Research the universe is flat in simple terms. Flat universe - infinite expansion, zero curvature

Doctor of Physical and Mathematical Sciences A. MADERA.

What do a piece of paper, a table surface, a donut and a mug have in common?

Two-dimensional analogues of Euclidean, spherical and hyperbolic geometries.

A Möbius strip with a point a on its surface, a normal to it and a small circle with a given direction v.

A flat sheet of paper can be glued into a cylinder and, by connecting its ends, you can get a torus.

A torus with one handle is homeomorphic to a sphere with two handles - their topology is the same.

If you cut out this figure and glue a cube out of it, it will become clear what a three-dimensional torus looks like, endlessly repeating copies of the green “worm” sitting in its center.

A three-dimensional torus can be glued together from a cube, just as a two-dimensional torus can be glued together from a square. Multi-colored “worms” traveling inside it clearly demonstrate which faces of the cube are glued together.

The cube, the fundamental region of a three-dimensional torus, is cut into thin vertical layers that, when glued together, form a ring of two-dimensional tori.

If two faces of the original cube are glued together with a 180-degree rotation, a 1/2-rotated cubic space is formed.

Rotating two faces 90 degrees gives a 1/4-rotated cubic space. Try these drawings and similar drawings on page 88 as inverted stereo pairs. “Worms” on non-rotated edges will gain volume.

If we take a hexagonal prism as a fundamental area, glue each of its faces directly to the opposite one, and rotate the hexagonal ends by 120 degrees, we get a 1/3-rotated hexagonal prismatic space.

Rotating the hexagonal face 60 degrees before gluing produces a 1/6-rotated hexagonal prismatic space.

Double cubic space.

Plate space occurs when the top and bottom sides of an infinite plate are glued together.

Tubular spaces - straight (A) and rotated (B), in which one of the surfaces is glued to the opposite one with a rotation of 180 degrees.

The distribution map of the microwave background radiation shows the distribution of matter density that was 300 thousand years ago (shown in color). Its analysis will make it possible to determine what topology the Universe has.

In ancient times, people believed that they lived on a vast flat surface, although covered here and there with mountains and depressions. This belief persisted for many thousands of years until Aristotle in the 4th century BC. e. I didn’t notice that a ship going out to sea disappears from sight not because as it moves away it shrinks to dimensions inaccessible to the eye. On the contrary, first the hull of the ship disappears, then the sails and, finally, the masts. This led him to the conclusion that the Earth must be round.

Over the past millennia, many discoveries have been made and enormous experience has been accumulated. And yet, fundamental questions still remain unanswered: is the Universe in which we live finite or infinite, and what is its shape?

Recent observations by astronomers and research by mathematicians show that the shape of our Universe should be sought among eighteen so-called three-dimensional orientable Euclidean manifolds, and only ten can lay claim to it.

OBSERVABLE UNIVERSE

Any conclusions about the possible shape of our Universe must be based on real facts obtained from astronomical observations. Without this, even the most beautiful and plausible hypotheses are doomed to failure. Therefore, let's see what the observational results say about the Universe.

First of all, we note that, no matter where we are in the Universe, around any point we can outline a sphere of arbitrary size containing the space of the Universe inside. This somewhat artificial construction tells cosmologists that the space of the Universe is a three-dimensional manifold (3-manifold).

The question immediately arises: what kind of diversity represents our Universe? Mathematicians have long established that there are so many of them that a complete list still does not exist. Long-term observations have shown that the Universe has a number of physical properties that sharply reduce the number of possible candidates for its shape. And one of the main properties of the topology of the Universe is its curvature.

According to the concept accepted today, approximately 300 thousand years after the Big Bang, the temperature of the Universe dropped to a level sufficient for electrons and protons to combine into the first atoms (see “Science and Life” Nos. 11, 12, 1996). When this happened, radiation that had initially been scattered by charged particles was suddenly able to pass unhindered through the expanding Universe. This radiation, now known as the cosmic microwave background, or relict radiation, is surprisingly uniform and reveals only very weak deviations (fluctuations) of intensity from the average value (see Science and Life No. 12, 1993). Such homogeneity can only exist in the Universe, the curvature of which is constant everywhere.

The constancy of curvature means that the space of the Universe has one of three possible geometries: flat Euclidean spherical with positive curvature or hyperbolic with negative. These geometries have completely different properties. For example, in Euclidean geometry, the sum of the angles of a triangle is exactly 180 degrees. This is not the case in spherical and hyperbolic geometries. If you take three points on a sphere and draw straight lines between them, then the sum of the angles between them will be more than 180 degrees (up to 360). In hyperbolic geometry, this sum is less than 180 degrees. There are other fundamental differences.

So which geometry should we choose for our Universe: Euclidean, spherical or hyperbolic?

The German mathematician Carl Friedrich Gauss understood in the first half of the 19th century that the real space of the surrounding world could be non-Euclidean. Carrying out many years of geodetic work in the Kingdom of Hanover, Gauss set out to explore the geometric properties of physical space using direct measurements. To do this, he chose three mountain peaks distant from one another - Hohenhagen, Inselberg and Brocken. Standing on one of these peaks, he directed the sun's rays reflected by the mirrors onto the other two and measured the angles between the sides of a huge triangle of light. Thus, he tried to answer the question: are the trajectories of light rays passing over the spherical space of the Earth bent? (By the way, around the same time, the Russian mathematician, rector of Kazan University Nikolai Ivanovich Lobachevsky proposed to experimentally study the question of the geometry of physical space using a star triangle.) If Gauss had discovered that the sum of the angles of the light triangle differs from 180 degrees, then the conclusion would have followed that the sides of the triangle are curved and real physical space is non-Euclidean. However, within the limits of measurement error, the sum of the angles of the “Brocken - Hohenhagen - Inselberg test triangle” was exactly 180 degrees.

