What shape does our universe have? What shape does the Universe have? Classical topology of space

> What is the shape of the Universe?

In what form does the Universe exist?: exploration of infinite space, WMAP cosmic microwave background map, geometry of the Universe and estimated shapes with photos.

Is it even worth thinking about what shape the Universe is? What are we dealing with? Sphere? Cone? Flat? And how to determine this?

The Universe is the only place in which we exist and beyond which we cannot escape (because there are none). Thanks to physical laws, natural constants and erupting heavy metals, we managed to create life on a small rocky ball, lost in one of many galaxies.

But don't you want to know where you live? Just get the opportunity to look at everything from the outside, as we did with our native planet Earth. For you to see? Endless darkness? Lots of bubbles? Snowball? A rat maze in the hands of aliens or something else? What is the shape of the Universe?

Well, the answer is much simpler, but also stranger. People began to think about the shape of the Universe back in ancient times. And people, due to lack of information, offered some pretty wonderful things. In Hindu texts it was an egg in the shape of a man. The Greeks saw an island floating in the void. Aristotle says that the Universe has the shape of an infinite sphere or simply a turtle.

Interestingly, Albert Einstein's contributions help test each of these models. Scientists have come up with three favorite shapes: positively curved, negatively curved and flat. We understand that the Universe exists in 4 dimensions and any of the figures border on crazy Lovecraftian geometry. So use your maximum imagination and let's go!

With the positively curved version, we get a four-dimensional sphere. This variety has an end, but does not have a clear border. More precisely, two particles would cross it before returning to the start. You can even test it at home. Take a balloon and draw a straight line until it returns to the starting point.

This species fits into three dimensions and appears if there is a huge amount of energy in space. To completely bend or close, the space would have to stop expanding. This will happen if a large-scale energy reserve appears that can create an edge. Current evidence shows that expansion is a never-ending process. So this scenario is out of the question.

The negatively curved shape of the Universe is a four-dimensional saddle. It is open, without boundaries in space and time. There is little energy here, so the Universe will not stop expanding. If you send two particles along straight lines, they will never meet, but will simply diverge until they go in different directions.

If a critical amount of energy fluctuates between extremes, then after infinity the expansion will stop. This is a flat Universe. Here the two particles will travel in parallel, but will never separate or meet.

It's easy to imagine these three shapes, but there are many more options. The soccer ball is reminiscent of the idea of ​​a spherical universe. The donut is technically flat, but connected at certain points. Some believe that huge warm and cool spots speak in favor of this option. You can see the supposed shapes of the Universe in the photo.

And now we come to the pipe. This is another type of negative curvature. One end will be narrowed, and the other will be wide. In the first half, everything seemed narrow and existed in two dimensions. And in a wide one, it would be possible to travel maximum distances, but you would have to return in the opposite direction (the direction changes at the bend).

What then? What are we dealing with? Bagel? Wind instrument? A giant head of cheese? Scientists have not yet ruled out options with a pipe and a saddle.

Grumps will argue that all this is pointless and we will never know the truth. But let's not be so categorical. Planck's latest data show that our Universe is... flat! Infinitely finite, completely uncurved and with a precise critical amount of energy.

It's unthinkable that not only can we find out what the Universe looks like, but there are people who are constantly trying to find even more information. If “flat” seems boring to you, then don’t forget that we don’t have enough information yet. So it's entirely possible that we could all exist in a giant donut.

Imagine a very large ball. Although it appears three-dimensional from the outside, its surface - a sphere - is two-dimensional, because there are only two independent directions of movement along the sphere. If you were very small and lived on the surface of this ball, you could well assume that you do not live on a sphere at all, but on a large flat two-dimensional surface. But if you were to accurately measure distances on a sphere, you would understand that you live not on a flat surface, but on the surface of a large sphere ( approx. translation It’s probably better to draw an analogy with the surface of the globe).
The idea of ​​the curvature of the surface of a ball can be applied to the entire Universe. This was a huge breakthrough in Einstein's General Theory of Relativity. Space and time were combined into a single geometric unit called space-time, and this space-time had geometry, it could be twisted, just like the surface of a huge ball is curved.
When you look at the surface of a large ball as a single thing, you feel the entire space of the sphere as a whole. Mathematicians love the surface of a sphere so that this definition describes the entire sphere, and not just part of it. One of the key aspects of describing the geometry of spacetime is that we need to describe all of space and all of time. This means that we need to describe “everything” and “always” “in one bottle.” The geometry of space-time is the geometry of all space plus all time together as one mathematical unit.

What determines the geometry of space-time?

