  Type Ia supernovae (SNe Ia) are likely the thermonuclear explosions of
carbon-oxygen (CO) white-dwarf (WD) stars, but their progenitor systems remain
elusive. Recently, Sharon & Kushnir (2022) used The Zwicky Transient Facility
Bright Transient Survey to construct a synthesized $^{56}$Ni mass,
$M_\text{Ni56}$, distribution of SNe Ia. They found that the rate of
low-luminosity ($M_\text{Ni56}\approx0.15\,M_{\odot}$) SNe Ia is lower by a
factor of $\sim10$ than the more common $M_\text{Ni56}\approx0.7\,M_{\odot}$
events. We here show that in order for the double-detonation model (DDM, in
which a propagating thermonuclear detonation wave, TNDW, within a thin helium
shell surrounding a sub-Chandrasekhar mass CO core triggers a TNDW within the
core) to explain this low-luminosity suppression, the probability of a low-mass
($\approx0.85\,M_{\odot}$) WD explosion should be $\sim100$-fold lower than
that of a high-mass ($\approx1.05\,M_{\odot}$) WD. One possible explanation is
that the ignition of low-mass CO cores is somehow suppressed. We use accurate
one-dimensional numerical simulations to show that if a TNDW is able to
propagate within the helium shell, then the ignition within the CO core is
guaranteed (resolved here for the first time in a full-star simulation), even
for $0.7\,M_{\odot}$ WDs, providing no natural explanation for the
low-luminosity suppression. DDM could explain the low-luminosity suppression if
the mass distribution of primary WDs in close binaries is dramatically
different from the field distribution; if the Helium shell ignition probability
is suppressed for low-mass WDs; or if multidimensional perturbations
significantly change our results.
