  We investigate the effect of dimensionality on the transition to explosion in
neutrino-driven core-collapse supernovae. Using parameterized hydrodynamic
simulations of the stalled supernova shock in one-, two- (2D), and three
spatial dimensions (3D), we systematically probe the extent to which
hydrodynamic instabilities alone can tip the balance in favor of explosion. In
particular, we focus on systems that are well into the regimes where the
Standing Accretion Shock Instability (SASI) or neutrino-driven convection
dominate the dynamics, and characterize the difference between them. We find
that SASI-dominated models can explode with up to ~20% lower neutrino
luminosity in 3D than in 2D, with the magnitude of this difference decreasing
with increasing resolution. This improvement in explosion conditions is related
to the ability of spiral modes to generate more non-radial kinetic energy than
a single sloshing mode, increasing the size of the average shock radius, and
hence generating better conditions for the formation of large-scale,
high-entropy bubbles. In contrast, convection-dominated explosions show a
smaller difference in their critical heating rate between 2D and 3D (<8%), in
agreement with previous studies. The ability of our numerical implementation to
maintain arbitrary symmetries is quantified with a set of SASI-based tests. We
discuss implications for the diversity of explosion paths in a realistic
supernova environment.