Basically, physicists work like this - they look for equations of motion whose solutions best describe the system that physicists want to describe. Einstein's equation represents classical equation of motion of space-time. It is classical because quantum effects were not taken into account when obtaining it. And thus, the geometry of space-time is interpreted as an exclusively classical concept, devoid of any quantum uncertainties. That is why it is the best approximation to the exact theory.
According to Einstein's equations, the curvature of spacetime in a given direction is directly related to the energy and momentum of everything in all spacetime that is not spacetime. In other words, Einstein's equations relate gravity to non-gravity and geometry to non-geometry. Curvature is gravity, and everything else is electrons and quarks, which make up atoms, which, in turn, make up matter, electromagnetic radiation, each particle - a carrier of interaction (except gravity) - “lives” in curved space-time and at the same time determines this curvature according to Einstein's equations.

What is the geometry of our space-time?

As just noted, a complete description of a given space-time includes not only all space, but also all the time. In other words, spacetime includes all events that have ever happened and that will ever happen.
True, now, if we are too literal in this concept, we may run into problems, because we will not be able to take into account all the smallest changes in the distribution of energy density and momentum in the Universe that have ever occurred and will still occur in the Universe. But, fortunately, the human mind is capable of operating with concepts such as abstraction And approximation Thus, we can build an abstract model that roughly describes the observable Universe quite well on large scales, say, on the scale of galaxy clusters.
But this is not enough to solve equations. It is also necessary to make certain simplifying assumptions regarding the curvature of spacetime. The first assumption we will make is that spacetime can be neatly divided into space and time. This, however, cannot always be done; for example, in some cases of spinning black holes, space and time "spin" together and thus cannot be neatly separated. However, there is no indication that our Universe can rotate in a similar way. Thus, we can easily make the assumption that space-time can be described as space changing over time.
The next important assumption coming from the Big Bang theory is that space looks the same in any direction at any point. The property of looking the same in any direction is called isotropy, and the property of looking the same at any point is called homogeneity. So we assume that our space homogeneous and isotropic. Cosmologists call this assumption maximum symmetry. This is considered to be a fairly reasonable assumption on large scales.
When solving Einstein's equations for the space-time geometry of our Universe, cosmologists consider three main types of energy that can and do bend space-time:
1. vacuum energy
2. radiation
3. ordinary substance
Radiation and ordinary matter are treated as a homogeneous gas filling the Universe, with some equation of state relating pressure to density.
Once the assumptions of homogeneity of energy sources and maximum symmetry are made, Einstein's equations can be reduced to two differential equations that can be easily solved using simple calculation methods. From the solutions we get two things: geometry of space and then how the dimensions of space change over time.

Open, closed or flat?

If at every moment of time space at every point looks the same in all directions, then such space must have constant curvature. If the curvature changes from point to point, then space will look different from different points and in different directions. Therefore, if the space is maximally symmetrical, then the curvature at all points must be the same.
This requirement somewhat narrows the possible geometries to three: space with constant positive, negative and zero curvature (flat). In the case when there is no vacuum energy (lambda term), there is only ordinary matter and radiation, curvature, in addition to everything else, also answers the question about the time of evolution:
Positive curvature: An N-dimensional space with constant positive curvature is an N-dimensional sphere. A cosmological model in which space has constant positive curvature is called closed cosmological model. In this model, space expands from zero volume at the time of the Big Bang, then at some point in time it reaches its maximum volume and begins to contract until the Big Crunch.
Zero curvature: A space with zero curvature is called flat space. Such a flat space is not compact, it extends infinitely in all directions, just as it is extended only open space. Such a Universe expands infinitely in time.
Negative curvature: An N-dimensional space with constant negative curvature is an N-dimensional pseudosphere. The only thing with which such a unique world can be more or less familiar is a hyperboloid, which is a two-dimensional hypersphere. A space with negative curvature is infinite in volume. In a space with negative curvature, it is realized open Universe. It, like a flat one, expands infinitely in time.
What determines whether the Universe will be open or closed? For a closed Universe, the total energy density must be greater than the energy density corresponding to a flat Universe, which is called critical density. Let's put it. Then in a closed universe w is greater than 1, in a flat universe w=1, and in open universe w is less than 1.
All of the above is true only in the case when only ordinary types of matter are taken into consideration - dust and radiation, and neglected vacuum energy, which may well be present. The vacuum energy density is constant, also called cosmological constant.

Where does dark matter come from?

