  We consider the evolution of perturbations to a flat FRW universe that arise
from a ``stiff source,'' such as a self-ordering cosmic field that forms in a
global symmetry-breaking phase transition and evolves via the Kibble mechanism.
Although the linear response of the normal matter to the source depends on the
details of the source dynamics, we show that the higher-order non-linear
perturbative equations reduce to a form identical to those of source-free
Newtonian gravity in the small wavelength limit. Consequently, the resulting
$n$-point correlation functions and their spectral counterparts will have a
hierarchical contribution arising from this gravitational evolution (as in the
source-free case) in addition to that possibly coming from non-Gaussian initial
conditions. We apply this formalism to the $O(N)$ nonlinear sigma model at
large $N$ and find that observable differences from the case of initially
Gaussian perturbations and Newtonian gravity in the bispectrum and higher-order
correlations are not expected on scales smaller than about $100\,h^{-1}{\rm
Mpc}$.