So, on a small (by astronomical standards) scale, the Universe appears as Euclidean (although, of course, it is impossible to extrapolate Gauss’s conclusions to the entire Universe).

Recent studies using high-altitude balloons flown over Antarctica also support this conclusion. When measuring the angular power spectrum of the CMB, a peak was detected, which the researchers believe can only be explained by the existence of cold black matter - relatively large, slowly moving objects - precisely in the Euclidean Universe. Other studies also support this conclusion, which sharply reduces the number of likely candidates for the possible shape of the Universe.

Back in the thirties of the 20th century, mathematicians proved that there are only 18 different Euclidean three-dimensional manifolds and, therefore, only 18 possible forms of the Universe instead of an infinite number. Understanding the properties of these manifolds helps to experimentally determine the true shape of the Universe, since a targeted search is always more effective than a blind search.

However, the number of possible forms of the Universe can be further reduced. Indeed, among the 18 Euclidean 3-manifolds, there are 10 orientable and 8 non-orientable. Let us explain what the concept of orientability is. To do this, consider an interesting two-dimensional surface - the Möbius strip. It can be obtained from a rectangular strip of paper, twisted once and glued at the ends. Now let’s take a point on the Möbius strip A, draw a normal (perpendicular) to it, and around the normal we draw a small circle with a counterclockwise direction when viewed from the end of the normal. Let's start moving the point along with the normal and the directed circle along the Möbius strip. When the point goes around the entire sheet and returns to its original position (visually it will be on the other side of the sheet, but in geometry the surface has no thickness), the direction of the normal will change to the opposite, and the direction of the circle will change to the opposite. Such trajectories are called orientation-reversing paths. And surfaces that have them are called non-orientable or one-sided. Surfaces on which there are no closed paths reversing the orientation, for example, a sphere, a torus and an untwisted ribbon, are called orientable or two-sided. Note by the way that the Möbius strip is a Euclidean non-orientable two-dimensional manifold.

If we assume that our Universe is a non-orientable manifold, then physically this would mean the following. If we fly from the Earth along a closed loop that reverses the orientation, then, of course, we will return home, but we will find ourselves in a mirror copy of the Earth. We will not notice any changes in ourselves, but in relation to us, the rest of the inhabitants of the Earth will have a heart on the right, all the clocks will go counterclockwise, and the texts will appear in a mirror image.

It is unlikely that we live in such a world. Cosmologists believe that if our Universe were non-orientable, then energy would be emitted from the boundary zones in which matter and antimatter interact. However, nothing like this has ever been observed, although theoretically it can be assumed that such zones exist outside the region of the Universe accessible to our view. Therefore, it is reasonable to exclude eight non-orientable manifolds from consideration and limit the possible forms of our Universe to ten orientable Euclidean three-dimensional manifolds.

POSSIBLE FORMS OF THE UNIVERSE

Three-dimensional manifolds in four-dimensional space are extremely difficult to visualize. However, we can try to imagine their structure if we apply the approach used in topology to visualize two-dimensional manifolds (2-manifolds) in our three-dimensional space. All objects in it are considered to be made of some kind of durable elastic material like rubber, allowing any stretching and curvature, but without tears, folds and gluing. In topology, figures that can be transformed from one to another using such deformations are called homeomorphic; they have the same internal geometry. Therefore, from a topological point of view, a donut (torus) and an ordinary cup with a handle are one and the same. But it is impossible to transform a soccer ball into a donut. These surfaces are topologically different, that is, they have different internal geometric properties. However, if you cut a round hole on a sphere and attach one handle to it, then the resulting figure will already be homeomorphic to a torus.

There are many surfaces that are topologically distinct from the torus and the sphere. For example, by adding a handle to the torus, similar to the one we see on the cup, we get a new hole, and therefore a new figure. A torus with a handle will be homeomorphic to a pretzel-shaped figure, which in turn is homeomorphic to a sphere with two handles. The addition of each new handle creates another hole and therefore a different surface. In this way you can get an infinite number of them.

All such surfaces are called two-dimensional manifolds or simply 2-manifolds. This means that a circle of arbitrary radius can be drawn around any point. On the surface of the Earth you can draw a circle containing its points. If we see only such a picture, it is reasonable to assume that it represents an infinite plane, a sphere, a torus, or indeed any other surface from an infinite number of tori or spheres with a varying number of handles.

These topological shapes can be quite difficult to understand. And in order to imagine them easier and more clearly, let’s glue a cylinder from a square sheet of paper, connecting its left and right sides. The square in this case is called the fundamental area for the torus. If you now mentally glue the bases of the cylinder together (the material of the cylinder is elastic), you will get a torus.

Let's imagine that there is some two-dimensional creature, say an insect, whose movement along the surface of the torus needs to be studied. This is not easy to do, and it is much more convenient to observe its movement in a square - a space with the same topology. This technique has two advantages. Firstly, it allows you to clearly see the path of an insect in three-dimensional space, following its movement in two-dimensional space, and secondly, it allows you to remain within the framework of well-developed Euclidean geometry on a plane. Euclidean geometry contains a postulate about parallel lines: for any straight line and a point outside it, there is a unique straight line parallel to the first and passing through this point. In addition, the sum of the angles of a plane triangle is exactly 180 degrees. But since the square is described by Euclidean geometry, we can extend it to the torus and claim that the torus is a Euclidean 2-manifold.

The indistinguishability of internal geometries for a variety of surfaces is associated with their important topological characteristic called developability. Thus, the surfaces of a cylinder and a cone look completely different, but nevertheless their geometries are absolutely the same. Both of them can be deployed in a plane without changing the lengths of the segments and angles between them, therefore Euclidean geometry is valid for them. The same applies to the torus, since it is a surface that develops into a square. Such surfaces are called isometric.

