  Using confocal microscopy, we measure the packing geometry and interdroplet
forces as a function of the osmotic pressure in a 3D emulsion system. We find
that the nonlinear elastic response of the pressure with density is not a
result of the anharmonicity in the interaction potential, but of the
corrections to the scaling laws of the microstructure away from the critical
point. The bulk modulus depends on the excess contacts created under
compression, which leads to the correction exponent \alpha=1.5.
Microscopically, the nonlinearities manifest themselves as a narrowing of the
distribution of the pressure per particle as a function of the global pressure.
