  The natural excitations of an interacting one-dimensional system at low
energy are hydrodynamic modes of Luttinger liquid, protected by the Lorentz
invariance of the linear dispersion. We show that beyond low energies, where
quadratic dispersion reduces the symmetry to Galilean, the main character of
the many-body excitations changes into a hierarchy: calculations of dynamic
correlation functions for fermions (without spin) show that the spectral
weights of the excitations are proportional to powers of
$\mathcal{R}^{2}/L^{2}$, where $\mathcal{R}$ is a length-scale related to
interactions and $L$ is the system length. Thus only small numbers of
excitations carry the principal spectral power in representative regions on the
energy-momentum planes. We have analysed the spectral function in detail and
have shown that the first-level (strongest) excitations form a mode with
parabolic dispersion, like that of a renormalised single particle. The
second-level excitations produce a singular power-law line shape to the
first-level mode and multiple power-laws at the spectral edge. We have
illustrated crossover to Luttinger liquid at low energy by calculating the
local density of state through all energy scales: from linear to non-linear,
and to above the chemical potential energies. In order to test this model, we
have carried out experiments to measure momentum-resolved tunnelling of
electrons (fermions with spin) from/to a wire formed within a GaAs
heterostructure. We observe well-resolved spin-charge separation at low energy
with appreciable interaction strength and only a parabolic dispersion of the
first-level mode at higher energies. We find structure resembling the
second-level excitations, which dies away rapidly at high momentum in line with
the theoretical predictions here.
