Flat universe. Why is there something and not nothing? Classical space topology

When astronomers and physicists say the universe is flat, they don't mean that the universe is flat like a leaf. We are talking about the property of three-dimensional flatness - Euclidean (non-curved) geometry in three dimensions. In Euclidean astronomy, the world is a convenient comparative model of the surrounding space. The substance in such a world is distributed uniformly, that is, the same amount of matter is contained in a unit volume, and isotropic, that is, the distribution of matter is the same in all directions. In addition, matter does not evolve there (for example, radio sources do not ignite and supernovae do not erupt), and space is described by the simplest geometry. This is a very convenient world to describe, but not to live in, since there is no evolution there.

It is clear that such a model does not correspond to observational facts. The matter around us is distributed inhomogeneously and anisotropically (somewhere there are stars and galaxies, but somewhere they are not), accumulations of matter evolve (change over time), and space, as we know from the experimentally confirmed theory of relativity, is curved.

What is curvature in 3D space? In the Euclidean world, the sum of the angles of any triangle is 180 degrees - in all directions and in any volume. In non-Euclidean geometry - in curved space - the sum of the angles of a triangle will depend on the curvature. Two classic examples are a triangle on a sphere where the curvature is positive, and a triangle on a saddle surface where the curvature is negative. In the first case, the sum of the angles of the triangle is greater than 180 degrees, and in the second case it is less. When we usually talk about a sphere or a saddle, we think of curved two-dimensional surfaces surrounding three-dimensional bodies. When we talk about the Universe, we must understand that we are moving to the concept of a three-dimensional curved space - for example, we are no longer talking about a two-dimensional spherical surface, but about a three-dimensional hypersphere.

So why is the Universe flat in a three-dimensional sense, if the space is curved not only by clusters of galaxies, our Galaxy and the Sun, but even by the Earth? In cosmology, the universe is viewed as a whole object. And as a whole object, it has certain properties. For example, starting from some very large linear scales (here one can consider both 60 megaparsecs [~180 million light years] and 150 Mpc), matter in the Universe is distributed uniformly and isotropically. On smaller scales, there are clusters and superclusters of galaxies and voids between them - voids, that is, the uniformity is broken.

How can one measure the flatness of the universe as a whole if information about the distribution of matter in clusters is limited by the sensitivity of our telescopes? It is necessary to observe other objects in a different range. The best that nature has given us is the cosmic microwave background, or , which, separated from matter 380 thousand years after the Big Bang, contains information about the distribution of this matter literally from the first moments of the existence of the Universe.

The curvature of the Universe is related to the critical density equal to 3H 2 /8πG (where H is the Hubble constant, G is the gravitational constant), which determines its shape. The parameter value is very small - about 9.3×10 -27 kg/m 3 or 5.5 hydrogen atoms per cubic meter. This parameter distinguishes the simplest cosmological models based on the Friedman equations, which describe: if the density is higher than the critical one, then the space has a positive curvature and the expansion of the Universe will be replaced by contraction in the future; if it is below critical, then the space has a negative curvature and the expansion will be eternal; if the critical density is equal, the expansion will also be eternal with a transition to the Euclidean world in the distant future.

The cosmological parameters describing the density of the Universe (and the main ones are the density of dark energy, the density of dark matter and the density of baryonic [visible] matter) are expressed as a ratio to the critical density. According to , obtained from measurements of the cosmic microwave background radiation, the relative density of dark energy is Ω Λ = 0.6879±0.0087, and the relative density of all matter (that is, the sum of the density of dark and visible matter) is Ω m = 0.3121±0.0087.

If we add up all the energy components of the Universe (densities of dark energy, all matter, as well as radiation densities and neutrinos that are less significant in our era), then we will get the density of all energy, which is expressed through the ratio to the critical density of the Universe and denoted by Ω 0 . If this relative density is equal to 1, then the curvature of the Universe is equal to 0. The deviation of Ω 0 from unity describes the energy density of the Universe Ω K associated with the curvature. By measuring the level of inhomogeneities (fluctuations) of the distribution of the relic background radiation, all density parameters, their total value and, as a consequence, the curvature parameter of the Universe are determined.

