 I've had outstanding lecturers from different fields of theoretical physics and mathematics, Nimar Kanihamed, William Bialek, Super Sashdev, Don Zagir, Sir Hoskin, Brian Hoskin, Sir Michael Berry, and so covering several areas of theoretical physics and mathematics. So today, we have a great pleasure, and we were very thankful for agreeing to come. It's a major figure in the world of science. I will say probably the most influential theoretical cosmology alive. And so Alan Guth made these important contributions to theoretical cosmology since the 80s, and there have been many thousands of papers reading and following his ideas and even text books and so on. And so he kindly agreed to give us three lectures on the field of what he started, which is the Inflationary Universe. So Alan is, I can say some words about Alan. He has started in MIT, and he has several post-doctoral positions in MIT, in Princeton, in Cornell, in Stanford. And now he's back in MIT as a full professor. He is the Victor Weisskopf Professor of Physics and Margaret McIver Faculty Fellow at the Massachusetts Institute of Technology. So he has received many prizes due to his work, where he is the person who started the Inflationary Universe scenario. And the prizes include our own Dirac Medal in 2002. That's one of the top recipients of the Dirac Medal, probably the first or second in cosmology, I would say. The medal he shared with Paul Steinhardt and Andre Lindy. And then he has also received other top prizes, including the Cavalry Prize, the Breakthrough Prize, a few years ago. He was one of the first group of eight people who received the Breakthrough Prize, and then also the prestigious Gruber Prize for cosmology. So essentially, it's a pleasure to have Alan here. Before welcoming Alan, I would like to say that this event is being funded by the Kuwait Foundation for Advice of Science, CAFAS. And so that's something we'd like to thank them for their support, the sense that they have been supporting our Salam lectures for several years. And in particular, we know that the Director General Dr. Shibab Elding is following this very excited about, because he likes very much cosmology, and he is following on his computer, because this is from the online. He will be here with us in a couple of days to also thank Alan. And we also have prestigious visitors today. We have in particular the President of the African University of Science and Technology, who is with us here. And so essentially, before starting, let me just ask you to give a warm welcome to Alan. Thank you, Fernando, for that wonderful introduction. And today, I'd like to tell you folks a little bit about inflationary cosmology. Might begin by saying that I haven't been here at the ICTP since 2003, when I accepted that direct medal. And the place seems to have developed a lot. It was great then. I'm very glad to be back. In today's talk, I'm going to assume that there's very little background of inflationary cosmology. So I'm going to talk about the three parts. In the first part, I will try to explain what inflation is and how it works. In the second part, I'll talk about the evidence that exists, or at least the strongest pieces of evidence, that our universe actually did undergo this process called inflation. And then in the third part, we'll get to this multiverse that's mentioned in the title. It turns out that inflation does not imply, but strongly suggests, that our universe is not unique, but is one of a class of universes, possibly infinite in size. So let me begin. I want to start by actually saying a few things about conventional cosmology. Maybe I'll come in front of this podium here so I can point better. I want to start by talking about the conventional Big Bang theory that is before inflation enters the picture. The Big Bang theory is basically the theory that the universe, as we know it, began some 13 to 15 billion years ago. And now, in fact, we have a very precise number, we believe, 13.82 plus or minus 0.05 billion years ago. The initial state, the state of the universe, 13.82 billion years ago, was a hot, dense, uniform soup of particles that filled space and was rapidly expanding. Now I should maybe clarify at this point that there's a popular cartoon picture of the Big Bang as a little egg that exploded into empty space. That is not and never was the scientific version of the Big Bang theory. I don't know of anything that's illogical about the egg theory, but it doesn't fit very well with what we see. When we look around the universe, what we see really looks uniform in all directions if we average over big enough splotches. Well, if there was an egg, unless we happened to be sitting right in the center of where the egg was, you'd expect that you'd see a hotspot when you looked in the direction of the egg and coldness in the opposite direction. But what we see is completely uniform. And therefore, the conventional version of the Big Bang theory started out by assuming that the universe was completely uniform from the very beginning. All space is filled with matter. The whole thing, space and the matter, are uniformly expanding. The theory explains a number of things. It explains how the early universe cooled and expanded. And it's actually a fairly straightforward, simple calculation to calculate how fast it would be expanding at any given time and how fast it would be cooling, just given some simple equations describing just how gases of free particles behave. The theory also explains how the light chemical elements formed because the Big Bang theory starts its description at a time when the universe was so hot that even the nuclei of atoms would not have been stable. So the nuclei have to be made as the universe evolves. These calculations were begun way back in the 1940s by George Gamoff and his collaborators. They discovered that they couldn't explain anything heavier than lithium and were very disappointed about the whole set of calculations and didn't really know what was going on. Now we know that they were right in their nuclear physics calculations. Only the lightest chemical elements were produced in the Big Bang. We are now confident. The heavier elements will all produce much later in the history of the universe in the interior of stars. And then when stars explode, they spew these heavier elements out into space ready to recollect into later generation stars like our own sun. But the classic Big Bang theory gives us enough information to calculate the expected abundances of these lightest isotopes of chemical elements. And they agree very well with what's observed. And this is one of the classic successful tests of the Big Bang theory. The theory also describes how the matter ultimately collected to form stars, galaxies, and everything else. That evolution theory is still somewhat a work in progress, but things seem to be working out very well. So I think now virtually all cosmologists are convinced that we have the basic principles correct in terms of understanding how galaxies formed. What I want to talk about next is what the conventional Big Bang theory does not describe. Because as you might suspect, that's where inflation enters the game. And that's what I really want to be talking about. The Big Bang theory says nothing about what caused the expansion. Although it's called the Big Bang theory, it is truly only the theory of the aftermath of a bang. It really says nothing about the bang itself. I like to say that it says nothing about what bangs, why it bangs, or what happened before it banged. It really is a bangless theory when you come right down to it. So that's something that we'd like to possibly fill in if we can. The classic Big Bang theory also says nothing about where the matter came from. And the classic Big Bang theory, for every particle in the universe today, there was at least a precursor particle that was present from the very beginning when the theory starts to describe what's happening. So that's also something which we might be able to do better. And inflation does offer possible answers to these questions, as I will try to explain. So OK, with that one slide introduction to conventional Big Bang theory, we'll now move on to inflation. Inflation is basically a theory of the bang of the Big Bang in the sense that the theory of what propelled the gigantic expansion of the universe that we call the Big Bang. And before inflation, there really was no other explanation. As I said, in the conventional Big Bang theory, it was really assumed that the universe was already expanding when the description of the conventional Big Bang theory begins. Inflation explains this expansion by means of a physical force, specifically gravitational repulsion. Now, the concept of gravitational repulsion might be familiar to some of you, but I suspect to many of you it may sound like an oxymoron. Gravity attracts, it doesn't repel. And certainly in the context of Newtonian gravity, gravity does only attract. There is no such thing as repulsive Newtonian gravitation. But there is in the context of general relativity. And from my point of view, this is something of a miracle of physics. And I like to describe inflation as being a consequence of two miracles of physics. So I should maybe first explain what I mean by a miracle of physics. I've been told many times that scientists aren't supposed to believe in miracles or depend on them in their scientific explanations. So when I talk about a miracle of physics, I'm not talking about anything supernatural, but rather I'm talking about a property of the laws of physics as we know them, which somehow have the aura of being miraculous. To be more precise, scientists are supposed to be precise, I will define a miracle of physics as a feature of the laws of physics that have two important properties. One, it should be something that was never taught to me when I was in school. That's one criteria to be a miracle of physics. And the gravitational repulsion was never taught to me when I was in school. And secondly, it should be so far reaching its consequences that it can really change our picture of how the universe works. And repulsive gravity is one of these miracles of physics and soon we'll come to the second miracle of physics which I think is necessary to understand how inflation works. So discussing gravitational repulsion itself, it's been something that's been known to be possible since the advent of general relativity, which is a new form of theory of gravitation which supersedes Newton's theory. And in particular, in general relativity, the key difference that's relevant here is that in Newtonian physics, the only source of a gravitational field is a mass. And it produces an attractive gravitational field. And we only know of positive masses so they all produce attractive gravitational fields. But in general relativity, it's the entire what is called the energy momentum tensor that produces gravitational fields. And this is really necessary just for the theory to be consistent with the theory of special relativity. Mass densities alone don't transform uniquely from one observer to another. What looks like a mass density to one observer will look like a combination of mass density and pressure to a moving observer. And similarly, something that's just a pressure and no mass density to a moving