# The Early Universe 1. Inflation Cosmology: Is Our Universe a Part of the Multiverse? Part 1

On the site of free lectures MIT OpenCourseWare posted a
course of lectures on the cosmology of
Alan Gus, one of the creators of the inflationary model of the universe. The course seemed interesting enough to me to start translating it.

We offer you a translation of the first lecture: “Inflationary Cosmology. Is our universe part of a multiverse? Part 1".

The title slide shows a photograph of the Planck satellite. This satellite was launched several years ago to measure cosmic background radiation. Cosmic background radiation is the most important key to understanding the history of the universe. The Planck is the third satellite that is fully designed to measure cosmic background radiation. The first satellite was called COBE, then was WMAP, now Planck.

The Planck is still in orbit. In fact, he completed the data collection, although the analysis of this data is far from complete. We will also discuss what exactly this satellite is observing.

I want to start by discussing the standard Big Bang theory, which will be the main theme of our course. We spend about 2/3 of the course discussing the standard theory of the Big Bang, and then move on to topics such as inflation. When we start studying inflation, it turns out that inflation is a fairly simple thing, if you understand the basic equations that arise in standard cosmology. It seems to me quite reasonable to spend two-thirds of the course on standard cosmology before moving on to inflation. By this time, we will deal with all the principles that we will use later, studying advanced topics, such as inflation.

The standard Big Bang model is the theory that the universe, as we know it, appeared 13-14 billion years ago. Today we can even more accurately name the age of the universe. The calculations are based on data from the Planck satellite, as well as some other information. Age is 13.82 ± 0.05 billion years. Thus, at present, the age of the universe since the Big Bang is pretty well established.

However, I did not in vain specify "the universe as we know it." Because we are not completely sure that the universe began with what we call the Big Bang. We have a very good description of the Big Bang and we are pretty sure that it was in fact, and we understand how it looked. But whether there was anything before him - this question is still completely open.

It seems to me that we should not assume that the universe began with the Big Bang. Later, at the very end of the course, when we study some of the consequences of inflation and the multiverse, we will see that there is good reason to believe that the Big Bang was not the beginning of the universe, but was just the beginning of our local universe, which is often called the pocket universe.

In any case, the Big Bang theory claims that at least our part of the universe 13.82 billion years ago was an extremely hot, dense, uniform substance of particles, which, according to the generally accepted standard model of the Big Bang, literally filled all of space. Now we are confident enough that it evenly filled all the space that is available to us for observation. I want to emphasize that this contradicts the widespread but incorrect visual picture of the Big Bang. According to this pictorial picture, the Big Bang looked like a small bomb of very dense substance, which then exploded and scattered into empty space. This is not a scientific picture of the Big Bang.

The reason is not the inconsistency of such a picture. It is difficult to say what is logical here and what is illogical. It just contradicts what we see. If it were a small bomb detonating in empty space, we would expect today that the universe would look different if you look in the direction where the bomb was and in the opposite direction. But we do not see any signs of this. When we look at the sky, the universe with very great accuracy looks exactly the same in all directions. Nowhere do we see any signs of an exploding bomb. On the contrary, it seems that the Big Bang happened evenly everywhere.

The Big Bang describes several important things, which we will talk more about in our course. He describes how the early universe expanded and cooled, and we will spend quite some time to understand the nuances that are hidden behind these words. In fact, the Big Bang is a very accurate model based on very simple assumptions. By and large, we assume that the early universe was filled with hot gas, which was in thermodynamic equilibrium, and that this gas expanded and contracted back due to gravity.

From these simple ideas, we can calculate, and we will learn how to calculate how quickly the universe expanded, what temperature it had, the density of matter at each moment in time. All the nuances can be calculated from these simple ideas, and to explore this is really interesting.

The Big Bang also explains how light chemical elements formed. This is the main theme of Steve Weinberg’s book, The First Three Minutes. Just around this period, chemical elements formed. It turns out that most of the chemical elements in the universe did not form during the Big Bang, but much later inside the stars. These elements were scattered into space during supernova explosions and from them stars of later generations were formed, one of which is our Sun.

Thus, the substance from which we are made was not actually created during the Big Bang, but was synthesized inside some distant star that exploded long ago. And maybe many stars whose remnants came together and formed our solar system. However, most of the matter in the universe, unlike most of the various kinds of elements, formed in the Big Bang. Most of the stuff in the universe is just hydrogen and helium.

About five different isotopes of hydrogen, helium and lithium were mainly formed in the Big Bang, and since we have a detailed picture of the Big Bang, which we will study in the future, we can calculate and predict the number of these different isotopes. These predictions are very well consistent with observations. This, of course, is one of the main confirmations that our picture of the Big Bang is correct. It can be predicted what the amount of helium-3 should be. This amount has been measured and is consistent with predictions. It is amazing.

Finally, the Big Bang explains how ultimately the matter gathered in clumps and stars, galaxies, and clusters of galaxies formed. We will talk about this a little bit, but we won’t dive very deeply into this topic, because it goes beyond our course. In principle, work in this direction is still ongoing. People do not understand everything about galaxies. But the general picture that it all started with an almost homogeneous Universe, and then the matter gathered in clumps that formed galaxies and other structures, is considered true. And from this very simple picture you can understand a lot about the universe.

Now I want to talk about what the usual theory of the Big Bang does not talk about, about the emergence of new ideas, such as inflation.

First, the usual Big Bang theory says nothing about what caused the expansion of the universe. In reality, this is only a theory of the consequences of an explosion. In the scientific version of the Big Bang in the emerging universe, everything expands, without explaining how this expansion began. This explanation is not part of the Big Bang theory. Thus, the scientific version of the Big Bang theory is not really an explosion theory. This is actually a theory of the consequences of an explosion.

Moreover, in a similar way, the usual Big Bang theory says nothing about where all the matter came from. The theory actually assumes that for every particle that we see in the Universe today, at the very beginning there was, if not the particle itself, then at least some kind of precursor particle, without explaining where all these particles came from. In short, I want to say that the Big Bang theory says nothing about what exploded, why it exploded, or what happened before it exploded. In the Big Bang theory, there really is no explosion. This is an unbroken theory, despite its name.

Inflation, it turns out, provides answers, very plausible answers, to many of these questions. Basically, we’ll talk about this today in the remaining time. As I said, from the point of view of the course, we will approach this topic in about the last third of the course.

What is space inflation? In essence, this is a minor modification, in terms of the overall picture, of the standard theory of the Big Bang. The best word to describe it is the word that I think was coined in Hollywood. Inflation is a prequel to the usual Big Bang theory. This is a brief description of what happened before, just before the Big Bang. Thus, inflation is indeed an explanation of the Big Bang explosion in the sense that it does provide a push theory that led the universe to this huge expansion process, which we call the Big Bang.

Inflation does it in such a way that I think of it as a miracle. When I use the word “miracle”, I use it in a scientific sense, just something so amazing that it deserves to be called a miracle, although it is part of the laws of physics. There are just a few features of the laws of physics that are crucial to inflation. I will talk about two of them, which I consider a miracle because when I was a student no one spoke about them at all. They simply were not part of the physics that people noticed and talked about.

The miracle of physics that I am talking about is something known since Einstein's general theory of relativity that gravity is not always attraction. Gravity can act as repulsion. Einstein described this in 1916, in the form of what he called the cosmological constant. The initial motivation for modifying the equations of the general theory of relativity was that Einstein considered the universe to be static. He realized that ordinary gravity would make the static universe contract. The universe cannot remain static. Therefore, he introduced this element, a cosmological constant, to compensate for the attraction of ordinary gravity and to be able to build a static model of the universe.

As you will soon find out, such a model is completely wrong. The universe looks very different. But the fact that the general theory of relativity may include this gravitational repulsion, which is compatible with all the principles of the general theory of relativity, is an important thing that Einstein himself discovered. Inflation takes this opportunity by allowing gravity to be the repulsive force that brought the universe into an expansion phase, which we call the Big Bang.

In fact, if we combine the general theory of relativity with some generally accepted ideas of elementary particle physics, there are clear signs, not quite a prediction, but rather clear signs that at very high energy densities there are states of matter that literally turn gravity upside down and attraction turns into repulsion. More specifically, as we will learn later, gravitational repulsion is created by negative pressure.

According to the general theory of relativity, it turns out that both pressure and energy density can create a gravitational field. In contrast to Newtonian physics, where only the density of the mass creates a gravitational field.

Positive pressure creates an attractive gravitational field. Positive pressure is a kind of normal pressure, and attractive gravity is a kind of normal gravity. Normal pressure creates normal gravity. But negative pressure is possible, and negative pressure creates repulsive gravity. This is the secret to what makes inflation possible.

Thus, inflation suggests that at least a small area of repulsive gravitational matter existed in the early universe. We do not know exactly when inflation occurred in the history of the universe, or in other words, we do not know exactly at what levels of energy it occurred. But a very plausible possibility of when inflation could occur is when the energy levels in the Universe were comparable to the energy levels in the theories of the Great Unification.

Theories of the Great Unification, which we will discuss a little later, are theories that combine weak, strong, and electromagnetic interactions into one single interaction. This association occurs at a typical energy of approximately 10

If this happened at such orders of energy, initially the site could be incredibly small - about 10

The gravitational repulsion created by this small stretch of repulsive gravitational matter has become the driving force behind the Big Bang, resulting in an exponential expansion of the stretch. With exponential expansion, there is a certain time in which the size of the plot doubles. If you wait the same amount, it will double again. If you wait the same amount, it will double again.

Since these doublings quickly accumulate, it does not take much time to create the entire Universe. After about 100 doubles, this tiny stretch of 10

If all this happens on the scale of the great theory of association, the doubling time is incredibly short, 10

The time it takes is only 10

Inflation ends because this repulsive gravitational matter is unstable. It decays, in the same sense as a radioactive substance. This does not mean that it rots like a decaying apple, it does mean that it turns into other types of matter. In particular, it turns into matter, which is no longer gravitationally repulsive. Thus, gravitational repulsion ends, and the particles created by the energy released at the end of inflation become hot matter in the ordinary Big Bang.

