Early Universe 2. Inflationary Cosmology: Is our universe part of the multiverse? Part 2

On the website of free lectures MIT OpenCourseWare posted a course of lectures on cosmology by Alan Gus, one of the creators of the inflationary model of the universe.

Your attention is invited to the translation of the second lecture: “Inflationary Cosmology. Is our universe part of the multiverse? Part 2".




Inflation and landscape string theory
I want to start with a brief repetition of what we discussed last time as part of a review lecture, which we will finish today. The summary of the last lecture is given in five slides. We began by discussing the standard Big Bang, by which I mean the Big Bang without taking inflation into account. I noticed that in fact, this theory describes only the consequences of the explosion. It begins with the description of the universe as a hot, dense substance of particles, which more or less evenly fills all the available space and expands.


Cosmic inflation is a prequel to the Big Bang. She describes how repulsive gravity, which in general relativity can be caused by negative pressure, drives a tiny portion of the early universe into a process of gigantic exponential expansion. Our visible universe is a consequence of such an event.

The total energy of such a site may be very small and may even be exactly zero. This is possible due to the fact that the gravitational field that fills space has a negative contribution to energy. As far as we can judge, in our real universe, the positive and negative contributions are approximately equal to each other. They can fully compensate each other. Thus, the total energy can be zero, which allows you to create a huge universe, starting with nothing, or with almost nothing.


The next item is evidence of inflation. Why do we think that there is a high probability that our universe has undergone inflation? I indicated three reasons. First, inflation can explain the uniformity of the universe on a large scale. The large-scale homogeneity of the universe is most pronounced in the cosmic microwave background radiation. We see that it is homogeneous with an accuracy of one hundred thousandth. If we make an amendment to the motion of the Earth, then its intensity over the whole sky is the same with an accuracy of one hundred thousand, regardless of direction.

Secondly, inflation can explain the remarkable fact of the Ω value, where Ω is the actual mass density of the universe divided by the critical mass density, i.e. density, which makes the universe completely flat. We know that in the first second after the Big Bang, their ratio was one with an accuracy of about 15 decimal places. Before inflation, we had no explanation for this fact at all. However, inflation brings Ω to unity and gives us an explanation why Ω at the beginning of the Big Bang was so close to unity.

In fact, inflation makes a prediction. We assume that if the theory of inflation is correct, then Ω should still be equal to 1. Ω was measured and a value of 1.0010 ± 0.0065 was obtained, which seems to me to be a remarkable result. Finally, inflation provides an explanation for the heterogeneities that we see in the universe. She explains them as quantum fluctuations that occurred during inflation. When inflation ended, quantum fluctuations forced inflation in some places to last a little longer than in others. So these heterogeneities appeared.

Currently we can measure these inhomogeneities with great accuracy. Heterogeneities, of course, are huge at the level of galaxies, here they are obvious, but they are difficult to associate with the early universe. Therefore, the most accurate comparison between what we observe and the theories of the early universe can be made by carefully studying the cosmic background radiation, which is not entirely uniform and has small fluctuations in intensity. These fluctuations are at the level of one hundred thousandth, and we can currently observe them.


Inflation gives a clear forecast of the spectrum of these fluctuations, how their intensity should vary depending on the wavelength. Last time I showed you a chart with data from the Planck satellite. The correspondence between prediction and theory is striking. We will return to this near the end of the course.


Finally, in a previous lecture, I began to talk about the possible consequences of inflation, such as the multiverse. That our universe can be built into a much larger entity consisting of many universes, which we call the multiverse. The key point is that most models tend to lead to perpetual inflation. Inflation, once started, never stops again.

The reason for this is that the metastable, gravitationally repulsive matter that causes inflation disintegrates, but at the same time it expands exponentially. For typical models, exponential expansion occurs much faster than decay. Thus, despite the fact that this unstable matter decays, its total volume does not actually decrease, but exponentially increases with time.

The disintegration of matter, however, occurs, and wherever decay occurs, what we call the pocket universe is formed. We live in one of these pocket universes. The number of pocket universes grows exponentially with time, as the whole system grows, which will continue, as far as we can judge, forever. This is the picture of the multiverse to which inflation leads.


At the very end of the lecture, I spoke about the problem, which is very important for our modern understanding of physics and cosmology. This is the discovery of dark energy. Around 1998, it was discovered that the expansion of the universe does not slow down under the influence of gravity, as one would expect, but, on the contrary, is accelerating. The universe is expanding faster and faster.