observer would look like it has a mass density. So they mixed under what are called Lorentz transformations. So general relativity in order to be relativistic, necessarily describes gravity as something that's created not only by mass densities but also by pressures. Now the thing about pressures is that they can have either sign. That fact is not maybe obvious either. We're mostly familiar with positive pressures. If you have a gas in a tube, the gas bounces against the side and pushes outward and that's a positive pressure. And with gases, probably all you can get is positive pressures. But matter consists of more possible states than just gases. So imagine instead of a gas, we're talking about a piece of rubber. If we have a piece of rubber, we can compress it and it will push back and that's a positive pressure just like the gas did. But with the piece of rubber, you can imagine cementing plates on the side so you could pull outward. And if you pull outward in all directions so that the rubber is stretched in all directions, it'll pull inward in all directions and that's a negative pressure. And according to general relativity, positive pressures produce attractive gravitational fields which you might think of as normal. A normal pressure is a positive one and a normal gravitational field is an attractive one. But negative pressures produce a repulsive gravitational field. Now with a piece of rubber, it's too weak to ever measure. In fact, the gravitational field of the mass of the rubber you can hardly measure, you can just barely measure it. But in principle, according to general relativity, stretching the rubber will produce a slightly repulsive component to the gravitational field. And there are other materials that particle physicists can invent which have much larger negative pressures where the negative pressure could really dominate and produce repulsive gravity. Einstein, going back into a little bit of history, Einstein used the possibility of a repulsive gravity very early in his exploration of general relativity. A year after Einstein invented general relativity, he wrote a paper described in what happens when we apply general relativity to the universe as a whole. At that time, Einstein was convinced that the universe was static because that's what everybody thought since Newton or earlier. There was not yet any evidence for expansion of the universe. So what Einstein tried to build was a static model of the universe where he just had a distribution of mass, infinite in size, filling all of space. But when Einstein tried applying his equations of general relativity to that system, he discovered that turns out, yes, gravity is actually attractive, and that means that that was not stable. Everything attracted to everything else and the whole system collapsed. The same thing is now believed to also be true for Newtonian mechanics, by the way. Newton actually did write about exactly this question and Newton convinced himself that an infinite distribution of static masses would be stable, but that was really a mistake, but it took Einstein to straighten it out. And Einstein realized that he had to do something if he wanted to preserve his static universe. So he added to his gravitational equations, the equations that describe how gravitational fields are created by matter, an extra term, which he called the cosmological term, which is consistent with all of the symmetries and properties that the equations were intended to have, where this extra term produced a gravitational repulsion, which if you adjusted the coefficient of that term just right, Einstein was able to build a static model of the universe where the gravitational repulsion just balanced the normal attraction of ordinary gravity. And this was Einstein's model of the universe. To Einstein, this extra term, what he called the cosmological term, was just an extra term in the equations that describe how gravitational fields are created. Modern physicists look at it differently. We look at exactly that term, same equation, and to us it looks like assigning a non-zero energy density to the vacuum, where the vacuum would have a positive energy density and a large negative pressure. And it's the negative pressure that Einstein was using to produce the gravitational repulsion to create this static model of the universe. And inflation works a lot like that actually. So in terms of modern particle physics, we now believe that it's super high energies of the kind that would have been present in the early history of the universe. There are actually many states that you can build that would have this property of a large negative pressure and therefore would instigate gravitational repulsion. In technical terms, those states are states there where the energy is dominated by the potential energy of a scalar field. And if you don't know what that means, don't worry about it, it won't matter to anything else. But we do have to believe that there are materials that produce repulsive gravity. Okay, once we have that, we could talk about the sequence of events that inflation talks about. Inflation proposes that at least a tiny patch of this repulsive gravity material existed in the early universe. It certainly doesn't have to fill the universe, it doesn't have to be the whole universe, we just need a tiny patch of this repulsive gravity material. How big it has to be depends on something which we don't really know. We don't really know the energy scale of inflation, which is another way of saying we don't really know exactly when in the sequence of the early universe inflation happened. So we don't know what the energy density was at that time. It turns out that all the successes of inflation happen whether inflation happened extremely early or just very, very early, certainly very, very early at least. So for numerical examples here, I'm going to assume that inflation is linked to grand unified theories, which is certainly historically how inflation was first discussed. Although now we realize there's a wider range of possibilities, but I just want to make it clear that the numbers that I'm giving you are tied to the assumption that inflation is driven by grand unified theories. But I should also say that if I took a different energy scale, the numbers would sound equally outlandish. The outlandishness of the numbers is invariant to this question of when exactly inflation happened. So if inflation did happen at the scale of grand unified theories, the initial patch would only have to be as big as 10 to the minus 28 centimeters. To put that in perspective, let me remind you that the size of a proton is about 10 to the minus 13 centimeters. So this is vastly smaller, more than a billion times smaller than the size of a single proton that we're talking about. So it's truly outlandish. Now since the patch is going to be enlarged fantastically by inflation, it doesn't matter if these patches are incredibly rare. As long as we can believe that there's at least one of them, it's all we really need to explain how our universe evolves from the system. Because once the inflation takes over, this patch is enormously magnified. So once one has such a patch, the process of gravitational repulsion causes that patch to expand. And the expansion is exponential. That is, it doubles in size and then doubles again and then doubles again. And the doubling time, again using numerical examples from grand unified theories, the doubling time would be about 10 to the minus 38 seconds. Incredibly fast. Now to build our universe, what we need for this patch, we need it to expand by a factor of at least about 10 to the 28th, which turns out to be about 100 doubling times. So we reduced the problem of how to get a huge universe to how to just get 100 of these doublings, which turns out to be not hard to do in terms of inventing particle physics models that would lead to that result. Now the 100 doesn't have to be hit exactly by any means, it's a minimum. If you have a model that produces 200 or 1,000 or 10,000 doublings, that's fine too. We just need at least 100 doublings to create a universe that looks like what we see. At the end of this period, notice I said we started at 10 to the minus 28 centimeters and then expanded by at least a factor of 10 to the 28. That means we have about a one centimeter size here. The full universe could be much larger, but at the end of inflation, the reason that's going to become the presently observed universe was about one centimeter across, about the size of a marble. It doesn't stop expanding when inflation is over it coasts and it's the expansion of this coasting period from the end of inflation until today, which takes the universe from the size of a marble to the size of tens of billions of light years, which is how we see the universe today. Now what ends inflation is the fact that this repulsive gravity material is not stable. So it decays in a fashion that's similar to a decay of a radioactive substance. Now when I use the word decay, I don't mean that it rots using the word decay in the sense of the same way uranium decays. That is it changes its form and in particular in this case, we're talking about the matter of the universe changing form from a repulsive gravity material to a normal material with attractive gravity. And that will happen because of the instability of the repulsive gravity material. And what it does it releases energy and that energy produces ordinary particles forming a hot dense primordial soup of exactly the type that had always been assumed as the starting point for the conventional Big Bang theory. So what inflation does is it sets up the initial conditions for the conventional Big Bang theory, which then plays out exactly as had been previously thought before inflation entered the picture. Now an important feature of this is that during the exponential expansion, the density of matter and energy in this expanding region did not thin out. That's crucially important. Just remember we're talking about expansion by at least a factor of 10 to the 28 and that's in linear size and radius the volume then goes like the cube of that so that would be 10 to the 84th power. If the matter thinned out as the volume increased by a factor of 10 to the 84, that would mean that really no matter what you started with you'd have nothing visible by the time the expansion was over. So it's really crucial for inflation that it's possible for this tiny patch to expand by a fantastic factor while maintaining a constant density of energy and mass inside the ever-enlarging volume. Now that certainly sounds like it must violate conservation of energy. More and more mass and energy are being created in this region as it expands but somehow I claim that total energy was in fact conserved. So how in the world can that happen? Well, you might suspect, since I told you there would be two miracles involved here that the second miracle of physics is about to unfold. The second miracle of physics is that although energy is conserved, something that is probably different from most of our intuitions about energy, energy is not always positive. There can be and are negative contributions to the energy as well as positive contributions and in fact, the energy of a gravitational field is strictly negative and that's a statement which is true both in