Early Universe 4. Kinematics of a Uniform Expanding Universe

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 fourth lecture: "Kinematics of a homogeneous expanding universe."

Isotropy and homogeneity of the universe

Last time we looked at the Doppler shift and talked a little about the special theory of relativity. Today we will start discussing cosmology. We consider the kinematic description of a homogeneous expanding universe. Our universe, we believe, is in a very good approximation.

In this lecture, we will look at some basic descriptive properties of the universe. The universe, of course, is a very complex object. For example, it keeps me with you, and we are quite complicated. But cosmology does not study all this. Cosmology is the study of the universe in general. We will consider the universe on the largest scale, where it is described by a very simple approximate model. In particular, on very large scales, the universe is fairly well described by three properties.

The first property is isotropy. This word comes from the Greek root, meaning the same in all directions. Of course, if you look around, the room does not look the same in all directions. The front of the audience is different from the back. The view of the city looks different than the view of the river. If you look further into space, in the direction of the Virgo cluster, which is the center of our Local Supercluster, it looks different than in the opposite direction.

But if you look at the universe on a very large scale, where in our case a very large scale means several hundred million light years, it starts to look very isotropic. If averaged, then on a very large scale it will be found that almost the same thing is seen regardless of the direction.

This becomes most apparent if you look at the cosmic background radiation, which is the furthest object we can see. This radiation appeared shortly after the Big Bang. It is worth recalling in a nutshell his story.

For about the first 400,000 years after its birth, the universe was filled with plasma. Inside the plasma, the photons cannot move freely. They move at the speed of light, but they have a very large cross section for scattering by free electrons that fill the plasma. Because of this, photons all the time change the direction of movement and their total movement in one direction is negligible.

Thus, the photons were locked in the substance, their average velocity relative to the plasma turned out to be zero. But according to our calculations, approximately 400,000 years after the Big Bang, the universe has cooled so much that the plasma has turned into a neutral gas, like the air in our audience. The air is transparent to photons, so the light moves in a straight line from me to your eyes and allows you to see my image.

Making analogies between the audience and the universe is a bit risky. The sizes are completely different. But in this case, the physics is absolutely the same. As soon as the universe was filled with neutral gas, it really became transparent to the photons of the cosmic background radiation. From now on, most of these photons travel freely in a straight line. When we look at them today, we, in fact, see an image of how the universe looked 400,000 years after the Big Bang.

In cosmology, the process of neutralizing gas in the universe is called recombination. In fact, this name is incorrect, because the prefix "re" implies a repeated action, and the gas was neutralized for the first time. Once I asked Jim Peebles, who may have used this name for the first time, why he chose him. He replied that the word “recombination” is used in plasma physics, so it was natural to use it in cosmology. But for cosmology, this name is wrong, the prefix "re" is completely superfluous here.

What do we see when studying cosmic background radiation? We see that it is extremely isotropic. The deviation in the background radiation temperature is about one thousandth.

$$

$$\ frac {δT} T = 10 ^ {- 3}$$

This is a very small number, but in fact the background radiation is even more isotropic.
This deviation of one thousandth has a certain angular distribution. Exactly such an angular distribution is obtained if we assume that we are moving through the cosmic background radiation. It is this movement of the solar system through the background radiation that we explain the deviation in $$$10 ^ {- 3}$ .

We have no independent way of measuring the speed of such a movement with sufficient accuracy. We simply customize it so as to get rid of deviations in the data as much as possible. This is a three-parameter fit, we can change the three components of speed. We have a complex angular pattern of radiation all over the sky, and three numbers that we can change.

After removing the deviations associated with our movement, residual deviations remain that are at the level of $$$10 ^ {- 5}$ , one hundred thousandth. The radiation is really extremely isotropic. Once I asked myself the question whether it is possible to polish a ball so that it becomes spherical with precision $$$10 ^ {- 5}$ . This can be done, but for this you need to use the technology used to create high-precision lenses, dealing with dimensions of the order of the light wavelength.

therefore $$$10 ^ {- 5}$ - really very high degree of isotropy. And that is what our universe looks like.

The second property of the universe is homogeneity. Isotropy means the same in all directions. Uniformity means the same in all places. Uniformity is more difficult to verify with high accuracy. To do this, for example, you need to find out whether the density of galaxies is the same at different distances. To check the isotropy, we looked at how the cosmic background radiation varies with the angle. But in order to test homogeneity, one needs to know how the distribution of galaxies varies with distance, and distances in cosmology are very difficult to measure.

