Just like the early-time cosmic acceleration associated with inflation, a negative pressure can be seen as a possible driving mechanism for the late-time accelerated expansion of the Universe as well. One of the earliest alternatives that could provide a mechanism producing such accelerating phase of the Universe is through a negative pressure produced by viscous or particle production effects. The viscous pressure contributions can be seen as small nonequilibrium contributions for the energy-momentum tensor for nonideal fluids.

Let us posit the thermodynamics of matter creation for a single fluid. To describe the thermodynamic states of a relativistic simple fluid we use the following macroscopic variables: the energy-momentum tensor T^{αβ} ; the particle flux vector N^{α}; and the entropy flux vector s^{α}. The energy-momentum tensor satisfies the conservation law, T^{αβ}_{;β} = 0, and here we consider situations in which it has the perfect-fluid form

T^{αβ} = (ρ+P)u^{α}u^{β} − P g^{αβ}

In the above equation ρ is the energy density, P is the isotropic dynamical pressure, g^{αβ} is the metric tensor and u^{α} is the fluid four-velocity (with normalization u^{α}u_{α} = 1).

The dynamical pressure P is decomposed as

P = p + Π

where p is the equilibrium (thermostatic) pressure and Π is a term present in scalar dissipative processes. Usually, it is associated with the so-called bulk pressure. In the cosmological context, besides this meaning, Π can also be relevant when particle number is not conserved. In this case, Π ≡ p_{c} is called the “creation pressure”. The bulk pressure, can be seen as a correction to the thermostatic pressure when near to equilibrium, thus, it should be always smaller than the thermostatic pressure, |Π| < p. This restriction, however, does not apply for the creation pressure. So, when we have matter creation, the total pressure P may become negative and, in principle, drive an accelerated expansion.

The particle flux vector is assumed to have the following form

N^{α} = nu^{α}

where n is the particle number density. N^{α} satisfies the balance equation N^{α}_{;α} = nΓ, where Γ is the particle production rate. If Γ > 0, we have particle creation, particle destruction occurs when Γ < 0 and if Γ = 0 particle number is conserved.

The entropy flux vector is given by

s^{α} = nσu^{α}

where σ is the specific (per particle) entropy. Note that the entropy must satisfy the second law of thermodynamics s^{α}_{;α} ≥ 0. Here we consider adiabatic matter creation, that is, we analyze situations in which σ is constant. With this condition, by using the Gibbs relation, it follows that the creation pressure is related to Γ by

p_{c} = − (ρ+p)/3H Γ

where H = a ̇/a is the Hubble parameter, a is the scale factor of the Friedmann-Robertson-Walker (FRW) metric and the overdot means differentiation with respect to the cosmic time. If σ is constant, the second law of thermodynamics implies that Γ ≥ 0 and, as a consequence, particle destruction (Γ < 0) is thermodynamically forbidden. Since Γ ≥ 0, it follows that, in an expanding universe (H > 0), the creation pressure p_{c} cannot be positive.