  We consider the optical conductivity in the one dimensional Hubbard model in
the metallic phase close to half filling. In this regime most of the spectral
weight is located at frequencies above an energy scale $E_{\rm opt}$ that tends
towards the optical gap in the Mott insulating phase for vanishing doping.
Using the Bethe Ansatz we relate $E_{\rm opt}$ to thresholds of particular
kinds of excitations in the Hubbard model. We then employ a mobile impurity
models to analyze the optical conductivity for frequencies slightly above these
thresholds. This entails generalizing mobile impurity models to excited states
that are not highest weight with regards to the SU(2) symmetries of the Hubbard
chain, and that occur at a maximum of the impurity dispersion.
