# Later it turned out that the neutrino mass is too small for this.

ρV = R * −4 (49)

This formula is designed so that the value of 1 / R * gives the mass of dark energy in a volume of space with the size R *. Then the density is found by dividing this mass by the volume R * 3. At R * ~ 10−3 cm, we have a density ρV ~ 10−120 MP4. A remarkable feature of relation (49) is that the density of dark energy does not depend on the "truly fundamental mass" and in this case the hierarchy really disappears.

Note also that expression (49) resembles the relation known from the quantum Casimir effect: the force of attraction between two parallel conducting plates per unit area of ​​the plates (in terms of dimension, this is the energy density) is given by a similar formula: ρCAS ~ d − 4, where d – the size of the small gap between the plates.

If we assume that dark energy (understood as an EG vacuum) should fill a full five-dimensional space, then its five-dimensional density in such a space, as is easy to see, will depend only on the size of the extra dimensions:

ρV5 ~ R * −6. (fifty)

As for the Einstein antigravitational force, in five-dimensional space it will take the form

FES ∝ xM * −4ρVR, (51)

retaining, as expected, a linear dependence on distance.

Using (51) and the above formulas, one can find, for example, an expression for the radius of zero gravitation in five-dimensional space. Instead of formula (32) (see Section 3), we now have

\$\$r_ {V} = \ left (\ frac {16 \ pi ^ {3}} {5} MR _ {*} ^ {6} \ right) ^ {1/5}. (52)\$\$

This radius does not exceed the size of the extra dimensions R *, only if the mass M is very small:

М ≤ МV ~ 10−17М *. (53)

The MV mass is 11 orders of magnitude less than the electron mass. This means that for all known elementary particles (not to mention macroscopic bodies), the corresponding values ​​of the zero-gravity radius go far beyond the limits of additional macroscopic dimensions (in the five-dimensional world).

Recall that, as it is assumed, quantum non-gravitational fields "do not live" in extra dimensions, they and their zero-point oscillations exist only on the three-dimensional brane. As applied to the considerations behind formulas (49) – (53), this should mean that the vacuum of five-dimensional space is not created by physical fields; it has a different, non-quantum-field nature. In multidimensional physics, as suggested by the above reasoning, the dark energy of the vacuum owes its existence precisely to the presence of extra dimensions in the world: its density depends only on the number and extent of these dimensions – without any connection with quantum fields on the brane. Then the observed EG-vacuum (as a three-dimensional shadow of a “truly fundamental” five-dimensional vacuum) should also have a “geometric” rather than “material” nature.

In multidimensional physics, Einstein’s cosmological constant Λ is also just a shadow of truly fundamental constants; for D argumentative essay outline = 5, the connection between them is as follows:

Λ = 8πR * −4MP − 2 = (M * R *) – 4R * −2. (54)

The hypothesis of additional macroscopic measurements promises new physics at energies close to the "truly fundamental" energy M * ~ MEW ~ 1 TeV. In particular, it predicts the production of gravitons, and possibly black holes in experiments at the Large Hadron Collider. We will not enter into further discussion of this large topic here and restrict ourselves to a reference to the review in Physics ± Uspekhi5.

### 4.4. Four energies

According to Aristotle, who is sometimes called the first physicist, everything in the world consists of four "basic elements" or elements – earth, water, fire and air. As far as is known, during the time of Aristotle, it was not discussed how much water or fire there is in the universe and how the quantities of the elements relate to each other. 1998-1999 in modern cosmology, there are also exactly four elements, or cosmic energies, of which everything in the world is made. The contribution of each of them to the total energy of the world has been measured fairly accurately (see Section 2). Recall that dark energy accounts for about 70%, dark matter – 25%, baryons – about 5%, radiation – a few hundredths of a percent of the total energy of the Universe.

The measured percentages refer to the current state of the world. In the course of the evolution of the Universe, the relative contribution of each of the energies changed due to the general cosmological expansion. For example, the fraction of vacuum was close to zero in the early Universe during the epoch of primordial nucleosynthesis when the world was a few minutes old, and the fraction of radiation was then approaching 100%. In the distant future, the contribution of dark energy will approach 100%, and the contributions of three non-vacuum energies will tend to zero.

A seemingly random and, moreover, time-varying recipe for a cosmic mixture may seem unnatural, monstrously complex, strange or even absurd – such definitions wander in scientific and popular literature. But in fact, behind an arbitrary, as it might seem, set of numbers hides a simple and time-independent pattern, which is a special kind of symmetry6.