This ends the prequel, and the main action begins - the usual theory of the Big Bang. The role of inflation is only to create the initial conditions for the usual theory of the Big Bang. There is a slight caveat. Inflation ends because matter is unstable, but ends almost everywhere, and not completely everywhere.

This repulsive gravitational matter decays, but it decays as a radioactive substance, exponentially, it has a half-life. But no matter how many half-lives there pass, there will always be a tiny small piece, there will be a little more of this matter. And this turns out to be important for the idea that in many cases inflation never ends completely. We will come back to this.

Now I want to talk about what happens during the exponential expansion phase. There is a very specific feature of inflation, this exponential expansion caused by repulsive gravity, which means that while it occurs, the mass density or energy density of this repulsive gravitational matter does not decrease. It would seem that if something doubles in size, then the volume should increase by 8 times, and the energy density should decrease by 8 times.

And this, of course, happens with ordinary particles. So, of course, it would happen if we had gas, ordinary gas, which we simply allowed to expand twice in size, the density would decrease by eight times, since the volume is equal to a cube of size. But this particular repulsive gravitational matter actually expands with constant density. It sounds like energy conservation is being violated, because it means that the total amount of energy inside this expanding volume is increasing. Energy per unit volume remains constant, and the volume becomes more and more exponentially.

I affirm that I have not lost my mind that this actually corresponds to the laws of physics that we know. And that this is consistent with energy conservation. Conservation of energy is indeed the sacred principle of physics. We do not know anything in nature that violates the principle of conservation of energy. Energy ultimately cannot be created or destroyed, the total amount of energy is fixed. There seems to be a contradiction here. How do we get rid of it?

A second miracle of physics is required here. Energy is truly conserved. The trick here is that energy is not necessarily positive. There are things that have negative energy. In particular, the gravitational field has negative energy. This statement, by the way, is true both in Newtonian physics and in the general theory of relativity. We will prove it later.

If you took a course in electromagnetism to calculate the energy density of an electrostatic field, you know that the energy density of an electrostatic field is proportional to the square of the electric field strength. It can be proved that this energy is exactly equal to the energy that must be added to the system in order to create an electric field of a given configuration. If we compare Newton’s law of gravity with Coulomb’s law, it becomes clear that this is actually the same law, except that they use different constants.

Both of them are laws of inverse squares and are proportional to two charges, where in the case of gravity these are masses that play the role of charges. But they have opposite signs. Two positive charges, as you know, repel, two positive masses are attracted to each other.

The same argument that allows you to calculate the energy density of the Coulomb field, allows you to calculate the energy density of the Newtonian gravitational field, still within the framework of Newtonian physics, while there remains a change in the sign of the force. This changes sign in all the calculations performed, and a negative value is obtained, which is the correct value for Newtonian gravity. The energy density of the Newtonian gravitational field is negative. The same is true in general relativity.

This means that in the framework of energy conservation, you can get more and more matter, more and more energy accumulated in the form of ordinary matter, which happens during inflation, as long as there is a compensating amount of negative energy created by the gravitational field, which fills an ever larger area of space. This is exactly what happens during inflation.

The positive energy of this repulsive gravitational substance, which grows and grows in volume, is exactly compensated by the negative energy of the gravitational field filling the area. Thus, the total energy remains constant, as it should, and there is a high probability that the total energy is exactly zero. Because everything we know is at least consistent with the possibility that these two energies are exactly equal to each other or very close.

Schematically, the picture is that the total energy of the Universe consists of huge positive energy in the form of matter and radiation, the matter we see, the matter with which we usually identify energy. But there is also a huge negative energy enclosed in a gravitational field that fills the universe. And, as far as we can judge, their sum can be equal to 0. At least this does not contradict anything.

In any case, during inflation, the black bar rises and the red bar goes down. And they rise and fall in an equal amount. Thus, the processes that occur during inflation preserve energy, since everything that complies with the laws of physics that we know about should conserve energy.

I want to talk about some evidence of inflation. Until now, I have described what inflation is, and today this description is enough. As I said, we will return and talk about all this in our course. Now let's move on to discussing some of the reasons why we believe that our Universe may have actually undergone this process called inflation, which I just talked about. There are three things that I want to talk about.

The first of these is the uniformity of the universe on a large scale. This is due to the fact that I told you at the beginning that if you look in different directions, then the Universe looks the same in all directions. An object whose dependence on direction can be measured with the greatest accuracy is cosmic background radiation, because we can measure it in any direction, and it is extremely homogeneous.

When this was done, it was found that cosmic background radiation is uniform with incredible accuracy - about 1/100000. This is an impressive level of uniformity. This means that the universe is indeed extremely homogeneous.

I want to make one reservation here to be completely accurate. If you just take and measure cosmic radiation, it turns out that there is an asymmetry that is greater than what I just said. An asymmetry of about 1/1000 can be detected, where one direction is hotter than the opposite. But we interpret this one-thousandth effect as our movement through cosmic background radiation, which makes it hotter in one direction and colder in the opposite direction. And this effect of our movement has a very definite angular distribution.

We have no other way of knowing what our speed is with respect to cosmic background radiation. We simply compute it from this asymmetry. But we cannot explain everything with this movement. We can calculate the speed. As soon as we calculate it, this will determine one of the asymmetries that we can subtract. After this, the residual asymmetries, asymmetries that we cannot explain, saying that the Earth has a certain speed with respect to cosmic background radiation, are at the level of one hundred thousandth. And this one thousandth, we attribute to the universe, and not to the movement of the Earth.

To understand the consequences of this incredible homogeneity, a little is needed about the history of this cosmic background radiation. Radiation in the early period of the Universe, when the Universe was plasma, was essentially trapped in matter. Photons moved at the speed of light, but the plasma has a very large cross section for scattering of photons by free electrons. This means that the photons moved with the substance, because, they could move freely only for a very short distance, then they scattered and moved in the other direction. Thus, with respect to matter, photons did not fly anywhere during the first 400,000 years of the history of the universe.

But then, according to our calculations, after about 400,000 years, the universe cooled down enough for the plasma to neutralize. And when the plasma is neutralized, it becomes a neutral gas, like air in this room. The air in this room seems completely transparent to us, and it turns out that the same thing happened in the universe.

The gas that filled the universe after its neutralization really became transparent. This means that the typical photon we see today in cosmic background radiation traveled in a straight line starting from about 400,000 years after the Big Bang. Which, in turn, means that when we look at cosmic background radiation, we, in fact, see an image of how the universe looked 400,000 years after the Big Bang. Just as the light from my face to your eyes gives you an idea of how I look.

So, we see the image of the universe at the age of 400,000 years, and it is homogeneous with an accuracy of one hundred thousandth. The question is, can we explain how the universe could become so homogeneous? If you are ready to simply assume that the universe was originally completely homogeneous for more than one hundred thousandth, then no one bothers you to do so. But if you want to try to explain this uniformity without assuming that it was from the very beginning, then using the usual theory of the Big Bang is simply not possible.

The reason is that in the framework of the evolutionary equations of the usual theory of the Big Bang, you can calculate, and we will calculate it later, in order to smooth everything over time, so that the cosmic background radiation looks smooth, you need to be able to move matter and energy about 100 times faster speed of light. Otherwise, it just won’t work out. We in physics do not know anything that happens faster than the speed of light. So, in the physics known to us and in the usual theory of the Big Bang, there is no way to explain this homogeneity, except to simply assume that it was there from the very beginning. For reasons we don’t know about.

On the other hand, inflation solves this problem very well. Inflation adds an exponential extension to the history of the universe. Due to the fact that this exponential expansion was so large, it follows that if you look at our universe before inflation occurred, it was significantly smaller than in ordinary cosmology, in which it did not have this exponential expansion.

Thus, in the inflationary model there was enough time for the observed part of the Universe to become homogeneous before the start of inflation, when it was incredibly small. And it became homogeneous, like air, which spreads evenly throughout the room, rather than gathering in one corner. After uniformity was achieved in this tiny region, inflation then stretched that region, which became large enough to include everything we see now, thereby explaining why everything we see looks so uniform. This is a very simple explanation, and it is possible only with the use of inflation, and not within the framework of the generally accepted theory of the Big Bang.

In inflationary models, the Universe begins with such a small size that uniformity is easily established. In the same way that air in a lecture hall evenly fills the lecture hall. Then inflation stretches the region, which is becoming large enough to include everything that we are currently observing. This is the first of my three proofs of inflation.

The second is what is called the problem of a flat universe. The question is why the early Universe was so flat? The question may immediately arise - what do I mean when I say that the early Universe was flat? One of the misconceptions that I sometimes encounter is that flat is often perceived as two-dimensional. This is not what I mean. Flat does not mean like a two-dimensional pancake. The universe is three-dimensional. Flat in our case means Euclidean, obeys the axioms of Euclidean geometry, in contrast to versions of Non-Euclidean geometry, which are admitted by the general theory of relativity.

The general theory of relativity allows three-dimensional space to be curved. We consider only uniform curvature. In reality, we do not see any curvature, but we know with greater accuracy that the universe is homogeneous than the fact that it is flat. So, imagine three possible options for the curvature of the universe, all of which will be considered homogeneous. Three-dimensional curved spaces are not easy to visualize, but all three of them are similar to two-dimensional curved spaces that are easier to imagine.

One option is closed geometry of the surface of a sphere. The analogy is that a three-dimensional universe is similar to a two-dimensional surface of a sphere. The number of dimensions changes, but important things remain. So, for example, if you place a triangle on the surface of a sphere, and this can be easily visualized, the sum of its three angles will be more than 180 degrees. Unlike Euclidean geometry, where the sum is always 180 degrees.

STUDENT: does bending of three-dimensional space occur in the fourth dimension? Just as two-dimensional models imply a different dimension?

TEACHER: good question. The question was, does three-dimensional curvature occur in the fourth dimension in the same way that two-dimensional curvature occurs in the third dimension? I think the answer is yes. But, I should clarify here a little. The third dimension from a purely mathematical point of view allows us to easily visualize the sphere. But the geometry of the sphere from the point of view of people studying differential geometry is a well-defined two-dimensional space without any need for a third dimension.

The third dimension is just a way for us to visualize curvature. But the same method works for three-dimensional space. In fact, by studying the three-dimensional curved space of a closed universe, we will do just that. We use the same method, imagine it in four dimensions, and it will be very close to the two-dimensional picture you are looking at.

