important only at incredibly small distances. We're not used to quantum fluctuations being relevant in the room or the things that we see classically. What makes it different for inflation basically is the exponential expansion again. This exponential expansion is constantly stretching quantum fluctuations at very, very small microscopic ways, once upwards, to get bigger and bigger into macroscopic and astronomical dimensions. And if this is right, it would mean that our whole galaxy is just a quantum fluctuation, which is a bizarre concept. But it seems to work, and I'll show you how well it works. Our best data comes from the WMAP satellite so far. The Planck satellite is now in the air and collecting data, but we don't have that data yet. Here's a picture of an artist's conception of the WMAP satellite. That's David Wilkerson for whom the satellite has been named. Here's the data. It's my favorite graph in the whole world. This is a graph of the temperature fluctuations as a function of the multiple. That is, they expand the temperature pattern that they see in the sky and spherical harmonics. And you talk about the intensity for each value of L that corresponds to the spherical harmonics. You can think of it as an angular wavelength. And I've tabulated on the top as an angular wavelength units with user degrees. That's 0.2 degrees, corresponding to a very high L. Low L corresponds to a large angular wavelength. So 180 degrees is way down here, corresponding to L equals 2. And the black lines and black dots represent the data, mostly from WMAP, but on the right hand side at shorter wavelengths, other experiments do better. So this includes the CBI, ACPAR, and boomerang experiments. The most impressive part is on this first peak, where those little dots with their sizes, their airbars, and they're just absolutely gorgeous fit between the experimental data. And the red curve, which is what you predict based on inflation, including lambda means including dark matter, dark energy, excuse me. Lambda stands in for dark energy. So this is a fantastic graph. Now, to make it a little easier to tell how impressive this graph really should be, we've also added to the graph curves representing other theories, just to make clear that it doesn't come out automatically that you get these things right. So if we had an open universe with omega being 0.3, like the astronomers thought 10 years ago, then you get this yellow line, which as you see doesn't fit the data at all. Another competing theory that for maybe a decade or so was viewed as a competing theory for inflation for the generation of these density fluctuations involves the creation of cosmic strings in the early universe, which would form at a phase transition. And the randomness of that phase transition would produce density fluctuations. That produces this purple curve, which again looks absolutely nothing like the data. And finally, we put in for one more comparison, what inflation itself would give if instead of having dark energy, you let omega equal one by putting in more dark matter. And that gives this slight green curve, which again, this bears no resemblance to the data. So the bottom line is that inflation fits the data absolutely gorgeously, and none of these other options do. For the sake of truth and advertising, I should mention that there is one theory that is actually a lot like inflation, but is sometimes viewed as an alternative to inflation. It can be something called the acperotic cyclic model, which at least according to some calculations, they're actually contradicting calculations. But some calculations gives exactly the same answers in relation to this problem. So inflation might not be the unique theory that does this. Okay, this is a more recent set of data from WMAP, just to show you that it continues to be a good fit. And this is the curve which shows the same WMAP data but with different high frequency data from other experiments, again showing a gorgeous fit. Okay, now I want to change gears. I've now tried to explain what inflation is, tried to explain why we think our universe, why at least some of us think, including me, that our universe began with the period of inflation. Now I want to talk about dark energy and the role that that plays in motivating this idea of a multiverse, which I'll also be talking about. So dark energy I mentioned before, it was discovered in 1998 by two large groups of astronomers that the universe has been accelerating for about the last five billion years out of the 13.7 billion years of its history. And an implication is that inflation of a sort is really happening today. That is, we have accelerated expansion presumably driven by some material with a negative pressure. However, it's not the same as the inflation in the early universe. It's much, much slower. The energy density of this negative pressure material is clearly much less than the energy density of the stuff that drove inflation. Now the stuff that is driving the expansion today is called dark energy. And you might ask, what is the dark energy? There is a very succinct answer to that, who knows. We don't know what it is, which is why we call it dark energy. But there is a simplest candidate and other options are really very similar. So we do know a lot about this dark energy even though we don't know what it is. The simplest option, which is the one I'll discuss, is the option that is just plain vacuum energy, which is synonymous with a cosmological constant. Einstein introduced this cosmological constant as an extra term in his field equations and he introduced it precisely for the purpose of describing a repulsive gravitational force. To Einstein, the vacuum was empty. He was not really a quantum physicist. He was really a classical physicist. So to Einstein it was not really conceivable that empty space should have weight, should have mass. So when he added this term, he just viewed it as an extra term in the equations that described how gravitational fields are produced. However, in the modern context, we look at it differently, looking at exactly the term that Einstein put in those equations, you can interpret it as an extra contribution of matter. And if you do, it's just energy density of empty space and it leads to exactly the same equations. And that's the modern interpretation that the cosmological constant is synonymous with vacuum energy. From the modern point of view, the vacuum is not empty. It's a quantum vacuum with quantum fluctuations going on all the time, particles appearing and disappearing. You have fields in the vacuum, Higgs fields with non-zero values. The modern vacuum, the vacuum known to modern particle physics is an extremely complicated state. And there's no reason why it should have zero energy density. So we're perfectly happy to attribute a non-zero energy density to it. So the cosmological constant becomes that. Nonetheless, even though we can glibly say that this is vacuum energy and feel that that's a plausible statement, it really is a nightmare when we try to understand it in detail. What's a nightmare is to try to understand the magnitude of this horrible dark energy. And here's the problem. The quantum vacuum, as I said, is far from empty. So the mere fact that we're assigning a non-zero energy density to the vacuum is no problem. That seems completely plausible. But we can ask what happens if we try to calculate what that vacuum energy density should be using our quantum picture of the vacuum. So in a quantum field theory, one can calculate the energy of quantum fluctuations and the problem is it diverges. The divergence comes from very short wavelength fluctuations. As you imagine talking about the quantum fields in some box where one makes up a box just to make the problem finite. Inside that box we can describe the electromagnetic field in terms of normal modes. Each normal mode contributes an energy, one half h bar omega, it acts like a harmonic oscillator, to the vacuum energy. And the problem is just that there's no shortest wavelength. There are always shorter and shorter wavelengths, higher and higher omega's, making the sum just grow and grow and grow in the context of a quantum field theory growing without bound. So in a quantum field theory, the vacuum energy just diverges. But one could say that that just means that the quantum field theory should not be trusted down to very, very short wavelengths. After all, we developed this quantum field theory by doing experiments and so on. And those experiments had some finite size. We don't know the physics down to arbitrary short size scales. So it's very natural to cut off that infinite sum at some short wavelength using the excuse that you don't believe that you have any right to extrapolate the physics beyond that short wavelength. So you get to choose what the wavelength is. A common number that people use because it seems to make some sense is to say that we believe physics, we believe that quantum field theories work except for quantum gravity effects. We're pretty well aware that we don't really understand how to put quantum gravity into these theories. And we could say that that's the place of ignorance where we should cut off our sum. So let me just pursue that. If you do that, it means cutting off at the Planck energy, 10 to the 19th GEV, which is the scale of quantum gravity. And if you use that cut off, you could then get a finite number for the vacuum energy density. But the problem is that the number you get is too large and it's not just a little bit too large, it's too large by about 120 orders of magnitude. And that is an embarrassment even for cosmologists. And this is the nightmare of dark energy. We have no way of understanding at the present time why the actual energy that we observe is so incredibly small compared to the estimate that we make naively of how much energy we might expect the vacuum to have. Now, even if you don't use the Planck scale, if you cut off even at the electroweak scale or any scale associated with particle physics, you get a number that's vastly too large, not necessarily 120 orders of magnitude, but nonetheless still vastly too large. The problem is that there's just a big mismatch between the energy scales of particle physics and energy scales of cosmology. This cosmological constant is very significant on the energy scales of cosmology, but that's still very, very low energy compared to any energy scale that's relevant to particle physics. So, let me leave that for now as a mystery and let me come back and talk about inflation because there's a feature of inflation that I have not yet talked about here that