  Strong coupling between cavity photons and molecular vibrations can lead to
the formation of vibron-polaritons. In a recent experiment with PVAc molecules
in a metal-metal microcavity [A.Shalabney et al., Ang.Chem.Int.Ed. 54 7971
(2015)], such a coupling was observed to enhance the Raman scattering
probability by several orders of magnitude. Inspired by this, we theoretically
analyze the effect of strong photon-vibron coupling on the Raman scattering
amplitude of organic molecules. This problem has recently been addressed in
[J.del Pino, J.Feist and F.J.Garcia-Vidal; J.Phys.Chem.C 119 29132 (2015)]
using exact numerics for a small number of molecules. In this paper we derive
compact analytic results for any number of molecules, also including the
ultra-strong coupling regime. Our calculations predict a division of the Raman
signal into upper and lower polariton modes,with some enhancement to the lower
polariton Raman amplitude due to the mode softening under strong coupling.
