  The majority of massive stars ($>8$ $\rm{M_{\odot}}$) in OB associations are
found in close binary systems. Nonetheless, the formation mechanism of these
close massive binaries is not understood yet. Using literature data, we
measured the radial-velocity dispersion ($\sigma_\mathrm{RV}$) as a proxy for
the close binary fraction in ten OB associations in the Galaxy and the Large
Magellanic Cloud, spanning an age range from 1 to 6 Myrs. We find a positive
trend of this dispersion with the cluster's age, which is consistent with
binary hardening. Assuming a universal binary fraction of $f_\mathrm{bin}$ =
0.7, we converted the $\sigma_\mathrm{RV}$ behavior to an evolution of the
minimum orbital period $P_\mathrm{cutoff}$ from $\sim$9.5 years at 1 Myr to
$\sim$1.4 days for the oldest clusters in our sample at $\sim$6 Myr. Our
results suggest that binaries are formed at larger separations, and they harden
in around 1 to 2 Myrs to produce the period distribution observed in few
million year-old OB binaries. Such an inward migration may either be driven by
an interaction with a remnant accretion disk or with other young stellar
objects present in the system. Our findings constitute the first empirical
evidence in favor of migration as a scenario for the formation of massive close
binaries.