Newtonian gravity and in general relativity. I think I can give you a pretty good, pretty convincing explanation in the context of Newtonian gravity that this is the case. So let me try. I'm gonna assume that most of you know something about the energy of a Coulomb field. It's a standard part of most undergraduate physics educations. The energy of a Coulomb field can be described by an energy density, which is a constant that depends on what units you use times the square of the electric field. And one can always equate the full energy of any system of charges to the energy of the electrostatic field that it produces if it is static by just integrating that energy density over the volume and saying that's the total energy. Now, if you think about it, Coulomb's law of electrostatics looks an awful lot like Newton's law of gravity. In fact, it looks exactly the same. They're both inverse square forces, but if you think about it a little more, the electrostatic law, Coulomb's law has the opposite sign for Newton's law. Two positive charges, we all know, repel each other while two positive masses, of course, attract each other. So if you imagine going through the calculation that you probably learned once about how to calculate the energy density of a Coulomb field, it's the same calculation for gravity, but with an opposite sign at step one and that negative sign carries through the whole calculation. And when you're done, you find that the energy density of a Newtonian gravitational field is a negative constant times the square of the gravitational field strength, whereby gravitational field strength, I mean, little g, the acceleration of gravity. So it's unambiguous that the energy density of a Newtonian gravitational field is negative and that does carry over into general relativity as well. It's not that obvious, but it does. So what that allows here is that the negative energy of gravity can cancel the huge positive energy of matter. So as this region expands, the total energy can still remain whatever it was at the beginning, which was presumably incredibly small and possibly even zero. There could be a perfect cancellation between the positive energy of matter and the negative energy of gravity. So I have a little schematic diagram here to illustrate that the total energy of the universe at least is consistent with being comprised of a huge positive shown in black, quantity of energy associated with matter and radiation and all the stuff that you notice around you. But there's also a huge negative contribution of just the gravitational field that fills the universe. And everything we know is consistent with the possibility that they cancel each other exactly producing a universe whose total energy is exactly zero. And this is really essential for inflation to work. It's not essential that it's zero. It's essential that it's small so that inflation can build the entire universe from this tiny speck we talked about that was only 10 to the minus 28 centimeters across. That tiny speck was very dense but it still had a total mass of only a gram or so. So all the rest of the matter that we see in the universe has to be produced by the inflation itself. And what makes that possible is the large negative energy of gravity so that the total energy of the universe is incredibly small and in fact consistent with being exactly zero as far as we know. Okay, I think that finishes part one of my three part lecture here. That's how inflation works. So let me move on now to the second area that I wanna discuss which is the evidence that our universe actually underwent this process called inflation. And I wanna talk about three pieces of evidence in particular. And the first piece of evidence I wanna talk about is the large scale uniformity of the universe. This large scale uniformity can be seen in the matter distribution. When you look out far and average over big regions, all the big regions look essentially the same. But the strongest evidence for uniformity is what we see in the cosmic microwave background radiation which can be measured to rather extraordinary precision and represents the furthest distance we can see actually. It's the light that comes from us directly from the Big Bang, it's the leftover heat of the Big Bang explosion. And that radiation has been measured very carefully and it turns out to be uniform in all directions, same intensity in all directions to an accuracy of a few parts in 100,000. So it really is rather remarkably uniform. Now that tells us something very significant and very simple really about the very early universe. To understand this we need to say a little bit about what we think the history of this radiation is. The radiation is really just a property of the heat and as far as we know it is certainly present at least from the end of inflation and in the conventional Big Bang theory it was present from the time zero. But for early times the photons that now make up the cosmic background radiation were frozen with the matter and that's because the matter was so hot that it ripped off the electrons from the atoms and formed just a gas of nuclei and free electrons, a gas that's called a plasma. And it turns out that a plasma is very opaque, very non-transparent to photons. So the photons, even though they were instantaneously at any given time moving at the speed of light, they were constantly scattering and changing directions. It's really the free electrons of this plasma that did that. So since the photons changed directions so frequently during this plasma phase they really didn't go anywhere. So you could think of each photon as really just being frozen with the matter that surrounded it. But all that changed at about 380,000 years into the history of the universe when, according to our calculations, the universe cooled enough so that the plasma neutralized became instead a gas of just neutral atoms, mostly hydrogen atoms, some helium traces of other things, but mostly hydrogen with small amounts of helium. Such a gas is very transparent to photons. So instead of the photons constantly scattering and changing directions, from that time onward from 380,000 years onward, the photons have, for the most part, just traveled on straight lines. So when we measure this cosmic background radiation today, what we are essentially seeing is a snapshot of what the universe looked like at 380,000 years after the Big Bang. That is, these photons give us an image of what the universe looked like at that time, just like the photons traveling on straight lines from my face to your eyes, let you know what my face looks like. So we're really seeing a picture of the universe at 380,000 years of age, and what we're concluding is that the universe was just an unbelievable uniform, uniformed to a few parts in 100,000 at the time of 380,000 years after the beginning. Okay, so what does this tell us about understanding the universe? The natural question is how did the universe get to be so uniform? Now we do know that the laws of physics do describe systems which smooth themselves out. The air in this room has smoothed itself out so it has an almost uniform density throughout the room, and if we weren't breathing and disturbing the air and by doing things like that, the air would acquire a very uniform temperature as well, uniform temperature and uniform density. That's sometimes called the zero-width law of thermodynamics. So we can ask ourselves, is the zero-width law of thermodynamics effective enough to describe why the universe became so uniform by 380,000 years after the beginning? The answer, however, turns out to be a resounding no, it's not possible, and what makes it not possible is the simple fact that nothing that we know of travels faster than the speed of light. And it turns out that if the universe at its birth was not uniform, for it to become uniform by 380,000 years would require the transport of matter and energy at about 100 times the speed of light. And this is by about a factor of 100. So you're still free to assume if you want that the universe just started out perfectly uniform. There's nothing, we don't really have a theory of the ultimate beginning of the universe so you're perfectly free if you want to attribute this uniformity to some unknown theory of the ultimate origin of the universe. But if one would prefer to have a theory that we can understand that would explain the uniformity of the universe, then the conventional big bank cannot do it, but inflation can. The difference with inflation is the expansion pattern of the universe. So what inflation does is it inserts into the history of the universe this period of gigantic exponential expansion. And what that means, you've got to have something like this written. What that means is that before the inflation, the region that's going to become the observed universe was vastly smaller than you would have thought in conventional cosmology without inflation. So there was plenty of time for this uniformity to be established in this tiny, tiny region before inflation started. The region I told you could be a smallest 10 to the minus 28 centimeters. Then once that uniformity is established by the ordinary zero with law of thermodynamics, then inflation takes over and stretches that tiny region to become huge, larger than everything that we see while maintaining the uniformity that was established before the inflation. So inflation gives a very natural and simple explanation for how the universe got to be so uniform. An explanation that is just not possible within the conventional big bank theory without inflation. Okay, so that was item one, the uniformity of the universe. Item number two that I want to discuss is something that's called the flatness problem. A word that might not make much sense yet, but hopefully it will by the time I finish talking about it. It's the question of why the early universe was so flat. And by flat, I don't mean two dimensional, as sometimes I find audiences interpret my words to mean, but by flat, I mean Euclidean. The point is that according to general relativity, space does not have to be Euclidean. Euclidean spaces are a very special case in the context of special relativity, excuse me, general relativity. General relativity is really a theory of curved spacetime. And in particular, if we assume that the universe is homogeneous, meaning the same in all places, and isotopic, meaning the same in all directions, and to a very good approximation, our universe has these two properties, if we assume that those properties are exact, then there are only three possible geometries, which go by the name of closed, open, and flat. And I'm talking here about the three dimensional geometry of our universe, but there are very good analogs and two dimensional surfaces, which are much easier to visualize. It's hard to visualize the curvature of a three dimensional space. Frankly, I can't do it, and I'm not sure I know anybody who can. But we can visualize curvature of two dimensional surfaces, and it's an excellent analogy, and we do have to write down equations that describe the three dimensional curvature, whether we can visualize them or not. So closed geometry is just like the surface of a sphere. And an important geometric property, which the three dimensional analog shares, is that if you draw a triangle