Of the four cosmic energies, most of all the above talked about dark energy – this is the main topic for us. We now give a brief summary of information on three non-vacuum energies.

As already mentioned, ordinary matter is nonrelativistic protons, neutrons and electrons; this type of cosmic energy is usually called baryonic. With this common substance, not everything is clear. The main question is: why are there protons and neutrons in the world, but antiprotons and antineutrons are not observed in the same quantities? Indeed, according to one of the general laws of physics, the equality of particles and antiparticles should be observed in nature. The same applies to electrons: their antiparticles, positrons, are very rare in natural conditions.

It is possible that the bias in favor of baryons originated in the early Universe during the era of very high temperatures, when these particles were relativistic. Under these conditions, particles and antiparticles were definitely in equal numbers. But if the symmetry between them was not strict, but weakly broken, then at a certain epoch a small excess of baryons with respect to antibaryons could have formed.

The hypothetical process of the formation of "extra" baryons is called cosmic baryogenesis. Later, when the temperature of the space environment decreased due to the cosmological expansion, the annihilation of the majority of baryons and antibaryons took place. But for the extra baryons, there were no partners-antibaryons for annihilation, and therefore they have been preserved in the Universe to this day. As a result, the initially very small excess of particles in relation to antiparticles turned into an almost one hundred percent predominance of baryons over antibaryons. This way of solving the problem was outlined by A.D.Sakharov7 and V.A.Kuzmin8. In their works, the necessary conditions for the effective course of baryogenesis were clarified, a number of important features of the process were studied; but a complete and final solution to the problem has not yet been reached.

The physics of radiation is much better known: radiation is a remnant, a relic, of a once dense and very hot state of matter in the early stages of the evolution of the Universe. The existence of the relict radiation was predicted by GA Gamov in the 1940s – 1950s and was confirmed by further observational discoveries. Radiation is photons that were in thermodynamic equilibrium with matter and were also very hot in the distant past of the Universe. Then, in the course of cosmological expansion, the radiation cooled down to the currently observed very low temperature – about 3 degrees above absolute zero temperature. At the same time, the photons themselves did not disappear, and their full number has survived to this day. There are a lot of these particles – in the modern era there are approximately 500 relic photons in every cubic centimeter of space. Radiation almost ideally evenly fills the entire volume of the Universe.

### Gamov Georgy Antonovich (1904-1968)

Soviet and American theoretical physicist, astrophysicist and popularizer of science.

Corresponding member of the USSR Academy of Sciences (from 1932 to 1938, posthumously restored in 1990). In 1933 he left the USSR, becoming a "defector". In 1940 he received US citizenship. Member of the US National Academy of Sciences (1953).

Gamow is known for his work on quantum mechanics, atomic and nuclear physics, astrophysics, cosmology, and biology. He is the author of the first quantitative theory of alpha decay, one of the founders of the theory of the "hot universe" and one of the pioneers of the application of nuclear physics to the issues of stellar evolution. He was the first to clearly formulate the problem of the genetic code. Gamow became widely known for his popular science works, in which he tells about modern scientific concepts in a living and accessible language.

The number of nonrelativistic baryons is also preserved during the expansion of the world, but their "piecewise" is much smaller – only about two particles per ten cubic meters of space. The ratio of the number of photons to the number of baryons is a large dimensionless "baryon number" B ~ 109. Mainly because of the ambiguity with antibaryons (see above), the physical nature of this number is one of the difficult mysteries of cosmology and microphysics. As long as baryons and antibaryons remained ultrarelativistic, there were about the same number of photons as there were photons. The reciprocal, 1 / V ~ 10-9, gives a quantitative measure of the weak symmetry breaking between particles and antiparticles in the early universe. The baryon number also serves as a measure of the entropy per baryon9; for this reason, the production of helium in primary nucleosynthesis at a world age of several minutes depends on it, in particular.

As for dark matter, the first hints of its existence appeared in the early 1930s; reliable information was obtained in the 1970s (see review (Einasto J., Einasto M., 2000) 10). In the 1980s, a hypothesis was discussed according to which dark matter is a gas of all kinds of nonrelativistic neutrinos and antineutrinos. Later it turned out that the neutrino mass is too small for this. It is now clear that none of the known elementary particles is suitable for the role of the carrier of dark matter.

