  The hydrodynamic part of the velocity autocorrelation function of a granular
fluid in the homogeneous cooling state has been calculated by using
mode-coupling theory for a finite system with periodic boundary conditions. The
existence of the shearing instability, leading to a divergent behavior of the
velocity flow fluctuations, is taken into account. A time region in which the
velocity autocorrelation function exhibits a power law decay, when time is
measured by the number of collisions per particle, has been been identified.
Also the explicit form of the exponential asymptotic long time decay has been
obtained. The theoretical prediction for the power law decay is compared with
molecular dynamics simulation results, and a good agreement is found, after
taking into account finite size corrections. The effects of approaching the
shearing instability are also explored.
