determination of the regulator $F$ in terms of the thermodynamical parameters. The two relevant regimes here addressed require different descriptions for the regulator expression. The most delicate case is the one associated to the matter creation, for which both the energy density and the Universe volume vanish asymptotically to the singularity. Then, the construction of a reliable ansatz for large asymptotic values of $F$ appears a non-trivial task. Nonetheless, a simple solution to this puzzle is offered by the condition of incompressibility defined by (\[incomp\]), which, as we shall see below, is also appropriate for the comparison to the original analysis in [@BK_Bianchi]. The isotropic flat Universe case can be instead easily faced by retaining the same choice as in [@Disc1], i.e. $F$ is provided by the ratio between the enthalpy and mass density of the Universe as stated in (\[F\]). This formulation appears well-grounded owing to the fact that the thermal history of the Universe naturally ensures that $F$ is large in the early Universe and it tends to unity in the present stage of evolution (say for a redshift $z<100$, when the pressure term becomes negligible).

Field equations {#field-equations .unnumbered}
===============

The line-element of the Bianchi I model 