content_image/347108.jpg content_image/347109.jpg content_image/347110.jpg content_image/347111.jpg
entropy fluxes are defined according to N i = nu i and S i = sN i − ( τ Π $^{2}$/ 2 ξT ) u $^{i}$, where n is the number density, s the specific entropy, T ≥ 0 the temperature, ξ the bulk viscosity coefficient, and τ ≥ 0 the relaxation coefficient for transient bulk viscous effect (i.e. the relaxation time), respectively. 

The evolution of the cosmological fluid is subject to the dynamical laws of particle number conservation N i ; i = 0 and Gibbs’ equation T dρ = d ( ρ/n ) + pd (1 /n ). In the following we shall also suppose that the energy-momentum tensor of the cosmological fluid is conserved, that is T k i ; = 0. 

The bulk viscous effects can be generally described by means of an effective pressure Π, formally included in the effective thermodynamic pressure p$_{eff}$ = p + Π [7]. Then in the comoving frame the energy momentum tensor has the components T 0 0 = ρ, T 1 1 = T 2 2 = T 3 3 = − p$_{eff}$ . For the line element given by Eq. (1), the Einstein field equations read 

<img src='content_image/347108.jpg'>

where one dot denotes derivative with respect to the time t , G is the gravitational constant and a is the scale factor. 

Assuming that the total matter content of the Universe is conserved, T j i ; j = 0, the energy density of the cosmic matter fulfills the conservation law: 

<img src='content_image/347109.jpg'>

where we introduced the Hubble parameter H = ˙ a/a . In presence of bulk viscous stress Π, the effective thermodynamic pressure term becomes p$_{eff}$ = p + Π. Then Eq. (6) can be written as 

<img src='content_image/347110.jpg'>

For the evolution of the bulk viscous pressure we adopt the causal evolution equation [7], obtained in the simplest way (linear in Π) to satisfy the H -theorem (i.e., for the entropy production to be non-negative, S i ; = Π $^{2}$/ξT ≥ 0 [3, 4]). According to the causal relativistic Israel-Stewart theory, the evolution equation of the bulk viscous pressure reads [7] 

<img src='content_image/347111.jpg'>