As far as we can judge, the universe is quite homogeneous, again, on scales of several hundred million light years, although it is difficult to say for sure. There is, however, a relationship between isotropy and homogeneity.

They are very similar to each other, however, logically they are different concepts, and it is worth spending a little time to understand how they are related to each other. In particular, the best way to understand what these properties mean is to consider examples where one property occurs without the other.

Suppose, for example, that we have a homogeneous, but not isotropic, universe. Is it possible, and if so, how? I want you to come up with such an example.

STUDENT: For example, a universe in which galaxies are distributed with a constant density, but they all rotate in a certain direction.

TEACHER: Indeed, the galaxies rotate, that is, they have an angular momentum. The angular moments of different galaxies can all look in a certain direction, and this will be an example of a homogeneous, but not isotropic universe.

Another simple example is a universe filled with cosmic background radiation, in which all photons flying in the z direction are more energetic than flying in the x or y directions. In this case, the universe would also be completely homogeneous, but not isotropic.

You can come up with many more such examples. Now let's try to think of an isotropic, but heterogeneous universe. The isotropy property, by the way, depends on the observer. Let's first think of a universe that is isotropic for us, but heterogeneous. Can anyone give an example?

STUDENT: Spherical shell around us.

TEACHER: Right. Spherical structure. Let me draw it.


If we are in the center, and matter is spherically symmetric distributed, then the universe will be isotropic for us, but not homogeneous.

Such a device of the universe, of course, seems strange, because we do not believe that we live in some special place of the universe. This is the essence of the Copernican revolution, which is deeply rooted in the psychology of scientists.

If the universe is isotropic for all observers, then it must be homogeneous. This is one of the reasons why we are sure that our universe is homogeneous. Since it is isotropic to us, we believe that it should be isotropic for everyone. Then it should be homogeneous.

I suggest you think about the following question: if the universe is isotropic with respect to two observers, can it be heterogeneous? This is actually a more subtle question than it might seem.

In the Euclidean space, isotropy for two different observers is enough to ensure uniformity. But for non-Euclidean spaces this is not always the case. We have not yet talked about non-Euclidean spaces, therefore, as long as you may not be able to work with them. As an example, you can take a curved surface in three-dimensional space.

Curved surfaces are very good examples of non-Euclidean two-dimensional geometries. Try to come up with a two-dimensional surface that would be isotropic for two points, but not homogeneous. This is your assignment for the next lecture.

Isotropy and homogeneity are two key properties that simplify our universe on very large scales. The third property is the expansion of the universe, which is described by the Hubble law.

Hubble law

Hubble’s law says that on average, all galaxies move away from us at a rate that is constant $$$H$ called the Hubble constant multiplied by the distance to the galaxy, $$$r$ . This law does not apply exactly to all galaxies. It is performed on average, as isotropy and uniformity are performed on average.

Now I want to talk about the units in which it is measured. This will lead us to the concept of "parsec". Astronomers measure the Hubble constant, which I will sometimes call the Hubble parameter, in kilometers per second per megaparsec: (km / s) / Mpc. This is the speed divided by the distance. Kilometers per second is speed, and speed per megaparsec is speed divided by distance, as it should be.

Notice, however, that kilometer and megaparsecs are units of distance. Between them, just a fixed relationship. Thus, the Hubble constant is in fact time minus the first degree. But the expression of the Hubble constant in the form of time in the minus of the first degree is rarely used. Instead, it is expressed in units that astronomers like to use. They measure speed, like ordinary people, in kilometers per second. But they measure distances in megaparsecs, where megaparsec is a million parsecs, and parsec is shown in the figure.


The base of this triangle is one astronomical unit, the average distance between the Earth and the Sun. The distance from which one astronomical unit is visible at an angle of one second is called a parsec. Parsec is about three light years. One parsec is 3.2616 light years. Megaparsec is a million parsec.

What is the Hubble constant? She has a very interesting story. It was first measured by George Lemaitre in 1927, and published in an article in French. The article at the time was ignored in the rest of the world. She was discovered later. Lemaitre was not an astronomer. He was a theoretical cosmologist. I have already said that he had a PhD from MIT in theoretical cosmology.


He used two different methods of calculation, using data from other scientists, and obtained slightly different results. The value he received in 1927 for the Hubble constant ranged from 575 to 625 (km / s) / Mpc. Two years later, in 1929, in his famous article, Hubble received a value of 500 (km / s) / Mpc.


