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Collisionless self-gravitating statistical systems of scalar interacting particles

This paper is devoted to consideration of the theory of collisionless statistical systems with interparticle scalar interaction. The mathematical model of such systems is constructed and the exact solution of Vlasov equation for isotropic homogenous model of the Universe is found. Asymptotic solutions of self-consistent Vlasov - Einstein model for conformally invariant scalar interactions are found.


Introduction
In series of Author's works [1], [2], [3] etc. the mathematical model of selfgravitating statistical system of scalar charged particles is formulated. In particular, in [4] this model is extended on sector of negative effective masses of scalar charged particles. However, in these papers as well as in other works of the Author with his pupils it was usually considered a locally equilibrium system. As is well-known, at inflationary stage thermodynamic equilibrium is violated [5] hence it is reasonable to consider collisionless situation realized at this stage which is described by corresponding Vlasov equations. This paper is devoted to this problem.
2 Self-consistent Mathematical Model of Collisionless Plasma of Scalar Charged Particles

Collisionless Kinetic and Transport Equations
Distribution functions f a (x, P ) of particles of "a" sort having scalar charge q a are defined by invariant collisionless kinetic equations [1] 1 : where [H a , f a ] is a Poisson bracket which can be written using covariant Cartan operator of differentiation ∇ in form 2 : 1 i.e. Vlasov equations.
2 Details see in [4] Hamilton function of a particle in scalar field and normalization ratio for generalized momentum have the next form: Distribution functions f a (x, P ) of particles of "a"-sort having scalar charge q a are defined through invariant collisionless kinetic equations [1]: where Hamilton function and normalization ratio for generalized momentum have the next form: [ Let us notice identity laws, valid for Hamilton function (6) which will be useful in future: where Ψ(x, P ) is an arbitrary function. Further, let us define macroscopic moments using distribution functions: -particle number density vectors which are identically conserved in consequence of kinetic equations (5); plasma energy-momentum tensor and plasma scalar charge density These macroscopic moments are related to each other by transport equations (energy-momentum conservation law).

Scalar Field Equation
First, let us consider Lagrangian function for a classical massive real scalar field Φ. In such a case the Lagrangian of real scalar field can be chosen in the next form: where m s is a mass of scalar field's quanta and it is ǫ 2 = 1 for a classical scalar field, ǫ 2 = −1 for a phantom scalar field; ǫ 1 = 1 for a field with repulsion of likecharged particles, ǫ 1 = −1 for a field with attraction of like-charged particles ǫ 1 = −1. Then energy-momentum tensor of a scalar field is: Let us write the equation of massive non-conformal scalar field with a source [1]: where Let us consider now the Lagrangian function for a classical massive real conformal scalar field Φ (see e.g., [4,5]; for a massive scalar field the conformal invariance is understood as asymptotic property at (m s → 0)): The Lagrangian function differs from the standard one (see e.g. [6]) in presence of factor 1/8π as well as introduced unit indicators ǫ α . In addition, Ricci tensor in the article is obtained by means of convolution of first and third indices of Riemann tensor R jl = g ik R ijkl . Components of energy-momentum tensor of a scalar field relative to Lagrangian function (18) are equal to [6,7]: assuming Φ ≡ Const, we get equation of massive scalar field with a source (see [7]):

Einstein Equations
Complete system of macroscopic equations comprises of kinetic equations (5), field equations (17) or (20) and Einstein equations: where T ik p is a defined above energy-momentum tensor of a statistical system and T ik s is an energy-momentum tensor of a scalar field (16) or (19). Covariant divergence of Einstein equations turns them into identities as a result of transport equation (14) and field equation (17) or (20).