turns out to be relevant to this issue, at least from the point of view of many people. And that leads us into this multiverse picture. I told you that this repulsive gravity material is unstable so that it decays. And it is indeed true that if you sit in any one location inside this repulsive gravity material and watch what happens with probability one, sooner or later, usually in a fairly short time, you will see a decay because the probability of it not decaying falls off exponentially with some time constant. And if you wait some numbers of those time constants, the probability that has not decay just gets unbelievably small. So in any one place, you expect inflation to end if you just sit in one place and wait. However, the catch is that while this repulsive gravity material is decaying, it's continuing to exponentially expand and the exponential expansion is in fact vastly faster than the decay. If that were not the case, you would not have a successful inflationary model in the first place. So if you wait, for example, for one half-life of the decay, half of it will decay, but the half that remains will be vastly larger than what you started with. So you're not making any progress towards ending inflation permanently. Rather, the inflating region just gets larger and larger and larger with time, even though it's decaying. So it's a very peculiar picture, but this is the picture of eternal inflation. Once inflation starts, it just never stops. The volume that is inflating increases with time, even though the inflating material is decaying because it's exponentially growing faster than it's exponentially decaying. So the inflation becomes eternal. Once it starts, it never stops. The inflating region never disappears, but instead pieces of it undergo decay. And when each piece decays, it produces a local universe, a local big bang, and each such local universe is called a pocket universe. And we would be living in one of these pocket universes. Pocket universes are not small, but there are many of them. And in fact, in this model, there'd be an infinite number of them. So instead of just getting one universe, inflation actually produces an infinite number of universes for the same price. So it's a real bargain. And that is the multiverse. And almost all models of inflation have this property that once they start, inflation never stops. And instead of producing just one universe, you produce a multiverse. Now that plays into another idea coming from another area. Michael promised in his introduction that I was gonna talk a little bit about string theory and he's right. So here's the string theory part. Now I don't know much string theory I should add. So all of this is really just hearsay that I learned from my colleagues. But nonetheless, my colleagues seem to agree pretty much on what the story is. Since the inception of string theory, theorists have sought the vacuum of string theory. But they were never able to find such an animal. And within the past 10 years or so, almost all string theorists have come to the belief that the reason they've been unsuccessful in finding the vacuum of string theory is that there is no such thing as the vacuum of string theory. Instead, string theory is believed to have a huge number and 10 to the 500 is the number that gets bandied about. I don't think you should take that number too seriously but some huge number of long-lived metastable states, any one of which could serve as a substrate for a pocket universe. And this huge set of metastable states is what's called the landscape of string theory. Now if you have eternal inflation, like we just described, eternal inflation can presumably produce an infinite number of pocket universes involving every one of these 10 to the 500 different kinds of vacua. So inflation populates the landscape, produces pocket universes connected with each type of vacuum. Now this leads to very peculiar consequences because even though we're assuming here that the laws of physics ultimately are the same everywhere governed by this fundamental string theory, because each pocket universe is built around a different vacuum, the low energy physics of each pocket universe, the physics that is like the physics that we measure would be completely different from pocket to pocket because the physics that we see is really just the physics of small perturbations about our vacuum. So things like the value of the Higgs field in our vacuum determine completely properties of particles that we measure. And all of the properties of the vacuum would play the same role as the Higgs fields do. So the low energy laws of physics could be completely different from one pocket to another. And in particular the vacuum energy could be different from one pocket to another because those are, vacuum energy is one of the consequences of the interactions of the particles which would vary from pocket to pocket. So that allows a new view of this cosmological constant problem, the problem of trying to understand why the vacuum energy that we measure is so vastly smaller than what we would expect it to be. And involves the idea of what's sometimes called the anthropic principle. People who like it and try to defend it sometimes call it environmental selection. Instead, that has a healthier sound to it than the anthropic principle