on the surface of the sphere, you can kind of tell from that picture that the sum of the three angles in the corners of the triangle will always be a little bit more than 180 degrees, where 180 degrees is what we learned in high school, is what happens for Euclidean triangles. But on the surface of a sphere, it's always a little more. The open geometry is similar to a saddle-shaped surface, and again, the picture illustrates that the sum of the three angles will always be a little bit less than 180 degrees for an open surface. And right in between these two is the special case of a surface which is flat, which obeys all the rules of Euclidean geometry, where the sum of the three angles is always exactly 180 degrees. And we'll think of this calculation of the sum of the angles of a triangle as the separator between open spaces, closed spaces, and flat spaces, where flat is just the borderline between the two others. In terms of cosmology, general relativity links the curvature of space to the presence of matter, one could think of matter as causing the curvature of space in general relativity. And according to general relativity, the flatness of the universe is just related to its mass density. Cosmologists talk about a quantity we call omega, capital Greek omega, which is just defined to be the actual average mass density of the universe divided by what's called the critical mass density. This critical mass density is not a universal constant. It depends on the expansion rate of the universe, but for any given expansion rate, you can calculate this critical mass density. The critical mass density is by definition that mass density that makes the universe exactly flat. So by definition then, if omega is one, that means the actual mass density is the critical density, and that means the universe is flat. If omega is bigger than one, that means the universe is closed. If omega is less than one, that means the universe is open. So it's important that we just get this vocabulary of omega under our belts. Now what makes flatness a mystery is that a universe with a critical density when one looks at how it evolves according to the basic equations of cosmology, turns out to have a very strange property. It's like a pencil balancing on its tip. In more technical terms, it's an unstable equilibrium point. The basic idea is that if the pencil is exactly straight up, it won't know which way to fall, and in principle it could remain straight up forever, at least according to classical physics. We'll ignore quantum physics for now when we talk about pencils. It's a purely classical pencil. It could stay straight up forever, but it was leaning just the tiniest bit in any direction. It would rapidly start to fall in the direction it was leaning. Omega is just like that. If omega in the early universe was exactly equal to one, it would stay exactly equal to one forever, but if omega was slightly below one, it would rapidly fall to zero, and it was, for reasonable numbers, it would fall to zero so quickly that there would never be time for anything like galaxies to form. The universe would end up not resembling our universe in any way. It would just be an empty universe. And if omega was slightly greater than one, it would rapidly rise towards infinity. When omega reaches infinity, that's the same as saying their closed universe, which has been expanding, slows down and comes to a halt, and then rapidly re-collapses. And again, for typical numbers, this would happen so fast that there would never be time for galaxies or anything complicated to form. So for the universe to look anything like the universe that we see, where there has been enough time for all of this complicated structure to form, omega in the early universe had to have started incredibly close to one. And so far, I've just been qualitative, but I can give you a number. To be even within the factor of 10 of the critical density today, which it certainly is, and I'm sort of thinking now myself of the history of all this, this all started when people were thinking about this around 1980. At that time, all we knew was that the universe was within about a factor of 10 of the critical density. But even with that very limited knowledge, one could still conclude that at one second after the Big Bang, omega must have been equal to one to the extraordinary accuracy of 15 decimal places. If omega was just one digit larger in the 15th decimal place, the universe would have reclapsed so fast that life could never have formed, galaxies could never have formed. And if omega was just one digit less in the 15th decimal place, then what we think it was, the universe would have just thinned out so quickly that no structures could form. It's unbelievably sensitive. And that's what was known as the flatness problem. Omega apparently had to be extra narrow close to one to get the universe that we see, but there was no mechanism, whatever, in the conventional Big Bang theory to explain why omega started out so extraordinarily close to one. Now again, if you are willing to just ascribe these features to an unknown theory of initial conditions, you could do that. You could assume that the universe, for some reason that we don't yet understand, started out with omega exactly equal to one. But otherwise, with the conventional Big Bang theory, there's just no way to explain it. But inflation gives a natural explanation. The key point is that during inflation, the evolution equations that describe the universe change completely because gravity is changing its sign. During inflation, gravity is instead of attractive and that of course changes the equations of evolution in a very dramatic way. And during inflation, it turns out that instead of omega being driven away from one, as it is throughout the rest of the history of the universe, during inflation, omega is driven towards one from either side and incredibly rapidly. So if we had the amount of inflation that I told you we needed about inflation by 100 doublings, if we had 100 doublings of inflation, then before that, omega didn't have to be one. It could have been two or a half or 10 or a tenth. Any of those numbers would be fine. If you want to start omega very far away from one, then you might need a little bit of extra inflation in order to drive omega to one successfully. But inflation always drives omega to one and incredibly rapidly, giving us a very natural again explanation of why the universe started out with omega, so incredibly close to one. This actually leads to a prediction. The mechanism that inflation uses to drive omega to one will almost always overshoot in the sense that it will drive omega closer to one than we can measure, unless inflation just happens to stop at just the right time. This could be avoided. But generically, inflation will always overshoot and produce a universe that's so, because omega's so close to one that it will be indistinguishable from one. So the prediction is that omega should be one even today. For many years, this was actually a problem for inflationary enthusiasts, of which I was one. Until about 1998, observation pointed to omega being about 0.2 or 0.3, missing by a factor of three to five from the magic value that inflation predicted. On a personal note, I might just mention that that made it fairly uncomfortable during this period for me to have dinner with astronomers. They were fairly ruthless and sneering at me and telling me that inflation was a very elegant theory, but it obviously wasn't right because it predicted omega should be one and it wasn't. However, things got better. Starting in 1998, omega started to look a lot more like one and now the most recent value from the Ponx satellite, combining their observations with other astronomical observations in a somewhat complicated way, comes up with the observational number that omega is 0.999 plus or minus 0.004 to 95% confidence. In other words, to simplify that, we now know that omega is equal to one to an accuracy of at least about a half of a percent. 0.004 translates to about a half of a percent. So, big success. I think it really is a spectacular success for inflation. The new ingredient, the thing that changed between 1998 and now, it was the discovery in 1998 that the universe is not slowing down under the influence of gravity, but it was discovered in 1998 by two important groups of astronomers that the universe actually has been accelerating for about the last five billion years or so. They discovered this by observing distant supernova explosions and using those distant supernova explosions as indicators to be able to track the expansion of the universe and discovered this surprising result. Now, the reason that affects omega is that if the universe is accelerating, it comes back to what I said earlier about general relativity and repulsive gravity. The only way we know how to explain it in the context of general relativity is to assume that the universe today is filled with some invisible kind of material with a negative pressure. And even though we're not completely sure what that material is, we can still calculate how much of it there needs to be in order to produce the observed repulsive gravity. And it's when that gets added in that we get this 0.999 instead of 0.2 or 0.3. So the astronomers were, in fact, they have to admit now essentially right about the matter that they were looking at, and it does only make up an omega of 0.2 or 0.3, but now we have this new ingredient which makes everything to fit together beautifully with the predictions of inflation. Okay, so that's the second of my three pieces of evidence for why I think we should believe the universe actually did undergo inflation. Let me now go on to number three, which is perhaps the most persuasive. I told you that on large scales, the universe was incredibly uniform, but obviously on smaller scales, there are a lot of non-uniformities. There's Trieste and the Earth and the Sun, and the Sun is part of a Milky Way galaxy that has a disk. We see a lot of different things happening in different places, so we need to explain these small-scale non-uniformities at the same time as we're trying to explain the large-scale uniformity, the fact that everything looks the same if we average over huge regions of space. In the context of conventional cosmology, one really didn't have a clue about how this happened, but in the context of inflation, we have in fact a spectacular explanation. Inflation attributes these non-uniformities to quantum fluctuations. So if this inflationary theory is right, or if this aspect of inflation is right, it means that things like the Andromeda galaxy and the Milky Way galaxy that we live in are in fact just quantum fluctuations, just random instances of quantum behavior. The key important feature of quantum mechanics here is the randomness of quantum mechanics. As you know, classical physics, which includes Newtonian physics and even general relativity, are completely deterministic theories. In the context of classical physics, inflation would just produce a perfectly smooth universe. The gigantic stretching that inflation involves would just smooth everything out. And if that were the case, there really would not be any possibility of galaxies forming because the universe would just be too uniform. There'd be no place for the