This indicates that space is currently filled with gravitationally repulsive matter, which we call dark energy. The simplest explanation for dark energy is simply the energy of a vacuum, the energy of empty space. Space has an energy density that has exactly the properties that we observe. Therefore, it seems natural to draw a link between dark energy and vacuum energy.

The vacuum energy, at first, may seem strange. If the vacuum is empty, why should it have an energy density? But in quantum field theory, this is not surprising, because in quantum field theory, the vacuum is actually not empty. In quantum field theory there is no such thing as real emptiness. Instead, constant quantum fluctuations of the fields arise in vacuum. In the modern standard model of elementary particle physics, there is even a field called the Higgs field, which, in addition to fluctuations, has a nonzero average value in vacuum.

Thus, the vacuum is a very complex state. The fact that it is in the state of the lowest possible energy density makes it vacuum, but this density does not have to be zero and there does not seem to be any reason why it should be zero. Therefore, there is no problem explaining the fact that a vacuum may have a non-zero energy density. The problem arises when we try to understand the magnitude of this vacuum energy. If a vacuum has an energy density, then, according to our assumptions, it should be much larger than what we observe in the form of acceleration of the expansion of the universe.

The typical order of magnitude for vacuum energy in particle physics is approximately 120 orders of magnitude greater than the number that is obtained according to the observed acceleration of the expansion of the universe. This is a big problem. We began to discuss a possible solution to this problem. This is just a possible solution, no one says that it is absolutely true. This solution is based on string theory and, in particular, on an idea called the string theory landscape.

Most string theory theorists believe that string theory does not have a unique vacuum. Instead, there is a colossal number, about ^10,500
various metastable states, which, despite the fact that they are metastable, are very long-lived, long-lived compared to the age of our universe. Thus, any of these ^10,500 different states can serve as a vacuum for one of the pocket universes.

At the same time, any vacuum state from the landscape can be realized in some pocket universe, thus embodying into reality all the possibilities that arise in string theory. Each type of vacuum has its own energy density, because in quantum field theory there are both positive and negative contributions.

The vacuum energy of a typical state can be both positive and negative. For these ^10,500 different vacuums, the range of energy densities varies from -10 ¹²⁰ to +10 ¹²⁰ observed values. The observed value is in this range, but is an extremely small part of the possible values.

STUDENT: Is the range from -10 ¹²⁰ to +10 ¹²⁰ chosen simply because we see a difference of 120 orders of magnitude, or are there other reasons?

TEACHER: When we talk about a difference of 120 orders of magnitude, a more accurate statement is that the estimate of the typical energy range is ^10,120 times higher than the observed value. In fact, 10 ¹²⁰ is accurate only within a few orders of magnitude, 10 ¹²³ , probably a slightly more accurate number. But for our purposes it is enough.

STUDENT: A general question about the properties of inflation. We believe that attracting gravity controls the movement of objects in space. Then why do we think repulsive gravity controls the expansion of space itself?

TEACHER: It behaves differently. The repulsive gravity that appears in the general theory of relativity is not just ordinary gravity with the opposite sign. If we have two bodies, then the usual gravity causes them to attract each other with a force proportional to the masses of these objects. Repulsive gravity is an effect caused by negative pressure in the space between them. Therefore, if there are two bodies, they will begin to accelerate from each other by an amount that is completely independent of their masses.

Repulsive gravity is not created by masses. This power is completely different, so we simply cannot compare them. In any case, when everything is moving away from each other, it is a question of a point of view whether to consider such a movement as an expansion of space or to consider it as the movement of objects through space. In the theory of relativity there is no way to insert a needle into space, pin it with a pin and say that it is motionless. So we cannot say whether space is moving or not.

In cosmology, a picture is usually simpler in which space expands with matter, and we will usually use such a picture. This gives a much simpler description of what is happening. Good question.

STUDENT: Why did the energy in the early universe seem to be close to zero? Are there theoretical models that can explain or predict that it is exactly zero?

TEACHER: Yes, there are such theories. This happens in the case of a closed universe. Even if the universe is almost flat, it can still be closed. If it is closed, it must have exactly zero energy.

STUDENT: Cosmic microwave background is the same in all directions. This implies that the cosmological principle is valid for the whole universe. Is it possible that in reality the universe is heterogeneous on a very large scale, that in reality it is kind of spotted, only the spots are very large? What we really are in such a spot, and it is different from other such spots, which are very far away?

