Information and Cosmological Objects

Estimation of the volume of information in cosmological objects, including stars of the Sun type, neutron stars, white dwarfs, black holes is necessary for generation of restrictions for their formation, development and interconversion. Information is an integral part of the Universe. The basic law of Zeilinger’s quantum mechanics postulates that the elementary physical system (in particular, fundamental particles: quark, electron, photon) bears one bit of information. By its physical essence information is heterogeneity of matter and energy. Therefore information is inseparably connected with matter and energy. An information approach along with a physical one allows to obtain new, sometimes more general data in relation to data obtained on the ground of physical rules only. The author’s works, testify about the practicality of information laws usage simultaneously with physical rules for cognition of the Universe. The results presented in this paper show the effectiveness of informational approach for studying the cosmological objects. In future the proposed models and estimations should undoubtedly be specified and presented in more detailed way. One should point out that informational approach allows formulating restrictions on the valuations of physical systems characteristics and physical processes while the physical methods and models can describe not only the restrictions but also concrete physical “mechanisms” of restriction formation, concrete valuations of physical systems characteristics.


Introduction
Estimation of the volume of information in cosmological objects, including stars of the Sun type, neutron stars, white dwarfs, black holes is necessary for generation of restrictions for their formation, development and interconversion.
Information is an integral part of the Universe. The basic principle of Zeilinger's quantum mechanics [1] postulates that the elementary physical system (in particular, fundamental particles: quark, electron, photon) bears one bit of information. By its physical essence information is heterogeneity of matter and energy. Therefore information is inseparably connected with matter and energy. The universal measure of physical heterogeneity of information is the Shannon information entropy [2,3]. It is important to note that the Neumann entropy cannot be applied as the universal measure of heterogeneity because it is equal to zero for structured pure state. An information approach along with a physical one allows to obtain new, sometimes more general data in relation to data obtained on the ground of physical rules only. The author's works, for instance the [4,5] testify about the practicality of information laws usage simultaneously with physical rules for cognition of the Universe. The estimates cited below are based on the foundational principle of Zeilinger quantum mechanics i.e. "an elementary system carries one bit of information" and prove it. The elementary systems are fundamental particles (quarks, leptons, photons). Let us evaluate the volume of information in the system of n q-bits. At first we consider the systems with equiprobable basic states.

Non Interacting Q-bits in the System
Suppose that the system contains n non-interacting q-bits. Let the q-bit be described by the wave function 1 ( 0 1 ) 2 ψ = + , where 0 , 1 -are the basic states of the q-bit [6]. While measuring the q-bit we'll obtain the basic states 0 , 1 with equal probabilities 1 / 2 . Uncertainty (information) of the q-bit in the state ψ is equal to 1 bit: Hence, the volume of information in the system containing n non-interacting q-bits with equiprobable basic states is proportional to the amount of q-bits and is equal to n bit.
This estimate determines the minimum volume of information in the system consisting of n q-bits with equiprobable basic states. It also explains the linear dependence of the volume of information on the mass or number of particles (elementary systems) in the usual substance (fundamental particles -quarks, leptons, photons).

The System Having N Basic States
Let us consider the case when the objects in the system are specified by the wave functions This dependence characterizes the neutron stars and white dwarfs [7]. The neutron stars and white dwarfs appear to be degenerate fermionic systems that fill in a zone of n states i e .

Q-bits Pairwise Interaction in the System
Suppose that a system contains n pairwise interacting q-bits with equiprobable basic states. The system consisting of n interacting particles (q-bits) can be described with the following wave function: i i -basic states of the i -th q-bit). Information on relationships between each pair of interacting q-bits , i j is equal to one unit (one bit) [8]. Let us demonstrate this. Under the linked (intricate) status of q-bits , i j the status of q-bit j is absolutely certain if the status of q-bit i is known, and vice versa the status of q-bit i is absolutely certain if the status of q-bit j is known.
The volume of information in the link of a system of n pairwise interacting q-bits with equiprobable basic states described with the wave function bit. The volume of information in the system of n pairwise interacting q-bits is formed from n bits of information available in the q-bits and ( 1) 2 n n ⋅ − bit of information on the link between the q-bits. The total volume of information in the system containing n pairwise interacting q-bits with equiprobable basic states is equal to ( 1) ( 1) 2 2 n n n n n I n ⋅ − ⋅ + = + = bit. This estimate determines the maximum volume of information in the system consisting of n q-bits with equiprobable basic states. In the system containing n pairwise interacting q-bits with equiprobable basic states the volume of information is proportional to squared number of q-bits and is equal to The volume of information in a system of n pairwise interacting q-bits with equiprobable basic states is n bit more than in a system consisting of 1 n − q-bit, 1 ( 1) ( 1)

Local Q-bit Interaction in the System
Let us consider a case when in the system of n q-bits there stand out n k groups of k q-bits, each of k q-bits interacting only with the q-bits of each group (suppose that n is divided by k ). Then the volume of information k I in the group of k pairwise interacting q-bits with equiprobable basic states is equal to bit. Hence the system under consideration consisting of n q-bits contains bit. This explains a linear dependence of the volume of information on the mass in the systems composed of compound particles of usual substance (for instance, in the systems composed of elementary particles -mesons, baryons, atoms, molecules, gas, … ). At 1 k = the system contains the minimum information content:

