In order to have a closed system from equations (4), (7) and (8) we have to add the equations of state for p and T . As shown in Appendix A, the equation of state, the temperature and the bulk viscosity of the quark-gluon plasma (QGP), can be determined approximately at high temperatures [25] from recent lattice QCD calculations [34], as and ω$_{0}$ ≃ 0 . 5 − 1 . 5 GeV. In the following we assume that αρ >> 9 /ω$_{0}$T 4 c , and therefore we take ξ ≃ αρ . In order to close the system of the cosmological equations, we have also to give the expression of the relaxation time τ , for which we adopt the expression [7], Eqs. (9) are standard in the study of the viscous cosmological models, whereas the equa- tion for τ is a simple procedure to ensure that the speed of viscous pulses does not exceed the speed of light. Eq. (11) implies that the relaxation time in our treatment is constant but strongly depends on EoS. These equations are without sufficient thermodynamical mo- tivation, but in the absence of better alternatives, we shall follow the practice of adopting them in the hope that they will at least provide some indication of the range of bulk viscous effects. The temperature law is the simplest law guaranteeing positive heat capacity. With the use of Eqs. (8), (9) and (11), respectively, we obtain the following equation describing the cosmological evolution of the Hubble function H We introduce the transformation u = ˙ H , so that Eq. (12) is transformed into a first order ordinary differential equation, III. AN APPROXIMATE SOLUTION P = ωρ, T = βρ $^{r}$, ξ = αρ + 9 ω$_{0}$ T 4 c , (9) α = 1 9 ω$_{0}$ 9 γ 2 − 24 γ + 16 γ − 1 , (10) with ω = ( γ − 1), γ ≃ 1 . 183, r ≃ 0 . 213, β ≃ 0 . 718, τ = ξρ − 1 ≃ α. (11) ¨ H + 3 2 [1 + (1 − r ) γ ] H ˙ H + 1 α ˙ H − (1 + r ) H − 1 ˙ H 2 + 9 4 ( γ − 2) H 3 + 3 2 γ α H 2 = 0 . (12) u du dH − (1 + r ) H − $^{1}$u2 + ( 3 2 [1 + (1 − r ) γ ] H + α − 1 ) u + 9 4 1 ( γ ) H 3 + 3 2 γ α H 2 = 0 . (13) 6