  We study a Fermi gas with strong, tunable interactions dispersively coupled
to a high-finesse cavity. Upon probing the system along the cavity axis, we
observe a strong optomechanical Kerr nonlinearity originating from the density
response of the gas to the intracavity field and measure it as a function of
interaction strength. We find that the zero-frequency density response function
of the Fermi gas increases by a factor of two from the
Bardeen-Cooper-Schrieffer to the Bose-Einstein condensate regime. The results
are in quantitative agreement with a theory based on operator-product
expansion, expressing the density response in terms of universal functions of
the interactions, the contact and the internal energy of the gas. This provides
an example of a driven-dissipative, strongly correlated system with a strong
nonlinear response, opening up perspectives for the sensing of weak
perturbations or inducing long-range interactions in Fermi gases.
