Density of Vacuum-Like Plasma and Hubble Constant


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.

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

1. Introduction

As shown in the article [1] , under certain conditions electron-positron plasma has the zero density of an enthalpy. For this plasma the ratio is fair


where is the energy density and is the pressure.

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 satisfies the conditions


the energy density and pressure of a random electromagnetic field are related to the temperature by the formulas



where and are the Fermi momenta of electrons and positrons; and are the mass and the electric charge of the electron; is the velocity of light in vacuum; is the Boltzmann’s constant; is the Planck’s constant divided by.

In order that plasma with such properties is vacuum-like [2] , it is necessary to impose the additional requirement of electro-neutrality [3] . In this case, the plasma becomes non-equilibrium. The sum of chemical potentials of the electrons and of the positrons is equal to zero


and their difference is expressed by the ratio


where is the fine-structure constant approximately equal to 1/137. In this case, Fermi momenta of electrons and positrons


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




and are the concentrations of electrons and positrons. If to assume that absolute value of plasma’s energy density is equal to the density of dark energy, the following estimations can be obtained:, , [1] .

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.

2. Equations of Particles Balance in Non-Equilibrium Plasma

The equations of electrons and positrons balance may be written in the following form [4] [5]



where and are the flows density of electrons and of positrons;; is a time and is a vector of spatial coordinates;, and, are the concentrations and spatial flows density of electrons and positrons; is the velocity of the generation and of the annihilation of particle-antiparticle pairs.

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



Here is electron momentum four-vector or positron momentum four-vector for which the zero component is represented in the form


is the Fermi-Dirac distribution function;


and are the hydrodynamic velocities of particles and antiparticles [4] , for which the relations are satisfied


The left parts of Equations (7.1) and (7.2) depend on coordinates and time using chemical potentials, and using hydrodynamic velocities,. Spatial components of hydrodynamic velocities may be defined by [5] [6] .


Components and are calculated from the expressions (9.3). Times of momentum relaxation for electron and for positron are positive values. They define the electric resistance of medium [6] [7] . Their physical nature and numerical values are determined by stochastic scattering of electrons and positrons [7] .

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 is a function of the difference between the chemical potentials of electrons and positrons, and equal zero in state of chemical equilibrium. We will choose this function in the form





are the invariant density of electrons and positrons; is the coefficient of two-photon annihilation of electron and positron, its dimension is cm3∙c−1. Such choosing of function provides a relaxation of solutions of Equations (7.1) and (7.2) to the state of local chemical equilibrium (6).

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 [8] for estimations. It follows that the expression for coefficient




is the classical radius of the electron, which is approximately equal to. The corresponding estimation for the coefficient is correct. According to Dirac [8] this result is true if we neglect the interaction between particles and antiparticles.

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






Time is approximately equal to and Compton wavelength is about. According to paper [1] the estimations

is correct. It means

Such big times and diffusion length are caused by a factor, i.e. by the low density of plasma.

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.

3. Expansion of the Universe and Non-Equilibrium Plasma

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 [6] , the dependence of concentration on time caused by the external reasons. The last hypothesis seems to be the most realistic.

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. We assume that the number of electrons and equal number of positrons of the researched plasma do not depend from space and time coordinates. In this case, the expansion of the Universe must lead to a change of particles’ concentration due to the increase of volume occupied by electrons and positrons.


Taking into account the weak dependence of from time, generated by sources (21) non-equilibrium may be considered practically stationary.

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



According to the expression (4.2) exceeds a unit at a size about, then we will find plasma density from the Formula (22.2)


If to use defined by the expression (18.1) as coefficient, then from relationship (23) we obtain

The absolute value of this density is about nine time less than experimental result [9] .

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 [1]




Here is a vector-potential of a random electromagnetic field;


is a tensor of this field; is Minkowski tensor;


is a difference between flows of electrons and positrons.

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




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

The tensor according to the Formula (24.1) may be represented by the sum




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 [10]


Considering that

we will find


The density of potential defines the contribution of a random electromagnetic field to the scattering matrix of spinor particles [10] .

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







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 [1] . Evaluations follow from Formulas (28.3)-(28.5)

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 attraction dominates at distances more than. i.e. defines the distance of stable equilibrium.

