  Due to the vast growth of the many-body level density with excitation energy,
its smoothed form is of central relevance for spectral and thermodynamic
properties of interacting quantum systems. We compute the cumulative of this
level density for confined one-dimensional continuous systems with repulsive
short-range interactions. We show that the crossover from an ideal Bose gas to
the strongly correlated, fermionized gas, i.e., partial fermionization,
exhibits universal behavior: Systems with very few up to many particles share
the same underlying spectral features. In our derivation we supplement quantum
cluster expansions with short-time dynamical information. Our nonperturbative
analytical results are in excellent agreement with numerics for systems of
experimental relevance in cold atom physics, such as interacting bosons on a
ring (Lieb-Liniger model) or subject to harmonic confinement. Our method
provides predictions for excitation spectra that enable access to
finite-temperature thermodynamics in large parameter ranges.
