  Stellar evolution theory predicts a "gap" in the black hole birth function
caused by the pair instability. Presupernova stars that have a core mass below
some limiting value, Mlo, after all pulsational activity is finished, collapse
to black holes, whereas more massive ones, up to some limiting value, Mhi,
explode, promptly and completely, as pair-instability supernovae. Previous work
has suggested Mlo is approximately 50 solar masses and Mhi is approximately 130
solar masses. These calculations have been challenged by recent LIGO
observations that show many black holes merging with individual masses, Mlo is
least some 65 solar masses. Here we explore four factors affecting the
theoretical estimates for the boundaries of this mass gap: nuclear reaction
rates, evolution in detached binaries, rotation, and hyper-Eddington accretion
after black hole birth. Current uncertainties in reaction rates by themselves
allow Mlo to rise to 64 solar masses and Mhi as large as 161 solar masses.
Rapid rotation could further increase Mlo to about 70 solar masses, depending
on the treatment of magnetic torques. Evolution in detached binaries and
super-Eddington accretion can, with great uncertainty, increase Mlo still
further. Dimensionless Kerr parameters close to unity are allowed for the more
massive black holes produced in close binaries, though they are generally
smaller.