There is an important difference between the articles of Lemaitre and Hubble. First, Hubble used mostly his own data, and Lemaitre used data from other scientists, mainly Hubble. In addition, Hubble argued that the data shows proportionality $$$v$ and $$$r$ . Lemaitre knew that this was true for a uniformly expanding universe. But he decided that the available data is not enough to prove this fact. However, he got the value for $$$H$ by taking the average speed of the galaxies and dividing it by the average distance.


The figure shows the Hubble data. Obviously, they were not very good. The maximum speed of galaxies reaches only about 1000 km / s. What is curious is that you can see that the vertical axis, where speed is set aside, should be measured in kilometers per second, but Hubble wrote kilometers on it. But this did not prevent the publication of the article in the collection of works of the National Academy of Sciences and became, of course, a famous work.

It can be seen that the data are scattered. Straight lines are drawn on the graph, but if you remove the lines, it is not obvious from the data itself that the connection is truly linear. However, Hubble decided that there was enough data. He later collected more data. Today there is no doubt about the existence of a linear relationship between speed and distance. At very large distances there are understandable deviations, but at least for moderate distances, the connection is linear.

It should be noted that the speed of the solar system through cosmic background radiation is also the speed of the solar system relative to the expanding universe. Therefore, both Hubble and Lemaitre had to make an estimate of the speed of the solar system and subtract it in order to obtain data that looked like a straight line.

Lemaitre estimated the speed of our solar system at 300 km / s, Hubble estimated it at 280 km / s. It was an important amendment, because the maximum speed of galaxies was only 1000 kilometers per second, and the amendment was about a third of the maximum speed.

STUDENT. What did they use to estimate the speed of the solar system?

TEACHER: I think that they simply picked up a speed at which the average expansion in all directions would be about the same. Honestly, I'm not sure. But it seems to me that this is the only thing they could use.

Hubble Permanent

Since those times, many measurements of the Hubble constant have been made, and the value has changed a lot. In the 40s-60s, there was a whole series of measurements in which Walter Baade and Allan Sandage played a major role. At the same time, the values ​​of the Hubble constant were constantly decreasing from the large values ​​obtained by Hubble and Lemaitre.

When I was a graduate student, everyone said that the Hubble constant is in the region from 50 to 100 (km / s) / Mpc. Still remained uncertain 2 times. But the value was much lower - 5 or 10 times lower than the value obtained by Hubble. And this value remained the main source of uncertainty in cosmology.

The value of the Hubble constant began to be specified in 2001. Then the Hubble Key Project was launched. The word Hubble here refers to the Hubble telescope, which was named after Edwin Hubble. The Hubble telescope was used to observe variable Cepheids in galaxies that were significantly farther than those in which Cepheids could have been observed before. Thereby it was possible to measure much better distances. Cepheids are crucial for determining distances in cosmology.

The resulting value was much more accurate: 72 ± 8 (km / s) / Mpc. Meanwhile, it was still controversial. I should note that when they said that the Hubble constant is in the region from 50 to 100, it was not meant that the size of the error was so great. The real situation was that there was a group of astronomers who claimed that the value was 50, and there were other groups of astronomers who claimed that the value was 100. Scientists who believed that the value of the Hubble constant was about 50 also studied time and also used data from the Hubble telescope. In the same year 2001, they got the value 60, with an accuracy of up to 10%.

In 2003, with the help of the WMAP satellite, which means Wilkinson Microwave Anisotropy Probe, the satellite dedicated to measuring the smallest variations of cosmic background radiation at the level of one hundred thousandth, received a value of 72 ± 5 (km / s) / Mpc. This value was based on data collected over one year.

In 2011, the same WMAP team, using data for 7 years, received the number 70.2 ± 1.4 (km / s) / Mpc, which is already very accurate. The latest value is obtained using a satellite similar to WMAP, but more modern and powerful, a satellite called Planck. It turned out a somewhat unexpectedly low value of 67.3 ± 1.2 (km / s) / Mpc.

Hubble constant value:
1927. Lemaitre: 575-625 (km / s) / Mps
1929 Hubble: 500 (km / s) / Mps
1940-70 Baade and Sandage: 50-100 (km / s) / Mpc
2001 Habble Key Project: 72 ± 8 (km / s) / Mpc
2001 Tamman and Sandage: 60 ± 6 (km / s) / Mpc
2003 WMAP: 72 ± 5
2011 WMAP: 70.2 ± 1.4 (km / s) / Mpc
2013 Plank: 67.3 ± 1.2 (km / s) / Mpc

STUDENT: Why did you get such a strong discrepancy in the value of the Hubble constant, measured in the last century and now?