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 is huge, it is about. Ratio of distances is huge too-about 1034. Physical interpretations of distance and distance are different. The distance corresponds to the boundary of a region outside of which electrons and positrons run up.

The distance is exactly three times less than the classical radius of electron defined by Formula (18.2). A simple replacement to leads to “proper” density value. But such simple change needs reasoning.

According to the condensed matter theory [11] [12] the Heisenberg operators of electron component and positron component of vacuum-like plasma are solutions of the equations



Here are the Dirac matrix and


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 is replaced to effective mass equal. If to use the solutions of Equations (34.1) and (34.2) with positive energy for description of quasiparticles states and for construction of causal Green function we will obtain the well known expression for cross section of two-photon quasiparticles annihilations [10] in which will replaced to. In non-relativistic limit for probability of this process we will obtain the formula


Exactly this result will be obtained if you make a change in Formula (18.1).

Using defined by Expression (35) from relationship (22.2) for non-equilibrium level of plasma we will obtain


Assuming that from Formulas (36), (23) and definitions of dark energy density and cosmological constant [3] we will find





Here is the Newtonian gravitation constant. From Expression (37.2) for concentration follows and from Formula (37.4) for cosmological constant follows.

From the Formula (37.2) new representations for distance can be obtained


And the ratio of distance to distance


where is the distance which may be interpreted as the characteristic size of the Universe; is very small mass that was call the “Hubble mass” [13] .

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. It corresponds to space in the form of a one-dimensional wire.

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.

4. Plasma Fluctuations and Acceleration

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

where is the radius of the world and is its second derivative on time,


is the critical matter density in the Universe that approximately equal. According to modern interpretation of astrophysical data the dark energy density is and we obtain

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 if is about megaparsec.

The positive density of dark energy follows from interpretation of astrophysical observations of supernovas Ia type in terms of ΛCDM model [14] . But it is not the only way of an interpretation of observed data. For instance, in article [15] was demonstrated that these observations may be interpreted in the terms of decelerated Universe expansion if assumed that in intergalactic space the index of light’s refraction is about 1.50.

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 [16] . For example, it may be the model, in which the hypothetical massive classical gravitons are introduced [13] .

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 we will present in form

Where according expressions (5.3) and (37.2) satisfies of formula

Value of corresponds to concentration of plasma particles. For from Equations (7.1) and (7.2) we obtain


If fluctuations are small

the simple equation is true




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 correlates with the characteristic time of cosmic gamma-ray burst [17] . For we obtain

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 in the time moment the fluctuation of chemical potential

has occurred. The positive correspond to productions of plasma quasiparticles pairs and the negative correspond to annihilations of quasiparticles pairs. Thus for we will obtain


is the error function. From these expressions for velocity and acceleration of fluctuations flow along the line we will obtain the formulas

Let’s estimate in the case

As the result we will obtain

Absolute value of acceleration is proportional of dark energy density and correlates with absolute value of acceleration if.

From assumption that supernova explosion is connected with vacuum-like plasma fluctuations and it produced the plasma quasiparticles annihilation follows that. It corresponds to accelerated motion of fluctuations to the center of fluctuations area and accelerated motion of fluctuation from external observer. By more complex hypotheses it would be possible to explain the dependence of acceleration of fluctuations from distance to the observer also. But we have not any reasons for such speculations. There is enough that a vacuum-like plasma fluctuations can lead to the accelerated movement of a source of electromagnetic radiation to the right direction.

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.

5. Conclusions

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 1021 cm, this repulsion exceeds a Coulomb attraction.

Thus, two spatial scales characterize the considered vacuum-like environment: repulsion () and attraction (). They differ by thirty-four orders ().

Analyses of small fluctuations in vacuum-like plasma demonstrated that acceleration of such fluctuations may be very big in time period about. It means that vacuum-like plasma fluctuations can lead to the accelerated movement of a source of electromagnetic radiation. Their influence on the radiation of supernova Ia type demands the additional research. It is well known that interaction between cosmic objects and electron-positron plasma may produce very interesting phenomena [18] .

Conflicts of Interest

The authors declare no conflicts of interest.


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