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The model in which expansion of the Universe leads to a generation of non-equilibrium vacuum-like electron-positron plasma is proposed and researched. The formulas that relate the Hubble’s constant with the concentration of plasma particles and the cosmological constant are obtained. The collective properties of vacuum-like plasma are investigated. It is shown, that the coefficient of a two-photon annihilation in such plasma is nine times less than for the free particles. A simple formula for dark energy density as a function of electron mass and charge is obtained. It was demonstrated that acceleration of plasma’s chemical potential fluctuations flow proportional of dark energy density.

As shown in the article [

where

Unusual properties of such material medium are caused by the existence of the random electromagnetic field generated by transitions between various quantum states of electrons and positrons. If a constant temperature

the energy density

where

In order that plasma with such properties is vacuum-like [

and their difference is expressed by the ratio

where

are equal

and are connected with temperature by the equation

Equation (4.5) is the consequence of the relation (1).

The electron-positron plasma, which satisfies the conditions (4.1)-(4.5), together with random electromagnetic field, is the vacuum-like material medium that has a zero enthalpy

and zero entropy

too. Due to these properties, it can play the role of dark energy. However, the energy density of researched medium is negative and its pressure is positive.

The total energy density and pressure are expressed in the following form:

For the energy densities of electrons, positrons and random electromagnetic field the formulas are true

where

If the electro-neutral plasma is in the state of chemical equilibrium, the following condition

is satisfied. Then the equality to zero of chemical potentials of electrons and positrons follows from the relationship (4.1) and equality to zero of their Fermi momenta from the relationship (4.3) follows too. If the density of plasma’s enthalpy is equal to zero then the relationship (4.5) is satisfied. In the case of the equilibrium state of plasma, Fermi momenta, the temperature, energy density, pressure, electron concentration and positron concentration are equal to zero. It means that vacuum-like plasma does not exist in an equilibrium state.

Thus, the necessary condition of existence of the vacuum-like electron-positron plasma is chemical non-equilibrium. The possible causes of non-equilibrium demand the clarification. The present article is devoted to this problem.

The equations of electrons and positrons balance may be written in the following form [

where

In the absence of external electromagnetic fields and the constant temperature density flow, expressions are true

Here

is the Fermi-Dirac distribution function;

The left parts of Equations (7.1) and (7.2) depend on coordinates and time using chemical potentials

Components

Taking into account definitions (10) left parts of the Equations (7.1) and (7.2) depend on the coordinates and time only through the chemical potentials and their derivatives. Electrons and positrons interact with each other by emitting and absorbing electromagnetic fields. As a result at a constant temperature and under the condition of absence of external interaction, the Equations (7.1) and (7.2) must describe the relaxation of the system to the state of the local chemical equilibrium, which is determined by equality (6). If chemical potentials do not depend on coordinates and time, global chemical equilibrium reached.

Using the analogy with the theory of charge transport in mesoscopic structures we will assume that

where

are the invariant density of electrons and positrons; ^{3}∙c^{−1}. Such choosing of function

The characteristic relaxation times of the concentration fluctuations determined by expressions

where indicated

For relaxation time of chemical potentials difference the formula are fair

By using, the definition (10) can be obtained the expression for characteristic distance of relaxation to zero of the difference of the chemical potentials

In researched vacuum-like plasma when the relationships

are fair we obtain

The Formula (16.3) was obtained from the expression (15) with assumption that

We will use the formula for a cross-section of two-photon annihilation that was obtained by Dirac at 1930 [

where

is the classical radius of the electron, which is approximately equal to

Using Formulas (18.1) and (18.2) from expressions (16.1)-(16.3) we will obtain

where

is correct. It means

Such big times

The characteristic relaxation time of the concentration fluctuations is about the age of the Universe. The time of restoration of chemical equilibrium exceeds the age of the Universe by eight orders.

