Einstein gravity of a diffusing fluid
We discuss the Einstein gravity for a fluid consisting of particles interacting with an environment of some other particles. The environment is described by a time-dependent cosmological term which compensates the lack of the conservation law of the energy–momentum of the dissipative fluid. The dissipation is approximated by a relativistic diffusion in the phase space. We are interested in a homogeneous isotropic flat expanding Universe described by a scale factor a. At an early stage the particles are massless. We obtain explicit solutions of the diffusion equation for a fluid of massless particles at finite temperature. The solution is of the form of a modified Jüttner distribution with a time-dependent temperature. Later, Universe evolution is described as a diffusion at zero temperature with no equilibration. We find solutions of the diffusion equation at zero temperature which can be treated as a continuation to a later time of the finite temperature solutions describing an early stage of the Universe. The energy–momentum of the diffusing particles is defined by their phase space distribution. A conservation of the total energy–momentum determines the cosmological term up to a constant. The resulting energy–momentum inserted into the Einstein equations gives a modified Friedmann equation. Solutions of the Friedmann equation depend on the initial value of the cosmological term. The large value of the cosmological constant implies an exponential expansion. If the initial conditions allow a power-like solution for a large time then it must be of the form a ≃ τ (no deceleration, τ is the cosmic time). The final stage of the Universe evolution is described by a non-relativistic diffusion of a cold dust.