Countless numbers of tori can be formed from other flat figures, for example from various parallelograms or hexagons, by gluing their opposite edges. However, not every quadrilateral is suitable for this: the lengths of its glued sides must be the same. This requirement is necessary to avoid, when gluing, extensions or compressions of the edges of the area, which violate the Euclidean geometry of the surface.

Now let's move on to varieties of higher dimensions.

REPRESENTATION OF POSSIBLE FORMS OF THE UNIVERSE

Let's try to imagine the possible forms of our Universe, which, as we have already seen, must be sought among ten orientable Euclidean three-dimensional manifolds.

To represent a Euclidean 3-manifold, we apply the method used above for two-dimensional manifolds. There we used a square as the fundamental region of the torus, and to represent a three-dimensional manifold we will take three-dimensional objects.

Let's take a cube instead of a square and, just as we glued the opposite edges of the square, glue together the opposite faces of the cube at all their points.

The resulting three-dimensional torus is a Euclidean 3-manifold. If we somehow ended up in it and looked forward, we would see the back of our heads, as well as copies of ourselves in each face of the cube - in front, behind, left, right, above and below. Behind them we would see an infinite number of other copies, just as if we were in a room where the walls, floor and ceiling are covered with mirrors. But the images in a three-dimensional torus will be straight, not mirrored.

It is important to note the circular nature of this and many other manifolds. If the Universe really had this shape, then if we left Earth and flew without any change in course, we would eventually return home. Something similar is observed on Earth: moving west along the equator, we will sooner or later return to our starting point from the east.

By cutting the cube into thin vertical layers, we get a set of squares. The opposite edges of these squares must be glued together because they make up the opposite faces of the cube. So a three-dimensional torus turns out to be a ring consisting of two-dimensional tori. Recall that the front and back squares are also glued together and serve as the faces of the cube. Topologists denote such a manifold as T 2 xS 1 , where T 2 means a two-dimensional torus and S 1 means a ring. This is an example of a bundle, or bundle, of tori.

Three-dimensional tori can be obtained not only using a cube. Just as a parallelogram forms a 2-torus, by gluing together opposite faces of a parallelepiped (a three-dimensional body bounded by parallelograms), we will create a 3-torus. From different parallelepipeds spaces are formed with different closed paths and angles between them.

These and all other finite manifolds are very simply included in the picture of the expanding Universe. If the fundamental area of ​​diversity is constantly expanding, the space formed by it will also expand. Each point in expanding space moves further and further away from the others, which exactly corresponds to the cosmological model. However, it must be taken into account that points near one face will always be adjacent to points on the opposite face, since, regardless of the size of the fundamental region, the opposite faces are glued together.

The next three-dimensional manifold, similar to a three-dimensional torus, is called 1/2 - rotated cubic space. In this space, the fundamental area is again the cube or parallelepiped. Four edges are glued as usual, and the remaining two, front and back, are glued with a 180-degree rotation: the top of the front edge is glued to the bottom of the back. If we found ourselves in such diversity and looked at one of these faces, we would see our own copy, but turned upside down, followed by an ordinary copy, and so on ad infinitum. Like a three-dimensional torus, the fundamental region of a 1/2-rotated cubic space can be sliced ​​into thin vertical layers so that when glued together, the result is again a bundle of two-dimensional tori, except this time the front and back tori are glued together with a 180-degree rotation .

A 1/4-rotated cubic space is the same as the previous one, but rotated 90 degrees. However, since the rotation is only a quarter, it cannot be obtained from any parallelepiped - its front and back parts must be squares to avoid curvature and skew of the fundamental area. In the front face of the cube, we would see another one behind our copy, rotated 90 degrees relative to it.

A 1/3-rotated hexagonal prismatic space uses a hexagonal prism rather than a cube as its fundamental region. To obtain it, you need to glue each face, which is a parallelogram, with its opposite face, and two hexagonal faces with a rotation of 120 degrees. Each hexagonal layer of this manifold is a torus, and thus the space is also a bundle of tori. In all hexagonal faces we would see copies rotated 120 degrees relative to the previous one, and copies in parallelogram faces are straight.

The 1/6-rotated hexagonal prismatic space is constructed similarly to the previous one, but with the difference that the front hexagonal face is glued to the back with a 60-degree rotation. As before, in the resulting bundle of tori the remaining faces - parallelograms - are glued directly to one another.

Double cubic space is radically different from previous manifolds. This finite space is no longer a bundle of tori and has an unusual gluing structure. Double cube space, however, uses a simple fundamental area, which is two cubes stacked on top of each other. When gluing, not all faces are directly connected: the top front and back faces are glued to the faces directly below them. In this space, we would see ourselves in a kind of perspective - the soles of our feet would be right in front of our eyes.

This ends the list of finite orientable Euclidean three-dimensional, so-called compact manifolds. It is likely that among them we need to look for the shape of our Universe.

Many cosmologists believe that the Universe is finite: it is difficult to imagine the physical mechanism for the emergence of an infinite Universe. Nevertheless, we will consider the four remaining orientable non-compact Euclidean three-dimensional manifolds until real data are obtained that exclude their existence.

The first and simplest infinite three-dimensional manifold is Euclidean space, which is studied in high school (it is denoted R 3). In this space, the three axes of Cartesian coordinates extend to infinity. In it we do not see any copies of ourselves, neither straight, nor rotated, nor inverted.

The next manifold is the so-called plate space, the fundamental region of which is an infinite plate. The upper part of the plate, which is an infinite plane, is glued directly to its lower part, also an infinite plane. These planes must be parallel to one another, but can be arbitrarily shifted when gluing, which is unimportant, given their infinity. In topology, this manifold is written as R 2 xS 1, where R 2 denotes a plane and S 1 a ring.

The last two 3-manifolds use infinitely long tubes as fundamental domains. The tubes have four sides, their cross-sections are parallelograms, they have neither top nor bottom - their four sides extend indefinitely. As before, the nature of the gluing of the fundamental domain determines the shape of the manifold.

