  Binary white dwarf (WD) coalescences driven by gravitational waves or
collisions in triple systems are potential progenitors of Type Ia supernovae
(SNe Ia). We combine the distribution of 56Ni inferred from observations of SNe
Ia with the results of both sub-Chandrasekhar detonation models and direct
collision calculations to estimate what mass WDs should be exploding in each
scenario to reproduce the observations. These WD mass distributions are then
compared with the observed Galactic WD mass distribution and Monte Carlo
simulations of WD-WD binary populations. For collisions, we find that the
average mass of the individual components of the WD-WD binary must be peaked at
~0.75Msun, significantly higher than the average WD mass in binaries or in the
field of ~0.55-0.60Msun. Thus, if collisions produce a large fraction of SNe
Ia, then a mechanism must exist that favors large mass WDs. On the other hand,
in an old stellar population, collisions would naturally result in a class of
low luminosity SNe Ia, and we suggest these may be related to 1991bg-like
events. For sub-Chandrasekhar detonations, we find that the average mass of the
exploding WDs must be peaked at ~1.1Msun. This is interestingly similar to the
average sum of the masses in WD-WD binaries, but it is not clear (and should be
further explored) whether double degenerate mergers would be sufficiently
efficient at synthesizing 56Ni to match the observed yields. If not, then
actual ~1.1Msun WDs would be needed for sub-Chandrasekhar detonations. Since
such high mass WDs are produced relatively quickly in comparison to the age of
the environments where SNe Ia are found, this would require either accretion
onto lower mass WDs prior to ignition or a long timescale between formation of
the ~1.1Msun WD and ignition (such as set by gravitational wave emission or
binary interactions).
