Quasi-Linear Evolution of Compensated Cosmological Perturbations: The Nonlinear Sigma Model

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 $100\,h^{-1}{\rm Mpc}$.