The obtained results show, that in normal scales of times small deviations from the state of chemical equilibrium of electron-positron plasma can be considered as stationary. However, it relates to small fluctuations for which the estimation is fair

The presence and long time existence of such fluctuations can't be the cause of high level of non-equilibrium plasma

defined by Formula (4.2).

Various external influence scan be sources of non-equilibrium plasma. For example, it may be the external electromagnetic radiation, current flow through the regions with high gradients of concentrations of plasma particles [

According to the modern vision, in the current period of the cosmic history, the spatial size of the Universe increases as a function of time. The rate of this expansion is measured by Hubble's constant

Taking into account the weak dependence of

From the Equations (7.1), (7.2) and the Formula (21) for vacuum-like plasma we obtain

According to the expression (4.2)

If to use

The absolute value of this density is about nine time less than experimental result

The electron-positron annihilation probability defined by the Formulas (18.1) and (18.2) doesn't consider the additional interaction between particles caused by random electromagnetic field and collective properties of plasma. In Introduction the formulas for the energy density of random electromagnetic field are presented. They include energy density of random field oneself and energy density of its interaction with electrons and positrons too.

The energy-momentum tensor of a random electromagnetic field, which includes the interaction with particles and antiparticles, can be represented in the form [

Here

is a tensor of this field;

is a difference between flows of electrons

Energy density and pressure of random electromagnetic field are calculated from relationships

where

is a tensor averaged by random processes of transitions between states of particle-particle, antiparticle-antiparticle, particle-antiparticle.

The tensor

where

are the average energy-momentum tensor of the electromagnetic field and of it’s interaction with electrons and positrons.

Using the relationships (25.1), (25.2) and (26.1), (26.2) it is possible to extract density of electromagnetic interaction potential from total expressions [

Considering that

we will find

The density of potential

Energy density and pressure of vacuum-like plasma may be represented in the form

where

are the contributions of transitions of electron-electron, positron-positron, electron-positron. The main contribution to the total energy density and the total pressure of a random electromagnetic field give the electron-electron and positron-positron transitions [

According to the structure of relationships (28.1) and (28.2) from (27.2), we obtain

These expressions demonstrate that the attraction between particles with an identical sign of the electric charge is the result of random electron-electron and positron-positron transitions. The density of attraction potential is three times more than energy density of particles.

Random electrons-positrons transitions lead to repulsion between particles with different sign of the electric charge. The density of repulsion potential is much less than the energy density of plasma components.

Strong attractive interaction must lead to unusual properties of electrons and positrons gasses. For example, for researched environment the characteristic distance can be defined

at which the attraction between electron and electron or between positron and positron is balanced by Coulomb repulsion between these particles. The repulsion dominates at distances less than

The characteristic distance

may be defined for interaction between electrons and positrons too. This is the distance at which force of Coulomb attraction force is equal to the force of repulsion force caused by random transitions. Distance ^{34}. Physical interpretations of distance

The distance

According to the condensed matter theory [

Here

Using the Expressions (4.1), (4.2) and (29.1), (29.2), (29.3) from Equations (32.1) and (32.2) we will obtain the approximate equations

Expressions (34.1) and (34.2) describe the quasiparticles of researched plasma. They are equal to the equations for a usual free electron and free positron operators in which the electron mass

Exactly this result will be obtained if you make a change

Using

Assuming that

Here

From the Formula (37.2) new representations for distance

And the ratio of distance

where

Obtained Formulas (37.1)-(37.4) and (38.1), (38.2) represent the characteristics of vacuum-like plasma as the functions of the fundamental constants and the mass of plasma components. It is interesting that obtained expressions for dark energy density and for cosmological constant do not include the Planck constant.

The Expression (37.1) for density of vacuum-like plasma can be represented in form

This representation is strong spatial anisotropic because

Thus, the expansion of the Universe must lead the electron-positron plasma to non-equilibrium state. Balance equations of plasma particles connect the concentrations of electrons and positrons with Hubble’s constant.