Based on the results of observations, taking into account only the CMB data (temperature, polarization and lensing), it was determined that the curvature parameter is very close to zero within small errors: Ω K = -0.004±0.015, and taking into account data on the distribution of galaxy clusters and measurements expansion rate according to type Ia supernova data parameter Ω K = 0.0008±0.0040. That is, the Universe is flat with high accuracy.

Why is it important? The flatness of the Universe is one of the main indicators of the very fast era described by the inflationary model. For example, at the time of birth, the Universe could have had a very large curvature, while now, according to the CMB data, it is known that it is flat. Inflationary expansion makes it flat in all observable space (meaning, of course, large scales on which the curvature of space by stars and galaxies is not significant) just as an increase in the radius of a circle straightens the latter, and with an infinite radius the circle looks like a straight line.

In ancient times, people thought that the earth is flat and stands 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 boundless and crooked, but also closed. The latest data obtained in the process of studying the background radiation indicate that the Universe may well be closed in on itself. It turns out that if you fly from the earth all the time, then at some point you will begin to approach it and eventually return back, bypassing the entire Universe and making a round-the-world trip, 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. Matter in the beginning was 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 (connection) of nuclei and electrons that began at that time, that is, the formation of neutral atoms of hydrogen and helium, radically changed the optical properties of the Universe. It has become 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 closed to us by a kind of “fire wall” of ionized matter. It is possible to look much deeper into the history of the Universe only if we learn how to register relic neutrinos, for which hot matter became transparent much earlier, and primary gravitational waves, for which matter of any density is not an obstacle, but this is a matter of the future, and far from it. the closest.

Since the formation of neutral atoms, our Universe has expanded by about 1,000 times, and the radiation of 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 when testing a large radio antenna, is practically the same in all directions. According to modern data, there are a hundred million times more relic photons than atoms, so our world simply bathes in streams of strongly reddened light emitted in the very first minutes of the life of the Universe.

Classical space topology

On scales larger than 100 megaparsecs, the part of the Universe that we see 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 that assume spherical symmetry and spatial homogeneity of matter distribution.

The classical cosmological solutions of Einstein's general relativity (GR) equations, which were found in 1922 by Alexander Friedman, have the simplest topology. Their spatial sections resemble planes (for infinite solutions) or spheres (for bounded solutions). But such universes, it turns out, have an alternative: a universe without edges and boundaries, a universe of finite volume closed on itself.

The first solutions found by Friedman described universes filled with only one sort of matter. Different pictures arose due to the difference in the average density of matter: if it exceeded the 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 contraction to a point. The universe with a density below the critical one had a negative curvature and expanded infinitely, its inflation rate tended to some constant value. This model is called open. The flat Universe, an intermediate case with a density exactly equal to the critical one, is infinite and its instantaneous spatial sections are flat Euclidean space with zero curvature. A flat one, like an open one, expands indefinitely, but the rate of its expansion tends to zero. Later, more complex models were invented, in which a homogeneous and isotropic universe was filled with a multi-component matter that changes with time.

Modern observations show that the Universe is now expanding with acceleration (see "Beyond the Universe's Event Horizon", 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 anti-gravity, which overcomes the attractive forces of ordinary matter on a large scale. First similar 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 builds up so rapidly that the universe becomes infinite in a finite amount of 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 represent it similar to our usual space. This picture is in good agreement with the data that astronomers receive with the help of telescopes that record infrared, visible, ultraviolet and X-ray radiation. And only the data of radio observations, namely a detailed study of the relict background, made scientists doubt that our world is arranged so straightforwardly.

Scientists will not be able to look behind the “wall of fire” separating 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 WMAP (Wilkinson Microwave Anisotropy Probe) space observatory, 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 has not been completed even today. Usually, physics is used to explain new cosmological data: the equations of state of matter, the laws of expansion, and the 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 exactly - topological.