galaxies to form. And historically, there was a period of six months to a year where those of us working on inflation were very worried about this. We had no idea how we could make the existence of galaxies, which is certainly a known fact about the universe, consistent with inflation. It looked like inflation was just gonna give us a completely smooth universe. But then it was realized that we should take into account quantum effects. And the key idea is simply that quantum physics is not totally predictive. It only predicts probabilities. So if the classical fixture was predicting a perfectly uniform density, then the quantum description will say that the density is almost perfectly uniform. But in any given place, it might be a little bit larger than that classically predicted density or a little bit lower. And that will happen differently in different places. And that gives you exactly a rippling pattern on top of this almost uniform background. And that's exactly what the early universe seems to have looked like. An almost perfectly uniform background, but with tiny ripples on top of that uniform, or almost uniform background. And then the way galaxies form from those tiny ripples is something that's well understood and was understood before inflation. A system like that is gravitationally unstable. And the places where the mass density is just slightly higher than average, even if it's only one part and a hundred thousand. That will create a slightly stronger gravitational field pulling matter into that region. And then the mass density will go up. The gravitational field will become even stronger and more matter will fall in. And it's unstable and it's those kinds of instabilities that lead to the complicated structure of galaxies and clusters of galaxies that we see in the universe today. So that part, one does understand purely on the basis of classical physics, although it takes rather large scale computer simulations to see how it actually plays out. But it ends up working very well. But for the origin of these fluctuations, cosmology before inflation had no explanation but inflation attributes these initial non-uniformities to simply the uncertainty of quantum mechanics and what the due calculations based on that assumption and those calculations end up giving spectacular results. Now I should confess to begin with that we don't know how to calculate what's called the amplitude of those fluctuations. That is how big the fluctuations are and the amount of density that changes. If we had a full theory of inflation where we really understood all the underlying dynamics, it would predict everything, including the magnitude or amplitude of these fluctuations. But we don't know that much where the details of inflation are still a matter of speculation. But what inflation predicts even without knowing the details is what's called the spectrum of the fluctuations. Where the word spectrum means pretty much the same thing that you would mean if you talked about the spectrum of a sound wave. As you probably know, you can take an arbitrary sound wave and break it up into components of definite wavelength. And then the amount of intensity you have of each wavelength is what the spectrum is. With sound waves, you might be more likely to talk about frequency but you could talk about wavelength if you wanted to. With these fluctuations in the early universe, they don't really oscillate so frequency doesn't work. Frequency doesn't have any meaning here but wavelength does. So you could talk about how the intensity of the fluctuations varies with the wavelength. What we can measure most accurately is the cosmic microwave background radiation where wavelengths are measured as angles on the sky. We're just seeing everything as a function of angle. No measurements of absolute distances and meters. So let me show you some of the results which are just gorgeous. This is the raw picture, the best raw picture that we have of the microwave sky and it's the fluctuations in its intensity. The fluctuations here are much exaggerated. This is not a telescope picture. This is a reconstruction using a false color code. The false color code is shown on the bottom. The actual differences in temperature are measured in micro kelvin, millions of a degree kelvin, about hundreds of those. They're still just 10 to the minus four kelvin. And this picture was developed by the Planck satellite which is the most recent satellite to measure the cosmic microwave background. There have been three satellites, by the way, that have been dedicated to measurements of the cosmic microwave background. First, COBE, Cosmic Background Explorer launched in 1990. Then WMAP launched in 2000 and now Planck. I forget the exact year of the launch, 2011 maybe, I'm not sure. And this is the really important picture, the picture of the spectrum. So what's shown here on the vertical axis is the magnitude of the fluctuations as a function of the wavelength. The wavelength is shown on two different scales. The longest wavelengths are at the left. The shortest wavelengths are at the right which might be the opposite of the way you would have plotted it, but it's the way they plotted it. The precise way that they define the wavelength comes from expanding the pattern of temperatures that are measured on the sky and what are called spherical harmonics, which I'm sure many of you know. If you do know, the L that appears here is the L of the spherical harmonic. But if you don't know about spherical harmonics, no problem, the effective wavelength measured in degrees is shown on the bottom graph. And the important thing is not really what the numbers are but how wonderfully these little dots fit the red line. The little dots are the actual measurements where the bars are the uncertainties in those measurements. You can see the bars on the left, the bars on the right are too small to be seen. The dots are already as big as the uncertainties of the measurement. The reason, by the way, why the errors are so much larger on the left than on the right is due to a fairly simple fact. The left is the large wavelengths going all the way down to or up to 90 degrees. If you think about a 90 degree wavelength, there aren't that many of them that fit on the sky. So if you're measuring the intensity of 90 degree wavelength waves, you just don't have a lot of samples on the sky. And because you have only a few samples, you can determine the average value as accurately. But if you're looking at a wave of 0.08 degrees, there are zillions of those on the sky. And that just gives you tons of data to average to beat down the errors. So the errors go way down on the right so that they're no longer visible. The dots show the actual data. The red lines are the predictions of an inflationary model. Now I should confess here that the inflationary model does have some free parameters in it, some numbers that are determined by the data. This is actually a six parameter fit. That is, there are six parameters that are fit to the data to make the fit look as good as possible. Of those six, there's really only one that's genuinely important to keep in mind. And that is the overall height of the curve. That is this amplitude I mentioned earlier is not predicted really by inflation, at least not by our current understanding of inflation. So the overall height here is fit to the data. So the fact that the height matches perfectly is not something that should impress you. That's just part of the way the data was analyzed. But the fact that the relative heights of all these different peaks fit perfectly, that really is the prediction of inflation that we're testing. And it works just unbelievably well. I regard this as one of the most amazing things that I've ever seen in my life, actually. It's really just mind-boggling that it's possible to predict properties of a cosmic microwave background fluctuations as intricate as this and go out and measure it and see if it fits perfectly. I find it just an amazing feat of science, really. So that was the third piece of evidence for inflation that I wanted to discuss. So now I wanna move on to the third of my segments of my lecture tonight. I wanna talk about the way in which inflation suggests that we might live in a multiverse rather than in a single universe. Sometimes people like to ask me if how would I define what I mean by a single universe and what do I mean by a multiverse? Probably I should start by giving you that definition. Sometimes universe is defined to mean everything that exists and it's pretty much a tautology that if you define universe to mean that, there's only one. There can't be two everythings that exist. But for purposes of cosmology, we define a universe essentially as everything that comes from a single big bang beginning. So if you have multiple big bangs, you have multiple universes by definition and that's really what we mean when we talk about having multiple universes. We mean having multiple big bangs. Okay, so the key point for this third part of my talk is that almost all detailed models of inflation lead to what's called eternal inflation and that in turn leads to a multiverse. Now by eternal inflation, what I mean is that once inflation starts in most of these models, it never stops. We're maligning the word eternal slightly because by its, I don't know about Italian, but by its English language definition, eternal would mean it existed for all time including the past and we're not saying that. We believe it did in fact start at some time in the past but it's eternal into the future. Once it starts, it never stops. And the reason for that can be described crudely by a fairly simple explanation. I told you earlier that inflation ends because this state that drives the inflation, the repulsive gravity material is unstable. So it decays like a radioactive substance decays. But that actually turns out to be a fairly accurate analogy and if you follow it through, you realize there's something peculiar going on. If it decays like a radioactive substance, it means it can be described by having some half-life. You can't predict when a particular uranium atom is going to decay. You could just define as half-life meaning that if you wait that time, the probability that decay will be a half. But you can't predict whether it will decay or not. The same is true for this inflating material. So if we take a chunk of it and follow it for a half-life, then indeed we should expect that at least on average, half of that region that we were looking at will no longer be inflating at the end of that half-life. But the catch is that in the meantime, the entire system has been exponentially expanding and the exponential expansion is essentially always vastly faster than the decay. So if you wait for a half-life, half of the region that was inflating will no longer be inflating. But the half that's still inflating will be vastly larger than the region you started with in the first place. So the amount of volume that's inflating has actually increased during that half-life, in fact by a huge factor, rather than decreased even though we were watching this unstable material decay. And that's what causes inflation to become eternal. Once this happens during one half-life, it just keeps repeating during each half-life and at least in the context of this