speed of sound. The parameter $\omega_o$ is a scale depending on the temperature $T$, and defines the validity of the underlying perturbation theory. In this relation, the viscosity is assumed to have a thermal part which can be determined through lattice calculations, and a vacuum contributing part, which can be fixed using quark and gluon condensates. The vacuum part would take the value $$16 |\epsilon_v| (1 + \frac{3}{8} \cdot 1.6) \simeq (560\ {\rm MeV})^4
\simeq (3 \,T_c)^4 \,$$ Our algorithm is the following. Using lattice QCD results on trace anomaly, $(\epsilon-3p)/T^4$, and other thermodynamical quantities, we can determine the bulk viscosity. To make use of the lattice QCD results, it is useful to make a suitable fit to the data at high temperatures. Then we obtain the following equations of state $$\begin{aligned}
\label{EoS}
p &=&\omega \rho, \hspace*{2cm} T =\beta \rho^r, \hspace*{2cm} c_s^2 = \omega \nonumber\end{aligned}$$ where $\omega=0.319$, $\beta=0.718\pm 0.054$ and $r=0.23\pm 0.196$. Using the equations of state, Eq. (\[EoS\]) in Eq. (\[ze\]), we obtain $$\label{zeta}
 \xi(\epsilon)=\frac{1}{9\omega_o}\frac{9\gamma^2-24\gamma+16}{\gamma-1}\rho+\frac{9}{\omega_o}T_c^4.$$

Appendix B: Estimations of $g(z)$ {#App:A .unnumbered}
=================================

For analytical purposes, the function $g(z)$, which is defined in $z$ parameter as $g(z)=F_0/F_1$ in Eq. (\[gofzz1\]), can be numerically estimated depending on the parameter $z$ 