Dark matter remains outside the framework of the standard model of elementary particle physics – this model does not provide anything of the kind, for it the existence of dark particles has been and remains a mystery. Dark matter still eludes direct physical experiment, despite many years of efforts in this direction. But it is reliably known that dark matter is about 4-6 times more in mass / energy than baryons. Dark particles fill huge volumes around galaxies, groups and clusters, forming dark corona, or halo.

According to the widespread point of view, the role of carriers of dark matter would be most suitable for so far unknown elementary particles with a rather large mass. They have already found a name – WIMPs (see section 2). Unlike protons and neutrons, these particles do not feel strong nuclear forces, but, like electrons, participate in the electroweak interaction. Dark particles are considered stable and persist during cosmological expansion. Perhaps these particles are, for example, the smallest in mass supersymmetric partners of such particles as a photon or a graviton (the latter is considered more plausible); then the dark particles would be fermions and, according to the rule accepted in particle physics, they would be called photino or gravitino.

As we can see, not so much is known about cosmic energies. Important questions about their physical nature remain largely unanswered. Nevertheless, each of the energies can be described macroscopically as a medium with a certain density and pressure. The densities of cosmic energies are measured in observations (see Section 2.6). The relationship between density and pressure, that is, the equation of state, for each of these media is also known. Baryons and dark matter are nonrelativistic (at least after the early era of nucleosynthesis); therefore, their pressure is very small compared to the energy density, so that it can simply be assumed to be zero. Radiation is an ultrarelativistic medium, and its pressure is one third of the energy density. In the case of a vacuum, the pressure, as we know, is negative and is equal to the density of dark energy, taken with a minus sign.

Knowing the equation of state of a given cosmic energy, one can determine how it behaves during the expansion of the Universe. This is indicated by one of the most general laws of nature – the law of conservation of internal energy, which appears in cosmology as the second Friedmann equation (see Section 2). It follows from this law that the total number of particles in a given expanding volume does not change with time (as we have already said). However, the latter is understandable: since this is a "accompanying" volume, and the particles are stable, it always contains the same conserved particles. The latter refers to particles of all three non-vacuum energies – baryons, photons and neutrinos, dark particles.

As for the dark energy of the vacuum, there are no (real) particles in it, and the conserved quantity is simply its density: the EG vacuum does not change at all during cosmological expansion.

### 4.5. Symmetry

The presence of physical characteristics of cosmic energies that persist in time makes it possible to write down the recipe for a cosmic mixture not in percentages that change during the cosmological expansion, but through constant values:

AV ~ AD ~ AB ~ AR ~ 1060 ± 1MP− !. (55)

Here, each of the four constants, called Friedmann’s integrals, represent a specific cosmic energy: vacuum dark energy (AV), dark matter (AD), baryons (AB), and radiation (AR). Friedmann’s integrals are approximately equal (within two orders of magnitude); their numerical value is given in "natural units", in which c = ħ = 1.

The place of Friedmann integrals in cosmology can be seen from the first Friedmann equation (11), which expresses the law of conservation of mechanical energy (see Section 2). If all four cosmic energies are explicitly taken into account in equation (11), then it looks like this:

\$\$\ dot R ^ 2 = \ left (\ frac {A_V} {R} \ right) ^ {- 2} + \ frac {A_B} {R} + \ frac {A_D} {R} + \ left (\ frac { A_R} {R} \ right) ^ 2. (56)\$\$

Equation (56) is the equation of the standard (ΛCDM) cosmological model with flat three-dimensional space and parabolic (E = 0) dynamics. Here R (z) is a scale factor (a function of time or redshift z), in proportion to which all cosmological distances change:

R (z) = R0 (1 + z) −1, R0 = 3 × 1060MP− !. (57)

The modern (z = 0) value of the scale factor in the normalization adopted here is close to the value of the Hubble radius R (z = 0) = R0 ~ H0−1. The expanding region of the world with a size of ~ R (z) is often called "our cosmic domain" or the Metagalaxy.

Friedmann integrals are integrals in the exact sense of the word: they are integration constants that arise when solving the second Friedmann equation (18). Recall that equation (18) expresses the law of conservation of internal energy, and cosmic energies are treated macroscopically as media with a certain relationship between pressure and density. For S = 0, Eq. (18) for each cosmic energy separately has the form

\$\$\ frac {\ dot {\ rho}} {\ rho \ left (1 + w \ right)} = – 3 \ frac {\ dot {R}} {R}, (58)\$\$

where ρ is the density of this energy, w is the ratio of the pressure of this energy to its density, so that w = −1, 0, 0, 1/3, respectively, for dark energy (EG vacuum), dark matter, baryons and radiation.

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