  We introduce a framework for calculating dynamical correlations in the
Lieb-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 and
time. The framework is based on a method for taking the thermodynamic limit of
sums of form factors that exhibit non integrable singularities. We expect our
framework to be applicable to any local operator.
  We determine the first three terms of this expansion and obtain an explicit
expression for the density-density dynamical correlations and the dynamical
structure factor at order $1/c^2$. We apply these to finite-temperature
equilibrium states and non-equilibrium steady states after quantum quenches. We
recover predictions of (nonlinear) Luttinger liquid theory and generalized
hydrodynamics in the appropriate limits, and are able to compute sub-leading
corrections to these.