There is a lot of different matter in the Universe, such as stars or hot gas or something else, that emits visible light or radiation at other wavelengths. And all this can either be seen with the eyes, or with the help of telescopes, or with some complex instruments. However, this is not all that is in our Universe - over the past two decades, astronomers have discovered evidence that there is a lot of invisible matter in the Universe.
For example, it turns out that visible matter in the form of stars and interstellar gas is not enough to keep galaxies gravitationally bound. Estimates of how much matter the average galaxy actually needs in order not to fly apart have led physicists and astronomers to the conclusion that most of the matter in the universe is invisible. This substance is called dark matter and it is very important for cosmology.
Since there is dark matter in the Universe, what could it be? What can it be “made” of? If it consisted of quarks, like ordinary matter, then the early Universe should have produced much more helium and deuterium than is now in our Universe. Particle physicists are of the opinion that dark matter consists of supersymmetric particles, which are very heavy, but interact very weakly with ordinary particles that are now observed at accelerators.
There is, therefore, much less visible matter in the Universe than is necessary even for a flat Universe. Therefore, if there is nothing else in the Universe, then it must be open. However, is there enough dark matter to “close” the Universe? In other words, if w B is the density of ordinary matter, and w D is the density of dark matter, then does the relation w B + w D = 1 hold? A study of the movements in galaxy clusters suggests that the total density is about 30% of the critical density, with visible matter accounting for about 5% and dark matter 25%.
But this is not the end - we still have one more source of energy in the Universe - the cosmological constant.

What about the cosmological constant?

Einstein did not like the results of his own work. According to his equations of motion, a Universe filled with ordinary matter should expand. But Einstein wanted a theory in which the Universe would always remain the same size. And to do this, he added a term to the equations, now known as cosmological term, which, when added to the energy density of ordinary matter and radiation, allowed the Universe to never expand and never contract, but to remain the same forever.
However, after Hubble discovered that our Universe is expanding, Einstein's cosmological term was forgotten and "abandoned". However, after some time, interest in it was awakened by relativistic quantum theories, in which the cosmological constant appears naturally dynamically from quantum oscillations of virtual particles and antiparticles. This is called the quantum zero energy level and is a very possible candidate for vacuum energy space-time. However, quantum theory has its own “problems” - how not to make this vacuum energy too large, and this is one of the reasons why physicists are exploring supersymmetric theories.
The cosmological constant can either speed up or slow down the expansion of the Universe, depending on whether it is positive or negative. And when the cosmological constant is added to space-time in addition to ordinary matter and radiation, the picture becomes much more complicated than the simplest cases of an open or closed Universe described above.

So what's the answer?

Almost immediately after the Big Bang, radiation dominance era, which lasted the first ten to one hundred thousand years of the evolution of our Universe. Now the dominant forms of matter are ordinary matter and vacuum energy. Judging by the latest observations by astronomers,
1. Our Universe is flat with good accuracy: The cosmic microwave background radiation is a relic left over from a time when the Universe was hot and filled with hot photon gas. Since then, however, due to the expansion of the Universe, these photons have cooled, and now their temperature is 2.73 K. However, this radiation is slightly inhomogeneous; their angular size of inhomogeneities, visible from our current position, depends on the spatial curvature of the Universe. So, observations of the anisotropy of the cosmic microwave background radiation indicate precisely that our The universe is flat.
2. There is a cosmological constant in the Universe: There is vacuum energy in the universe, or at least something that acts as vacuum energy, which causes the universe to expand at an accelerated rate. Evidence of the accelerated expansion of the Universe is data on the redshifts of distant supernovae.
3. Most of the matter in the Universe is in the form of dark matter: The study of the movement of galaxies leads to the conclusion that ordinary matter in the form of stars, galaxies, planets and interstellar gas makes up only a small fraction of the total matter in the Universe.
As of today's era


So now in the Universe the energy density of vacuum is more than twice as high as the energy density of dark matter, and at the same time the contribution of baryonic visible matter can simply be neglected. So our flat Universe should expand forever.

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The next version of the structure of the Universe was put forward by physicist Frank Steiner from the University of Ulm, who, together with his colleagues, re-analyzed the data collected by the Wilkinson Microwave Anisotropy Probe (WMAP) space probe, which was once launched for detailed photography of the cosmic microwave background radiation.

However, do not rush to talk about the edges of the Universe. The fact is that this polyhedron is closed on itself, that is, having reached one of its faces, you will simply get back inside through the opposite side of this multidimensional “Möbius loop”.

Interesting conclusions follow from this presentation. For example, that by flying on some “high-speed” rocket in a straight line, you can eventually return to the starting point, or, if you take a “very large” telescope, you can see the same objects in different directions of space, only due to finitude speed of light - at different stages of life.