Restrictions on the Volume of Informationin the System with Equiprobable and Non-equiprobable Basic States
In the general case, the volume of information n I in the system of n q-bits with equiprobable basic states is no less than n bit and no more than Uncertainty (information) of the q-bits in the state ψ is equal to In the general case, the volume of information n I in the system of n q-bits is larger or equal to zero bits and is not  [9]. The information volume contained in the black hole is proportional to its squared mass. How to explain it? Let us assume that a black hole contains n pairwise interacting particles (q-bits). Then the quadratic dependence of the volume of information in the black hole on its mass can be explained by the fact that each interaction forms 1 bit of information. The black hole is described with the wave Therefore, a black hole is the aggregate of particles (let us call them black particles) each having a mass equal to 0.23th of Planck mass) and interacting with all other black particles that form a black hole [5,10].

Emission and Absorption of Usual Substance by a Black Hole
Suppose that at the initial instant of time a black hole consisting of n black particles has the mass of For further estimates we implement the law of conservation of uncertainty (information) [4,5] and energy conservation principle. According to the law of conservation of uncertainty (information) a change in the system "a black hole with the mass 0 n M n m = ⋅ -external environment" on emission of one black particle must be balanced by the occurrence of n particles containing 1 bit each: In the case of a black hole containing one black particle the radiation frequency is maximal and in inverse proportion to Planck time unit. Similar dependences are true for absorption of photons by black holes. Having the estimates of black holes distribution by mass one can calculate the intensity of aggregated distribution of black holes radiation by frequencies and compare them with the experiment results. From the obtained radiation frequency expression one can draw the estimate of black hole radiation temperature.
Let us calculate the radiation temperature From the above chart it follows that: 1. The mass of a black hole formed during the supernova explosion is close to the mass of an optimal black hole in the system "a black hole -radiation". One can expect that during the supernova explosions the black holes be formed having the masses under which the information volume in the adjacent space is close to minimum.
2. For formation of the black holes with the mass equal to million masses of the Sun the volume of information exceeding the volume of information of the Universe (1090 bit) is required [4,5,9,14]. Significant volumes of information are needed for formation of the black holes with the same mass as of the Sun's. In such case the volume of information of about 1076 is required. It means that locally (in the zone of black hole formation) there must take place the intensive physical processes of radiation formation. For instance, the supernova explosions and accelerated motion of relativistic particles.

The Volume of information in the Neutron Stars and White Dwarfs. The Information Model of a Neutron Star and White Dwarf
When evaluating the volume of information in the neutron star we must take into account the volume of information in the structure of the star and in the neutrons. In the neutron fermi-gas during its complete degeneration all the low energy levels are filled in up to the Fermi level while all the subsequent ones remain empty. Temperature rise can change distribution of neutrons in the levels to only a small extent: a small fraction of neutrons sitting in the levels close to the Fermi level pass to the empty levels possessing bigger energy and clear the levels from which the migration took place.
The volume of information in the white dwarfs and neutron stars. When evaluating the volume of information in the neutron star we must take into account the volume of information in the structure of the star and in the neutrons. The volume of information in the structure of degenerate fermi-gas is equal to N*log2 (N) bit. Here N -is the number of filled in energy levels (number of neutrons). In the neutron star N=(M/m), where M -is the mass of neutron star, m -is the mass of neutron. The volume of information in one neutron is about 9,422 bit. The volume of information in the neutrons of the star is equal to 9,422*N. Aggregated volume of information in the neutron star (without considering the information in the crust nickel and iron) is approx. 9,422*(M/m) +(M/m)*log2(M/m) bit. The volume of information in the neutron star is proportional to the mass multiplied by the logarithm of dyadic mass. In a similar way one can evaluate the volume of information in the white dwarfs. Recall for the sake of comparison that the volume of information in the usual substance (non-interacting particles, gas) is proportional to mass, the volume of information in the black hole is proportional to squared mass. ). The most commonly encountered are the white dwarfs consisting of carbon and oxygen with helium-hydrogen shell [7,15]. Under the masses of white dwarfs 0.6 Мsun -1.44Msun, with the radiuses equal to the Earth's the surface temperature can be relatively high (from 100,000 K to 200,000 K). The main feature of their construction is a nucleus, its gravity equilibrium being supported by degenerate electron gas whose properties do not allow any further modifications of its structure. The degenerate gas pressure puts into equilibrium the gravitation (under prescribed mass) and the loss of heat resulting from non-degenerate component of the substance does not change this pressure and the losses alone are relatively insignificant.
The fate of the supergiant star remnants depends on the mass of remaining nucleus. When hydrostatic equilibrium breaks down there occurs gravitation collapse (lasting for seconds of fractions of seconds) and if Мkern<1.4Мsun, then the nucleus will shrink up to the Earth dimensions and a white dwarf is produced. If 1.4Мsun<Мkern<3Мsun, then the pressure of incumbent layers will be so strong that the electrons are "forced into" protons thus generating neutrons and emitting neutrino e p e n ν + − + → + . The so-called a degenerate neutron gas is generated. The pressure of degenerate neutron gas halts the subsequent shrinkage of the star. However, it appears that part of the neutron stars are formed during supernova outbursts and appear to be the remnants of massive stars that had exploded as the Supernova of the second type.
For evaluating the volume of information available in the white dwarf of the mass of the Sun we assume that the white dwarf contains more or less equal number of atoms of carbon and oxygen. Then the total number of atoms in the white dwarf of the mass of the Sun is equal to  Note that the given estimate does not take into account the availability of other elements in the neutron stars that decrease the radiation temperature. Similar dependencies are true for the cases when the neutrons are absorbed by neutron stars. The estimate of radiation temperature of the white dwarf of the mass of the Sun was derived by using information method, which is similar to estimations of black hole and neutron star radiation temperature and is equal to about 3E+07K.
According to the law of conservation of uncertainty (information) the change in the system "neutron star of the The frequency of each absorbed photon must be equal to

