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We introduce a framework for calculating dynamical correlations in theLieb-Liniger model in arbitrary energy eigenstates and for all space and time,that combines a Lehmann representation with a $1/c$ expansion. The $n^{\rm th}$term of the expansion is of order $1/c^n$ and takes into account all $\lfloor\tfrac{n}{2}\rfloor+1$ particle-hole excitations over the averaging eigenstate.Importantly, in contrast to a 'bare' $1/c$ expansion it is uniform in space andtime. The framework is based on a method for taking the thermodynamic limit ofsums of form factors that exhibit non integrable singularities. We expect ourframework to be applicable to any local operator.  We determine the first three terms of this expansion and obtain an explicitexpression for the density-density dynamical correlations and the dynamicalstructure factor at order $1/c^2$. We apply these to finite-temperatureequilibrium states and non-equilibrium steady states after quantum quenches. Werecover predictions of (nonlinear) Luttinger liquid theory and generalizedhydrodynamics in the appropriate limits, and are able to compute sub-leadingcorrections to these.

A systematic $1/c$-expansion of form factor sums for dynamical correlations in the Lieb-Liniger model