  A closed mathematical model of the statistical self-gravitating system of
scalar charged particles for conformal invariant scalar interactions is
constructed on the basis of relativistic kinetics and gravitation theory.
Asymptotic properties of the model are investigated in the ultrarelativistic
limit. It is shown, that scalar charge density automatically generates scalar
field effective mass and the value of this mass is found. In the paper it is
proved the asymptotic conformal invariance of constitutive equations in case of
homogenous isotropic Universe. Also it is proved the asymptotic conformal
invariance of field equations at the early stages of cosmological evolution.