Thus, one of the possibilities is closed geometry, where the sum of the three angles of a triangle is always greater than 180 degrees. Another possibility is what is commonly called the saddle shape, or space of negative curvature. In this case, the sum of the three angles, as they narrow, becomes less than 180 degrees. And only for the flat case, the sum of the three angles is exactly 180 degrees, which is the case of Euclidean geometry.

The geometry on the surfaces of these objects is not Euclidean, although if we consider the three-dimensional geometry of objects embedded in three-dimensional space, it is still Euclidean. But the geometry on two-dimensional surfaces is not Euclidean on the upper two surfaces, and Euclidean on the lower surface.

This is how it works in the general theory of relativity. There are closed universes with positive curvature and a sum of angles of more than 180 degrees. There are open universes where the sum of the three angles is always less than 180 degrees. And there is a case of a flat universe, which is located on the border of these two, in which the Euclidean geometry works. In our universe, Euclidean geometry works very well. That's why we all taught her at school. We have very good evidence that the early Universe was unusually close to this flat case of Euclidean geometry. This is what we are trying to understand and explain.

In accordance with the general theory of relativity, the geometry of the universe is determined by the density of the mass. There is a certain value of the mass density, called the critical density, which depends on the rate of expansion, by the way, this is by no means a universal constant. But for a given expansion rate, the critical density can be calculated, and this critical density is the density that makes the universe flat. Cosmologists define a number called Ω (Omega). Ω is simply the ratio of the actual mass density to the critical mass density. So, if Ω equals 1, then the actual density is equal to the critical density, which means a flat universe. If Ω is greater than 1, then we get a closed universe, and if Ω is less than 1, there will be an open universe.

The evolution of the value of Ω is special in that Ω equal to 1, during the development of the Universe in ordinary cosmology, behaves very much like a pencil balancing at its tip. This is an unstable equilibrium point. In other words, if Ω were exactly equal to 1 in the early Universe, it would remain exactly equal to 1. Just like a pencil that is ideally mounted at its tip, it will not know where to fall and, in principle, will remain in this position forever. At least in classical mechanics. We will not consider quantum mechanics for our pencil. For an analogy, we use a pencil of classical mechanics.

But if the pencil tilts slightly in any direction, it will quickly begin to fall in that direction. Similarly, if Ω in the early universe were just over 1, it would quickly increase to infinity. This is a closed universe. Infinity really means that the universe reaches its maximum size, then begins to shrink and collapse. If Ω were slightly less than 1, it would quickly decrease to 0, and the universe would simply become empty, as it would expand rapidly.

Therefore, the only way for Ω to be close to 1 today, and as far as we can say, Ω today is 1, is initially to be incredibly close to 1. It’s like that pencil that stood for 14 billion years and has not fallen. Численно, для Ω, чтобы сегодня быть где-то в допустимом диапазоне очень близким к 1, означает, что Ω через одну seconds. после Большого Взрыва должна была быть равна 1 с невероятной точностью в 15 десятичных знаков. Это делает значение плотности массы вселенной через одну seconds. после Большого Взрыва, вероятно, самым точным числом, которое мы знаем в физике. We really know it with an accuracy of 15 decimal places. If it were not in this range, then it would not be close to 1 today due to the amplification effect during the evolution of the universe.

The question is, how did this happen? In the usual Big Bang theory, theoretically the initial value of Ω could be anything. To correspond with what we are currently observing, it should have been in this incredibly narrow range, but in theory there is nothing that would make it be there. The question is, why was Ω initially so incredibly close to 1? As in the previously mentioned problem of homogeneity, we can simply assume that it initially turned out to be what it should have been, i.e. equal to 1. You can do this. But if you want to have any explanation of why this happened, in ordinary cosmology there is nothing that can explain it. However, inflation allows us to explain this.

In the inflationary model, the evolution of Ω changes, because gravity turns into a repulsive force instead of an attractive one, and this makes Ω change in a different way. It turns out that during inflation, Ω does not move away from 1, as it was throughout the rest of the history of the universe, but, on the contrary, quickly moves to 1, exponentially fast. With such a rate of inflation, which we talked about, inflation is about 10

The farther the initial Ω was from 1, the longer inflation will be needed to bring it closer to 1. But for Ω significantly different from 1, inflation will not take much longer, since inflation brings omega closer to 1 exponentially. This is a very powerful force, bringing omega closer to 1. And it gives us a very simple explanation of why Ω in the early universe seemed to be extremely close to 1.

In fact, a prediction follows from this. Since inflation is so close to bringing Ω closer to 1, we expect that today Ω should really be 1, or within the range of measurable accuracy. You can imagine inflationary models, where Ω would be, say 0.2, this is what it was previously thought to be, but for this inflation should end right at the right time before it even approaches 1. Because each exponential increase makes it an order of magnitude closer to 1. This is a very fast effect. Therefore, without a very careful adjustment, most of any inflationary models will bring Ω so close to 1 that today we see it as 1.

Previously, it seemed that this was not so. Until 1998, astronomers were convinced that Ω was only 0.2 or 0.3, while inflation had a fairly clear prediction that Ω should be 1. Personally, this caused me quite a bit of inconvenience. Whenever I had lunch with astronomers, they said that inflation is a beautiful theory, but it cannot be correct, because Ω is 0.2, and inflation predicts Ω is 1. And this is simply a mismatch.

Everything changed in 1998. Now the most accurate number for Ω that we have, obtained from the Planck satellite along with some other measurements, is 1.0010, ± 0.0065. 0.0065 is an important thing. The number is very, very close to 1, and the error is greater than this difference. Thus, today we know that with an accuracy of 0.5% or maybe 1%, Ω is 1, which is what inflation predicts.

The new component that made all this possible, which changed the measured omega value from 0.2 to 1, is a new component of the universe’s energy balance, the discovery of what we call dark energy. We learn a lot about dark energy during the course. The discovery in 1998 consisted in the fact that the expansion of the Universe does not slow down under the influence of gravity, as was expected before that time, but instead, the expansion of the Universe actually accelerates.

This acceleration must be due to something. What causes this acceleration is called dark energy. Even though there are significant gaps in knowledge of dark energy, we can still calculate how much it should be in order to create the acceleration that we observe. And when all this comes together, we get a number that is much better aligned with inflation than the previous one.

STUDENT: Was the accelerating Universe an unknown factor at the time, due to which it was incorrectly believed that Ω was 0.2 or 0.3?

TEACHER: Yes, it is. This was entirely due to the fact that it was not known about acceleration at that time. In fact, the visible substance was accurately measured. This gave only 0.2 or 0.3. And this new component, the dark energy that we only know about because of acceleration, makes the necessary difference.

STUDENT: is this data that makes Ω equal to 0.2 or 0.3, is it really just a component of the universe that we see through telescopes?

TEACHER: right. Including dark matter. In fact, we do not see everything. Without going into details now, we will discuss them later in the course, there is something called dark matter that is different from dark energy. Despite the fact that matter and energy are essentially the same thing, in our case they are different. Dark matter is matter, the conclusion about the existence of which we draw because of its influence on other matter. Looking, for example, at the speed of rotation of galaxies, one can calculate how much substance must be inside these galaxies so that the orbits are stable. It turns out that substances are needed much more than we actually see. This invisible matter is called dark matter, and it gives a contribution of 0.2 or 0.3. Visible matter is only about 0.04.

The next point that I want to talk about is the heterogeneity of the universe on a small scale. On the largest scales, the universe is incredibly homogeneous - accurate to one hundred thousandth, but on a smaller scale, the universe today is extremely heterogeneous. Earth is a big bunch in the distribution of the mass density of the universe. Earth is about 10 to

We are sure that these clumps evolved from the very minor disturbances that we see in the early universe, most clearly visible through cosmic background radiation. The mass density in the early universe, in our opinion, was homogeneous with an accuracy of about one hundred thousandth. But at the level of one hundred thousandth, we see that inhomogeneities exist in the cosmic background radiation.

Objects such as the Earth have formed because these small heterogeneities in mass density are gravitationally unstable. In places where there is a slight excess in the density of matter, this excess of density creates a gravitational field that attracts even more matter to these areas, which, in turn, produces an even stronger gravitational field that attracts even more matter. The system is unstable, it forms complex clusters that we see, such as galaxies, stars, planets and so on.

This is a complicated process. But it all starts with these very weak heterogeneities that we believe existed shortly after the Big Bang. We see these inhomogeneities in cosmic background radiation. Their measurement tells us a lot about the conditions under which the universe existed then and allows us to build theories that explain how such a universe turned out. It is for measuring these inhomogeneities that satellites such as COBE, WMAP and Planck are created with very high accuracy.

Inflation answers the question of where the heterogeneity came from. There was no explanation in the usual Big Bang theory. It was simply assumed that there were heterogeneities and added them artificially, but there was no theory where they could come from. In inflationary models, where all matter is created by inflation, heterogeneities are also controlled by this inflation and appear due to quantum effects.

It is hard to believe that quantum effects can be important for the large-scale structure of the universe. The Andromeda Galaxy doesn't look like it's a quantum oscillation. But if you consider this theory quantitatively, it really works very well. The theory is that the vibrations that we see in cosmic background radiation were indeed purely a consequence of quantum theory, mainly the uncertainty principle, which states that it is impossible to have something completely homogeneous. This is not consistent with the principle of uncertainty.

When we use the basic ideas of quantum mechanics, we can calculate the properties of these vibrations. To do this, we need to know more about very high energy physics, physics that was relevant during the inflation period, in order to be able to predict the amplitude of these oscillations. We cannot predict the amplitude. In principle, inflation would allow us to do this if we knew enough about the underlying particle physics, but we know too little about it. Therefore, in practice, we cannot predict the amplitude.

However, inflationary models provide a very clear prediction of the spectrum of such fluctuations. By this I mean a change in the intensity of the vibrations depending on the wavelength. The spectrum here means the same thing as for sound, except that you need to consider the wavelength, not the frequency, because these waves do not actually oscillate. But they have wavelengths just like sound waves, and if we talk about the intensity of different wavelengths, the idea of the spectrum is really the same as in sound.

