  Non-thermal fixed points in the evolution of a quantum many-body system
quenched far out of equilibrium manifest themselves in a scaling evolution of
correlations in space and time. We develop a low-energy effective theory of
non-thermal fixed points in a bosonic quantum many-body system by integrating
out long-wave-length density fluctuations. The system consists of $N$
distinguishable spatially uniform Bose gases with $\mathrm{U}(N)$-symmetric
interactions. The effective theory describes interacting Goldstone modes of the
total and relative-phase excitations. It is similar in character to the
non-linear Luttinger-liquid description of low-energy phonons in a single
dilute Bose gas, with the markable difference of a universal non-local coupling
function depending, in the large-$N$ limit, only on momentum, single-particle
mass, and density of the gas. Our theory provides a perturbative description of
the non-thermal fixed point, technically easy to apply to experimentally
relevant cases with a small number of fields $N$. Numerical results for $N=3$
allow us to characterize the analytical form of the scaling function and
confirm the analytically predicted scaling exponents. The fixed point which is
dominated by the relative phases is found to be Gaussian, while a non-Gaussian
fixed point is anticipated to require scaling evolution with a distinctly lower
power of time.
