  It has been suggested based on analytic theory that even in non-rotating
supernova progenitors stochastic spin-up by internal gravity waves (IGWs)
during the late burning stages can impart enough angular momentum to the core
to result in neutron star birth spin periods below 100ms, and a relatively firm
upper limit of 500ms for the spin period. We here investigate this process
using a 3D simulation of oxygen shell burning in a $3M_\odot$ He star. Our
model indicates that stochastic spin-up by IGWs is less efficient than
previously thought. We find that the stochastic angular momentum flux carried
by waves excited at the shell boundary is significantly smaller for a given
convective luminosity and turnover time than would be expected from simple
dimensional analysis. This can be explained by noting that the waves launched
by overshooting convective plumes contain modes of opposite angular wave number
with similar amplitudes, so that the net angular momentum of excited wave
packets almost cancels. We find that the wave-mediated angular momentum flux
from the oxygen shell follows a random walk, but again dimensional analysis
overestimates the random walk amplitudes since the correlation time is only a
fraction of the convective turnover time. Extrapolating our findings over the
entire life time of the last burning stages prior to collapse, we predict that
the core angular momentum from stochastic spin-up would translate into long
birth spin periods of several seconds for low-mass progenitors and no less than
100ms even for high-mass progenitors.