The tubular space is formed by gluing together both pairs of opposite sides. After gluing, the original parallelogram-shaped section becomes a two-dimensional torus. In topology, this space is written as the product T 2 xR 1.

By rotating one of the bonded surfaces of the tubular space by 180 degrees, we obtain a rotated tubular space. This rotation, taking into account the infinite length of the tube, gives it unusual characteristics. For example, two points located very far from one another, at different ends of the fundamental region, after gluing will be nearby.

What is the shape of our Universe after all?

In order to choose one of the above ten Euclidean 3-manifolds as the form of our Universe, additional data from astronomical observations is needed.

The easiest way would be to find copies of our Galaxy in the night sky. Having discovered them, we will be able to establish the nature of the gluing of the fundamental region of the Universe. If it turns out that the Universe is a 1/4-rotated cubic space, then straight copies of our Galaxy will be visible from four sides, and rotated by 90 degrees from the remaining two. However, despite its apparent simplicity, this method is of little use for establishing the shape of the Universe.

Light travels at a finite speed, so when we observe the Universe, we are essentially looking into the past. Even if we one day discover an image of our Galaxy, we will not be able to recognize it, because in its “young years” it looked completely different. It is too difficult to recognize a copy of ours from the huge number of galaxies.

At the beginning of the article it was said that the Universe has constant curvature. The homogeneity of the cosmic microwave background radiation directly indicates this. However, it has slight spatial variations of about 10 -5 kelvin, indicating that there were minor fluctuations in the density of matter in the early Universe. As the expanding Universe cooled, the matter in these regions eventually created galaxies, stars and planets. The map of microwave radiation allows you to look into the past, to the times of initial irregularities, to see the outlines of the Universe, which was then a thousand times smaller. To appreciate the meaning of this map, consider a hypothetical example: the Universe in the form of a two-dimensional torus.

In the three-dimensional Universe, we observe the sky in all directions, that is, within a sphere. Two-dimensional inhabitants of a two-dimensional Universe would be able to observe it only within a circle. If this circle were smaller than the fundamental region of their Universe, they could get no indication of its shape. If, however, the circle of vision of two-dimensional creatures is larger than the fundamental region, they would be able to see intersections and even repetitions of patterns in the Universe and try to find points with the same temperatures that correspond to the same region. If there were enough such points in their vision circle, they could conclude that they live in a torus Universe.

Even though we live in a three-dimensional universe and see a spherical region, we face the same problem as two-dimensional creatures. If our sphere of vision is smaller than the fundamental region of the Universe 300,000 years ago, we will not see anything unusual. Otherwise, the sphere will intersect it in circles. By finding two circles that have the same variations in microwave radiation, cosmologists can compare their orientations. If the circles are arranged crosswise, this will mean there is gluing, but without rotation. Some of them, however, can be combined according to a quarter or half turn. If enough of these circles can be discovered, the mystery of the fundamental region of the Universe and its gluing together will be revealed.

However, until an accurate map of microwave radiation appears, cosmologists will not be able to draw any conclusions. In 1989, researchers from NASA attempted to create a map of the cosmic microwave background radiation. However, the angular resolution of the satellite was about 10 degrees, which did not allow accurate measurements to be made that would satisfy cosmologists. In the spring of 2002, NASA made a second attempt and launched a probe that mapped temperature fluctuations with an angular resolution of about 0.2 degrees. In 2007, the European Space Agency plans to use the Planck satellite, which has an angular resolution of 5 arc seconds.

If the launches are successful, then within four to ten years accurate maps of CMB fluctuations will be obtained. And if the size of the sphere of our vision turns out to be large enough, and the measurements are sufficiently accurate and reliable, we will finally know what shape our Universe has.

Based on materials from the magazines "American Scientist" and "Popular Science".

In ancient times, people thought that the earth was flat and stood on three whales, then it turned out that our ecumene is round and if you sail all the time to the west, then after a while you will return to your starting point from the east. Views of the Universe changed in a similar way. At one time, Newton believed that space was flat and infinite. Einstein allowed our World to be not only limitless and crooked, but also closed. The latest data obtained during the study of cosmic microwave background radiation indicate that the Universe may well be closed on itself. It turns out that if you fly away from the earth all the time, then at some point you will begin to approach it and eventually return back, going around the entire Universe and traveling around the world, just as one of Magellan’s ships, having circled the entire globe, sailed to the Spanish port of Sanlúcar de Barrameda.

The hypothesis that our Universe was born as a result of the Big Bang is now considered generally accepted. The matter was initially very hot, dense, and expanded rapidly. Then the temperature of the Universe dropped to several thousand degrees. The substance at that moment consisted of electrons, protons and alpha particles (helium nuclei), that is, it was a highly ionized gas - plasma, opaque to light and any electromagnetic waves. The recombination (combination) of nuclei and electrons that began at this time, that is, the formation of neutral hydrogen and helium atoms, radically changed the optical properties of the Universe. It became transparent to most electromagnetic waves.

Thus, by studying light and radio waves, one can see only what happened after recombination, and everything that happened before is covered by a kind of “wall of fire” of ionized matter. We can look much deeper into the history of the Universe only if we learn to register relic neutrinos, for which hot matter became transparent much earlier, and primary gravitational waves, for which matter of any density is no barrier, but this is a matter of the future, and far from it. the closest one.

Since the formation of neutral atoms, our Universe has expanded approximately 1,000 times, and the radiation from the recombination era is today observed on Earth as a relic microwave background with a temperature of about three degrees Kelvin. This background, first discovered in 1965 during tests of a large radio antenna, is virtually the same in all directions. According to modern data, there are a hundred million times more relict photons than atoms, so our world is simply bathed in streams of strongly reddened light emitted in the very first minutes of the life of the Universe.