It can be measured. These are not the last measurements, these are the last measurements for which I have a graph. The red line is theoretical prediction. Black dots are real measurements. This is seven year WMAP data. It is difficult to convey how happy I was when I saw this curve.

I also have graphs of what other theories predict. For some time, for example, people were very serious about the idea that the inhomogeneities that we see in the Universe, these fluctuations, were possibly caused by the random formation of the so-called cosmic strings that formed in phase transitions in the early Universe. This, of course, was a viable idea at one time, but as soon as this curve was measured, it turned out that the prediction of cosmic strings did not look at all like that. Since then, they have been excluded as a source of density fluctuations in the universe. Various other models are also shown here. I will not spend time on them, because there are other things that I want to talk about.

In any case, this is a definite success. And this is the latest data. This is data from the Planck satellite, which was launched in March last year. I don’t have it on the chart on the same scale, but again you see a theoretical curve based on inflation and points that show data with tiny little dash of errors. Absolutely clear correspondence.

STUDENT: what happened to the theory of inflation after they discovered dark energy? Has she changed significantly?

TEACHER: has the theory changed?

STUDENT: there was another curve in the previous chart.

TEACHER: Regarding inflation without dark energy. I think that not so much the theory of inflation is different for these two curves, but the curve that you see today is the result of inflation and the evolution that has occurred since then. And it is the evolution that has occurred since then that makes a big difference between these curves.

Thus, inflation theory should not have changed much. And she really did not change. But, of course, the curve looks much better after dark energy was detected, because the correct mass density became known, and gradually we got more and more data on these fluctuations, which fit perfectly into what inflation predicts.

Now I want to move on to the idea of the multiverse. I will try to quickly go over it so that we can finish it. We still won’t try to understand all the details now, so I’ll talk about some of them in the remaining 10 minutes of the lecture. I want to talk a little about how inflation leads to the idea of a multiverse. We will return to this at the very end of the course, and this is certainly an exciting aspect of inflation.

The gravitationally repulsive material that creates inflation is metastable, as we said. It is breaking up. This means that if you are in a place where inflation occurs and you wonder what the probability is that it will occur a little later, this probability decreases exponentially - it falls by half for every doubling, each half-life. But at the same time, the volume of any area that is swelling also grows exponentially, growing due to inflation. In fact, in any reasonable inflationary model, growth is much faster than decay. If you look at the area that is swelling, if you wait for the half-life, half of the volume of this area will no longer swell, according to the definition of half-life. But the remaining half will be significantly larger than the volume with which we started. That is the whole point.

This is a very unusual situation, because it seems to have no end. The area that is swelling gets bigger and bigger, even when it splits, because expansion is faster than decay. This leads to the phenomenon of perpetual inflation. The size of the swelling region increases with time, despite the fact that the swelling matter decays. This leads to what we call perpetual inflation. Eternal here means eternal in the future, as far as we can judge, but not eternal in the past. Inflation begins at some finite time, but then, as soon as it begins, it will continue forever.

Whenever part of this swelling region undergoes a phase transition and becomes normal, it locally looks like a Big Bang. Our Big Bang is one of these local events, and the universe formed by any of these local events, where an expanding region decays, is called a pocket universe. Pocket simply because there are many such universes on the scale of this multiverse. They are in some ways small, although they are the same size as the universe in which we live. And our universe is one such pocket universe.

Thus, instead of a single universe, inflation produces an infinite number of them. This is what we call the multiverse. It is worth noting that the word multiverse is also used in other contexts and other theories, but inflation, as it seems to me, is the most plausible way to build a multiverse. This is what most cosmologists mean when they talk about the multiverse.

What is the place of dark energy here? She plays a very important role. In 1998, two groups of astronomers independently discovered that the universe is now expanding with acceleration. We now know that the universe has expanded rapidly over the past five billion years out of 14 billion years of the history of the universe. There was a period when expansion slowed down to five billion years ago. The consequence of this is that inflation is actually happening today. This accelerated expansion of the universe that we see is very similar to inflation, and we really interpret it as a similar kind of physics. We believe that it was caused by some kind of negative pressure, just as inflation was caused by negative pressure.

This matter, which, apparently, fills space and has negative pressure, we call dark energy. Dark energy is simply, by definition, something, whatever it is, that causes this acceleration. One may ask, what is dark energy really? The surest answer to this is that no one knows. However, there is the most likely candidate. The most likely candidate, and other candidates are not much different from him, just that dark energy is vacuum energy. The energy of the void. It may be surprising that emptiness can have energy. But I'll tell you about it, and this is not so surprising.

But if dark energy is simply vacuum energy, it is fully consistent with everything we know about the nature of the expansion of the universe that we can measure.

STUDENT: why only in the last five billion years did the universe begin to expand rapidly?

TEACHER: Now I can explain this. Now that I have said that there is probably vacuum energy, I can give you an answer. The answer is that the energy of the vacuum does not change with time, because it is simply the energy of the vacuum. This is the same as what I said about energy density during inflation. It is just a constant. At the same time, ordinary matter becomes more discharged as the universe expands, decreasing its density in proportion to the cube of the size of the universe.

It so happened that before about the past five billion years, ordinary matter dominated the universe, which created attractive gravity and caused the universe to slow down. But then, about five billion years ago, the matter in the universe became so discharged that ordinary matter ceased to dominate the vacuum energy, and the vacuum energy began to cause acceleration. The energy of the vacuum was all the time, causing repulsion, but it was dominated by the attracting gravity of ordinary matter until the last five billion years.

Now I want to talk, why can there be something in the void? Why can emptiness have energy? The answer is actually quite clear to physicists today. Quantum vacuum, in contrast to the classical vacuum, is a very complex state. It is not empty at all. This is actually a complex set of vacuum vibrations. We think that there is even a field called the Higgs field, which you probably heard about, which on average has a non-zero value in a vacuum. Things like an electromagnetic field constantly fluctuate in a vacuum due to the uncertainty principle, which leads to the presence of energy density in these fluctuations.

So, as far as we can tell, there is no reason for the vacuum energy to be zero. But this does not mean that we understand what its significance is equal to. Today, the real problem from the point of view of fundamental physics is not finding out why the vacuum can have a non-zero energy density. The problem is to understand why it is so small. Why is this a problem? The quantum field theory, which we will not study in detail, says that, for example, the electromagnetic field is constantly oscillating. This is due to the principle of uncertainty. These vibrations can have any wavelength. And each wavelength contributes to the energy density of vacuum fluctuations.

However, there is no shortest wavelength. In a box of any size, there is the longest wavelength, but not the shortest wavelength. It turns out that when we try to calculate the vacuum energy density in quantum field theory, it diverges from the side of short wavelengths. It becomes literally endless, since a formal calculation shows that all wavelengths contribute, and the shortest wavelength does not exist.

What does this mean in real physics? We believe that this is not necessarily a problem with our understanding of quantum field theory. In fact, we believe that this is only a limitation on the range in which our assumptions are true. Of course, quantum theory works extremely well when it is tested in the laboratory. We think that at very short wavelengths, something should limit this infinity. A good candidate for limiting infinity at short wavelengths is the effects of quantum gravity, which we do not understand.

Thus, one way of estimating the true energy density predicted by quantum field theory is to limit the wavelengths on Planck scales, the energy scale, and the length scale associated with quantum gravity, which is about 10

I must say that there is still a way out. The energy that we calculated here is only one of the contributions to the total energy of the vacuum. There are also negative contributions. If we calculate the fluctuation of the electron field, then its contribution to the energy will be negative. In principle, it is possible that these contributions compensate each other accurately or almost exactly, but we do not know why they should do this. Thus, there is a big question about the theoretical prediction of vacuum energy density.

Now I want to talk a little about the landscape of string theory, which can be a possible explanation for the smallness of the energy of the vacuum. This is only a possible explanation, everything is very speculative here. But one possible explanation for this very small vacuum energy that we are observing combines the idea of perpetual inflation and string theory. It is based on the idea that string theory does not have a unique vacuum. For many years, theorists have unsuccessfully tried to find a vacuum in string theory. They simply could not understand how string theory should look like a vacuum.

And then, a little over 10 years ago, many string theorists began to unite around the idea that maybe they could not find a vacuum, because there is no unique vacuum for string theory. Instead, they now claim that there is a huge number, they consider numbers like

If you combine this with the idea of perpetual inflation, you can come to the conclusion that during perpetual inflation, most likely, all these

With this assumption, string theory is the supposed law of physics that governs everything. But if you lived in one of these pocket universes, you would actually see the laws of physics that were very different from the laws in other pocket universes. The fact is that the physics that we actually see and measure is low-energy physics compared to the energy scale of string theory. We see only small fluctuations in the structure of the vacuum in which we live.

So those particles that we see - electrons and quarks, which combine into protons and neutrons, may be characteristic of our particular pocket universe. In other pocket universes, there may be completely different types of particles, which are vibrations of other types of vacuum. So, even if the laws of physics are the same everywhere - the laws of string theory, in practice the observed laws of physics can vary greatly from one pocket universe to another. In particular, due to the fact that in different universes there is a different vacuum, the energy density of a vacuum can be different in different universes. And this gives a possible answer to why the observed vacuum energy is so small.

We’ll talk about this next time.

We offer you a translation of the first lecture: “Inflationary Cosmology. Is our universe part of a multiverse? Part 1".

The title slide shows a photograph of the Planck satellite. This satellite was launched several years ago to measure cosmic background radiation. Cosmic background radiation is the most important key to understanding the history of the universe. The Planck is the third satellite that is fully designed to measure cosmic background radiation. The first satellite was called COBE, then was WMAP, now Planck.

The Planck is still in orbit. In fact, he completed the data collection, although the analysis of this data is far from complete. We will also discuss what exactly this satellite is observing.

I want to start by discussing the standard Big Bang theory, which will be the main theme of our course. We spend about 2/3 of the course discussing the standard theory of the Big Bang, and then move on to topics such as inflation. When we start studying inflation, it turns out that inflation is a fairly simple thing, if you understand the basic equations that arise in standard cosmology. It seems to me quite reasonable to spend two-thirds of the course on standard cosmology before moving on to inflation. By this time, we will deal with all the principles that we will use later, studying advanced topics, such as inflation.

