  We address the problem of calculating finite-temperature response functions
of an experimentally relevant low-dimensional strongly-correlated system: the
integrable 1D Bose gas with repulsive \delta-function interaction (Lieb-Liniger
model). Focusing on the observable dynamical density-density function, we
present a Bethe Ansatz-based method allowing for its accurate evaluation over a
broad range of momenta, frequencies, temperatures and interaction parameters,
in finite but large systems. We show how thermal fluctuations smoothen the zero
temperature critical behavior and present explicit quantitative results in
experimentally accessible regimes.
