content_image/340648.jpg
<img src='content_image/340648.jpg'>

viscous fluid results in different a ( t )-behaviors with increasing t . At t = 0, a ( t ) remains finite. Correspondingly, H ( t ) remains also finite. In general, the dependence on t is much more complicated than in Eq. (46). Here we have an A/ P root of an exponential function. If exp( − Bt/ P ) >> A , a remains constant. 

Fig. 2 illustrates the dependence of the two thermodynamical quantities, ρ and T , on the comoving time. The non-viscous Universe shows a singular behavior in ρ at vanishing t , as shown in the left panel of Fig. 2. This is not obvious in the case where we have taken into consideration a finite viscosity coefficient, i.e., ρ is finite at t = 0. In both cases, ρ is decreasing with increasing t , reflecting that the Early Universe was likely expanding. Also the life time of the thermal viscous Universe seems to be shorter than for the non-viscous Universe. Almost the same behavior is observed in the right panel of Fig. 2. The temperature T seems to be finite at vanishing t in the viscous Universe. The T -singularity is only present, if we assume that the background matter is non-viscous ideal gas. 

In left panel of Fig. 3, we show the dependence of the bulk viscous pressure Π on t . Π takes negative values at very small t . Then it switches to positive values at some values of t . After reaching the maximum value, Π decays exponentially with increasing t . At larger t , Π entirely vanishes. The deceleration parameter q , given by Eq. (28), is depicted in the right panel of Fig. 3, and it is compared with q for a non-viscous fluid, q = − 3. 