TEACHER: Of course it can, if the picture of the multiverse is correct. She predicts exactly that. Other pocket universes can be seen as other spots using your terminology, and they will be very different from what we see.

Thus, inflation changes the attitude to this issue. Before, before inflation, the homogeneity of the universe had no explanation, so it was a postulate. No one has postulated that the universe is homogeneous on a certain scale. If a postulate is made, then it is simply stated that the universe is homogeneous, and such a postulate was used.

But now, when we believe that the homogeneity of the universe is caused by a dynamic process, inflation, then it is natural to ask the question, what uniformity does inflation create. This, of course, is a size that is much larger than we can observe. Thus, we really do not intend to see the heterogeneity caused by different foci of inflation. But the inflationary model makes it very likely that we would see them if we could see far enough.

STUDENT: If the universe is expanding and we are also expanding, how can we observe a change in distances?

TEACHER: Very good question. It may seem that if the universe expands, then everything should expand. And if everything expands, then measuring something with a ruler, we get the same length. How do we even see that everything expands? The answer to this question is that the expansion of the universe does not really mean that everything expands. When they say that the universe is expanding, it means that galaxies are getting farther from each other, but individual atoms do not grow.

The length of the ruler, determined by the number of atoms and their size, does not increase with the universe. At the moment, expansion is partly due to repulsive gravity, which causes the universe to expand rapidly. But basically the expansion right now is just the residual speed from the Big Bang. The substance in this case simply moves in space, and this movement does not cause the atoms to become larger.

STUDENT: What is the future of our universe? Will it expand indefinitely, or will it stop at some point?

TEACHER: As you probably guess, nobody really knows. But the models I'm talking about give a definite answer at the level of our pocket universe and at the level of the whole multiverse. At the level of our pocket universe, our universe will become thinner. Life eventually becomes impossible, because the density of matter will become too small.

Perhaps the universe will fall apart. Our vacuum may not be entirely stable. Very few things are stable in string theory, if string theory is the correct theory. But even if the vacuum decays, it will expand even faster than it disintegrates. So disintegration will lead to holes in our universe. It will look like Swiss cheese. But the universe as a whole will simply expand exponentially, as far as we can judge, forever.

The multiverse is a more interesting object. The multiverse, as I said, will constantly create new pocket universes. The multiverse will live forever, even if each pocket universe in the multiverse is formed, and then eventually dies, dies from complete thinning and becoming nothing.

STUDENT: In addition to the previous question. Do you allow the possibility of a cyclic process? Those. the universe expands, reaches its maximum, then begins to shrink, collapses, and then begins to expand again, and everything repeats?

TEACHER: Such an opportunity certainly exists, and there are people who take this very seriously. I do not see any evidence of this. In addition, there has never been and still does not have a reasonable theory of rebound, which should be part of this theory.

STUDENT: How are, apart from the cosmological constant, various vacuums still differ?

TEACHER: They can vary in so many ways. They are fundamentally different from each other in how their internal structure is arranged in space. If you don’t go into details that I myself may not fully understand, string theory asserts that space has nine dimensions, not three that we observe. Nine dimensions become three due to the fact that the extra dimensions are twisted into tiny nodules that are too short to be seen.

However, there are many different ways of twisting these extra dimensions, and this leads to a very large number of possible vacuums. Additional measurements may be twisted in different ways. This means that the low-energy physics in these vacuums can be very different. Practically everything can be different, even the dimension of space can be different, since it is possible to have a different number of twisted dimensions.

The set of particles may be completely different, because what we consider a particle is actually just a vacuum oscillation.It can be completely different. It can be noted that it can be observed.

If you haven’t been in the world?

TEACHER: The number of particles may not be saved. When one of the regions expands exponentially during inflation, the energy in it is not very well described in the language of particles. It is described in terms of fields. Fields sometimes behave like particles, but not always. In principle, there is a description in terms of particles, but it is not as obvious as a description in terms of fields.

Thus, there is energy, consisting in various fields, while the region is growing. The energy stored in these fields increases as the region expands. Energy density remains approximately constant. This seems to be a violation of the law of conservation of energy, but as we said, the expanding region is filled with a gravitational field, which takes up an increasing amount and volume, and the gravitational field has a negative energy density. Thus, the total energy that needs to be stored remains very small and possibly zero. At the same time the region can grow without restrictions, still having this very small or zero total energy.

