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The aim of this work is to construct a class of self-consistent cosmological solutions to the combined Einstein-Boltzmann system, consisting of Einstein's equations and the Boltzmann equation. The Boltzmann equation is first rewritten in tetrad form, using a frame comoving with the fluid. In this way, the integrations with respect to the momenta become the same as in flat spacetime, and hence are independent of the metric.

For the collision term we use a relativistic BGK-type kinetic theory model, [1], which generalizes the term to a form suitable for using a particle (Eckart) frame, [2], together with its extension to polyatomic gases, [3].

Then, on using the Chapman-Enskog expansion to first order, a near equilibrium configuration, consistent with the Eckart theory, is constructed. This is used to calculate the energy-momentum tensor, from which the bulk and shear viscosity coefficients are read off, as well as to determine the particle current density, from which the coefficient of heat conductivity is found.

The so constructed energy-momentum tensor is then used to find self-consistent solutions to the Einstein-Boltzmann system with viscosity and heat flow. For this we consider a class of homogeneous and tilted locally rotationally symmetric cosmological models of Bianchi type VIII, and hence extend an earlier work, [4], where Robertson-Walker cosmologies were studied. The equations are rewritten as an integrable system of first order ordinary differential equations, suitable for numerical integration. The time evolution of some of the models are then studied, with focus on the dissipative terms.

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