Classical topology of space

On scales larger than 100 megaparsecs, the part of the Universe visible to us is quite homogeneous. All dense clumps of matter - galaxies, their clusters and superclusters - are observed only at shorter distances. Moreover, the Universe is also isotropic, that is, its properties are the same along any direction. These experimental facts underlie all classical cosmological models, which assume spherical symmetry and spatial homogeneity of the distribution of matter.

Classical cosmological solutions to the equations of Einstein's general theory of relativity (GTR), which were found in 1922 by Alexander Friedman, have the simplest topology. Their spatial sections resemble planes (for infinite solutions) or spheres (for limited solutions). But such universes, it turns out, have an alternative: a universe of finite volume that has no edges or boundaries, closed on itself.

The first solutions found by Friedman described universes filled with only one type of matter. Different pictures arose due to differences in the average density of matter: if it exceeded a critical level, a closed universe with positive spatial curvature, finite dimensions and lifetime was obtained. Its expansion gradually slowed down, stopped and was replaced by compression to a point. The Universe with a density below the critical one had a negative curvature and expanded indefinitely, the rate of its inflation tended to some constant value. This model is called open. The flat Universe, an intermediate case with density exactly equal to the critical one, is infinite and its instantaneous spatial sections are flat Euclidean space with zero curvature. A flat one, just like an open one, expands indefinitely, but the speed of its expansion tends to zero. Later, more complex models were invented in which a homogeneous and isotropic universe was filled with multicomponent matter that changed over time.

Modern observations show that the Universe is now expanding at an accelerating rate (see “Beyond the Horizon of Universal Events”, No. 3, 2006). This behavior is possible if space is filled with some substance (often called dark energy) with a high negative pressure, close to the energy density of this substance. This property of dark energy leads to the emergence of a kind of antigravity, which overcomes the gravitational forces of ordinary matter on large scales. The first such model (with the so-called lambda term) was proposed by Albert Einstein himself.

A special mode of expansion of the Universe arises if the pressure of this matter does not remain constant, but increases with time. In this case, the increase in size increases so quickly that the Universe becomes infinite in a finite time. Such a sharp inflation of spatial dimensions, accompanied by the destruction of all material objects, from galaxies to elementary particles, is called the Big Rip.

All these models do not assume any special topological properties of the Universe and present it as similar to our familiar space. This picture agrees well with the data that astronomers obtain using telescopes that record infrared, visible, ultraviolet and X-ray radiation. And only radio observation data, namely a detailed study of the cosmic microwave background, made scientists doubt that our world is structured so straightforwardly.

Scientists will not be able to look beyond the “wall of fire” that separates us from the events of the first thousand years of the life of our Universe. But with the help of laboratories launched into space, every year we learn more and more about what happened after the transformation of hot plasma into warm gas

Orbital radio receiver

The first results obtained by the space observatory WMAP (Wilkinson Microwave Anisotropy Probe), which measured the power of the cosmic microwave background radiation, were published in January 2003 and contained so much long-awaited information that its understanding is not completed today. Physics is usually used to explain new cosmological data: equations of state of matter, expansion laws and spectra of initial perturbations. But this time the nature of the detected angular inhomogeneity of the radiation required a completely different explanation - a geometric one. More precisely, topological.

The main goal of WMAP was to build a detailed map of the temperature of the cosmic microwave background radiation (or, as it is also called, the microwave background). WMAP is an ultra-sensitive radio receiver that simultaneously detects signals coming from two almost diametrically opposite points in the sky. The observatory was launched in June 2001 into a particularly calm and “quiet” orbit, located at the so-called Lagrangian point L2, one and a half million kilometers from Earth. This 840 kg satellite is actually in orbit around the sun, but thanks to the combined action of the gravitational fields of the Earth and the Sun, its orbital period is exactly one year, and it does not fly away from the Earth. The satellite was launched into such a distant orbit so that interference from earthly man-made activity would not interfere with the reception of cosmic microwave background radiation.

Based on the data obtained by the space radio observatory, it was possible to determine a huge number of cosmological parameters with unprecedented accuracy. Firstly, the ratio of the total density of the Universe to the critical density is 1.02±0.02 (that is, our Universe is flat or closed with very little curvature). Secondly, the Hubble constant, which characterizes the expansion of our World on large scales, 72±2 km/s/Mpc. Thirdly, the age of the Universe is 13.4 ± 0.3 billion years and the red shift corresponding to the recombination time is 1088 ± 2 (this is the average value, the thickness of the recombination boundary is significantly greater than the indicated error). The most sensational result for theorists was the angular spectrum of disturbances of the relict radiation, more precisely, the value of the second and third harmonics was too small.

Such a spectrum is constructed by representing the temperature map as a sum of various spherical harmonics (multipoles). In this case, from the general picture of disturbances, variable components are isolated that fit on the sphere an integer number of times: quadrupole 2 times, octupole 3 times, and so on. The higher the number of the spherical harmonic, the more high-frequency background oscillations it describes and the smaller the angular size of the corresponding “spots”. Theoretically, the number of spherical harmonics is infinite, but for a real observation map it is limited by the angular resolution with which the observations were made.

To correctly measure all spherical harmonics, a map of the entire celestial sphere is needed, and WMAP receives its verified version within a year. The first such not very detailed maps were obtained in 1992 in the Relic and COBE (Cosmic Background Explorer) experiments.

