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

Finite temperature correlations in the Lieb-Liniger 1D Bose gas