which has been mired in mud for many years. But here's the idea. And I'm gonna advocate this as something that you should consider, something which could be the right answer. I'm not gonna try to persuade you that it is the right answer. But I will try to persuade you it's a possible answer. I really think that there's nothing wrong with it. It could be the right answer. And I wanna begin by talking about an alternative situation where I think we really would believe that the anthropic principle is the right answer. So let me instead of talking about the cosmological constant talk about the local value of the mass density. And by local I mean take a sphere around yourself of 10 kilometers and call that local and ask what's the average mass density in that sphere that's 10 kilometers about you. The answer is certainly on the order of one gram per centimeter cubed. That's sort of the density of matter around the earth. If you compare that with the average density of matter in the universe, it turns out that that's actually about 10 to the 30 times larger than the average density of the universe. So it's fair to ask yourself when I measure the mass density why don't I get a typical value for the universe? Why do I instead get a value that's 10 to the 30 times larger than what I know is typical for my universe? Well why is that so? Is it chance? Is it luck? Is it divine providence? Well I think the answer here as I hinted before is the anthropic answer. I think most of us would presumably accept this as a selection effect. That is we would assume that life can only evolve or at least strongly preferentially evolves in those rare regions of the universe where the density of matter is unusually high. So we don't worry about the fact that the mass density we see around us is 10 to the 30 larger than the average mass density. We just say that that's a natural consequence of the fact that life forms in those regions. And if you accept that, you've accepted the anthropic principle, whether you like it or not. So how does that play out for the cosmological constant or the vacuum energy density? Well the history goes back to 1987 when it was actually Steve Weinberg who pointed out that perhaps the cosmological constant could be explained this way. The idea is that the cosmological constant, otherwise known as the vacuum energy density, maybe it is huge in most pocket universes. However, we should remember that this vacuum energy density really does affect the evolution of these universes. A very large positive cosmological constant causes the universe to fly apart. It produces a strongly pulsive force. A very large negative cosmological constant, it could have either sign, would cause the universe to rapidly implode and collapse. So if, for example, we insist that our universe be capable of producing galaxies like the galaxies in our universe, then there's only a very, very narrow range of vacuum energies that allow that to happen. And that's basically what Weinberg was talking about, who was using galaxies as a criterion for the formation of life. If there's no galaxies, there's no life, he assumed. And if you grant yourself that assumption, then you're forced to conclude that life would form only an incredibly small fraction of these universes, only in those very rare universes, those very rare pocket universes, in which the vacuum energy density was incredibly small compared to the average. In 1998, Steve teamed with Martell and Shapiro to do a more serious calculation of galaxy formation in universes with different values of the vacuum energy. And they concluded that within a factor of maybe five or so, they could explain why the cosmological constant is as small as what we measure. So this is the anthropic explanation of the cosmological constant or the dark energy. Now, I won't say that this is without controversy. There's a lot of controversy about this. Many physicists regard this anthropic arguments as totally ridiculous, having nothing to do with science, they think. I disagree. However, I would recommend that this anthropic explanation should be considered the explanation of last resort. It's certainly not something you really want to jump on to until we really understand the landscape of string theory, which we don't, and until we really understand how to know where life will form and where it won't, which we do not understand now. The best we can do is to give plausibility arguments for these anthropic explanations. And therefore, I would say that an anthropic argument becomes attractive only when the search for a more deterministic explanation has failed, as has so far been the case, for the cosmological constant. I might mention that anthropic arguments are also discussed for other quantities in physics besides the cosmological constant. For example, the Higgs mass, the Topcork mass, and the magnitude of the density perturbations in the early universe. So this idea has gotten bigger than just an explanation for the cosmological constant. So is it time to accept what I'm referring to as an explanation of last resort? Well, I would just like to say that your guess is as good as mine. I'm not gonna try to tell you what you should believe. But for the cosmological constant, I would argue that it's only time to take this idea seriously, I think, just because we have