Light scattering in inhomogeneous Tomonaga-Luttinger liquids

We derive the dynamical structure factor for an inhomogeneous Tomonaga-Luttinger liquid (TLL) as can be formed in a confined strongly interacting one-dimensional gas. In view of the current experimental progress in the field, we provide a simple analytic expression for the light-scattering cross section, requiring only the knowledge of the density dependence of the ground-state energy as it can be extracted, e.g., from exact or quantum Monte Carlo (QMC) techniques and a Thomas-Fermi description. We apply the result to the case of one-dimensional quantum bosonic gases with dipolar interaction in a harmonic trap, using an energy functional deduced from QMC computations. We find a universal scaling behavior peculiar to the TLL, a signature that can be probed eventually by Bragg spectroscopy in experimental realizations of such systems.