The main purpose of WMAP was to build a detailed map of the temperature of the cosmic microwave background (or, as it is also called, the microwave background). WMAP is an ultra-sensitive radio receiver that simultaneously registers 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 the Earth. This 840 kg satellite is actually in orbit around the Sun, but due to the combined action of the gravitational fields of the Earth and the Sun, its period of revolution is exactly one year, and it does not fly away from Earth anywhere. The satellite was launched into such a distant orbit so that interference from terrestrial man-made activity would not interfere with the reception of relict radio emission.

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 one is 1.02 ± 0.02 (that is, our Universe is flat or closed with a very small curvature). Secondly, the Hubble constant, which characterizes the expansion of our World on a large scale, is 72±2 km/s/Mpc. Thirdly, the age of the Universe is 13.4 ± 0.3 billion years and the redshift corresponding to the recombination time is 1088 ± 2 (this is an average value, the thickness of the recombination boundary is much larger than the indicated error). The most sensational result for theoreticians was the angular spectrum of relic radiation disturbances, more precisely, the too small value of the second and third harmonics.

Such a spectrum is constructed by representing the temperature map as a sum of various spherical harmonics (multipoles). In this case, variable components are distinguished from the general picture of perturbations that fit on the sphere an integer number of times: a quadrupole - 2 times, an octupole - 3 times, and so on. The higher the number of the spherical harmonic, the more high-frequency oscillations of the background 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.

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

How does a bagel look like a coffee cup?
There is such a branch of mathematics - topology, which explores the properties of bodies that are preserved under any of their deformations without gaps and 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, and to do the same with a torus, you can only make two cuts. Topologists simply adore all kinds 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 a large number measurements. So our three-dimensional Universe, closed on itself, can be easily imagined only by living in a six-dimensional space. Cosmic topologists do not encroach on the time yet, leaving it with the opportunity to simply flow linearly, without locking 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.

The final CMB temperature map is based on a painstaking analysis of maps showing the intensity of radio emission in five different frequency ranges.

An unexpected decision

For most spherical harmonics, the obtained experimental data coincided with model calculations. Only two harmonics, quadrupole and octupole, turned out to be clearly below the level expected by theorists. Moreover, the probability that such large deviations could occur by chance is extremely small. Quadrupole and octupole suppression was noted as early as in the COBE data. However, the maps obtained in those years had poor resolution and large noise, so the 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 turned out to be so small, at first it was completely incomprehensible. It has not yet been possible to come up with a physical mechanism for their suppression, since it must act on the scale of the entire observable Universe, 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 ways and found a topological answer to the question that arose. The mathematical solution of 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 the background radiation. This effect arises due to repeated observation of the same region of the recombining plasma through different parts of the closed dodecahedral space. It turns out that low harmonics, as it were, extinguish themselves due to the passage of a 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 near and on the opposite face, since these regions 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 that is only slightly larger than the horizon of visible events.

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

The magnitude of the dipole anisotropy associated with the motion of the Earth is 3.353*10-3 K. This corresponds to the motion of the Sun relative to the background radiation 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, where it belongs, moves relative to the relic at a speed of about 600 km/s.

All other perturbations (starting 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 radio emission from 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 own radiation of the Milky Way (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 Thor

The simplest body with a topology more complex than a sphere or a plane is a torus. Anyone who held a donut in their hands can imagine it. Another more correct mathematical model of a flat torus is demonstrated by the screens of some computer games: this is a square or rectangle, the opposite sides of which are identified, and if the moving object goes down, then 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 that has a finite volume and does not have any boundaries.

In three-dimensional space, a similar procedure can be done with a cube. If you identify its opposite faces, then a three-dimensional torus is formed. If you look 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 quite finite. In such a world, there are no boundaries, but there are three selected directions parallel to the edges of the original cube, along which periodic rows of the 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 facets, the inhabitant of such a world will see his head, and not his face, as in the earthly room of laughter. A more correct model would be a room equipped with 6 TV cameras and 6 flat LCD monitors, which display the image taken by the film camera located opposite. In this model visible world closes on itself due to the exit to another television dimension.