The standard Big Bang model is the theory that the universe, as we know it, appeared 13-14 billion years ago. Today we can even more accurately name the age of the universe. The calculations are based on data from the Planck satellite, as well as some other information. Age is 13.82 ± 0.05 billion years. Thus, at present, the age of the universe since the Big Bang is pretty well established.

However, I did not in vain specify "the universe as we know it." Because we are not completely sure that the universe began with what we call the Big Bang. We have a very good description of the Big Bang and we are pretty sure that it was in fact, and we understand how it looked. But whether there was anything before him - this question is still completely open.

It seems to me that we should not assume that the universe began with the Big Bang. Later, at the very end of the course, when we study some of the consequences of inflation and the multiverse, we will see that there is good reason to believe that the Big Bang was not the beginning of the universe, but was just the beginning of our local universe, which is often called the pocket universe.

In any case, the Big Bang theory claims that at least our part of the universe 13.82 billion years ago was an extremely hot, dense, uniform substance of particles, which, according to the generally accepted standard model of the Big Bang, literally filled all of space. Now we are confident enough that it evenly filled all the space that is available to us for observation. I want to emphasize that this contradicts the widespread but incorrect visual picture of the Big Bang. According to this pictorial picture, the Big Bang looked like a small bomb of very dense substance, which then exploded and scattered into empty space. This is not a scientific picture of the Big Bang.

The reason is not the inconsistency of such a picture. It is difficult to say what is logical here and what is illogical. It just contradicts what we see. If it were a small bomb detonating in empty space, we would expect today that the universe would look different if you look in the direction where the bomb was and in the opposite direction. But we do not see any signs of this. When we look at the sky, the universe with very great accuracy looks exactly the same in all directions. Nowhere do we see any signs of an exploding bomb. On the contrary, it seems that the Big Bang happened evenly everywhere.

The Big Bang describes several important things, which we will talk more about in our course. He describes how the early universe expanded and cooled, and we will spend quite some time to understand the nuances that are hidden behind these words. In fact, the Big Bang is a very accurate model based on very simple assumptions. By and large, we assume that the early universe was filled with hot gas, which was in thermodynamic equilibrium, and that this gas expanded and contracted back due to gravity.

From these simple ideas, we can calculate, and we will learn how to calculate how quickly the universe expanded, what temperature it had, the density of matter at each moment in time. All the nuances can be calculated from these simple ideas, and to explore this is really interesting.

The Big Bang also explains how light chemical elements formed. This is the main theme of Steve Weinberg’s book, The First Three Minutes. Just around this period, chemical elements formed. It turns out that most of the chemical elements in the universe did not form during the Big Bang, but much later inside the stars. These elements were scattered into space during supernova explosions and from them stars of later generations were formed, one of which is our Sun.

Thus, the substance from which we are made was not actually created during the Big Bang, but was synthesized inside some distant star that exploded long ago. And maybe many stars whose remnants came together and formed our solar system. However, most of the matter in the universe, unlike most of the various kinds of elements, formed in the Big Bang. Most of the stuff in the universe is just hydrogen and helium.

About five different isotopes of hydrogen, helium and lithium were mainly formed in the Big Bang, and since we have a detailed picture of the Big Bang, which we will study in the future, we can calculate and predict the number of these different isotopes. These predictions are very well consistent with observations. This, of course, is one of the main confirmations that our picture of the Big Bang is correct. It can be predicted what the amount of helium-3 should be. This amount has been measured and is consistent with predictions. It is amazing.

Finally, the Big Bang explains how ultimately the matter gathered in clumps and stars, galaxies, and clusters of galaxies formed. We will talk about this a little bit, but we won’t dive very deeply into this topic, because it goes beyond our course. In principle, work in this direction is still ongoing. People do not understand everything about galaxies. But the general picture that it all started with an almost homogeneous Universe, and then the matter gathered in clumps that formed galaxies and other structures, is considered true. And from this very simple picture you can understand a lot about the universe.

Now I want to talk about what the usual theory of the Big Bang does not talk about, about the emergence of new ideas, such as inflation.

First, the usual Big Bang theory says nothing about what caused the expansion of the universe. In reality, this is only a theory of the consequences of an explosion. In the scientific version of the Big Bang in the emerging universe, everything expands, without explaining how this expansion began. This explanation is not part of the Big Bang theory. Thus, the scientific version of the Big Bang theory is not really an explosion theory. This is actually a theory of the consequences of an explosion.

Moreover, in a similar way, the usual Big Bang theory says nothing about where all the matter came from. The theory actually assumes that for every particle that we see in the Universe today, at the very beginning there was, if not the particle itself, then at least some kind of precursor particle, without explaining where all these particles came from. In short, I want to say that the Big Bang theory says nothing about what exploded, why it exploded, or what happened before it exploded. In the Big Bang theory, there really is no explosion. This is an unbroken theory, despite its name.

Inflation, it turns out, provides answers, very plausible answers, to many of these questions. Basically, we’ll talk about this today in the remaining time. As I said, from the point of view of the course, we will approach this topic in about the last third of the course.

What is space inflation? In essence, this is a minor modification, in terms of the overall picture, of the standard theory of the Big Bang. The best word to describe it is the word that I think was coined in Hollywood. Inflation is a prequel to the usual Big Bang theory. This is a brief description of what happened before, just before the Big Bang. Thus, inflation is indeed an explanation of the Big Bang explosion in the sense that it does provide a push theory that led the universe to this huge expansion process, which we call the Big Bang.

Inflation does it in such a way that I think of it as a miracle. When I use the word “miracle”, I use it in a scientific sense, just something so amazing that it deserves to be called a miracle, although it is part of the laws of physics. There are just a few features of the laws of physics that are crucial to inflation. I will talk about two of them, which I consider a miracle because when I was a student no one spoke about them at all. They simply were not part of the physics that people noticed and talked about.

The miracle of physics that I am talking about is something known since Einstein's general theory of relativity that gravity is not always attraction. Gravity can act as repulsion. Einstein described this in 1916, in the form of what he called the cosmological constant. The initial motivation for modifying the equations of the general theory of relativity was that Einstein considered the universe to be static. He realized that ordinary gravity would make the static universe contract. The universe cannot remain static. Therefore, he introduced this element, a cosmological constant, to compensate for the attraction of ordinary gravity and to be able to build a static model of the universe.

As you will soon find out, such a model is completely wrong. The universe looks very different. But the fact that the general theory of relativity may include this gravitational repulsion, which is compatible with all the principles of the general theory of relativity, is an important thing that Einstein himself discovered. Inflation takes this opportunity by allowing gravity to be the repulsive force that brought the universe into an expansion phase, which we call the Big Bang.

In fact, if we combine the general theory of relativity with some generally accepted ideas of elementary particle physics, there are clear signs, not quite a prediction, but rather clear signs that at very high energy densities there are states of matter that literally turn gravity upside down and attraction turns into repulsion. More specifically, as we will learn later, gravitational repulsion is created by negative pressure.

According to the general theory of relativity, it turns out that both pressure and energy density can create a gravitational field. In contrast to Newtonian physics, where only the density of the mass creates a gravitational field.

Positive pressure creates an attractive gravitational field. Positive pressure is a kind of normal pressure, and attractive gravity is a kind of normal gravity. Normal pressure creates normal gravity. But negative pressure is possible, and negative pressure creates repulsive gravity. This is the secret to what makes inflation possible.

Thus, inflation suggests that at least a small area of repulsive gravitational matter existed in the early universe. We do not know exactly when inflation occurred in the history of the universe, or in other words, we do not know exactly at what levels of energy it occurred. But a very plausible possibility of when inflation could occur is when the energy levels in the Universe were comparable to the energy levels in the theories of the Great Unification.

Theories of the Great Unification, which we will discuss a little later, are theories that combine weak, strong, and electromagnetic interactions into one single interaction. This association occurs at a typical energy of approximately 10

^{ 16 }GeV, where GeV is approximately the mass or energy equivalent to the mass of the proton. We are talking about energies that are about 10^{ 16 }times greater than the equivalent mass energy of a proton. With such energies, it is very possible that there will be states that create repulsive gravity.If this happened at such orders of energy, initially the site could be incredibly small - about 10

^{ -28 }centimeters in order, in the end, to lead to the creation of all that we see at great distances. And the universe, which we see today, is completely a consequence of such a site.The gravitational repulsion created by this small stretch of repulsive gravitational matter has become the driving force behind the Big Bang, resulting in an exponential expansion of the stretch. With exponential expansion, there is a certain time in which the size of the plot doubles. If you wait the same amount, it will double again. If you wait the same amount, it will double again.

Since these doublings quickly accumulate, it does not take much time to create the entire Universe. After about 100 doubles, this tiny stretch of 10

^{ -28 }centimeters may become large enough not to become a universe, but become the size of a small ball that will eventually become an observable universe after it continues to expand after inflation ends.If all this happens on the scale of the great theory of association, the doubling time is incredibly short, 10

^{ -37 }seconds, which is very fast. The plot expands exponentially at least at 10^{ 28 }times, which, as I already mentioned, takes only about 100 doublings, and can expand much more. There are no restrictions. If it has expanded more than is necessary to create our universe, it simply means that the part of the universe in which we live is larger than we see. There is nothing to worry about. Everything that we see looks homogeneous, and how far this extends, we are not able to find out. Thus, large inflation rates are fully consistent with what we see.The time it takes is only 10

^{ -35 }seconds, which is 100 times 10^{ -37 }seconds. The site, which is destined to become our currently observable Universe, at the end of inflation becomes the size of a ball with a diameter of about a centimeter.Inflation ends because this repulsive gravitational matter is unstable. It decays, in the same sense as a radioactive substance. This does not mean that it rots like a decaying apple, it does mean that it turns into other types of matter. In particular, it turns into matter, which is no longer gravitationally repulsive. Thus, gravitational repulsion ends, and the particles created by the energy released at the end of inflation become hot matter in the ordinary Big Bang.

This ends the prequel, and the main action begins - the usual theory of the Big Bang. The role of inflation is only to create the initial conditions for the usual theory of the Big Bang. There is a slight caveat. Inflation ends because matter is unstable, but ends almost everywhere, and not completely everywhere.