Then, in the end, the region falls apart. When it breaks up, new particles are born, a huge number of new particles. This is the substance of which we are made. New particles are much larger than the number of particles that was in the region when inflation began.

STUDENT: So, everything that happens during inflation is determined by the law of conservation of energy?

TEACHER: It seems to me that this is an exaggeration, because if nothing happened, the energy would also be saved. Therefore, to describe the development of the universe you need more than just energy conservation.


The anthropic principle
Let's continue. I stopped at the landscape of string theory and how it forms all of these possible vacuums. In string theory there are ^10,500 different vacuums. We do not really know the exact amount, but it is about the same huge number. And only 10 ^-120 vacuums of the total number have very little energy. Therefore, the energy density is distributed from +10 ¹²⁰ to -10 ¹²⁰ from the vacuum energy that we observe.

This means that the energy that we observe is only in a narrow section in the middle, which occupies 10 ^-120widths of the entire distribution. All this, of course, very rough estimates. What matters is not quantity, but whether you agree with the idea. We assume that approximately 10 ^-120 different vacuums will have a rather low energy density.

But at the same time there will still be a huge number of such vacuums, because 10 ^-120 times 10 ⁵⁰⁰ multiplied by 10 ³⁸⁰ . Even though such vacuums will be very rare, there are ^10,380different types of vacuums, all of which have the observed vacuum energy density. Thus, in the string theory landscape there is no problem in finding a vacuum, whose energy density is as small as the one we are observing. But then the question arises, if they are so incredibly rare, is it not a miracle that we live in one of these unusual vacuums with such an extremely low energy density.

This leads to what is sometimes called the anthropic principle or selection effect. To show how this works, so that it doesn’t sound as crazy as it may seem, I want to start with an example where I think you can really say that this effect is happening. Let's just look at our position in our own visible universe and pay attention, for example, to mass density.

The place where we live is very unusual in many ways, but one of the parameters, which is simple and quantitative, is mass density. The density of objects around this room is about one gram per cubic centimeter, maybe 10 times more or less. Factor 10 is not very important for what I will talk about.

The fact is that the average mass density of the visible universe is about 10 ^-30 grams per cubic centimeter. It's incredible how empty the universe is. This is a much lower density than that which we can achieve in laboratories on Earth with the best vacuum systems.

In the place where we live, the mass density is 10 ³⁰times greater than the average density of the visible universe. So we do not live in a typical place of our visible universe. We live in a very atypical place. One may wonder how to explain this? Is it just an accident that we live in an area with such a high mass density? If this is a matter of chance, then it does not seem very likely. This is luck? Is it divine providence, or what?

I think most of you will agree that this is most likely a selection effect. This is the place where life arises. Life does not occur in most of the visible universe. It appears in rare places, such as the surface of our planet, which is special in many respects, but just mass density is enough to make it extremely special. We differ in 10 ³⁰ times the average of our environment.


If we explain why we live in such an unusual place of our visible universe, simply the requirements of life, then it is not so difficult to spread this idea further. Steve Weinberg first drew attention to this in 1987. Of course, he was not the first to express this thought, but he was the first to whom the rest at least believed a little.

He noted that the low energy density of the vacuum can be explained in a similar way. If we live in an atypical place within our visible universe, then, similarly, there is no reason to expect that we should live in a typical place of the multiverse. Perhaps only a small fraction of the various types of pocket universes can support life. Perhaps the only way to have life is to have a very small value of the vacuum energy density.

There is some physics behind it. Recall that the vacuum energy density accelerates expansion. Therefore, if the vacuum energy density were significantly greater than we observe, the universe would expand incredibly quickly and spread out before time arose for something interesting, for example, for the formation of galaxies. Weinberg based his arguments on the assumption that galaxies are a necessity for life.

If the vacuum energy density were significantly greater than what we are observing, the universe would scatter so quickly that galaxies could never form. Consequently, there would be no planets, nothing that is connected with life known to us.

On the contrary, if the vacuum energy density were negative, but would have a greater value compared to what we observe, then a large negative acceleration would occur. Such universes simply shrink, collapse in a very short time, too fast to form the life of any type we know. Thus, there is a physical argument that states that life is formed only when the vacuum energy density is very small.