theoretical model, it goes on forever producing larger and larger regions of inflating material. Inflation keeps stopping in some places as we go along in this picture. Wherever inflation stops, we produce a ordinary universe, the matter has become normal, just like the history I described for you of what we think our universe underwent. So we produce what we call pocket universes in this context and the pocket universes go on getting produced and produced and produced forever so that ultimately we have an infinite number of these pocket universes. And we would be living in one of these infinity of pocket universes if this picture is right. These pocket universes would be just like our universe and one of them would in fact be our universe. Now you might think that this does not connect to anything in physics. So even though it's a cute story and might even be true, is there any reason why physicists should care about it? It turns out that now there is a reason why we should care about it. And it comes down to this dark energy material that I told you about that was discovered in 1998. And there's the energy that's needed to explain how the universe has been accelerating, what's been causing it to accelerate. So in 1998, as I mentioned already, two groups of astronomers discovered that for the past five to six billion years, the expansion of the universe has been accelerating and some of those folks won the Nobel Prize in 2011. According to general relativity, the only way to explain this is that there is some kind of repulsive gravity that is negative pressure material that's filling the universe, driving this expansion of the universe, which does in fact appear to be exponential. And that stuff, whatever it is, is called the dark energy. However, the simplest explanation for what this dark energy is, and astronomers have been testing this better and better as time goes on and everything that has been measured is completely consistent with this simplest explanation. The simplest explanation is that it's simply vacuum energy, energy of empty space. Now you might wonder why anyone would believe that empty space should have an energy density that's non-zero, but there is a simple answer to that. From the point of view of particle physics, the vacuum is not empty. It's really a very complicated state. The only thing that defines the vacuum as being the vacuum is that it's in principle the state of lowest possible energy. But in the vacuum, there are complicated things that are going on all the time because of quantum theory, the electromagnetic field inside the vacuum is constantly fluctuating. There's just no way that the electromagnetic field can be still and be consistent with the uncertainty principles of quantum theory. There's also, according to current particle physics, a field associated with every kind of elementary particle, electrons, for example. And the electron field is also constantly undergoing fluctuations just like the electromagnetic field. And besides that, there's also at least one field that we know of. The field is called the Higgs field, the particle of which was discovered famously at CERN in 2012. The Higgs particle is associated with a Higgs field and the Higgs field has a non-zero value throughout space and also has fluctuations due to quantum effects. So there are all kinds of things that go on in the vacuum and there's no reason to expect the energy of the vacuum to be zero. So particle physicists have no reason to be the least bit surprised that the energy turned out to be non-zero, but if we try to understand the actual value of the energy density that was measured, that turns out to be a huge mystery. The value that was measured does not make any sense to particle physicists. The point is that we don't actually calculate this vacuum energy density. If we could, then we'd either be right or wrong and that would sort of end the story one way or the other. But even though we cannot calculate it, we can estimate it by understanding the contributions coming from particular places like the ones I mentioned. We just don't know all the places where the energy can come from. We really don't know exactly how to calculate the energy of the things I mentioned either, but we can estimate it, as I said. And the natural estimate is what is called the Planck scale. It's the energy scale at which the effects of quantum gravity are expected to become important. And the mystery is that what's actually measured is much larger than this expected value. This Planck value is larger by a full 120 orders of magnitude, 10 to the 120th power. So this might be called probably the worst estimate that physicists have ever made in the entire history of physics, probably is. And it certainly means that there's something very funny going on that we don't understand about the vacuum energy density, the energy density of empty space. So physicists since 1998 and actually some before, although since 1998 it's become a passion for almost everybody. Physicists have been trying to invent explanations for why the vacuum energy is so small. And if you look through the literature, you do find explanations, but you can't find any explanation that's believed by more than the authors and maybe an equal number of other physicists who've read it somehow are sympathetic. But by and large, there's no accepted explanation for why the vacuum energy is so small. But one possibility is an explanation that involves eternal inflation and the multiverse. The multiverse, I say, offers a possible, although I admit, controversial explanation of why the energy density of the vacuum is 120 orders of magnitude smaller than the expected Planck scale. And here I mentioned that the energy density of the vacuum is synonymous with what Einstein called the cosmological constant. What the Einstein was just a constant added to the equations that describe how gravitational fields are created to us is a vacuum energy density. It's the same equations. Nobody got no controversy about that. It's just a question of what you call that term. But it has exactly the same form as it would if the vacuum had a non-zero energy density and that's the way modern physicists look at it. So how does a multiverse get us out of this problem? And the answer hinges on the possibility that there might be more than one kind of vacuum, even there might be even many more than one kind of vacuum. And in particular, according to string theory, the vacuum is not at all unique. There are zillions of states in string theory that look like vacua. And the number is said to be 10 to the 500 or more. And with the or more might be an or much more, 10 to the 5,000, 10 to the 10,000, possibly infinite numbers that are talked about these days. So 10 to the 500 really is a bare minimum from the point of view of string theorists. So you have many, many different kinds of vacua. And each one would have its own vacuum energy that would be calculated according to the properties of that particular vacuum. They would typically be of the order of the so-called Planck scale, this huge scale, much larger than what we observe, but they could be positive or negative. And they would tend to be spread presumably from sort of plus the Planck scale to minus the Planck scale. And everything in between, very densely packed. And if that were the case, there would then be plenty of kinds of vacuum with energy densities in the range that we observe. These would still be incredibly rare in the set of all vacua. There would be a tiny fraction of a huge number. So the tiny fraction is still a huge number of vacua. But it would still only be a tiny fraction of the full set, which leads to the question of how can you believe that we'd be living in such a rare kind of vacuum? And if you can't make it plausible that we'd be living in one of these very rare vacua, you don't really have an explanation. But there is actually a reason why we might find ourselves in such an extremely rare type of vacuum. And that is that the vacuum energy is not just some number that you can only find by looking under a rock. The vacuum energy really affects the evolution of the universe in a dramatic way. It drives the acceleration. That's how we learned about vacuum energies in the first place. So if the vacuum energy really were of the order of this Planck scale, that kind of universe would just blow apart by the gravitational repulsion essentially instantly. And there would never be time for anything like life, even if you're very speculative about what kinds of life might exist throughout the multiverse. Nothing, no life is gonna form if the universe blows apart in 10 to the minus 42 seconds. Similarly, if the vacuum energy were negative and of this Planck scale, the universe would simply implode again in about 10 to the minus 43 seconds or something like that. So the only kinds of universes that would allow billions of years for evolution would be universes with incredibly small values of the vacuum energy. So the hypothesis is that life only forms in those pocket universes with incredibly small vacuum energies because those are the only universes that can live long enough for complicated things like galaxies and people to form. So the conclusion then would be that if we could see the whole multiverse, most of it would have vacuum energies of the order of the Planck scale, huge compared to what we observe, but we would still argue that almost all living things that exist in that multiverse would see a very, very small value of the vacuum energy because living things would only form in those regions where the vacuum energy is incredibly small. So this does provide a possible explanation for the very small vacuum energy. I'll be talking a little more about the details of it tomorrow. It is controversial because people aren't completely in agreement of whether this kind of an argument suffices and even people who believe that there's kind of argument suffices, there are still of course numerical issues in terms of how small does this really explain for the value of the vacuum energy density. But I will say that I'm pretty sure that nobody in physics would really, well, with a few exceptions, I think very few physicists would claim that there's any better explanation that we know than this one. I wanna close by just giving you a few quotations that give you some sort of anecdotal sense of how seriously physicists take this issue of multiverse. And so a few people I'd like to quote. The first is Sir Martin Rees, he's been knighted. If you do any work in astrophysics, you would know this name, of course, but he's astronomer Royale of Great Britain. He's also a former president of the Royale Society. He's also a former master of Trinity College at Cambridge. He's been knighted. He has most every honor you can imagine other than the Nobel Prize, very highly respected. And he said at a conference a few years ago that he's sufficiently confident about the multiverse to bet his dog's life on it. Now I have to admit I've never met Martin Rees' dog. So I don't know exactly how cute he is or what exactly these stakes are, but people usually love their dogs pretty much. So these are pretty high stakes. Now at the same conference, Andre Linda was present of Stanford University. He's the first who's played a very major role in developing