The picture of the suppression of low-frequency harmonics described above is correct if the time for which the light crosses the initial volume is sufficiently small, that is, if the dimensions of the initial body are small compared to cosmological scales. If the dimensions of the part of the Universe accessible for observation (the so-called horizon of the Universe) turn out to be smaller sizes of the initial topological volume, then the situation will be no different from what we see in the usual infinite Einsteinian Universe, and no anomalies will be observed in the CMB spectrum.

The maximum possible spatial scale in such a cubic world is determined by the size of the original body - the distance between any two bodies cannot exceed half the main diagonal of the original cube. The 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 the lowest large-scale angular fluctuations, which means that by studying the relict background, one can estimate the size of our Universe.

3D mosaics

A flat topologically complex three-dimensional Universe can only be built on the basis of cubes, parallelepipeds and hexagonal prisms. In the case of curved space, a wider class of figures possesses such properties. In this case, the angular spectra obtained in the WMAP experiment best agree with the dodecahedral model of the Universe. 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 small positive curvature, regular dodecahedrons can fill the entire space without holes and mutual intersections. With a certain ratio between the size of the dodecahedron and the curvature, 120 spherical dodecahedrons are needed for this. Moreover, this complex structure of hundreds of “balls” can be reduced to a topologically equivalent one, consisting of only one dodecahedron, in which opposite faces rotated by 180 degrees are identified.

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

For an ordinary inhabitant of the Earth, all these topological intricacies at first glance do not have much meaning. But for physicists and philosophers - a completely different matter. Both for the worldview as a whole and for a unified theory explaining 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 CMB fluctuation spectrum, 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 data obtained in the WMAP experiment, and only for the highest harmonics, the results of the CBI (balloon) and ACBAR (ground Antarctic) studies are added. On the normalized plot of the angular spectrum of fluctuations of the relic radiation, several maxima are seen. 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 time of recombination. The maximum amplitude of oscillations falls on the size of the causally related region (sound horizon) at the moment of recombination. On smaller scales, the plasma oscillations were attenuated by photon viscosity, while on large scales, the perturbations were independent of each other and were not in phase. Therefore, the maximum fluctuations observed in the modern era fall 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 time 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 the theory, corresponds with high accuracy to a flat Euclidean Universe.

A lot of information about the cosmological parameters is contained in the second and subsequent acoustic peaks. Their very existence reflects the fact of "phasing" of acoustic oscillations in plasma in the era of recombination. 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 of fluctuations on different scales to occur, these (very far from each other) regions must have been able to interact with each other. It is this 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 weighty confirmations of this scenario.

The relict radiation was observed in a 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 has 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 form 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 cosmic microwave background radiation 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 the same and symmetrical only for an observer who is stationary relative to the background background. The Earth, on the other hand, moves relative to it at a sufficiently high speed, due to which a significant dipole component appears in the background radiation. In this case, the circles turn into ellipses, their size, 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 to help figure out whether they are or are not there.

Multilinked 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 of 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 rather inhomogeneous. Then, in the process of inflation, when the Universe expanded according to a law close to exponential, its initial dimensions increased by many orders of magnitude. Today we see only a small part of the Big Universe, in which heterogeneities still remain. True, they have such a large spatial extent that they are invisible inside the area accessible to us. The inflationary scenario is by far the best developed cosmological theory.

For a multiply connected universe, such a sequence of events is not suitable. 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 multiply connectedness of our Universe is confirmed are already clear: these are non-inflationary models and the so-called models with weak inflation, in which the size of the universe during inflation increases only a few times (or tens of times). 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 the WMAP data drew attention to the fact that the quadrupole and octupole components of the cosmic microwave background radiation have close orientations to each other and lie in a plane that almost coincides with the galactic equator. The conclusion of this group is that there was an error when subtracting the background of the Galaxy from the data of observations of the microwave background and the real magnitude of the harmonics is completely different.

