Universal correlations along the BEC-BCS crossover

We show that the long-distance behavior of the two-body density correlation functions and the Cooper-pair probability density of a balanced mixture of a two-component Fermi gas at $T=0$, is universal along the BEC-BCS crossover. Our result is demonstrated by numerically solving the mean-field BCS model for different finite short-range atomic interaction potentials. We find an analytic expression for the correlation length in terms of the chemical potential and the energy gap at zero momentum.