TEACHER: In the early measurements, scientists made a big mistake in estimating distances. It seems to me that this was due to the incorrect identification of cefid. They used the same way two different types of cefid, which should be interpreted differently. I’m not quite sure about the details, but they’re definitely wrong in estimating distances. The speed is quite easy to measure, and the error they got is very big.

STUDENT: The last obtained values ​​of 70.2 ± 1.4 and 67.3 ± 1.2 do not fall within the limits of each other’s error.

TEACHER: Why is this so? No one knows for sure. I note that the error means standard deviation - σ. The result does not have to be within one σ error. With probability 2/3, the answer lies within the limits of σ, but with probability 1/3 it lies outside the limits of σ.

Values ​​differ by approximately 2.5 σ. This means that with a probability of about 1%, the value of the Hubble constant satisfies both dimensions. It is still debated whether this is acceptable or not. In experimental physics, and especially in cosmology, such discrepancies appear regularly, and people often have different opinions about whether this indicates something very important or whether these discrepancies will disappear with time.

I want to add that the initial large overestimation of the Hubble constant had a very significant impact on the history of cosmology. Scientists using the Big Bang model tried to estimate the age of the universe. The result depended on the model, the density of the substance and things like that. However, the Hubble constant is an important parameter. The faster galaxies scatter now, the less time they needed to move to the current distance, and the younger our universe. With a very good degree of accuracy, any estimate of the age of the universe is inversely proportional to the Hubble constant.

Since the initial value of the Hubble constant differed from the current value by 7 times, the age of the universe was also 7 times smaller. Scientists received that, according to the Big Bang model, the age of the universe is 2 billion years instead of 14 billion years, as it is considered now.

However, as early as the 1920s and 1930s there was considerable geological evidence that the Earth was much older than 2 billion years. Scientists also knew something about the evolution of stars, and it was clear that many stars are also older than 2 billion years. Therefore, the universe could not be only 2 billion years old. This led to very serious problems with the development of the Big Bang theory. In particular, it was regarded as an additional proof of the so-called theory of the stationary universe. According to this theory, the universe exists infinitely long, and as it expands, a new substance is created that fills a new space, so that the density of matter remains unchanged.

In his article of 1927, Lemaitre himself built a very complicated theory, in my opinion, so that it would not contradict the known age of the universe. Instead of the Big Bang, his model began with static equilibrium, where a positive cosmological constant creating repulsive gravity, which we talked about in the introductory lecture, compensates for normal gravitational attracting ordinary matter. That is, it was a static universe of exactly the same type that Einstein originally proposed.

But in the Lemaeter universe, the mass density was slightly less than that of Einstein, so that it gradually expanded more and more. Normal gravity was not enough to keep it in place. Over time, the expansion of the universe picked up speed and made it possible to get a universe that is much older than the one obtained in the simple Big Bang model.

Expansion of the universe

Now I want to discuss what follows from the Hubble expansion law. At first glance it seems that the Hubble law implies that we are the center of the universe. All galaxies are moving away from us, so we are in the center. In fact, it is not. If you look more closely, as shown in the figure, it turns out that if the Hubble law is valid for one observer, it is also true for any other observer, as long as there is no possibility of measuring the absolute velocity.

We believe that we are at rest, but this is just our definition of a reference system. If we lived in another galaxy, we might as well have thought that this galaxy rested. The figure shows the expansion in only one direction, but this is enough to illustrate the idea.

In the upper figure, we believe that we live in galaxy A. Other galaxies are moving away from us with speeds proportional to distance. We evenly placed these galaxies in the picture. The neighboring galaxy is moving away from us with speed$$$v$ . The next galaxy is removed at a speed of 2 $$$v$ . Next with speed 3 $$$v$and so on, ad infinitum.

Now we want to go from galaxy A to galaxy B. Suppose that we live in galaxy B and consider galaxy B to be at rest. Now we will describe our picture in the reference system of the galaxy B. Galaxy B has no speed because it is at rest relative to its reference system.