The WMAP observations were carried out at 5 different frequencies specifically in order to correctly separate the cosmological and local backgrounds. And the core WMAP team believes that the processing of the observations was done 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 the beginning of 2006, WMAP has been making continuous observations for four years, which should be enough to double the accuracy, but these data have not yet been published. We need to wait a bit, and perhaps our assumptions about the dodecahedral topology of the Universe will take on a completely conclusive nature.

Mikhail Prokhorov, Doctor of Physical and Mathematical Sciences

Checking the validity of the cosmological model of the Universe, according to which about 72% of its mass falls on dark energy, using a new method, confirmed that the Universe is "flat", and the so-called cosmological constant, which Albert Einstein called his major mistake, may be the explanation for its accelerated expansion, according to the authors of the paper, which will be published in the journal Nature on Thursday.

Albert Einstein added a cosmological constant characterizing the properties of vacuum to his own equations of general relativity to allow for a stable universe that does not contract or expand. However, some time after that, the American astronomer Edwin Hubble showed that the universe is actually expanding, and Einstein himself called the cosmological constant his "biggest mistake".

The cosmological constant remained a subject of scientific interest, but until the 1990s, it was thought to be slightly different from zero. In 1998-1999, observations of supernovae showed that the Universe is expanding with acceleration, and then data from the WMAP probe (Wilkinson Microwave Anisotropy Probe), which studies the cosmic microwave background, the "echo" of the Big Bang, led scientists to assume that mysterious dark energy is "pushing" the Universe , which accounts for about 72% of its mass. These findings sparked a new interest in the cosmological constant.

Christian Marinoni and Adeline Buzzi from the University of Provence (France) proposed a new method for verifying the validity of ideas about the structure and properties of the Universe, based on the geometry of pairs of galaxies with a large redshift, that is, very distant from the observer. They took advantage of the fact that modern ideas, the "shape" of the Universe depends on its "content", which means that geometric measurements can be used to determine the composition of the Universe and, in particular, the amount of dark energy in it.

The scientists used a modification of the Elcock-Paczynski test, developed by American and Polish astronomers more than 30 years ago. This test is based on considering symmetrical objects in outer space as "standard spheres", any distortion of which will be due to the distortion of space caused by the expansion of the Universe.

This test was repeatedly tried to apply, for example, to galaxy clusters, but this lacked the accuracy of measurements. Marinoni and Buzzi studied the distribution of the mutual orientation of pairs of galaxies orbiting each other. In a Universe without dark energy, this distribution would be spherically symmetrical - that is, the number of pairs oriented in either direction would be the same.

Observations showed that, in fact, the farther from the Earth the pairs of galaxies are, the more asymmetric was the distribution of their orientation - more pairs were located along the line of sight from the Earth. This, as scientists note, corresponds to the model of a flat universe.

The flat Universe is such a model of the development of the Universe, according to which its expansion is infinite, and the curvature of space is zero, that is, it is flat. In such a model, the life of the Universe ends either with the "Big Frost" (Big Freeze), when the expanding Universe experiences thermal death - in such a system with a uniformly distributed energy, no mechanical work or movement is possible, or "Big Rip" (Big Rip), when the acceleration of the expansion will "overpower" the electromagnetic, weak and gravitational interactions, and the Universe will simply "tear". Previously, the data of the same WMAP pointed to the "plane" of the Universe.
with dark energy.

In addition, as the researchers note, they were able to show that the most successful explanation for the phenomenon of dark energy can be precisely the Einstein cosmological constant, which denotes the energy of vacuum. Scientists, according to them, have received the most accurate estimate of the magnitude of this constant to date.

World science faces a number of questions, the exact answers to which it, apparently, will never receive. The age of the universe is just one of them. Up to a year, a day, a month, a minute, it, apparently, will never be able to be calculated. Though...

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

And now NASA is proud to announce that the age of the universe has been determined with an error of “only” 0.2 billion years. And this age is equal to 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 it installed?