had so much trouble finding any other explanation. So I would say it's hard to deny that as of now, the selection effect explanation is by far the most plausible explanation that is known for the very small vacuum energy that we have been observing. Now this, to me at least, is very disappointing because I had always hoped that pure thought and theoretical physics would be able to ultimately predict what our universe should look like. And there was a strong hope for that in the early days of string theory, where people hoped that string theory would allow us to ultimately calculate all of the parameters of the standard model of particle physics. That would be great if it happened. But if this landscape picture is correct, it could really mean that all of the parameters of the standard model have the values that they have just out of historical accidents. It's the vacuum that we happen to fall into and in different vacuums they would have different values and there would be no way to predict what the value would be. And that would be disappointing. It has been pointed out by enthusiasts of the anthropic principle that this would not be the first time such a thing has happened in physics. An analogy that's often used is to go back to Kepler. Kepler thought that the radii of the orbits of planets was a fundamental quantity that should be calculable. And he thought about models of regular solids embedded in regular solids. And that made sense from his point of view because from his point of view, the solar system was the ultimate universe. It was the work of some God. And it should obey mathematical rules. So it should be possible to calculate the radii of the planets. But now we regard all of that as total nonsense. We do regard the radii of planets as just being historical accidents. And we've gotten over that. We don't fret anymore about the fact that we can't predict the radius of the orbit of Pluto. We just take the measured value from our tables and accept that and go on and do different problems in physics. So it's possible that the mass of an electron is the same way. That eventually we'll get used to the idea that the mass of an electron is not something fundamental that we should try to predict, but rather just some historical accident like the radius of the orbit of Pluto that we just have to measure and accept. So that's really all I wanna say, but now I'll summarize just to make it clear what it is that at least I think that I've said. So what I think that I've said is first of all that the inflationary paradigm is in great shape. Inflation can explain why the universe is so smooth and homogeneous when averaged over large regions. It can explain why the mass density of the universe is so close to the critical value. And it can also explain the ripples that we see in the cosmic background radiation. So it explains the large-scale uniformity. It explains why the mass density ratio omega was equal to one to 15 decimal places at one second after the Big Bang. And it predicts correctly that omega is equal to one today where we know in fact that that is true to within an accuracy of about 1%. And finally it explains very accurately the ripples that we see in the cosmic background radiation explaining those ripples as quantum fluctuations that took place during inflation. Then we come to this idea of the multiverse. And if you've counted while I was talking, basically I would summarize what I was saying by saying that there are three wins that are blowing us towards believing in the multiverse. None of these prove that there's a multiverse. I'm not claiming that we could prove that there's a multiverse. But there are really three different arguments, all of which seem to be leading in the same direction. First, from inflation itself, almost all inflationary models are eternal into the future. That is once inflation starts, it never stops, but instead goes on forever producing pocket universe after pocket universe. On a completely separate thread starting from string theory, string theorists almost all agree that string theory has no unique vacuum, but instead a landscape of perhaps 10 to the 500 long-lived metal stable states, any one of which could be our vacuum. And finally, the thing that I think really drove large fractions of the physics community to think seriously about these anthropic arguments is number three, the discovery by astronomers, that the universe today is accelerating, which forces us to deal with this issue of a vacuum energy. And a vacuum energy, which is non-zero, but unbelievably small compared to anything that we understand. So one convenient explanation is this anthropic argument, which says that perhaps we see a small vacuum energy density simply because conscious beings only form in those parts of the multiverse where the vacuum energy density is incredibly small. And therefore, it's not surprising that we as conscious beings observe a very small vacuum energy density. The ultimate bottom line of this, though, is I think we don't really understand all this. This could be the right answer. Might not. But we've never had a model of the universe that works so well and that we could explain so many features of the universe, but at the same time, it's so mysterious, because we really can't make sense out of this dark energy. Thank you.