  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.