It turns out that with the help of a single device, appearing under the name MAP - Microwave Anisotropy Probe (Microwave Anisotropy Probe).

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

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 indicators of 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 CMF.

COBE has 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 received an image with a resolution 35 times better than its predecessor.

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

Star map showing the temperature fluctuations of the cosmic microwave background.

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

As a result of the collisions, the light became polarized, and this imprint 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 ratio of "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 value of the density of matter and energy in the Universe is less than the critical one, then the cosmos is open and concave like a saddle.

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

A diagram showing the ratio of matter forms in the Universe.

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

The MAP readings confirmed this.

Another extremely interesting circumstance also turned out: 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. Those were formed as a result of the explosions of old stars, the residual matter of which fell on the surface of other stars and, in the process of thermonuclear fusion, formed heavier compounds.

Doctor of Physical and Mathematical Sciences A. MADERA.

What do a piece of paper, a tabletop, 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, a torus can be obtained.

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

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

If two faces of the original cube are glued 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 faces will gain volume.

If we take a hexagonal prism as the fundamental area, glue each of its faces with the opposite one directly, 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.

A lamellar 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 with a rotation of 180 degrees.

The distribution map of the microwave background radiation shows the density distribution of matter, which 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 in some places covered with mountains and depressions. This belief persisted for many thousands of years, until Aristotle in the 4th century BC. e. did not notice that the ship going to sea disappears from sight, not because, as it moves away, it decreases to sizes inaccessible to the eye. On the contrary, the ship's hull disappears first, 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 colossal experience has been accumulated. Nevertheless, fundamental questions still remain unanswered: is the Universe, inside which we live, finite or infinite, and what is its form?

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 claim it.

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. So let's see what the results of observations say about the Universe.

First of all, we note that, no matter where in the Universe we are, around any of its points, you can outline a sphere of arbitrary size, containing inside the space of the Universe. 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 does our Universe represent? Mathematicians have long established that there are so many of them that complete list still does not exist. Long-term observations have shown that the Universe has a number of physical properties, which drastically reduce the number of possible contenders for its form. And one of the main such 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 to combine electrons and protons into the first atoms (see "Science and Life" Nos. 11, 12, 1996). When this happened, the radiation, which was initially scattered by charged particles, suddenly became able to pass unhindered through the expanding universe. This radiation, now known as the cosmic microwave background or relic radiation, is surprisingly homogeneous and reveals only very slight 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. So, 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 key differences as well.

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

The German mathematician Carl Friedrich Gauss understood already 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 investigate the geometric properties of physical space using direct measurements. For this, he chose three mountain peaks remote from one another - Hohenhagen, Inselberg and Brocken. Standing on one of these peaks, he directed the sun's rays reflected by the mirrors to the other two and measured the angles between the sides of the huge light triangle. Thus, he tried to answer the question: are the trajectories of light rays passing over the spherical space of the Earth curved? (By the way, at about the same time, the Russian mathematician, the rector of Kazan University, Nikolai Ivanovich Lobachevsky, proposed experimentally investigating the question of the geometry of physical space using the star triangle.) If Gauss had discovered that the sum of the angles of the light triangle differs from 180 degrees, then the conclusion would follow that the sides of the triangle are curved and the real physical space is non-Euclidean. However, within the limits of the measurement error, the sum of the angles of the "test triangle Brocken - Hohenhagen - Inselberg" was exactly 180 degrees.

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

Recent studies conducted with high-altitude balloons raised over Antarctica also support this conclusion. When measuring the angular power spectrum of the CMB, a peak was registered, which, as 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 drastically reduces the number of likely contenders for the possible shape of the universe.

Back in the 1930s, mathematicians proved that there are only 18 different Euclidean three-dimensional manifolds and, therefore, only 18 possible forms of the Universe instead of their 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 take a point on the Möbius strip a, draw a normal (perpendicular) to it, and draw a small circle around the normal with a counterclockwise direction, if viewed from the end of the normal. Let's start moving the point along with the normal and the directed circle along the Möbius band. 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 orientation, such as a sphere, a torus, and an untwisted tape, are called orientable or two-sided. By the way, we note 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 gaze. Therefore, it is reasonable to exclude eight non-orientable manifolds from consideration and restrict possible forms our Universe by ten orientable Euclidean three-dimensional manifolds.