Weinberg and his staff calculated what the requirements for the formation of galaxies should be. It turned out that in order for galaxies to form, the vacuum energy density should not be about 5 times the observed energy density. This may be a possible explanation. Although this, of course, is not a generally accepted explanation and it is very debatable.


Some physicists accept this idea of ​​selection. I tend to accept it. But many physicists consider it absolutely ridiculous, saying that anything can be explained with such arguments. And there is some truth in this. You can, if you want, much to explain, simply claiming that it is necessary for the emergence of life.

Therefore, in my opinion, the arguments of the effect of selection or the anthropic principle should always be considered as arguments of the last hope. That is, as long as we do not understand the landscape of string theory, and we do not understand it in detail, and until we really understand what is required to create life, we really can’t do anything more. how to give plausible arguments of the anthropic principle.

But these arguments sound reasonable. I think that there is nothing illogical about them; they may well be explanations for some things. As I noted earlier, this explains why we live in such an unusual place in our own visible universe. The selection effect arguments become very attractive when the search for more direct explanations has failed. In the case of an attempt to explain a very low vacuum energy density, other explanations were not crowned with success. We have no quantitative, direct understanding of why the vacuum energy should be so small.


Is it time to accept this explanation of the last hope that the density of vacuum energy is so small, simply because it is necessary for life to develop? I really do not know. But I will say that in the case of a low vacuum energy density, people have been trying very, very hard for some years to find an explanation for this in particle physics, and no one has found anything that others would find acceptable. So this is definitely a very serious problem. I think it’s time to take the last hope argument seriously. That the vacuum energy density is small only because in those parts of the multiverse where this is not the case, no one lives. It seems to me that the selection effect is the most plausible of any explanation that is currently known.


Let's summarize what we have learned. I showed that the inflationary paradigm is now in excellent condition. She explains the uniformity of the universe on a large scale. It predicts the mass density of the universe with an accuracy of 1% and explains the oscillations that we see in the cosmic background radiation, treating them as the result of quantum fluctuations that occurred in the very early universe.

The inflationary pattern leads to three ideas that indicate the possibility of the existence of the multiverse. This, of course, is not proof that we live in the multiverse, but nonetheless. First, this is the statement that almost all inflationary models lead to the idea of ​​perpetual inflation, that the exponential expansion of inflationary matter is ahead of the decay of this matter, so that its volume grows forever and exponentially.


The second point is that in 1998, astronomers discovered the surprising fact that the expansion of the universe does not slow down as it expands, but rather accelerates. This indicates that there must be some special matter in the universe, different from the substance that we already know, and this particular matter is called dark energy. We do not have a simple interpretation of what it is, but, most likely, it is vacuum energy. If this is so, then this immediately leads to the important question of why this energy matters, which we observe. Apparently, it is much smaller than one would expect.

And third, theorists studying string theory give us an interesting explanation. They say that, possibly, according to the laws of physics, there is no single vacuum, but there is a huge number of different vacuums that string theory predicts. If this is the case, then we assume that among many different vacuums there will be a large number of those that have a very small energy density. They constitute a tiny fraction of the total number of different vacuums, but, nevertheless, there are quite a lot of them. Then the idea of ​​the selection effect can provide a possible explanation for why we live in one of these very unusual vacuums that this incredibly small energy density has.

I want to end with a little story. How much do physicists really take all this seriously? I will tell you about the conversation that took place at the conference several years ago. I'll start with Martin Reese. This is an astronomer from the UK, the former president of the Royal Society, the former head of a college in Trinity, a very respected and, by the way, good person. He said he was confident enough in the multiverse to put the life of his dog on her.

Andrei Linde from Stanford, a true enthusiast of the multiverse idea, is also one of the founders of the theory of inflation, said he is confident enough in the multiverse to put his life on it. Steve Weinberg was not at this conference, but he wrote an article that became known later, commenting on this discussion. Do you think he was ready to deliver? He said that he was so confident in the multiverse that he was ready to put on her the life of Andrey Linde and the life of Martin Reese's dog.

This concludes the overview. Are there any questions before we start to the beginning, to the present beginning of our course?

STUDENT: The selection effect states that Ω is equal to 1, and the vacuum energy is much less than it can be, just because life exists within these limitations, that life can exist only in this way. But we look at carbon based life. What if there are any other life forms that allow you to have another energy, density, and so on?