inflationary cosmology. In fact, he shared the Dirac Prize with me back in 2002 and also the Gruber Prize and also the fundamental physics prize and also the Cavalry Prize. And he's a big enthusiast of the multiverse. So he was at the same conference that Martin Rees made the statement we just quoted and he said rather quickly that he's so confident that he would bet his own life on the existence of the multiverse. Of course, he didn't say exactly how the bet would be determined and things like that was getting a little sticky. But nonetheless, he was fully expressing enormous confidence in the presence of the multiverse. Now personally, what I always regard is the voice of reason is Steve Weinberg, winner of the 1979 Nobel Prize and author of many books, including The First Three Minutes, which is a marvelous explanation of basic cosmology written before inflation, but otherwise it has all the details right. And he's somebody who I always trusted. He was not at this conference, but he read about these comments later. They were quoted by some journalists. And he said that he has just enough confidence in the multiverse to bet. Try to think of something really clever here because Steve is very clever. He said he has just enough confidence to bet, but why is it both Andre Lindey and Martin Rees' dog? I'll stop there. Thank you. Thank you very much. I think it's a great presentation. Any questions? I just wanted to, I'm complete outside the condensed metaphysicist. I just wanted to ask about this map of the density of microwave radiation. So if my, I did not betray me, I saw not only fluctuations, but also correlations, the angular correlations. Am I right? Because it was like the clusters of, let's say, blue color and the clusters of... Yeah. Yeah, I think you're right. I'll try to get back to the picture so I can... Everybody can know what you're talking about. You see, the word brown, like a cluster, it does not look like completely random in triangle. That's right. Is there any explanation for that? Yeah, no, that's exactly... Well, the correlations you're talking about are the fact that there are some wavelengths. I think I'm amplified. The correlations you're talking about are simply the fact that there are some wavelengths that have much stronger fluctuations than other wavelengths. And when you see fluctuations on the same wavelength, it looks like a correlation. And that's exactly what's shown more explicitly on this curve. And as you can see, there's one particular wavelength of a little bit less than half of a degree, or about a half of a degree, that stands out enormously over the rest. And that's what you're seeing. So you're absolutely right. Those correlations are there. And they are part of the theory. I'm Adrian Tomkins from the Earth System Physics Group here. So I'm completely an outsider as well. I was just wondering, earlier in your talk, you mentioned that there was the positive feedback that drove omega towards one. And then towards the end, you were talking about the multiverse and the vacuum energy and basically the observational selection theory. Couldn't one also invoke the observation selection theory also for the omega being close to one? As you mentioned yourself, if it's not close to one, then basically the expansion or the collapse is extremely quick. So I was just wondering why that was discounted as an alternative theory for the omega being basically one in the universe that we exist. Okay, very good question. I think there is an answer. The answer has to do with the scale of the region that has a uniform value of omega. It's certainly true that if you insist that the universe be exactly homogeneous, that omega is the same everywhere, then omega has to be very close to one for life to form. On the other hand, if you're saying we don't have inflation, we don't have a theory, we're just gonna talk about what might happen at random. Then it's true that life would only form where omega is very near one, but there's absolutely no reason, and in fact it's extraordinarily improbable for that reason to be nearly as large as what we see. To produce life, it completely suffices to have a small patch. A few galaxies size or something like that, cluster size maybe, where omega is one in that patch, but no reason for it to also be one outside. So the possible anthropic argument for omega equals one doesn't seem to suffice at all to explain the fact that omega appears to be one throughout the visible universe. Very good question here, Ellen. Yes, thank you very much for the wonderful talk. If I understood correctly, you're saying that the fundamental hypothesis is to do with what happened right before the classical Big Bang theory kicks in. So this inflation theory takes it a little bit further back, but there's still a gap, right, from the very beginning. I mean, there's still something to do. No, that's right, that's right. I should maybe emphasize it more. You're calling attention to it. You're absolutely right. Inflation is not a theory of the ultimate origin of the universe. Inflation starts with there already being a small universe and we have to make some mild assumptions about the properties of that small universe for inflation to start. But inflation does explain how we go from a tiny speck in a preexisting universe to a universe that looks like what we see. It explains the many of the properties of the universe that we see, but it does not explain how that initial universe came into existence. Well, I would consider an extension of the classical Big Bang theory, not an alternative, because the classical Big Bang theory does play out in full in the context of this inflationary picture. Inflation sets up the initial conditions for the conventional Big Bang theory while previously those initial conditions, which were very extreme really in terms of how fine-tuned they had to be, previously those initial conditions just had to be assumed. In terms of trying to understand the ultimate origin of the universe, maybe I should spring from your question, cut some comments about that. Theoretical cosmologists do attempt to address that question, although we certainly do not yet have a consensus theory of how the universe ultimately originated. But the kinds of theories that people have been playing with are theories that involve vague ideas about quantum gravity. We don't really have a complete theory of quantum gravity. But if we believe that one can imagine a theory of quantum gravity and one can imagine some of its properties, then one can imagine that in the space of all possible states of existence, there would be a state in this theory that could be called nothingness, a state where there's no space, no time, no anything. And that would be the natural starting point. And then you can ask, is it possible to have quantum mechanical tunneling from that state of nothingness to a state of a small universe that would then inflate? And there have been a number of papers along those lines over the years. But since we don't really have a solid theory of quantum gravity, all of that is quite speculative at this point. More questions? Yes. I have a question about the cosmological constant problem. We said at the beginning that the energy is constant in the universe, but not necessary it is positive. Because gravitational energy lowers the energy because it gives negative contribution. So I'm asking if maybe a possible alternative solution to the cosmological constant problem needs to quantize gravity and see if in some theories the orders of magnitude of different at least gets down to a best agreement to the one observed. Quick, so you're asking if the smallness of our observed vacuum energy might just be the effect of gravitational fields which are decreasing the total energy? I think the answer is that the way these things are measured, one does have a clear separation between the energy of the state itself and the energy of the gravitational field. So I don't think that that possibility exists. People have certainly talked about similar things, maybe you'd even consider this identical, but if we modified our theory of gravity, gave up on general relativity as being the exact theory, but considered alternatives, then people have talked about possibilities that somehow gravitational fields can be screened so that the effect of these vacuum energies would look small, even if the vacuum energy really is large. People have been exploring things like that, but I think I'm pretty safe in saying that nobody has found anything yet that has been convincing. One questions? Let me ask a question myself. Go for it. Go for it, okay. Well, I have two questions. So let me see. There has been some debate about inflation and how much is a physical theory. Can you make some comments about that because there's some debate on the scientific American and the internet and so on. Right, right, yeah, some of you may be aware, I could fill in some details of what Fernando was saying, some of you may be aware that there was an article in Scientific American, I think even a cover article in Scientific American last February by Anna E. Joss, Paul Steinhard and Avi Loeb, which made very strong claims that inflation is not really a theory at all, it's not, festival cannot be tested by the methods of traditional science they claimed and they even claim that people who espouse inflation know that and are sometimes advocating a new version of non-empirical science they accused us of. In my opinion, all that is total nonsense and I don't really think it extends very far beyond the three authors of that article. When the article came out, I was somewhat incensed and so were many of my friends. So four of us got together actually and decided to answer by writing a letter to the editor of Scientific American and to try to make it clear that it really was a community opinion that we were discussing and not just several other physicists besides the several physicists who wrote the article, we solicited co-signers of our letter. We wanted to keep it, we didn't want to make it a broad petition because Scientific American doesn't publish petitions. We wanted to make it look like a letter with some authors. So we started thinking about 20 authors, but we ended up with 30 because there was no way that we found that we could limit it because a lot of friends were asked to sign it. So we ended up submitting a letter to Scientific American which you could find, I think it came out in something like July, both in the print edition and you could find it on the web. And I was very pleased when we went around looking for co-signers, we went to essentially all of the leaders in the field, at least people we recognized as leaders of the field and literally every one of them agreed with us that the claims in the Scientific American article that inflation is untestable are just wrong. Almost everybody we contacted signed the letter, a few people did not for various personal reasons. But anyway, we think that that article is essentially nonsense. One key point which I think the article maligns is what it means to say that a theory is falsifiable. Everybody, at least they like to quote Popper. I think Popper is actually not so popular these days in the philosophy of science. I think most philosophers of science today recognize that scientific decisions