When moving from one reference system to another, we will use Galilean transformations. Models that take into account the theory of relativity will be discussed later. When moving from one reference system to another, all we need to do to convert the speeds is to add a fixed speed to each initial speed, equal to the speed difference between the two reference systems.

To go from the top to the bottom image, we add speed to each speed.$$$v$directed to the left. For galaxy B, the initial velocity was$$$v$and was sent to the right. After folding with speed$$$v$directed to the left, we get 0. This is what we need. We are doing a transformation that will bring galaxy B to rest.

After we add$$$v$ to the speed of the galaxy Z, which was moving with speed $$$v$ left we get speed 2 $$$v$left. When we add$$$v$ to galaxy y, we get speed 3 $$$v$left. When adding$$$v$ to the speed of the galaxy C, we get speed for it $$$v$to the right. This leads us to the bottom picture.

If we look from the point of view of Galaxy B, then the neighboring galaxies move away from it at a speed$$$v$ . The following galaxies are removed at a speed of 2 $$$v$ and so on. . , , , .

- , . . . , . .

, - , . .

: , , ?

: . , , -, , , .

However, there is no way to mark the space. According to the principle of relativity, it is impossible to say whether you are moving relative to space or not. It makes no sense to talk about movement relative to space. It also does not make sense to talk about the movement of space relative to you.

Therefore, both points of view are correct. However, in some cases, for example, in the case of a closed universe, if you look at the universe globally, you can ask yourself: does the volume of the closed universe increase during expansion. In this case, the answer is yes, the volume does increase.

Therefore, we will assume that the universe itself is expanding. But with local observations, there is no difference between the expansion of the universe and the statement that galaxies simply move in space.

STUDENT: Why don't the galaxies themselves expand?

TEACHER: Soon after the Big Bang, the universe was filled with an almost completely homogeneous gas, which simply expanded evenly. But the gas was not completely homogeneous. Its density had tiny fluctuations. We see similar fluctuations in the cosmic background radiation today, which were caused by gas density fluctuations in the early universe.

These vibrations eventually turned into galaxies because they are gravitationally unstable. Wherever there is a small excess mass, a slightly stronger gravitational field is created. It attracts even more substance to itself, which creates an even stronger gravitational field. As a result, this almost uniform distribution of gas with small density deviations equal to one hundred thousandth turns into huge clots of matter in the form of galaxies.

Gravity, forming a galaxy, overpowers the expansion of the universe. The substance that forms the galaxy expands in the early universe. But the gravitational pull of the galaxy draws it back. Thus, the galaxy reaches its maximum size, then begins to decrease and reaches equilibrium, where the rotational movement compensates for gravity and determines its final size.

Scale factor and associated coordinate system

The picture shows the expansion of the universe. Small spots depict galaxies. The physical distance between a pair of galaxies is small in the left figure and much more in the right one. A more convenient way to describe a uniformly expanding system is to introduce a coordinate system that expands with it. We call these coordinates divisions (in English - notch (notch, label)).

The divisions are artificial coordinates, you can consider them as marks on the map. With a uniform expansion, we can take any of these drawings and take it as a map of our region of the universe. Then we can go to any other drawing by simply converting the units on the map to physical distances with another scale factor.

, , . . , 1 7 , , 1 8 , 1 9 .

, , . . , , .

Galaxies have approximately constant coordinates in the associated coordinate system. The scale factor shows what the physical distance of a unit of the accompanying distance is, and increases with time. In the following lectures, we will use the accompanying coordinate system to describe the expanding universe.

So physical distance$$$l_p$ (p from English phisical - physical) between any two points on the map and the time-dependent scale factor $$$a(t)$ multiplied by the accompanying distance $$$l_c$ (c from English comoving coordinates - accompanying coordinates).

$$

$$l_p(t) = a(t) \cdot l_c$$

By physical distance, I mean distance in the real world. If we are talking about Massachusetts, then this is the distance in kilometers between real physical objects.
For collateral distances, I'm going to use a definition that is slightly different from what is often used. In most books, the accompanying distance, like the physical distance, is measured in ordinary units of length, meters. Therefore, the scale factor is dimensionless. It simply shows how many times you need to stretch the map so that it matches the actual physical distances.

, , , , , , . , , , , , .

, . - , . . — , . , , .

It turns out that the scale factor is measured in meters per division instead of being dimensionless. The main advantage of this is that when you finish your calculations, the answer should not contain any divisions, since you are calculating something real. Thus, there is a good check of the dimension, that the divisions should disappear from any calculation of a physical quantity.