TEACHER: Yes, what you point out, of course, is the great weakness of the selection effect argument. We really know carbon-based life, a life similar to ours, and we can talk about what conditions are necessary for such a life. But perhaps there is a life that is completely different from ours, about which we know nothing, and which can exist in completely different conditions. This is really a weakness.

Nevertheless, I want to say, although this too can be argued, and not everyone will agree with me, but a similar situation arises if we want to explain the unusual features of that part of the universe in which we live. For example, using the example I used earlier, that we live in a place where the mass density 10 ³⁰ times greater than average. If we are prepared to use the arguments of the anthropic principle to explain this, then I think the same problems arise here.

If in the universe, in reality, there is an abundance of another life, flourishing in a vacuum, then we would have a much better chance of being one of them than becoming an extremely unusual creature living on the surface of the planet. Therefore, I think that this is a possible weakness that needs to be borne in mind, but I do not think that this should completely prohibit us from using these arguments. Although this is certainly a reason for skepticism.

STUDENT: You mentioned last time that the various pocket universes that make up the multiverse are separate from each other, although they arise as small regions in the original vacuum. Due to what they are separated from each other? If they all form in the same space, don't they stay in that space?

TEACHER: They do remain, but the space in which they are formed expands very quickly. Thus, in most cases, although in fact not always, the two pocket universes will form far enough from each other to never touch each other as they grow, because the space between them expands too fast to allow them to meet.

However, the pocket universes collide if two pocket universes form close enough to each other. The expansion of the space between them will not be enough to separate them from each other, and they will collide. How often this happens is an extremely difficult question to which no one knows the answer. There are at least one article by a group of astronomers who were looking for possible signs of a collision of universes in the past. They have not found anything definite. But this is what you need to think about, and this is what people think about. There are actually quite a few publications about collisions of universes.

STUDENT: When did you say “long-lived” what time did you mean by that?

TEACHER: I have used the word “long-lived” in at least two contexts. I talked about the long-lived metastable vacuum. Here, under the long-lived, I meant a long compared to the age of our universe from the time of the Big Bang. Here long means long compared to 10 ¹⁰ years.

I also said that if the vacuum energy of the universe were large and negative, the universe would collapse very quickly. This can occur in 10 to ²⁰ seconds. This can happen very quickly depending on how large the cosmological constant is.

STUDENT: I ​​read that there is an effect when different observers can see the vacuum differently. For example, if an observer sees a vacuum in an inertial system, another observer, who is accelerating with respect to this observer, will see particles, a warm gas. How strongly do we see this effect due to the fact that the universe is expanding rapidly, and we may be accelerating with respect to some vacuum?

TEACHER: You actually touch a very controversial issue. You said that you heard that if you take an accelerating observer moving through a vacuum, then this accelerating observer would see something unlike vacuum. He would see particles that look as if they have a temperature that can be calculated and which is determined by the acceleration.

The question is that from what we see, it really exists in reality, and what is caused by our own movement. I do not know the exact answer to this question. But when such questions arise, we, as a rule, believe that an observer who moves freely, actually means an observer moving freely in a gravitational field, or as the geodesic observer sometimes says. Such an observer essentially defines what can be called reality. Then you can calculate what the accelerating observers see in relation to this reality.

We are practically geodesic observers. The earth is pressing on us, which slightly violates our inertia. But on a cosmic scale, where everything is compared with the speed of light, we are essentially inertial or geodesic observers.

STUDENT: I ​​have a philosophical question. We cannot observe other universes. Suppose we have a theory, such as inflation, which makes many predictions. And she also makes a prediction about the existence of the multiverse. But we cannot empirically test whether this is true or not, most likely we will never get an answer. If we are going to be strict empiricists, should we even address this issue?

TEACHER: This one is also discussed in the scientific community, and people accept both points of view. There is a point of view to which I am inclined that not every aspect of our theories can be verified. If you take any theory, even Newtonian gravity, you can imagine the consequences of Newtonian gravity, which no one has ever tested.

Therefore, I think that in practice we must accept theories that have made quite a few predictions we have verified in order for the theory to become convincing. In this case, at the same time, we must also take seriously the consequences of the theory that cannot be directly verified.

As for other pocket universes. Although unlikely, very unlikely, it is extremely unlikely that we will ever find direct observational evidence for the existence of another pocket universe, in theory, this is not impossible, because pocket universes can, in principle, collide. Thus, we can, in principle, find evidence that our universe in the past had contact with another pocket universe.