The Information Volume of the Sun. Information Model of a Star of the Sun Type
The Sun consists for the most part from hydrogen (~74% of its mass) and helium (~25% of its mass). The number of hydrogen atoms  Table 3 presents the masses of black holes. Estimations were based on the assumption that the black holes had been formed from the neutron star, white dwarf of the Sun type.  kg. One can see from the table 3 that the mases of the black holes that can be generated during formation of black holes from neutron star, white dwarf, star of the Sun type are close to the mass of the optimal black hole (in the system "black hole -radiation") -8,08E+22kg.

Black Hole Merger
Let us consider a system consisting of the two black holes. Suppose that the black holes have the mass of 1 . Apparently the volume of information in a new black hole does not coincide with the aggregated volume of information in the original black holes. We give a system of equations for estimation of the mass of a black hole under formation when the two black holes are merged.
Let us suppose that the black holes before merging possess the mass of  x Us = . The average mass of a particle of usual substance is equal to m .
From the law of conservation of uncertainty (information) it follows: We believe that the main energy of black holes and usual substance is concentrated in the mass. Then from the energy conservation principle it follows: Thus we have a two-equations system  «Fourteen billion years ago at birth of the Universe it was enclosed in the point with the radius of 10 -33 cm, which is incommensurably smaller then the proton radius -10 -13 cm. In that volume all the information about the future of the Universe had already been built-in. The Big Bang occurred» (A.Cherepashuk [19]). 10 -33 cm -is the size of a Planck particle. The Planck particle contains one Nat of information (≈ 1,45 bit) while the information about the Universe contains not less than ≈ 10 14 bit of classical information. Hence, all the information about the future of the Universe was enclosed in the Universe segment of the radius larger than 10 -33 cm. It is not impossible that similar informational considerations would allow to prove either independent development of the Universe determined by the information (heterogeneity) contained in it, or the presence of additional external control executed from outside of the Universe.

Conclusion
The results presented in this paper show the effectiveness of informational approach for studying the cosmological objects.
1. We consider the following system of q-bits: non interacting q-bits in the system, the system having n basic states, q-bits pairwise interaction in the system, local q-bit interaction in the system.
2. It is shown that in the general case, the volume of information I n in the system of n q-bits is larger or equal to zero bits and is not larger than n(n+1)/2 bit.
3. It is shown that the information volume contained in the black hole is proportional to its squared mass. A black hole is the aggregate of particles (black particles) each having a mass equal to 0.23th of Planck mass and interacting with all other black particles that form a black hole. The estimates of black holes distribution by mass one can calculate the intensity of aggregated distribution of black holes radiation by frequencies and compare them with the experiment results. From the obtained radiation frequency expression one can draw the estimate of black hole radiation temperature. Are given information characteristics of black holes of different masses.
4. The neutron star with the mass of the Sun contains 59 10 32 , 2 ⋅ ≈ bit. 5. The white dwarf with the mass of the Sun contains 59 1, 24 10 ≈ ⋅ bit. Estimate of dependence of the temperature of neutron star radiation on its mass (number of neutrons in the star).
6. The Sun contains 58 10 3 , 1 ⋅ ≈ bit of information. 7. The mases of the black holes that can be generated during formation of black holes from neutron star, white dwarf, star of the Sun type are close to the mass of the optimal black hole (in the system "black hole -radiation"). 8. A black hole cannot be created by means of merging some black holes and only some black holes. Merging of black holes can occur only with the absorption and emission of usual substance. The mass of a black hole that was formed as a result of the merger of 2 black holes of the same masses without using any additional usual substances is 2 times less than the sum of masses of merging black holes. The remaining mass is dissipated in the space. 9. All the information about the future of the Universe was enclosed in the Universe segment of the radius larger than 10 -33 cm. 10. In future the proposed models and estimations should undoubtedly be specified and presented in more detailed way.