Next, I want to show that this ratio leads to the Hubble law and to understand what the Hubble constant is equal to when the scale factor changes. This is a fairly simple calculation. Physical distance to any object$$$l_p$ is given by the formula $$$l_p(t) = a(t) \cdot l_c$and we want to know what his speed is. His speed$$$v_p$ by definition, just equal to the time derivative of $$$l_p(t)$ :

$$

$$v_p = \frac d{dt}l_p(t) = \frac d{dt} (a \cdot l_c) = \frac {da}{dt} \cdot l_c$$

insofar as $$$l_$is constant. On average, our galaxies are resting in the associated coordinate system.

You can rewrite this equation in a slightly more useful way by dividing and multiplying by$$$a$ :

$$

$$v_p = \frac {da}{dt} \cdot l_c = (\frac 1{a} \frac {da}{dt}) \cdot a\cdot l_c = \frac 1{a} \frac {da}{dt} \cdot l_p(t)$$

The advantage of multiplication and division is that $$$a(t) \cdot l_c$ just equal $$$l_p$, physical distance. It turns out that the speed of any remote object is equal to$$$\frac 1a \frac {da}{dt}$multiplied by the distance to this object. This is the Hubble law. In this case, the Hubble constant, which itself will be a function of time, is equal to:

$$

$$H(t) = \frac 1{a(t)} \frac {da(t)}{dt}$$

If we know how to change $$$a$depending on the time, we know how the Hubble constant changes. Hubble's constant is completely determined by the function.$$$a(t)$ . We can also check the dimensions of which I spoke. $$$a$ , , , . .

. $$$a$ .The original scale factor was introduced by Alexander Friedman, who first came up with an equation describing the expansion of the Universe in the early 1920s. He used the letter R. Lemaitre also used the letter R. It seems to me that Einstein also probably used R. Closer by now, Steve Weinberg wrote a book on gravity and cosmology, which still used the letter R. It was the last major work in which R was used for the scale factor.

The disadvantage of using the letter R is that in R in general theory of relativity it also means another concept. This is the standard symbol for the so-called scalar curvature. Therefore, to avoid confusion between these two quantities, nowadays almost everyone designates the scale factor as$$$a$ .

Light spread

If we want to study our expanding universe, we need to understand how light rays propagate through it. It is quite simple. Let be $$$x$ - this is the accompanying coordinate, which is measured in divisions, and there is a light beam moving in the direction $$$x$ . I can describe how such a light beam moves, if I can write a formula for $$$dx/dt$, that is, how fast the light beam moves in the associated coordinate system.

The basic principle that we will use is that the light always moves at the speed of light.$$$c$ . But $$$c$ - this is the physical speed of light, the speed measured in meters per second. BUT $$$dx/dt$- this is the speed measured in divisions per second, because our associated coordinate system is marked not in meters, but in divisions. This is very important, because the ratio of meters and divisions is constantly changing, and we want to measure the values ​​in divisions so that we get a good picture of the description of the universe using the accompanying coordinates, with which we can work.

Therefore, we want to know what is$$$dx/dt$ , but it’s just a unit conversion problem. $$$dx/dt$- the speed of light in ticks per second. We know the speed of light in meters per second, which is equal to$$$c$ .Thus, to convert the meters into divisions you just need to divide by the scale factor. Again, it is convenient to measure the accompanying length in divisions, because we can check which units we get.

$$

$$\frac {dx}{dt} = \frac c{a(t)}$$

You can make sure that everything is correct by checking our dimensions. To indicate units of measurement, I will use square brackets. So, we will check which units are obtained if$$ divide by $$$a(t)$ . This is, of course, a trivial problem, but we will make sure that we get the right answer.

$$$c$ Of course, measured in meters per second. $$$a(t)$as we said, measured in meters per division. Meters are reduced, and we get divisions per second.

$$

$$[\frac c{a(t)}] = \frac {/}{/} = \frac {}$$

I said that we should never get divisions for physical quantities. But the answer is not a physical quantity. This is the speed of light in related coordinates and depends on which coordinates we have chosen. Therefore, there should be divisions per second, because$$$x$ measured in divisions a $$$t$measured in seconds. So we put$$$a(t)$in the right place. It should be in the denominator, not in the numerator.

STUDENT: Why do we not take into account in calculations that when the universe expands, the light source moves away from the observer?