STUDENT: What determines the stability of a particular vacuum state? Simply, are higher energy vacuums less stable than low energy vacuums?

TEACHER: As far as I know, there is indeed a tendency that high-energy vacuums are less stable, and low-energy vacuums are more stable. But everything is not so simple. There are many parameters not dependent on energy density.

STUDENT: If our universe has such a small energy density relative to the average, does this mean that it will also be much longer-lived than average?

TEACHER: I think so. But this does not change the picture of Swiss cheese, which I described for our final future. It just changes the frequency of decays. But since the future of the pocket universe, if this picture is correct, will be infinite, decays will occur no matter how small the probability. In fact, an infinite amount of decays will occur.

We must move on, even if there are more questions. We still have a whole semester ahead to discuss all this.


So, we will start the course with a discussion of the Hubble law, although the Hubble law will quickly lead us to the question of Doppler shift, which I will mainly talk about by the end of today and most of the next lecture. The Hubble law is a simple equation v = H ∙ r , where v is the removal rate of any typical galaxy.

The Hubble law is not an exact law; individual galaxies deviate from the Hubble law. But in principle, the Hubble law says what the galaxy removal rate is equal to, at least with reasonable accuracy. H is often called the Hubble constant. Sometimes it is called the Hubble parameter.

The problem with the name "Hubble constant" is that it is not a constant during the lifetime of the universe. It is constant throughout the life of an astronomer, but not constant throughout the life of the universe. We will mainly talk about universes, not astronomers. Even throughout our history, this is not a constant, because the estimate of the Hubble constant has changed about 10 times since the initial estimate of Hubble.

r in the equation is the distance to the galaxy. If you look at the lecture notes two years ago, they begin with the fact that the Hubble Act was discovered by Hubble in 1929. When I began to review my notes this year, I realized that I had heard that this statement had become controversial. Almost everything in cosmology is debatable, and even this statement is debatable.

There is an opinion that in fact Lemaitre and not Hubble deserves the honor of opening the Hubble's law. There are some reasons for this statement. Some amateur historians, I think they are often mentioned in the press, claim that we know about Lemetra's works mainly from the translation made in 1931 of his work in 1927, where he wrote about the basics of cosmology.

It turned out that, apparently, several significant paragraphs from the French article of 1927, paragraphs on the Hubble constant, for some reason did not fall into the English translation of 1931. For a while it seemed like a dirty game, there were accusations that Hubble or Hubble's friends did not include these items during the translation of the article.

True, it was finally found a couple of years ago by a physicist by the name of Mario Livio, who studied the archives of monthly astronomical letters. It turned out that Lemaitre himself removed these items.

The items basically gave a numerical estimate of the Hubble constant, but by 1931 the Hubble article had already been published. Lemaitre realized that in his article there was only a less accurate estimate of the same value that Hubble indicated, so he cut it out of his translation. However, it is certainly true that Lemaitre knew about the Hubble law from theoretical considerations, since Lemaeter built a model of an expanding Universe.

I don’t know if he was really the first person to realize that the expanding model of the universe creates a linear relationship between speed and distance, but of course he knew about it, understood Hubble’s law and gave an estimate based on observational data. However, he did not try to use observational data to show that there is a linear relationship. In those paragraphs that were not translated, Lemaitre simply looked at a large group of galaxies, calculated the average value for v , the average value for r, and determined H , dividing the two average values. But he admitted that in fact there was not enough good data to say whether the connection is linear.

I think it is fair to say that Hubble is the person who really led the argument, at first rather weak, but then with time becoming more compelling that there is astronomical evidence of a linear relationship between speed and distance. So, most likely, the law will continue to be called the Hubble law. If you look at what it is called on Wikipedia, you will see that at the moment both options are acceptable, but Wikipedia articles are changing rapidly, so let's see what she writes next year. Also, we should probably be proud of Lemaeter. It is often written that Lemaitre was a Belgian priest, but he was also a student at MIT, he had a MIT Ph.D. degree that he received in 1927.

You can read his dissertation. When I was writing my book, I remember going to the MIT archive, taking his dissertation and reading it. In fact, it is not very easy to write, but interesting. Although he received his doctorate from MIT, it turned out that he did most of his work at the observatory at Harvard College. But the Harvard College Observatory at the time did not give degrees. It was just an observatory. He wanted a degree, so he enrolled in MIT, wrote a thesis and received a Ph degree. D.