This repulsive gravitational matter decays, but it decays as a radioactive substance, exponentially, it has a half-life. But no matter how many half-lives there pass, there will always be a tiny small piece, there will be a little more of this matter. And this turns out to be important for the idea that in many cases inflation never ends completely. We will come back to this.

Now I want to talk about what happens during the exponential expansion phase. There is a very specific feature of inflation, this exponential expansion caused by repulsive gravity, which means that while it occurs, the mass density or energy density of this repulsive gravitational matter does not decrease. It would seem that if something doubles in size, then the volume should increase by 8 times, and the energy density should decrease by 8 times.

And this, of course, happens with ordinary particles. So, of course, it would happen if we had gas, ordinary gas, which we simply allowed to expand twice in size, the density would decrease by eight times, since the volume is equal to a cube of size. But this particular repulsive gravitational matter actually expands with constant density. It sounds like energy conservation is being violated, because it means that the total amount of energy inside this expanding volume is increasing. Energy per unit volume remains constant, and the volume becomes more and more exponentially.

I affirm that I have not lost my mind that this actually corresponds to the laws of physics that we know. And that this is consistent with energy conservation. Conservation of energy is indeed the sacred principle of physics. We do not know anything in nature that violates the principle of conservation of energy. Energy ultimately cannot be created or destroyed, the total amount of energy is fixed. There seems to be a contradiction here. How do we get rid of it?

A second miracle of physics is required here. Energy is truly conserved. The trick here is that energy is not necessarily positive. There are things that have negative energy. In particular, the gravitational field has negative energy. This statement, by the way, is true both in Newtonian physics and in the general theory of relativity. We will prove it later.

If you took a course in electromagnetism to calculate the energy density of an electrostatic field, you know that the energy density of an electrostatic field is proportional to the square of the electric field strength. It can be proved that this energy is exactly equal to the energy that must be added to the system in order to create an electric field of a given configuration. If we compare Newton’s law of gravity with Coulomb’s law, it becomes clear that this is actually the same law, except that they use different constants.

Both of them are laws of inverse squares and are proportional to two charges, where in the case of gravity these are masses that play the role of charges. But they have opposite signs. Two positive charges, as you know, repel, two positive masses are attracted to each other.

The same argument that allows you to calculate the energy density of the Coulomb field, allows you to calculate the energy density of the Newtonian gravitational field, still within the framework of Newtonian physics, while there remains a change in the sign of the force. This changes sign in all the calculations performed, and a negative value is obtained, which is the correct value for Newtonian gravity. The energy density of the Newtonian gravitational field is negative. The same is true in general relativity.

This means that in the framework of energy conservation, you can get more and more matter, more and more energy accumulated in the form of ordinary matter, which happens during inflation, as long as there is a compensating amount of negative energy created by the gravitational field, which fills an ever larger area of space. This is exactly what happens during inflation.

The positive energy of this repulsive gravitational substance, which grows and grows in volume, is exactly compensated by the negative energy of the gravitational field filling the area. Thus, the total energy remains constant, as it should, and there is a high probability that the total energy is exactly zero. Because everything we know is at least consistent with the possibility that these two energies are exactly equal to each other or very close.

Schematically, the picture is that the total energy of the Universe consists of huge positive energy in the form of matter and radiation, the matter we see, the matter with which we usually identify energy. But there is also a huge negative energy enclosed in a gravitational field that fills the universe. And, as far as we can judge, their sum can be equal to 0. At least this does not contradict anything.

In any case, during inflation, the black bar rises and the red bar goes down. And they rise and fall in an equal amount. Thus, the processes that occur during inflation preserve energy, since everything that complies with the laws of physics that we know about should conserve energy.

I want to talk about some evidence of inflation. Until now, I have described what inflation is, and today this description is enough. As I said, we will return and talk about all this in our course. Now let's move on to discussing some of the reasons why we believe that our Universe may have actually undergone this process called inflation, which I just talked about. There are three things that I want to talk about.

The first of these is the uniformity of the universe on a large scale. This is due to the fact that I told you at the beginning that if you look in different directions, then the Universe looks the same in all directions. An object whose dependence on direction can be measured with the greatest accuracy is cosmic background radiation, because we can measure it in any direction, and it is extremely homogeneous.

When this was done, it was found that cosmic background radiation is uniform with incredible accuracy - about 1/100000. This is an impressive level of uniformity. This means that the universe is indeed extremely homogeneous.

I want to make one reservation here to be completely accurate. If you just take and measure cosmic radiation, it turns out that there is an asymmetry that is greater than what I just said. An asymmetry of about 1/1000 can be detected, where one direction is hotter than the opposite. But we interpret this one-thousandth effect as our movement through cosmic background radiation, which makes it hotter in one direction and colder in the opposite direction. And this effect of our movement has a very definite angular distribution.

We have no other way of knowing what our speed is with respect to cosmic background radiation. We simply compute it from this asymmetry. But we cannot explain everything with this movement. We can calculate the speed. As soon as we calculate it, this will determine one of the asymmetries that we can subtract. After this, the residual asymmetries, asymmetries that we cannot explain, saying that the Earth has a certain speed with respect to cosmic background radiation, are at the level of one hundred thousandth. And this one thousandth, we attribute to the universe, and not to the movement of the Earth.

To understand the consequences of this incredible homogeneity, a little is needed about the history of this cosmic background radiation. Radiation in the early period of the Universe, when the Universe was plasma, was essentially trapped in matter. Photons moved at the speed of light, but the plasma has a very large cross section for scattering of photons by free electrons. This means that the photons moved with the substance, because, they could move freely only for a very short distance, then they scattered and moved in the other direction. Thus, with respect to matter, photons did not fly anywhere during the first 400,000 years of the history of the universe.

But then, according to our calculations, after about 400,000 years, the universe cooled down enough for the plasma to neutralize. And when the plasma is neutralized, it becomes a neutral gas, like air in this room. The air in this room seems completely transparent to us, and it turns out that the same thing happened in the universe.

The gas that filled the universe after its neutralization really became transparent. This means that the typical photon we see today in cosmic background radiation traveled in a straight line starting from about 400,000 years after the Big Bang. Which, in turn, means that when we look at cosmic background radiation, we, in fact, see an image of how the universe looked 400,000 years after the Big Bang. Just as the light from my face to your eyes gives you an idea of how I look.

So, we see the image of the universe at the age of 400,000 years, and it is homogeneous with an accuracy of one hundred thousandth. The question is, can we explain how the universe could become so homogeneous? If you are ready to simply assume that the universe was originally completely homogeneous for more than one hundred thousandth, then no one bothers you to do so. But if you want to try to explain this uniformity without assuming that it was from the very beginning, then using the usual theory of the Big Bang is simply not possible.

The reason is that in the framework of the evolutionary equations of the usual theory of the Big Bang, you can calculate, and we will calculate it later, in order to smooth everything over time, so that the cosmic background radiation looks smooth, you need to be able to move matter and energy about 100 times faster speed of light. Otherwise, it just won’t work out. We in physics do not know anything that happens faster than the speed of light. So, in the physics known to us and in the usual theory of the Big Bang, there is no way to explain this homogeneity, except to simply assume that it was there from the very beginning. For reasons we don’t know about.

On the other hand, inflation solves this problem very well. Inflation adds an exponential extension to the history of the universe. Due to the fact that this exponential expansion was so large, it follows that if you look at our universe before inflation occurred, it was significantly smaller than in ordinary cosmology, in which it did not have this exponential expansion.

Thus, in the inflationary model there was enough time for the observed part of the Universe to become homogeneous before the start of inflation, when it was incredibly small. And it became homogeneous, like air, which spreads evenly throughout the room, rather than gathering in one corner. After uniformity was achieved in this tiny region, inflation then stretched that region, which became large enough to include everything we see now, thereby explaining why everything we see looks so uniform. This is a very simple explanation, and it is possible only with the use of inflation, and not within the framework of the generally accepted theory of the Big Bang.

In inflationary models, the Universe begins with such a small size that uniformity is easily established. In the same way that air in a lecture hall evenly fills the lecture hall. Then inflation stretches the region, which is becoming large enough to include everything that we are currently observing. This is the first of my three proofs of inflation.

The second is what is called the problem of a flat universe. The question is why the early Universe was so flat? The question may immediately arise - what do I mean when I say that the early Universe was flat? One of the misconceptions that I sometimes encounter is that flat is often perceived as two-dimensional. This is not what I mean. Flat does not mean like a two-dimensional pancake. The universe is three-dimensional. Flat in our case means Euclidean, obeys the axioms of Euclidean geometry, in contrast to versions of Non-Euclidean geometry, which are admitted by the general theory of relativity.

The general theory of relativity allows three-dimensional space to be curved. We consider only uniform curvature. In reality, we do not see any curvature, but we know with greater accuracy that the universe is homogeneous than the fact that it is flat. So, imagine three possible options for the curvature of the universe, all of which will be considered homogeneous. Three-dimensional curved spaces are not easy to visualize, but all three of them are similar to two-dimensional curved spaces that are easier to imagine.

One option is closed geometry of the surface of a sphere. The analogy is that a three-dimensional universe is similar to a two-dimensional surface of a sphere. The number of dimensions changes, but important things remain. So, for example, if you place a triangle on the surface of a sphere, and this can be easily visualized, the sum of its three angles will be more than 180 degrees. Unlike Euclidean geometry, where the sum is always 180 degrees.

STUDENT: does bending of three-dimensional space occur in the fourth dimension? Just as two-dimensional models imply a different dimension?

TEACHER: good question. The question was, does three-dimensional curvature occur in the fourth dimension in the same way that two-dimensional curvature occurs in the third dimension? I think the answer is yes. But, I should clarify here a little. The third dimension from a purely mathematical point of view allows us to easily visualize the sphere. But the geometry of the sphere from the point of view of people studying differential geometry is a well-defined two-dimensional space without any need for a third dimension.

The third dimension is just a way for us to visualize curvature. But the same method works for three-dimensional space. In fact, by studying the three-dimensional curved space of a closed universe, we will do just that. We use the same method, imagine it in four dimensions, and it will be very close to the two-dimensional picture you are looking at.