TEACHER: The point is that the special theory of relativity says that all inertial observers are equivalent and that the speed of light does not depend on the speed of the source that produced the light beam. If I am at rest with respect to the accompanying coordinate system, then we can assume that I am an inertial observer. If the light beam flies past me, then for me its speed is equal to c, regardless of where this beam was fired, regardless of what happened in the past.

, , . , . , , , . , . .

, $$$a(t)$ . , . , . .

Now I want to talk a little about clock synchronization in the cosmological associated coordinate system. In the special theory of relativity, as you know, it is difficult to talk about synchronizing clocks over long distances. Clock synchronization depends on the speed of the observer. This is one of the principles of the special theory of relativity, which I talked about in the last lecture.

In the special theory of relativity there is no universal way to synchronize clocks. You can synchronize the clocks for one observer, but then they will not be synchronized for another observer moving in relation to the first one. In our case, it seems still more difficult. The clocks, fixed in the accompanying coordinate system, move with the flying galaxies. All these clocks move relative to each other according to the Hubble law.

The idea of ​​synchronizing such a clock seems overwhelming. It turns out, however, that we can synchronize such a clock, and we can introduce the concept of so-called cosmological time, that is, time, which would be the same on all these clocks. I consider clocks that are fixed with respect to local galaxies. In other words, the clock is fixed with respect to the accompanying, expanding coordinate system.

Our basic assumption, which simplifies everything, is that the universe we are considering is homogeneous. This means that what I see does not depend on where I am. If I lived in some galaxy, got a stopwatch and noticed how much time passed between changing the Hubble constant from one value to another, I would get exactly the same time interval as in any other galaxy. Otherwise, the universe would not be homogeneous. Uniformity means everyone sees the same thing.

Thus, we all, no matter where we live in such a universe, have a common history. The only thing we do not know yet is how to synchronize our watches initially. So that the time on my watch matches the time on your watch. But if we can send signals to each other, we could just agree - let's set our clock to zero when the Hubble constant is equal to, for example, 500 (km / s) / Mpc. And then we will have a clear synchronization.

As soon as we synchronize our clocks in this way, for each of us the Hubble constant changes over time in the same way, according to the principle of homogeneity. When measuring time intervals, we get the same result. Now we need to measure only time intervals, because we have agreed that all our clocks are set to the same time at a certain value of the Hubble constant.

One may wonder what options we have for synchronizing clocks. I mentioned the Hubble constant. This is certainly one of the parameters that can in principle be used to synchronize clocks in our model of the universe.

Can we use the scale factor itself to synchronize time? No, we can not, because of the ambiguity of division. I have no way to compare my division with yours. We can compare physical distances because they are related to physical properties. For example, the size of a hydrogen atom has a certain physical size, regardless of where it is located in our universe.

We could use hydrogen atoms to define a meter, and we would all use the same meters. We could use meters to determine the units of time — how long the light travels one meter. Thus, we can agree on meters and seconds, because they are associated with physical phenomena that are the same everywhere in our homogeneous universe. But with divisions it is not. Everyone can have their own division. This is just the size of the map he draws.

Thus, we cannot compare the scale factors and agree that we will set our clocks for a certain time, when both our scale factors have a certain value. We will get different synchronization depending on how we chose the division. Thus, the scale factor cannot serve as a synchronization mechanism, in contrast to the Hubble constant.

If we recall the cosmic background radiation, then it has a temperature that decreases as the universe expands. Therefore, it can also be used to synchronize the clock.

I want to make one interesting note. For our universe, the Hubble constant varies with time, the background radiation temperature varies with time. No problem using them for synchronization. But if we consider other mathematical models of the universe, we can imagine a universe where the Hubble coefficient is constant. In fact, such models were studied soon after the appearance of the general theory of relativity. This is the so-called de Sitter space. Something like this happens during inflation, so we will talk about de Sitter space later.

In the de Sitter space, the Hubble coefficient is absolutely constant, so at least one of the mechanisms that I mentioned for clock synchronization disappears. Also, in the pure space of de Sitter there is no cosmic microwave background radiation, so this mechanism also disappears. Is there anything else left? It turns out no. There is no way to synchronize clocks in de Sitter space. It can be shown that if you synchronize the clock in de Sitter space in any way, then you can do the transformation that will make the clock not synchronized. At the same time the space will be the same as before.

Thus, the concept of synchronization is not so simple. It depends on whether the Hubble constant changes over time. In the case of our real universe, it really changes.