Perfect fluid cosmology with geodesic world lines

It is shown that for a perfect fluid with an equation of state $p=p\left(\rho \right)$, if the world lines are geodesics, then they are hypersurface orthogonal and the scalars $p$, $\rho$, ${\sigma }^2$, and ${\theta }^2$ are all constants over these hypersurfaces, irrespective of any spatial-homogeneity assumption. However, an examination of some simple cases does not reveal any spatially nonhomogeneous solution with these properties.