How is a bagel similar to a coffee cup?
There is a branch of mathematics - topology, which studies the properties of bodies that are preserved under any deformation without breaks or gluing. Imagine that the geometric body we are interested in is flexible and easily deformed. In this case, for example, a cube or a pyramid can be easily transformed into a sphere or a bottle, a torus (“donut”) into a coffee cup with a handle, but it will not be possible to turn a sphere into a cup with a handle if you do not tear and glue this easily deformable body. In order to divide a sphere into two unconnected pieces, it is enough to make one closed cut, but you can do the same with a torus only by making two cuts. Topologists simply love all sorts of exotic constructions such as a flat torus, a horned sphere or a Klein bottle, which can only be correctly depicted in a space with twice the number of dimensions. Likewise, our three-dimensional Universe, closed on itself, can be easily imagined only by living in six-dimensional space. For a while, cosmic topologists have not yet encroached, leaving it the opportunity to simply flow linearly, without being locked into anything. So the ability to work in the space of seven dimensions today is quite enough to understand how complex our dodecahedral Universe is structured.

The final CMB temperature map is built from painstaking analysis of maps showing the intensity of radio emission in five different frequency ranges

Unexpected decision

For most spherical harmonics, the experimental data obtained coincided with model calculations. Only two harmonics, quadrupole and octupole, were clearly below the level expected by theorists. Moreover, the likelihood that such large deviations could arise by chance is extremely small. Suppression of the quadrupole and octupole was noted in the COBE data. However, the maps obtained in those years had poor resolution and great noise, so discussion of this issue was postponed until better times. For what reason the amplitudes of the two largest-scale fluctuations in the intensity of the cosmic microwave background radiation turned out to be so small was completely unclear at first. It has not yet been possible to come up with a physical mechanism to suppress them, since it must act on the scale of the entire Universe we observe, making it more homogeneous, and at the same time stop working on smaller scales, allowing it to fluctuate more strongly. This is probably why they began to look for alternative paths and found a topological answer to the question that arose. The mathematical solution to the physical problem turned out to be surprisingly elegant and unexpected: it was enough to assume that the Universe is a dodecahedron closed on itself. Then the suppression of low-frequency harmonics can be explained by spatial high-frequency modulation of background radiation. This effect occurs due to repeated observation of the same region of recombining plasma through different parts of a closed dodecahedral space. It turns out that low harmonics seem to cancel themselves due to the passage of the radio signal through different facets of the Universe. In such a topological model of the world, events occurring near one of the faces of the dodecahedron turn out to be close to the opposite face, since these areas are identical and in fact are one and the same part of the Universe. Because of this, the relict light coming to Earth from diametrically opposite sides turns out to be emitted by the same region of the primary plasma. This circumstance leads to the suppression of the lower harmonics of the CMB spectrum even in a Universe only slightly larger in size than the visible event horizon.

Anisotropy map
The quadrupole mentioned in the text of the article is not the lowest spherical harmonic. In addition to it, there are a monopole (zero harmonic) and a dipole (first harmonic). The magnitude of the monopole is determined by the average temperature of the cosmic microwave background radiation, which today is 2.728 K. After subtracting it from the general background, the largest is the dipole component, which shows how much higher the temperature in one of the hemispheres of the space surrounding us is than in the other. The presence of this component is caused mainly by the movement of the Earth and the Milky Way relative to the relict background. Due to the Doppler effect, the temperature in the direction of movement increases, and in the opposite direction it decreases. This circumstance will make it possible to determine the speed of any object in relation to the cosmic microwave background radiation and thus introduce the long-awaited absolute coordinate system, locally at rest in relation to the entire Universe.

The magnitude of dipole anisotropy associated with the Earth's motion is 3.353*10-3 K. This corresponds to the motion of the Sun relative to the CMB background at a speed of about 400 km/s. At the same time, we “fly” in the direction of the border of the constellations Leo and Chalice, and “fly away” from the constellation Aquarius. Our Galaxy, together with the local group of galaxies in which it belongs, moves relative to the relic at a speed of about 600 km/s.

All other disturbances (from the quadrupole and above) on the background map are caused by inhomogeneities in the density, temperature and velocity of matter at the recombination boundary, as well as by the radio emission of our Galaxy. After subtracting the dipole component, the total amplitude of all other deviations turns out to be only 18 * 10-6 K. To exclude the Milky Way’s own radiation (mainly concentrated in the plane of the galactic equator), observations of the microwave background are carried out in five frequency bands in the range from 22.8 GHz to 93 .5 GHz.

Combinations with a torus

The simplest body with a topology more complex than a sphere or plane is a torus. Anyone who has held a bagel in their hands can imagine it. Another more correct mathematical model of a flat torus is demonstrated by the screens of some computer games: it is a square or rectangle, the opposite sides of which are identified, and if a moving object goes down, it appears from above; crossing the left border of the screen, it appears from behind the right, and vice versa. Such a torus is the simplest example of a world with a non-trivial topology, which has a finite volume and does not have any boundaries.

In three-dimensional space, a similar procedure can be done with a cube. If we identify its opposite faces, a three-dimensional torus is formed. If you look from inside such a cube at the surrounding space, you can see an infinite world, consisting of copies of its one and only and unique (non-repeating) part, the volume of which is completely finite. In such a world there are no boundaries, but there are three distinct directions parallel to the edges of the original cube, along which periodic rows of original objects are observed. This picture is very similar to what can be seen inside a cube with mirrored walls. True, looking at any of its faces, an inhabitant of such a world will see the back of his head, and not his face, as in an earthly funhouse. A more correct model would be a room equipped with 6 television cameras and 6 flat LCD monitors, on which the image captured by the film camera located opposite is displayed. In this model, the visible world closes on itself thanks to access to another television dimension.

The picture of suppression of low-frequency harmonics described above is correct if the time it takes for light to cross the initial volume is sufficiently short, that is, if the dimensions of the initial body are small compared to cosmological scales. If the dimensions of the observable part of the Universe (the so-called horizon of the Universe) turn out to be smaller than the dimensions of the original topological volume, then the situation will be no different from what we will see in the usual infinite Einstein Universe, and no anomalies in the spectrum of the cosmic microwave background radiation will be observed.