Thus, one of the possibilities is closed geometry, where the sum of the three angles of a triangle is always greater than 180 degrees. Another possibility is what is commonly called the saddle shape, or space of negative curvature. In this case, the sum of the three angles, as they narrow, becomes less than 180 degrees. And only for the flat case, the sum of the three angles is exactly 180 degrees, which is the case of Euclidean geometry.

The geometry on the surfaces of these objects is not Euclidean, although if we consider the three-dimensional geometry of objects embedded in three-dimensional space, it is still Euclidean. But the geometry on two-dimensional surfaces is not Euclidean on the upper two surfaces, and Euclidean on the lower surface.

This is how it works in the general theory of relativity. There are closed universes with positive curvature and a sum of angles of more than 180 degrees. There are open universes where the sum of the three angles is always less than 180 degrees. And there is a case of a flat universe, which is located on the border of these two, in which the Euclidean geometry works. In our universe, Euclidean geometry works very well. That's why we all taught her at school. We have very good evidence that the early Universe was unusually close to this flat case of Euclidean geometry. This is what we are trying to understand and explain.

In accordance with the general theory of relativity, the geometry of the universe is determined by the density of the mass. There is a certain value of the mass density, called the critical density, which depends on the rate of expansion, by the way, this is by no means a universal constant. But for a given expansion rate, the critical density can be calculated, and this critical density is the density that makes the universe flat. Cosmologists define a number called Ω (Omega). Ω is simply the ratio of the actual mass density to the critical mass density. So, if Ω equals 1, then the actual density is equal to the critical density, which means a flat universe. If Ω is greater than 1, then we get a closed universe, and if Ω is less than 1, there will be an open universe.

The evolution of the value of Ω is special in that Ω equal to 1, during the development of the Universe in ordinary cosmology, behaves very much like a pencil balancing at its tip. This is an unstable equilibrium point. In other words, if Ω were exactly equal to 1 in the early Universe, it would remain exactly equal to 1. Just like a pencil that is ideally mounted at its tip, it will not know where to fall and, in principle, will remain in this position forever. At least in classical mechanics. We will not consider quantum mechanics for our pencil. For an analogy, we use a pencil of classical mechanics.

But if the pencil tilts slightly in any direction, it will quickly begin to fall in that direction. Similarly, if Ω in the early universe were just over 1, it would quickly increase to infinity. This is a closed universe. Infinity really means that the universe reaches its maximum size, then begins to shrink and collapse. If Ω were slightly less than 1, it would quickly decrease to 0, and the universe would simply become empty, as it would expand rapidly.

Therefore, the only way for Ω to be close to 1 today, and as far as we can say, Ω today is 1, is initially to be incredibly close to 1. It’s like that pencil that stood for 14 billion years and has not fallen. Численно, для Ω, чтобы сегодня быть где-то в допустимом диапазоне очень близким к 1, означает, что Ω через одну seconds. после Большого Взрыва должна была быть равна 1 с невероятной точностью в 15 десятичных знаков. Это делает значение плотности массы вселенной через одну seconds. после Большого Взрыва, вероятно, самым точным числом, которое мы знаем в физике. We really know it with an accuracy of 15 decimal places. If it were not in this range, then it would not be close to 1 today due to the amplification effect during the evolution of the universe.

The question is, how did this happen? In the usual Big Bang theory, theoretically the initial value of Ω could be anything. To correspond with what we are currently observing, it should have been in this incredibly narrow range, but in theory there is nothing that would make it be there. The question is, why was Ω initially so incredibly close to 1? As in the previously mentioned problem of homogeneity, we can simply assume that it initially turned out to be what it should have been, i.e. equal to 1. You can do this. But if you want to have any explanation of why this happened, in ordinary cosmology there is nothing that can explain it. However, inflation allows us to explain this.

In the inflationary model, the evolution of Ω changes, because gravity turns into a repulsive force instead of an attractive one, and this makes Ω change in a different way. It turns out that during inflation, Ω does not move away from 1, as it was throughout the rest of the history of the universe, but, on the contrary, quickly moves to 1, exponentially fast. With such a rate of inflation, which we talked about, inflation is about 10

^{ 28 }times, it is enough that the value of Ω before inflation is not very limited. Ω before inflation could be not 1, but could be 2 or 10 or 1/10 or 100 or 1/100.The farther the initial Ω was from 1, the longer inflation will be needed to bring it closer to 1. But for Ω significantly different from 1, inflation will not take much longer, since inflation brings omega closer to 1 exponentially. This is a very powerful force, bringing omega closer to 1. And it gives us a very simple explanation of why Ω in the early universe seemed to be extremely close to 1.

In fact, a prediction follows from this. Since inflation is so close to bringing Ω closer to 1, we expect that today Ω should really be 1, or within the range of measurable accuracy. You can imagine inflationary models, where Ω would be, say 0.2, this is what it was previously thought to be, but for this inflation should end right at the right time before it even approaches 1. Because each exponential increase makes it an order of magnitude closer to 1. This is a very fast effect. Therefore, without a very careful adjustment, most of any inflationary models will bring Ω so close to 1 that today we see it as 1.

Previously, it seemed that this was not so. Until 1998, astronomers were convinced that Ω was only 0.2 or 0.3, while inflation had a fairly clear prediction that Ω should be 1. Personally, this caused me quite a bit of inconvenience. Whenever I had lunch with astronomers, they said that inflation is a beautiful theory, but it cannot be correct, because Ω is 0.2, and inflation predicts Ω is 1. And this is simply a mismatch.

Everything changed in 1998. Now the most accurate number for Ω that we have, obtained from the Planck satellite along with some other measurements, is 1.0010, ± 0.0065. 0.0065 is an important thing. The number is very, very close to 1, and the error is greater than this difference. Thus, today we know that with an accuracy of 0.5% or maybe 1%, Ω is 1, which is what inflation predicts.

The new component that made all this possible, which changed the measured omega value from 0.2 to 1, is a new component of the universe’s energy balance, the discovery of what we call dark energy. We learn a lot about dark energy during the course. The discovery in 1998 consisted in the fact that the expansion of the Universe does not slow down under the influence of gravity, as was expected before that time, but instead, the expansion of the Universe actually accelerates.

This acceleration must be due to something. What causes this acceleration is called dark energy. Even though there are significant gaps in knowledge of dark energy, we can still calculate how much it should be in order to create the acceleration that we observe. And when all this comes together, we get a number that is much better aligned with inflation than the previous one.

STUDENT: Was the accelerating Universe an unknown factor at the time, due to which it was incorrectly believed that Ω was 0.2 or 0.3?

TEACHER: Yes, it is. This was entirely due to the fact that it was not known about acceleration at that time. In fact, the visible substance was accurately measured. This gave only 0.2 or 0.3. And this new component, the dark energy that we only know about because of acceleration, makes the necessary difference.

STUDENT: is this data that makes Ω equal to 0.2 or 0.3, is it really just a component of the universe that we see through telescopes?

TEACHER: right. Including dark matter. In fact, we do not see everything. Without going into details now, we will discuss them later in the course, there is something called dark matter that is different from dark energy. Despite the fact that matter and energy are essentially the same thing, in our case they are different. Dark matter is matter, the conclusion about the existence of which we draw because of its influence on other matter. Looking, for example, at the speed of rotation of galaxies, one can calculate how much substance must be inside these galaxies so that the orbits are stable. It turns out that substances are needed much more than we actually see. This invisible matter is called dark matter, and it gives a contribution of 0.2 or 0.3. Visible matter is only about 0.04.

The next point that I want to talk about is the heterogeneity of the universe on a small scale. On the largest scales, the universe is incredibly homogeneous - accurate to one hundred thousandth, but on a smaller scale, the universe today is extremely heterogeneous. Earth is a big bunch in the distribution of the mass density of the universe. Earth is about 10 to

^{ 30 }degrees denser than the average density of matter in the universe. This is an incredibly dense clot. The question is how did these clots form? Where did they come from?We are sure that these clumps evolved from the very minor disturbances that we see in the early universe, most clearly visible through cosmic background radiation. The mass density in the early universe, in our opinion, was homogeneous with an accuracy of about one hundred thousandth. But at the level of one hundred thousandth, we see that inhomogeneities exist in the cosmic background radiation.

Objects such as the Earth have formed because these small heterogeneities in mass density are gravitationally unstable. In places where there is a slight excess in the density of matter, this excess of density creates a gravitational field that attracts even more matter to these areas, which, in turn, produces an even stronger gravitational field that attracts even more matter. The system is unstable, it forms complex clusters that we see, such as galaxies, stars, planets and so on.

This is a complicated process. But it all starts with these very weak heterogeneities that we believe existed shortly after the Big Bang. We see these inhomogeneities in cosmic background radiation. Their measurement tells us a lot about the conditions under which the universe existed then and allows us to build theories that explain how such a universe turned out. It is for measuring these inhomogeneities that satellites such as COBE, WMAP and Planck are created with very high accuracy.

Inflation answers the question of where the heterogeneity came from. There was no explanation in the usual Big Bang theory. It was simply assumed that there were heterogeneities and added them artificially, but there was no theory where they could come from. In inflationary models, where all matter is created by inflation, heterogeneities are also controlled by this inflation and appear due to quantum effects.

It is hard to believe that quantum effects can be important for the large-scale structure of the universe. The Andromeda Galaxy doesn't look like it's a quantum oscillation. But if you consider this theory quantitatively, it really works very well. The theory is that the vibrations that we see in cosmic background radiation were indeed purely a consequence of quantum theory, mainly the uncertainty principle, which states that it is impossible to have something completely homogeneous. This is not consistent with the principle of uncertainty.

When we use the basic ideas of quantum mechanics, we can calculate the properties of these vibrations. To do this, we need to know more about very high energy physics, physics that was relevant during the inflation period, in order to be able to predict the amplitude of these oscillations. We cannot predict the amplitude. In principle, inflation would allow us to do this if we knew enough about the underlying particle physics, but we know too little about it. Therefore, in practice, we cannot predict the amplitude.

However, inflationary models provide a very clear prediction of the spectrum of such fluctuations. By this I mean a change in the intensity of the vibrations depending on the wavelength. The spectrum here means the same thing as for sound, except that you need to consider the wavelength, not the frequency, because these waves do not actually oscillate. But they have wavelengths just like sound waves, and if we talk about the intensity of different wavelengths, the idea of the spectrum is really the same as in sound.

