  Temporal relaxation of density fluctuations in supercooled liquids near the
glass transition occurs in multiple steps. The short-time $\beta$-relaxation is
generally attributed to spatially local processes involving the rattling motion
of a particle in the transient cage formed by its neighbors. Using molecular
dynamics simulations for three model glass-forming liquids, we show that the
$\beta$-relaxation is actually cooperative in nature. Using finite-size scaling
analysis, we extract a growing length-scale associated with $\beta$-relaxation
from the observed dependence of the $\beta$-relaxation time on the system size.
Remarkably, the temperature dependence of this length scale is found to be the
same as that of the length scale that describes the spatial heterogeneity of
local dynamics in the long-time $\alpha$-relaxation regime. These results show
that the conventional interpretation of $\beta$-relaxation as a local process
is too simplified and provide a clear connection between short-time dynamics
and long-time structural relaxation in glass-forming liquids.
