  The rate of non-radiative decay between two molecular electronic states is
succinctly described by the energy gap law, which suggests an
approximately-exponential dependence of the rate on the electronic energy gap.
Here, we inquire whether this rate is modified under vibrational strong
coupling, a regime whereby the molecular vibrations are strongly coupled to an
infrared cavity. We show that, under most conditions, the collective
light-matter coupling strength is not large enough to counter the entropic
penalty involved with using the polariton modes, so the energy gap law remains
unchanged. This effect (or the lack thereof) may be reversed with deep strong
light-matter couplings or large detunings, both of which increase the upper
polariton frequency. Finally, we demonstrate how vibrational polariton
condensates mitigate the entropy problem by providing large occupation numbers
in the polariton modes.