The maximum possible spatial scale in such a cubic world is determined by the dimensions of the original body; the distance between any two bodies cannot exceed half the main diagonal of the original cube. Light coming to us from the recombination boundary can cross the original cube several times along the way, as if reflected in its mirror walls, because of this the angular structure of the radiation is distorted and low-frequency fluctuations become high-frequency. As a result, the smaller the initial volume, the stronger the suppression of lower large-scale angular fluctuations, which means that by studying the CMB, we can estimate the size of our Universe.

3D mosaics

A flat topologically complex three-dimensional Universe can be built only on the basis of cubes, parallelepipeds and hexagonal prisms. In the case of curved space, a wider class of figures has such properties. At the same time, the best angular spectra obtained in the WMAP experiment are consistent with a model of the Universe having the shape of a dodecahedron. This regular polyhedron, which has 12 pentagonal faces, resembles a soccer ball sewn from pentagonal patches. It turns out that in a space with a slight positive curvature, regular dodecahedrons can fill the entire space without holes or mutual intersections. Given a certain ratio between the size of the dodecahedron and the curvature, this requires 120 spherical dodecahedrons. Moreover, this complex structure of hundreds of “balls” can be reduced to a topologically equivalent one, consisting of just one single dodecahedron, whose opposite faces are identified, rotated by 180 degrees.

The universe formed from such a dodecahedron has a number of interesting properties: it has no preferred directions, and it describes the magnitude of the lowest angular harmonics of the CMB better than most other models. Such a picture arises only in a closed world with a ratio of the actual density of matter to the critical density of 1.013, which falls within the range of values ​​​​allowable by today's observations (1.02 ± 0.02).

For the average inhabitant of the Earth, all these topological intricacies at first glance do not have much significance. But for physicists and philosophers it’s a completely different matter. Both for the worldview as a whole and for a unified theory that explains the structure of our world, this hypothesis is of great interest. Therefore, having discovered anomalies in the spectrum of the relic, scientists began to look for other facts that could confirm or refute the proposed topological theory.

Sounding plasma
On the spectrum of CMB fluctuations, the red line indicates the predictions of the theoretical model. The gray corridor around it is the permissible deviations, and the black dots are the results of observations. Most of the data is obtained from the WMAP experiment, and only for the highest harmonics results from the CBI (balloon) and ACBAR (ground-based Antarctic) studies are added. The normalized graph of the angular spectrum of CMB fluctuations shows several maxima. These are the so-called “acoustic peaks”, or “Sakharov oscillations”. Their existence was theoretically predicted by Andrei Sakharov. These peaks are due to the Doppler effect and are caused by the movement of the plasma at the moment of recombination. The maximum amplitude of oscillations occurs within the size of the causally related region (sound horizon) at the moment of recombination. On smaller scales, plasma oscillations were weakened by photon viscosity, and on large scales the disturbances were independent of each other and were not phased. Therefore, the maximum fluctuations observed in the modern era occur at the angles at which the sound horizon is visible today, that is, the region of the primary plasma that lived a single life at the moment of recombination. The exact position of the maximum depends on the ratio of the total density of the Universe to the critical density. Observations show that the first, highest peak is located approximately at the 200th harmonic, which, according to theory, corresponds with high accuracy to a flat Euclidean Universe.

A lot of information about cosmological parameters is contained in the second and subsequent acoustic peaks. Their very existence reflects the fact that acoustic oscillations in plasma are “phased” during the recombination era. If there were no such connection, then only the first peak would be observed, and fluctuations on all smaller scales would be equally probable. But in order for such a causal relationship between oscillations on different scales to arise, these (very distant from each other) regions had to be able to interact with each other. This is precisely the situation that naturally arises in the inflationary Universe model, and the confident detection of the second and subsequent peaks in the angular spectrum of CMB fluctuations is one of the most significant confirmations of this scenario.

Observations of the cosmic microwave background radiation were carried out in the region close to the maximum of the thermal spectrum. For a temperature of 3K it is at a radio wavelength of 1mm. WMAP conducted its observations at slightly longer wavelengths: from 3 mm to 1.5 cm. This range is quite close to the maximum, and it contains lower noise from the stars of our Galaxy.

Multifaceted world

In the dodecahedral model, the event horizon and the recombination boundary lying very close to it intersect each of the 12 faces of the dodecahedron. The intersection of the recombination boundary and the original polyhedron forms 6 pairs of circles on the microwave background map, located at opposite points of the celestial sphere. The angular diameter of these circles is 70 degrees. These circles lie on opposite faces of the original dodecahedron, that is, they coincide geometrically and physically. As a result, the distribution of CMB fluctuations along each pair of circles should coincide (taking into account the rotation by 180 degrees). Based on the available data, such circles have not yet been detected.

But this phenomenon, as it turned out, is more complex. The circles will be identical and symmetrical only for an observer stationary relative to the relict background. The Earth moves relative to it at a fairly high speed, which is why a significant dipole component appears in the background radiation. In this case, the circles turn into ellipses, their sizes, location in the sky and the average temperature along the circle change. It becomes much more difficult to detect identical circles in the presence of such distortions, and the accuracy of the data available today becomes insufficient; new observations are needed that will help figure out whether they exist or not.

Multiply related inflation

Perhaps the most serious problem of all topologically complex cosmological models, and a considerable number of them have already arisen, is mainly of a theoretical nature. Today, the inflationary scenario for the evolution of the Universe is considered standard. It was proposed to explain the high homogeneity and isotropy of the observable Universe. According to him, at first the Universe that was born was quite heterogeneous. Then, during the process of inflation, when the Universe expanded according to a law close to exponential, its original size increased by many orders of magnitude. Today we see only a small part of the Big Universe, in which inhomogeneities still remain. True, they have such a large spatial extent that they are invisible within the area accessible to us. The inflationary scenario is the best developed cosmological theory so far.