Scientists tried to make such observations, but nothing similar to “mirror reflections” was found. Either because the model is incorrect, or because the “range” of modern observational astronomy is not enough. Nevertheless, the discussion about the shape and size of the Universe continues.

Now Steiner and his comrades have added new wood to the fire.

Planck weighs about two tons. It should cruise around the Lagrange point L2. As the satellite rotates around its axis, it will gradually capture a complete map of the microwave background with unprecedented accuracy and sensitivity (illustrations by ESA/AOES Medialab and ESA/C. Carreau).

The German physicist compiled several models of the Universe and checked how microwave background density waves are formed in them. He claims that the closest match to the observed cosmic microwave background radiation is provided by the donut universe, and even calculated its diameter. The “donut” turned out to be 56 billion light years across.

True, this torus is not quite ordinary. Scientists call it a 3-torus. Its actual form is difficult to imagine, but researchers explain how to at least try.

First, imagine how a regular “donut” is formed. You take a piece of paper and roll it into a tube, gluing two opposite edges together. Then you roll the tube into a torus, gluing its two opposite “exits” together.

With a 3-torus, everything is the same, except that the starting ingredient is not a sheet, but a cube, and you need to glue not the edges of the planes, but each pair of opposite faces. Moreover, glue it in such a way that, having left the cube through one of its faces, you will find that you again got inside through its opposite face.

Several experts who commented on Steiner's work noted that it does not definitively prove that the Universe is a “multidimensional donut”, but only that this shape is one of the most likely. Some scientists also add that the dodecahedron (which is often compared to a soccer ball, although this is incorrect) is still a “good candidate.”

Frank's answer to this is simple: the final choice between forms can be made after more accurate measurements of the cosmic microwave background radiation than those carried out by WMAP. And such a survey will soon be carried out by the European satellite Planck, which is scheduled to launch on October 31, 2008.

“From a philosophical point of view, I like the idea that the Universe is finite and one day we might be able to fully explore it and know everything about it. But since questions in physics cannot be solved by philosophy, I hope that Planck will answer them,” says Steiner.

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, or 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 one. 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 map of the microwave background, 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 geometrically and physically coincide. 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

Einstein's general theory of relativity studies the geometry of 4-dimensional space-time. However, the question of the shape (geometry) of three-dimensional space itself remains unclear to this day.

By studying the distribution of galaxies, scientists have come to the conclusion that our Universe, with a high degree of accuracy, is spatially homogeneous and isotropic on large scales. This means that the geometry of our world is the geometry of a homogeneous and isotropic three-dimensional manifold. There are only three such manifolds: a three-dimensional plane, a three-dimensional sphere and a three-dimensional hyperboloid. The first manifold corresponds to the usual three-dimensional Euclidean space. In the second case, the Universe has the shape of a sphere. This means that the world is closed, and we could get to the same point in space simply by moving in a straight line (like traveling around the Earth). Finally, space in the shape of a hyperboloid corresponds to an open three-dimensional manifold, the sum of the angles of a triangle in which is always less than 180 degrees. Thus, studying only the large-scale structure of the Universe does not allow us to unambiguously determine the geometry of three-dimensional space, but significantly reduces the possible options.

The study of cosmic microwave background radiation, the most accurate cosmological observable at the moment, allows progress in this issue. The fact is that the shape of three-dimensional space has a significant impact on the propagation of photons in the Universe - even a slight curvature of the three-dimensional manifold would significantly affect the spectrum of the cosmic microwave background radiation. Modern research on this topic says that the geometry of the Universe is flat with a high degree of accuracy. If space is curved, then the corresponding radius of curvature is 10,000 greater than the causally connected region in the Universe.

The question of the geometry of three-dimensional manifold is closely related to the evolution of the Universe in the future. For space in the form of a three-dimensional hyperboloid, the expansion of the Universe would last forever, while for spherical geometry the expansion would give way to compression, followed by the collapse of the Universe back into a singularity. However, based on modern data, the rate of expansion of the Universe today is determined not by the curvature of the three-dimensional manifold, but by dark energy, a certain substance with a constant density. Moreover, if the density of dark energy remains constant in the future, its contribution to the total density of the Universe will only increase with time, and the contribution of curvature will decrease. This means that the geometry of the three-dimensional manifold will likely never have a significant impact on the evolution of the Universe. Of course, it is impossible to make any reliable predictions about the properties of dark energy in the future, and only more accurate studies of its properties can shed light on the future fate of the Universe.