are more complicated than just testing a theory and deciding if it's falsified by the observations. For example, when the Michelson-Morley experiment was done, one might think that that falsified Newtonian mechanics and proved that relativity had to be right. But that was not the initial action that scientists had to choose between that explanation and the alternative explanation that the null effect seen by the Michelson-Morley experiment was due to ether drag, which was another possible explanation. So in general, I think modern philosophers of science recognize that it's a complicated judgment call among scientists to decide what combination of ideas best fits the observations rather than just falsifying. Anyway, let me forget that. The key point really is that even if you take Popper literally, Popper says we should only consider theories that make predictions that could be falsified. It's certainly true that every model of inflation makes very specific predictions that can be falsified. And if I can make an analogy here, it's I think very much like the situation in quantum field theory. There are many different quantum field theories that make many different predictions. We believe the standard model of particle physics is correct in its sphere of influence, in its sphere of validity, because it works so well. And we never ask when we talk about the standard model of particle physics, are quantum field theories falsifiable or not? The standard model of particle physics is certainly falsifiable and it works very well. With inflation, there are again many different models. Inflation is not a detailed theory by itself. It's really a class of theories. Each one of those make definite predictions, just like definite quantum field theories make definite predictions. And each one of them can be falsified. What the authors of the Scientific American Article were essentially claiming is that if I give them a particular model of inflation, they say, I don't wanna test that. I wanna ask what class of theories it belongs to and ask whether the class can be falsified. I don't think Popper ever said anything like that. And I think it's a totally crazy thing to say. And if you applied it to quantum field theory, the whole history of the standard model would be different. I think it's just not the right way to look at science at all. Certainly inflation does make very testable predictions that is individual models do and whole classes of model make very common predictions. And they've been tested and it worked extraordinarily well. And I think it really is a fact that if you look at any observational cosmology paper, it ends up concluding that they've done some kind of test of inflation and it worked to the last one. So the ideas in the Scientific American Article really have not had any influence in the scientific community. Although I do worry that they do have a lot of play in Scientific American which is, I think I'm fortunate. And as following up on this question, inflation started in the early 90s, early 80s. And then up to 1998, as you say, people thought that Omega was between 0.2 and 0.3. What kept you going for 15 years or so? Because that's a good lesson for all many of us who work on things that people say, oh, maybe this is not yet tested, but... Right. Well, I was, from the beginning, extraordinarily impressed by the flatness problem, the fact that we know by conventional cosmology that Omega must have been one to 15 decimal places at one second after the Big Bang. And I should maybe add, since now this is the main topic, that we had pretty good reason to trust conventional cosmology back to one second. Because I mentioned at the very beginning of my talk that one of the big successes of conventional cosmology was its predictions for the Big Bang nuclear synthesis, for the predicting the abundances of the light chemical elements. And those processes began at about one second into the history of the universe. So a good reason to trust the model back to one second and the model told us that Omega must have been one to 15 decimal places, which I think also means that the value of the mass density of the universe at one second after the Big Bang was, in fact, the best known number in all of physics since we knew it to 15 decimal places. And there needs to be some explanation for that. Because inflation got it right on the nose, I thought that was a very strong piece of evidence. And as far as Omega, it's certainly true that most astronomers thought Omega was 0.2 or 0.3. But the numbers varied. There were a few astronomers who thought Omega was one. And it was clear, obviously, not something that was completely clear, Kat. Very good, very good. So, oh, there's the last question over there, yes? Sorry, just one last question. I would like to thank you for making this extremely interesting lecture accessible for everybody, including me. So thank you for that. And I don't know how to phrase this question correctly, but I would like to understand what's so special about the vacuum energy that is calculated to have 120 orders of magnitude discrepancy with dark energy. And so what is the assumption that drops and that makes us have this even more bubbling number of 10 to the 500 different type of vacuoi and why would there not be infinite vacuoi? So I don't know if that makes sense. Oh, it does. Let's see, there are different parts there. For one, the number of vacuoi might be infinite. There's nothing that we know of that guarantees that it's finite. String theorists are still debating about that. The number 10 to the 500 was a number that comes from an early discussion by Busso and Pochinsky, or the people who more or less introduced this idea of what's called the landscape of string theory. Slightly different from the idea of the multiverse. The multiverse is the statement that these island universes actually exist. This landscape of string theory is just a statement about the possible states of string theory. And it was Busso and Pochinsky who first loudly announced that there might be a huge number of such states and they gave an estimate of 10 to the 500. It's based on the number of different ways you can combine the various ingredients of string theory in building a vacuum. The vacuum actually ends up looking somewhat like something constructed from tinker toys with different things stuck on here and there. And there's a number of ways of combining things that led to the 10 to the 500. Now you also asked about basically the Planck scale. Why do physicists think that the vacuum energy should be 120 waters of magnitude larger than what we observe? That comes, as I mentioned qualitatively, it comes from the energy scale at which quantum gravity is believed to become important. I didn't explain anything about why quantum gravity has anything to do with this or why that scale should be the scale. So let me try to answer you in some detail. When we calculate the energy, for example, of just the quantum fluctuations of electric and magnetic fields in the vacuum, we think of those as waves. And the waves have to be there because otherwise they violate the uncertainty principles of quantum mechanics. And each wave of each wavelength contributes to the energy density associated with the fluctuations of electromagnetic fields. But the total number we get actually turns out to be infinite the way we know how to calculate it. And the infinity comes from the fact that there is no shortest wavelength. If you take into account all the waves down to 10 to the minus 10 meters, there are waves shorter than that. That's still contributing. You go to 10 to the minus 20 meters, there's still waves shorter than that. And it doesn't end. At least we don't know what the ending is. And that's where the idea of the Planck scale comes from. The idea is that we decide that since we get infinity, if we believe the physics that we understand goes down to zero length, so that we get all infinity of different possible wavelengths, we assume that something must change at very short distance scales, at very short distances, so that the physics that we know no longer applies. And there's good reason to believe that that might happen because everything we say about extremely short length scales is just an extrapolation. We can't do experiments at incredibly short length scales. So we're just extrapolating. And when we extrapolate, we get an infinity, so we assume the extrapolation must break down. So that leads to the question of where do we expect the extrapolation to break down? And here's where quantum gravity comes in. The only thing that we know of, although we could be wrong, the only thing that we know of that's likely to change things when we look at smaller and smaller scales would be the appearance of quantum gravity as being an important factor. So we therefore assume, and we don't think that this is an assumption we can count on, but it's the natural assumption, we assume that our infinity gets cut off at the Planck scale where quantum gravity becomes important. And that predicts this very large energy, including all the way once down to the Planck scale, which gives us this huge number that's 120 orders of magnitude larger than what we observe. I can maybe add to that an alternative way of looking at the problem, which does give the same qualitative answer, we could ask what the string theory predict. What I gave you was the quantum field theory description. But if you say quantum field theory isn't right, let's assume string theory is right for the very small scale behavior. The natural scale of string theory, since it is a quantum theory of gravity, is the same Planck scale that I've been talking about. And if you ask what are the energies of the different vacua, the 10 to the 500 different vacua that are predicted by string theory, we don't know how to write down all 10 to the 500, but if we write down one that seems typical, the energies are of the order of the Planck scale. So it's all consistent. And it's consistently, outrageously wrong. Thank you. So before we finish, I have an announcement to make. Being today the birthday of Abdul Salam, there's a tradition now in STP, we started a few years ago, is that the family of Abdul Salam, they have created a new prize, which is called the Spirit of Salam Award. And they gave it every year, and it's announced on Salam's birthday. So I'm very pleased to announce the two winners this year. They're two scientists from Latin America, one physicist and one other one's a mathematician. They are Professor Santiago Alberto Berhoschi from Mexico, and Professor Victor Latore Aguilar from Peru. They have been both very close to STP, and very close to Professor Salam, and they have both contributed to develop science in their own country and have closed the region. So we are very pleased to announce and to congratulate them for this award. So now to finish, let's standard the tradition now in STP. Before we thank Aaron, everybody will be invited to have some refreshments outside, and except for the diploma students that can come and talk more private to Alan. I suggest that this conversation today will be a little bit short because they have two more days to ask more questions. And then we have been going for quite a long. So let's thank Alan again for this wonderful lecture. Thank you.