For a multiconnected universe, such a sequence of events does not fit. In it, all of its unique part and some of its closest copies are available for observation. In this case, structures or processes described by scales much larger than the observed horizon cannot exist.

The directions in which cosmology will have to be developed if the multiconnectedness of our Universe is confirmed are already clear: these are non-inflationary models and so-called models with weak inflation, in which the size of the Universe increases only a few times (or tens of times) during inflation. There are no such models yet, and scientists, trying to preserve the familiar picture of the world, are actively looking for flaws in the results obtained using a space radio telescope.

Processing artifacts

One of the groups that conducted independent studies of WMAP data drew attention to the fact that the quadrupole and octupole components of the CMB have a close orientation to each other and lie in a plane almost coinciding with the galactic equator. The conclusion of this group: an error occurred when subtracting the Galactic background from the microwave background observation data and the real value of the harmonics is completely different.

WMAP observations were carried out at 5 different frequencies specifically in order to correctly separate the cosmological and local background. And the core WMAP team believes that the observations were processed correctly and rejects the proposed explanation.

The available cosmological data, published back in early 2003, were obtained after processing the results of only the first year of WMAP observations. To test the proposed hypotheses, as usual, an increase in accuracy is required. By early 2006, WMAP had been continuously observing for four years, which should be enough to double its accuracy, but the data has yet to be published. We need to wait a little, and perhaps our assumptions about the dodecahedral topology of the Universe will become completely demonstrative.

Mikhail Prokhorov, Doctor of Physical and Mathematical Sciences

The planet was once considered flat, and this seemed a completely obvious fact. Today we also look at the “shape” of the Universe as a whole.

The WMAP probe looks into space

In the case of the Universe, “flatness” implies the seemingly obvious fact that light and radiation propagate in it in a strictly straight line. Of course, the presence of matter and energy makes its own adjustments, creating distortions in the space-time continuum. But still, in a flat Universe, strictly parallel beams of light never intersect, in full accordance with the planimetric axiom.

If the Universe is curved along a positive curve (like a huge sphere), the parallel lines in it should eventually come together. In the opposite case - if the Universe resembles a giant “saddle” - the parallel lines will gradually diverge.

The question of the plane of the Universe was studied, in particular, by the space probe WMAP, about the main achievements of which we wrote in the article “Mission: in progress.” Having used it to collect data on the distribution of matter and dark energy in the young Universe, scientists analyzed them and came to the almost unanimous conclusion that it is still flat. Let us note - almost unanimously. For example, this view of things was recently challenged by a group of Oxford physicists led by Joseph Silk, who showed that the WMAP results may well have been misinterpreted.

World science faces a number of questions to which it will apparently never receive accurate answers. The age of the Universe is one of these. Up to a year, a day, a month, a minute, it will apparently never be possible to calculate it. Although...

At one time, it seemed that narrowing the estimated age to 12-15 billion years was a great achievement.

And now NASA proudly announces: the age of the Universe has been established with an error of “only” 0.2 billion years. And this age is 13.7 billion years.

In addition, it was possible to find out that the first stars began to form much earlier than expected.

How was this established?

It turns out that with the help of one single device, which appears under the name MAP - Microwave Anisotropy Probe.

It was recently renamed the Wilkinson Microwave Anisotropy Probe (WMAP) in honor of Princeton University astrophysicist David Wilkinson, who died in 2002.

The late Professor David Wilkinson, after whom the WMAP probe was named.

This probe, located at a distance of about 1.5 million kilometers from Earth, recorded the cosmic microwave background (CMB) throughout the sky for a whole year.

Ten years ago, another similar device, Cosmic Microwave Background Explorer (COBE), made the first spherical survey of the CMB.

COBE detected microscopic temperature fluctuations in the microwave background that correspond to changes in the density of matter in the young Universe.

MAP, equipped with much more sophisticated equipment, peered into the depths of space for a year, and obtained an image with a resolution 35 times better than its predecessor.

The cosmic microwave background is cosmic microwave background radiation left over from the Big Bang. These are, relatively speaking, photons remaining after the burst of light radiation that occurred as a result of the explosion, and cooled over billions of years to a microwave state. In other words, this is the oldest light in the Universe.

“Membrane” already wrote that in the fall of 2002, the Degree Angular Scale Interferometer radio telescope, located at the South Pole, discovered that cosmic background microwave radiation polarized.

A star map showing temperature variations in the cosmic microwave background.

Polarization in space was one of the key predictions of standard cosmological theory. According to it, the young Universe was filled with photons that constantly collided with protons and electrons.

The collisions polarized the light, an imprint that remained even after the charged particles formed the first neutral hydrogen atoms.

It was expected that this discovery would help explain exactly how the Universe expanded in a fraction of a second and how the first stars were formed, as well as clarify the relationship between “ordinary” and “dark” types of matter and dark energy.

The amount of dark matter and energy in the Universe plays a key role in determining the shape of the cosmos—more precisely, its geometry.

Scientists proceed from the assumption that if the density of matter and energy in the Universe is less than critical, then space is open and concave like a saddle.

If the value of the density of matter and energy coincides with the critical value, then space is flat, like a sheet of paper. If the true density is higher than what is considered critical in theory, then space should be closed and spherical. In this case, the light will always return to the original source.

A diagram showing the relationship between the forms of matter in the Universe.

The expansion theory, a kind of consequence of the Big Bang theory, predicts that the density of matter in the Universe is as close as possible to critical, which means the Universe is flat.

The readings from the MAP probe confirmed this.

Another extremely interesting circumstance was revealed: it turns out that the first stars began to appear in the Universe very quickly - just 200 million years after the Big Bang itself.

In 2002, scientists conducted a computer simulation of the formation of the most ancient stars, in which metals and other “heavy” elements were completely absent. They were formed as a result of explosions of old stars, the residual matter of which fell onto the surface of other stars and, in the process of thermonuclear fusion, formed heavier compounds.