In frameworks of ΛCDM model of cosmology for acceleration of Universe expansion the formula is true

where

and

is the critical matter density in the Universe that approximately equal

that correspond to accelerated expansion of Universe.

For researched vacuum-like electron-positron plasma we obtain the another values

that correspond to decelerated expansion of Universe. In all cases

The positive density of dark energy follows from interpretation of astrophysical observations of supernovas Ia type in terms of ΛCDM model [

The phenomenon of supernovas positive acceleration can be explained not only with the help of dark energy concept. This result may be obtained by the different versions of the Gravity Theory which are alternative to General Relativity [

In this context, the hypothesis of non-gravitational nature of supernovas Ia acceleration may be considered too. For example, this acceleration may be produced by fluctuations of vacuum-like plasma that were connected with supernova explosion. The fluctuations flow is the function of the time. It means that the fluctuations move with acceleration. Of course, the source of electromagnetic radiation in this case move with acceleration too.

Let’s assume that fluctuations not destroy the plasma electroneutrality and condition

is true. The chemical potential

Where according expressions (5.3) and (37.2)

Value of

If fluctuations are small

the simple equation is true

where

is a coefficient of diffusion in vacuum-like electron-positron plasma.

For degenerate nonequilibrium plasma it is logical to assume that

This value is about 4.52 s. It is interesting, that value of

From Equation (41) follows that the characteristic time of relaxation of plasma chemical potential in spatial independent case is

In stationary case for characteristic length of a relaxation the formula is true

Let’s assume that in the spatial area

has occurred. The positive

where

is the error function. From these expressions for velocity

Let’s estimate

As the result we will obtain

Absolute value of acceleration

From assumption that supernova explosion is connected with vacuum-like plasma fluctuations and it produced the plasma quasiparticles annihilation follows that

In the conclusion of this paragraph we note that in the case of “the big fluctuation” when

Equation (40) becomes unstable. Its solutions increase with increase of time. This situation needs the special research outside the frameworks of this article.

Presented theoretical results demonstrate that the expansion of the Universe must lead to vacuum-like electron-positron plasma generation. This plasma may exist under the condition of violated chemical equilibrium between electron gas and positron gas only. The difference between chemical potentials of electrons and of positrons describes the level of plasma non-equilibrium. This difference turns out to be proportional to the Hubble’s constant and inversely proportional to the plasma temperature in two degrees.

Researched plasma is the material medium that consists of the electron gas, positron gas and a random electromagnetic field caused by transitions between different possible states of particles and antiparticles. Under special conditions, the absolute value of the energy density of random electromagnetic field is more than energy densities of electron gas and positron gas. In this case, the components of plasma acquire the collective properties different from the properties of an ideal gas.

The probability of two-photon annihilation of quasiparticles in this vacuum-like medium is nine times less than for free electrons and positrons. Due to this circumstance, the absolute value of plasma energy density is equal to the density of dark energy obtained as a result of interpretation of the astrophysical measurements. In researched model, the density of dark energy relates with electron mass, electron charge and Hubble constant by very simple formula

The cause of decreasing of the annihilation probability is the attraction that created by a random electromagnetic interaction between particles with the identical sign of the electric charge. The region of the prevalence of Coulomb repulsion is reduced in three times due to this attraction.

Between particle and antiparticle having electric charges of different signs, the random electromagnetic interaction creates the repulsion. It doesn’t exert a noticeable impact on the probability of annihilation. However, under the huge distances between an electron and a positron which is about 10^{21} cm, this repulsion exceeds a Coulomb attraction.

Thus, two spatial scales characterize the considered vacuum-like environment: repulsion (

Analyses of small fluctuations in vacuum-like plasma demonstrated that acceleration of such fluctuations may be very big in time period about

Obukhov, I.A. (2017) Density of Vacuum-Like Plasma and Hubble Constant. Journal of High Energy Physics, Gravitation and Cosmology, 3, 572-587. https://doi.org/10.4236/jhepgc.2017.34044