Hubble's law is an indication that the universe is expanding. Initially, Einstein proposed a model of the universe, which was static. And it was Hubble who convinced Einstein that, according to observations, the universe is not static, but obeys its law of expansion.

This created the theory of an expanding universe. Today I want to talk about how to measure v , the speed in the Hubble law. There are also many discussions about how to measure r , distance. I think this is quite well described in the book of Steve Weinberg. I want to provide you with an independent study of the book by Steve Weinberg, to find out how distances to distant galaxies are estimated. Roughly speaking, they are estimated by finding objects in distant galaxies, the brightness of which, as you think you know, in one way or another.

The difficulty lies in understanding for which objects we are sure that we know their brightness. For such objects there is a common name - standard candles. A standard candle is an object whose brightness we know. As soon as we find an object, the brightness of which, we think, we know, we can tell how far the object is located by measuring how bright it looks. This becomes a very simple way to estimate distances, and this is the only way to estimate distances to distant galaxies. In fact, this is a much more complex topic; you can read about it in Weinberg’s book.

Doppler shift
The removal rate of galaxies is measured using the Doppler shift, and I will talk about the remaining few minutes of today's lecture. In the upcoming lectures, we plan to study how the Doppler shift is calculated in the nonrelativistic and relativistic cases. We will study the simplest cases: when the observer is motionless, and the source moves in a straight line; the source is motionless, and the observer moves.

I will start with the option when the observer is still and the source is moving, which we usually consider in the case of distant galaxies. We are in our own frame of reference, so we are still and the galaxy is moving. We need to calculate the redshift. However, I must tell you that the cosmological redshift is actually a bit different from what we calculate in this and the next lecture.

In the next lectures, we calculate the redshift in the special theory of relativity. But cosmology is not governed by the special theory of relativity, because special theory of relativity does not describe gravity, and gravity plays an important role in cosmology. We will talk about the cosmological red shift a bit later. At the moment, we, like Hubble, ignore gravity, which is normal for nearby stars. The further they are, the more important is the gravitational influence. Ignoring gravity, one can simply use special relativity or even Newtonian kinematics to calculate the relationship between$$ $ inline $ v $ inline $ and red shift.


So, the first task that we are going to solve, I think that I will simply formulate it, that is all that we will have time, this is a task, where there is a radiation source, which in our figure moves to the right with speed$$ $ inline $ v $ inline $ and an observer who is motionless.

Of course, all these statements depend on the frame of reference. We work in a frame of reference in which the observer is motionless. For the non-relativistic case, we also assume that the air, and we will speak of a sound wave, is motionless in this frame of reference. Thus, our reference system is not only an observer's reference system, but also a reference system in which air is stationary, and we consider the non-relativistic case of a sound wave.

Let's define our notation. Let be$$ $ inline $ u $ inline $ equal to the speed of the sound wave. It is usually measured relative to air, but the air is at rest in our drawing, therefore$$ $ inline $ u $ inline $ will be the speed of the sound wave relative to the figure.$$ $ inline $ v $ inline $ - source speed. We will be interested in two time periods.$$ $ inline $ Δt_s $ inline $ where$$ $ inline $ s $ inline $ denotes the source (from the English source - source), is the period of the sound wave at the source, that is, it is the same as the period of the wave, which is measured by the source.

$$$ inline $ Δt_o $ inline $ - this is the period of the sound wave at the observer (from the English observer - the observer) or the observed period. The subscript is the letter$$ $ inline $ o $ inline $ , not null. An important point, which may be qualitatively obvious, is that these two periods, or time intervals, are not equal to each other. The reason is that the source is moving. I identified$$ $ inline $ v $ inline $ positive, as astronomers would define for moving away objects. As the source moves away from us, each subsequent wave that travels from the source to us must travel a little more distance.

This means that each wave crest is slightly delayed when arriving at the receiver, compared with the situation when the source is stationary. If each wave crest is delayed, this means that the time between the arrivals of the crests is longer. I.e$$ $ inline $ Δt_o $ inline $ will be more than$$ $ inline $ Δt_s $ inline $ due to the extra distance that every crest of the wave must travel. We will deal with these calculations in the next lecture.

Then we will do the calculations for the case when the observer moves and the source is stationary. Then we will talk a little about the special theory of relativity and we will repeat both calculations taking into account the special theory of relativity, we will talk about light rays and speeds that can be comparable to the speed of light.