Kinetic Equations
Let us assume scalar potential to be function only on time Φ(η) for a space-flat Friedmann Universe: Let us also assume distribution function depending only on time and squared space momentum: so that kinetic energy of particles is equal to Calculating Poisson bracket (10), let us reduce kinetic equations (5) to the next form: Thus, homogenous isotropic solution of collisionless kinetic equation in Friedmann metrics does not explicitly depend on time variable which was also true for a gas of neutral particles [8,9]:

Sewing of the Collisionless Solution and Equilibrium Solution
To find certain distribution in plasma where previously thermodynamic equilibrium was previously maintained, let us apply next method [8,9]. Let us assume that till certain instant η 0 local thermodynamic equilibrium was maintained in plasma and it was instantly violated at the same instant. Thus, at η < η 0 distribution functions were locally equilibrium i.e.: (µ are chemical potentials, θ is a temperature), and at η >= η 0 they were described by collisionless kinetic equations. Therefore at instant η = η 0 : The single option to fulfill this condition is possible if P 2 0 in equilibrium distribution is substituted by integral: Thus, the exact solution of collisionless kinetic equations (5), which becomes equilibirum one at η = η 0 , is: where the following denotations are introduced: m 0 = a 0 qΦ 0 ; a 0 = a(η 0 ); µ 0 = µ(η 0 ); (31) Thus, m 0 is an effective particle mass at instant of equilibrium violation. Further for the sake of brevity we will call the solution (30) quasi-equilbrium solution.

Moments of Quasi-equilibrium Distribution Function
It is easily seen that quasi-equilibrium distribution function's moments (11) and (12) which are interesting for us, coincide in the algebraic structure with corresponding moments of locally-equilibrium distribution: where u i is a unit timelike vector of a synchronous observer. However, macroscopic scalars n, P, E, σ relative to quasi-equilibrium distribution (30) do not coincide with corresponding scalars relative to equilibrium distribution function. Calculating, we find 3 : so that: which ensures conservation law of particle number. Thus, the expression for particle number density coincides with standard equilibrium one with constant mass, temperature and chemical potential within the accuracy of multiplication by scale factor (a 0 /a) 3 . Expressions for remaining three scalars do not coincide with equilibrium ones: The behavior of these scalars is substantially defined by the behavior of invari- Let us note that atΦ = Const integrals in expressions for all three scalars are constant values.

Field Equations
In case of conformally non-invariant field (15) field equation (17) in Friedmann metrics takes form: In case of conformally invariant scalar field (18) this equation needs to be substituted by the following one: Let us consider conformally invariant case of massless scalar field. Let us rewrite scalar charge density (39) in the following form: whereσ Thus, conformally invariant equation (46) for a massless scalar field takes form: i.e. equation for scalar conformally invariant massless field with a source becomes autonomous ordinary second-order integro-differential equation. We can find the solutions of this equation in different extreme cases. Let us first consider a ultrarelativistic case: In such a case equation (46) becomes a linear second-order differential equation with constant coefficients: where In a case of like-charged particles repulsion (e 1 = +1) this equation has its general solution: and in a case of attraction (e 1 = −1) it is: Let us notice that exactly the same solution was found in [10] for ultrarelativistic equilibrium plasma. This coincidence is easily explained since quasi-equilibrium distribution (30) coincides with equilibrium one in the ultrarelativistic limit. Let us consider now a non-relativistic case 4 : Then instead of homogenous equation (48) we get inhomogeneous equation: where The solutions of this equation are:

Einstein Equation
It can easily be seen that energy-momentum tensor of conformally invariant massless scalar field Φ(η) has a form of energy-momentum tensor of the ideal flux (e 1 = +1) where so that: The single non-trivial Einstein equation in Friedmann metrics takes the next form: Substituting the expressions for energy density from (37) and (57) into this equation, we reduce it to: Thus, cosmological model in case of massless conformally invariant scalar field with a collisionless source is reduced to two ordinary integro-differential equations of the second order (46) and first order (61) relative to functions a(η) and Φ(η). In particular, for the ultrarelativistic case we haveĒ pl (Φ) =Ē 0 = Const. Choosing a convergent solution of scalar field equation (50) and substituting it into Einstein equation (61), we reduce it to the next form a ′2 = 8π 3 φ 2 0 ω 2 0 cos 2 ω 0 η + 8π 3Ē 0 .
The solution of this ordinary differential equation has the following form: E(x, k) is an elliptic integral of the second kind: