  I investigate the tunneling decay rate of a polaritonic system formed by a
strong coupling between a vacuum cavity mode and $N$ metastable systems. Using
a simple model potential, I find the instanton solutions controlling the
low-temperature tunneling rate. The resulting rate modification due to the
cavity is proportional to the mean of the second power of the light-matter
coupling. No collective effect that would enhance the rates by a factor of
$\sqrt{N}$ is present, which is in line with the results in the thermal
activation regime.
