On homothetic cosmological dynamics

We consider the homogeneous and isotropic cosmological fluid dynamics which is compatible with a homothetic, timelike motion, equivalent to an equation of state $\rho +3P=0$. By splitting the total pressure $P$ into the sum of an equilibrium part $p$ and a non-equilibrium part $\mathrm{\Pi }$, we find that on thermodynamical grounds this split is necessarily given by $p=\rho$ and $\Pi = - (4/3)\rho$, corresponding to a dissipative stiff (Zel’dovich) fluid.