Three-dimensional manifolds in four-dimensional space are extremely difficult to visualize. However, one can try to imagine their structure by applying 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, which allows any stretching and curvature, but without tears, folds and gluing. In topology, figures that can be transformed one into another with the help of such deformations are called homeomorphic; they have the same internal geometry. Therefore, from the point of view of topology, a bagel (torus) and an ordinary cup with a handle are one and the same. But it is impossible to transfer a soccer ball into a bagel. These surfaces are topologically different, that is, they have different internal geometric properties. However, if a round hole is cut out on the sphere and one handle is attached to it, then the resulting figure will already be homeomorphic to a torus.

There are many surfaces that are topologically different from the torus and the sphere. For example, by adding a handle to the torus, similar to the one we see by the cup, we get a new hole, and hence a new figure. A torus with a handle will be homeomorphic to a figure resembling a pretzel, which in turn is homeomorphic to a sphere with two handles. The addition of each new handle creates another hole, and thus another 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 of their points. 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 is an infinite plane, sphere, torus, or in general any other surface from an infinite number of tori or spheres with a different number of handles.

These topological forms can be quite difficult to understand. And in order to visualize them more easily and 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 domain for the torus. If we now mentally glue the bases of the cylinder (the material of the cylinder is elastic), we get a torus.

Imagine that there is some two-dimensional creature, say an insect, whose movement on the surface of the torus needs to be investigated. It is not easy to do this, and it is much more convenient to observe its movement in a square - a space with the same topology. This approach has two advantages. Firstly, it allows you to visually see the path of an insect in three-dimensional space, following its movement in two-dimensional space, and secondly, it allows you to stay within the framework of a well-developed Euclidean geometry on a plane. Euclidean geometry contains the postulate of parallel lines: for any straight line and a point outside it, there is a single line parallel to the first and passing through this point. Also, the sum of the angles of a flat triangle is exactly 180 degrees. But since the square is described by Euclidean geometry, we can extend it to the torus and say that the torus is a Euclidean 2-manifold.

The indistinguishability of internal geometries for various surfaces is associated with their important topological characteristic, called developability. So, the surfaces of a cylinder and a cone look completely different, but nevertheless their geometries are exactly the same. Both of them can be deployed in a plane without changing the lengths of the segments and the angles between them, so 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.

An innumerable number of tori can also be formed from other flat figures, for example, from various parallelograms or hexagons, by gluing their opposite edges together. However, not every quadrilateral is suitable for this: the lengths of its glued sides must be the same. Such a requirement is necessary in order to avoid lengthening or contraction of the region edges during gluing, which violate the Euclidean geometry of the surface.

We now turn to manifolds of higher dimension.

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 together, we glue the opposite faces of the cube together at all their points.

The resulting 3D torus is a Euclidean 3-manifold. If we somehow ended up in it and looked forward, we would see our back of the head, as well as our copies 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 images in a 3D torus will be straight, not mirrored.

It is important to note the circular nature of this and many other varieties. If the universe really had such a shape, then, having left the Earth and flying without any changes 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 the 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 need to be glued together because they make up the opposite faces of the cube. So the 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 faces of the cube. Topologists refer to such a manifold as T 2 xS 1 , where T 2 means a two-dimensional torus and S 1 is a ring. This is an example of a bundle, or bundle, of tori.

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

These and all other finite manifolds are very easily included in the picture of the expanding Universe. If the fundamental region of the manifold is constantly expanding, the space formed by it will also expand. Each point in expanding space is moving further and further away from the rest, which exactly corresponds to the cosmological model. In this case, 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, opposite faces are glued.