It can be measured. These are not the last measurements, these are the last measurements for which I have a graph. The red line is theoretical prediction. Black dots are real measurements. This is seven year WMAP data. It is difficult to convey how happy I was when I saw this curve.

I also have graphs of what other theories predict. For some time, for example, people were very serious about the idea that the inhomogeneities that we see in the Universe, these fluctuations, were possibly caused by the random formation of the so-called cosmic strings that formed in phase transitions in the early Universe. This, of course, was a viable idea at one time, but as soon as this curve was measured, it turned out that the prediction of cosmic strings did not look at all like that. Since then, they have been excluded as a source of density fluctuations in the universe. Various other models are also shown here. I will not spend time on them, because there are other things that I want to talk about.

In any case, this is a definite success. And this is the latest data. This is data from the Planck satellite, which was launched in March last year. I don’t have it on the chart on the same scale, but again you see a theoretical curve based on inflation and points that show data with tiny little dash of errors. Absolutely clear correspondence.

STUDENT: what happened to the theory of inflation after they discovered dark energy? Has she changed significantly?

TEACHER: has the theory changed?

STUDENT: there was another curve in the previous chart.

TEACHER: Regarding inflation without dark energy. I think that not so much the theory of inflation is different for these two curves, but the curve that you see today is the result of inflation and the evolution that has occurred since then. And it is the evolution that has occurred since then that makes a big difference between these curves.

Thus, inflation theory should not have changed much. And she really did not change. But, of course, the curve looks much better after dark energy was detected, because the correct mass density became known, and gradually we got more and more data on these fluctuations, which fit perfectly into what inflation predicts.

Now I want to move on to the idea of the multiverse. I will try to quickly go over it so that we can finish it. We still won’t try to understand all the details now, so I’ll talk about some of them in the remaining 10 minutes of the lecture. I want to talk a little about how inflation leads to the idea of a multiverse. We will return to this at the very end of the course, and this is certainly an exciting aspect of inflation.

The gravitationally repulsive material that creates inflation is metastable, as we said. It is breaking up. This means that if you are in a place where inflation occurs and you wonder what the probability is that it will occur a little later, this probability decreases exponentially - it falls by half for every doubling, each half-life. But at the same time, the volume of any area that is swelling also grows exponentially, growing due to inflation. In fact, in any reasonable inflationary model, growth is much faster than decay. If you look at the area that is swelling, if you wait for the half-life, half of the volume of this area will no longer swell, according to the definition of half-life. But the remaining half will be significantly larger than the volume with which we started. That is the whole point.

This is a very unusual situation, because it seems to have no end. The area that is swelling gets bigger and bigger, even when it splits, because expansion is faster than decay. This leads to the phenomenon of perpetual inflation. The size of the swelling region increases with time, despite the fact that the swelling matter decays. This leads to what we call perpetual inflation. Eternal here means eternal in the future, as far as we can judge, but not eternal in the past. Inflation begins at some finite time, but then, as soon as it begins, it will continue forever.

Whenever part of this swelling region undergoes a phase transition and becomes normal, it locally looks like a Big Bang. Our Big Bang is one of these local events, and the universe formed by any of these local events, where an expanding region decays, is called a pocket universe. Pocket simply because there are many such universes on the scale of this multiverse. They are in some ways small, although they are the same size as the universe in which we live. And our universe is one such pocket universe.

Thus, instead of a single universe, inflation produces an infinite number of them. This is what we call the multiverse. It is worth noting that the word multiverse is also used in other contexts and other theories, but inflation, as it seems to me, is the most plausible way to build a multiverse. This is what most cosmologists mean when they talk about the multiverse.

What is the place of dark energy here? She plays a very important role. In 1998, two groups of astronomers independently discovered that the universe is now expanding with acceleration. We now know that the universe has expanded rapidly over the past five billion years out of 14 billion years of the history of the universe. There was a period when expansion slowed down to five billion years ago. The consequence of this is that inflation is actually happening today. This accelerated expansion of the universe that we see is very similar to inflation, and we really interpret it as a similar kind of physics. We believe that it was caused by some kind of negative pressure, just as inflation was caused by negative pressure.

This matter, which, apparently, fills space and has negative pressure, we call dark energy. Dark energy is simply, by definition, something, whatever it is, that causes this acceleration. One may ask, what is dark energy really? The surest answer to this is that no one knows. However, there is the most likely candidate. The most likely candidate, and other candidates are not much different from him, just that dark energy is vacuum energy. The energy of the void. It may be surprising that emptiness can have energy. But I'll tell you about it, and this is not so surprising.

But if dark energy is simply vacuum energy, it is fully consistent with everything we know about the nature of the expansion of the universe that we can measure.

STUDENT: why only in the last five billion years did the universe begin to expand rapidly?

TEACHER: Now I can explain this. Now that I have said that there is probably vacuum energy, I can give you an answer. The answer is that the energy of the vacuum does not change with time, because it is simply the energy of the vacuum. This is the same as what I said about energy density during inflation. It is just a constant. At the same time, ordinary matter becomes more discharged as the universe expands, decreasing its density in proportion to the cube of the size of the universe.

It so happened that before about the past five billion years, ordinary matter dominated the universe, which created attractive gravity and caused the universe to slow down. But then, about five billion years ago, the matter in the universe became so discharged that ordinary matter ceased to dominate the vacuum energy, and the vacuum energy began to cause acceleration. The energy of the vacuum was all the time, causing repulsion, but it was dominated by the attracting gravity of ordinary matter until the last five billion years.

Now I want to talk, why can there be something in the void? Why can emptiness have energy? The answer is actually quite clear to physicists today. Quantum vacuum, in contrast to the classical vacuum, is a very complex state. It is not empty at all. This is actually a complex set of vacuum vibrations. We think that there is even a field called the Higgs field, which you probably heard about, which on average has a non-zero value in a vacuum. Things like an electromagnetic field constantly fluctuate in a vacuum due to the uncertainty principle, which leads to the presence of energy density in these fluctuations.

So, as far as we can tell, there is no reason for the vacuum energy to be zero. But this does not mean that we understand what its significance is equal to. Today, the real problem from the point of view of fundamental physics is not finding out why the vacuum can have a non-zero energy density. The problem is to understand why it is so small. Why is this a problem? The quantum field theory, which we will not study in detail, says that, for example, the electromagnetic field is constantly oscillating. This is due to the principle of uncertainty. These vibrations can have any wavelength. And each wavelength contributes to the energy density of vacuum fluctuations.

However, there is no shortest wavelength. In a box of any size, there is the longest wavelength, but not the shortest wavelength. It turns out that when we try to calculate the vacuum energy density in quantum field theory, it diverges from the side of short wavelengths. It becomes literally endless, since a formal calculation shows that all wavelengths contribute, and the shortest wavelength does not exist.

What does this mean in real physics? We believe that this is not necessarily a problem with our understanding of quantum field theory. In fact, we believe that this is only a limitation on the range in which our assumptions are true. Of course, quantum theory works extremely well when it is tested in the laboratory. We think that at very short wavelengths, something should limit this infinity. A good candidate for limiting infinity at short wavelengths is the effects of quantum gravity, which we do not understand.

Thus, one way of estimating the true energy density predicted by quantum field theory is to limit the wavelengths on Planck scales, the energy scale, and the length scale associated with quantum gravity, which is about 10

^{ -33 }centimeters. If you do this, then you can calculate the energy density of the electromagnetic field of the vacuum and get a finite number. But it is too big. It differs not by some small number, but very much. It is more than 120 orders of magnitude. Thus, we do not understand why the vacuum energy is what it is, because our simple estimates say that it should be 120 orders of magnitude more.I must say that there is still a way out. The energy that we calculated here is only one of the contributions to the total energy of the vacuum. There are also negative contributions. If we calculate the fluctuation of the electron field, then its contribution to the energy will be negative. In principle, it is possible that these contributions compensate each other accurately or almost exactly, but we do not know why they should do this. Thus, there is a big question about the theoretical prediction of vacuum energy density.

Now I want to talk a little about the landscape of string theory, which can be a possible explanation for the smallness of the energy of the vacuum. This is only a possible explanation, everything is very speculative here. But one possible explanation for this very small vacuum energy that we are observing combines the idea of perpetual inflation and string theory. It is based on the idea that string theory does not have a unique vacuum. For many years, theorists have unsuccessfully tried to find a vacuum in string theory. They simply could not understand how string theory should look like a vacuum.

And then, a little over 10 years ago, many string theorists began to unite around the idea that maybe they could not find a vacuum, because there is no unique vacuum for string theory. Instead, they now claim that there is a huge number, they consider numbers like

^{ 10,500 }, a huge number of metastable states that live long, any of which may look like a vacuum for a long period of time, even if it can eventually decay or go to one of the other metastable states. This is called the string theory landscape. This huge set of vacuum states, any of which can be a vacuum, which, for example, fills some kind of pocket universe.If you combine this with the idea of perpetual inflation, you can come to the conclusion that during perpetual inflation, most likely, all these

^{ 10,500 }or more types of vacuum will arise . That is, different pocket universes within themselves will have different types of vacuum created at random. Then we will have a multiverse, which will consist of many, up to^{ 10,500 }degrees or more different types of vacuums in different pocket universes.With this assumption, string theory is the supposed law of physics that governs everything. But if you lived in one of these pocket universes, you would actually see the laws of physics that were very different from the laws in other pocket universes. The fact is that the physics that we actually see and measure is low-energy physics compared to the energy scale of string theory. We see only small fluctuations in the structure of the vacuum in which we live.

So those particles that we see - electrons and quarks, which combine into protons and neutrons, may be characteristic of our particular pocket universe. In other pocket universes, there may be completely different types of particles, which are vibrations of other types of vacuum. So, even if the laws of physics are the same everywhere - the laws of string theory, in practice the observed laws of physics can vary greatly from one pocket universe to another. In particular, due to the fact that in different universes there is a different vacuum, the energy density of a vacuum can be different in different universes. And this gives a possible answer to why the observed vacuum energy is so small.

We’ll talk about this next time.