  By selectively probing the center of a trapped gas, we measure the local, or
homogeneous, contact of a unitary Fermi gas as a function of temperature. Tan's
contact, C, is proportional to the derivative of the energy with respect to the
interaction strength, and is thus an essential thermodynamic quantity for a gas
with short-range correlations. Theoretical predictions for the temperature
dependence of C differ substantially, especially near the superfluid
transition, Tc, where C is predicted to either sharply decrease, sharply
increase, or change very little. For T/T_F>0.4, our measurements of the
homogeneous gas contact show a gradual decrease of C with increasing
temperature, as predicted by theory. We observe a sharp decrease in C at
T/T_F=0.16, which may be due to the superfluid phase transition. While a sharp
decrease in C below Tc is predicted by some many-body theories, we find that
none of the predictions fully accounts for the data.