The next 3-manifold similar to a 3-torus is called 1/2 - rotated cubic space. In this space, the fundamental domain is again a cube or a parallelepiped. Four faces are glued as usual, and the remaining two, front and back, are glued with a 180 degree rotation: the top of the front face is glued to the bottom of the back. If we found ourselves in such a variety 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 3D torus, the fundamental region of a 1/2-rotated cubic space can be sliced ​​into thin vertical slices, so that when glued together, you again get a bunch of 2D tori, with the only difference being that this time the front and back tori are glued rotated by 180 degrees. .

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

The 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 fiber of this manifold is a torus, and thus the space is also a sheaf of tori. In all hexagonal faces, we would see copies rotated 120 degrees relative to the previous one, and copies in the faces - parallelograms - 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 rotation of 60 degrees. As before, in the resulting bundle of tori, the remaining faces - parallelograms - are glued directly to one another.

The double cubic space is radically different from the previous manifolds. This finite space is no longer a sheaf of tori and has an unusual gluing structure. The double cubic space, however, uses a simple fundamental region, which is two cubes stacked one on top of the other. When gluing, not all faces are connected directly: the top front and back faces are glued to the faces directly below them. In this space, we would see ourselves in a peculiar 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 quite probable that among them it is necessary to look for the form of our Universe.

Many cosmologists believe that the Universe is finite: it is difficult to imagine the physical mechanism for the creation of an infinite Universe. Nevertheless, we consider the four remaining orientable non-compact Euclidean 3-manifolds until real evidence is obtained that excludes their existence.

The first and simplest infinite 3-manifold is the 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 of our copies, neither straight, nor rotated, nor inverted.

The next manifold is the so-called plate space, whose fundamental domain is an infinite plate. The upper part of the plate, which is an infinite plane, is glued directly to its lower part, which is also an infinite plane. These planes must be parallel to one another, but can be arbitrarily shifted during gluing, which is not essential, given their infinity. In topology, this manifold is written as R 2 xS 1 , where R 2 denotes a plane and S 1 an annulus.

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

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

By rotating one of the surfaces of the tubular space to be glued by 180 degrees, we obtain a rotated tubular space. This twist, given 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 close.

What is the shape of our Universe?

In order to choose one of the above ten Euclidean 3-manifolds as the shape of our Universe, additional data from astronomical observations are 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-turned cubic space, then direct 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 not very suitable 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 one day we find 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 a huge number of galaxies.

At the beginning of the article it was said that the Universe has a constant curvature. The homogeneity of the cosmic microwave background radiation directly indicates this. However, it has slight spatial variations, about 10 -5 Kelvin, indicating that there were slight 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 microwave radiation map allows you to look into the past, at the time of the initial inhomogeneities, to see the blueprints of the Universe, which was then a thousand times smaller. To appreciate the significance of this card, consider a hypothetical example: the Universe as a two-dimensional torus.

In a 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 not get any indication of its shape. If, however, the circle of vision of two-dimensional creatures is larger than the fundamental region, they could see intersections and even repetition of images of the Universe and try to find points with the same temperatures that correspond to the same region of it. If there were enough such points in their circle of vision, they could conclude that they live in a torus universe.

Despite the fact that we live in a three-dimensional universe and see a spherical region, we face the same problem as two-dimensional creatures. If our field of vision is smaller than the fundamental region of the universe 300,000 years ago, we won't see anything out of the ordinary. Otherwise, the sphere will intersect it in circles. By finding two circles that have the same variations in microwave radiation, cosmologists can compare their orientation. If the circles are criss-crossed, this will mean that there is a gluing, but no rotation. Some of them, however, can be combined according to a quarter or half turn. If enough of these circles can be found, the secret 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, NASA researchers attempted to map the cosmic microwave background radiation. However, the angular resolution of the satellite was about 10 degrees, which did not allow making accurate measurements 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 already on the order of 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, accurate maps of the CMB fluctuations will be obtained within four to ten years. And if the size of the sphere of our vision is large enough, and the measurements are accurate and reliable enough, we will finally know what shape our Universe has.