Nebular H$\alpha$ Limits for Fast Declining Type Ia Supernovae
  One clear observational prediction of the single degenerate progenitor
scenario as the origin of type Ia supernovae (SNe) is the presence of
relatively narrow ($\approx$1000 km s$^{-1}$) H$\alpha$ emission at nebular
phases, although this feature is rarely seen. We present a compilation of
nebular phase H$\alpha$ limits for SN Ia in the literature and demonstrate that
this heterogenous sample has been biased towards SN Ia with relatively high
luminosities and slow decline rates, as parameterized by $\Delta$m$_{15}(B)$,
the difference in $B$-band magnitude between maximum light and fifteen days
afterward. Motivated by the need to explore the full parameter space of SN~Ia
and their subtypes, we present two new and six previously published nebular
spectra of SN Ia with $\Delta$m$_{15}(B)$$ > $1.3 mag (including members of the
transitional and SN1991bg-like subclasses) and measure nondetection limits of
$L_{H\alpha}$$~<~$0.85--9.9$\times$10$^{36}$ ergs s$^{-1}$, which we confirmed
by implanting simulated H$\alpha$ emission into our data. Based on the lastest
models of swept-up material stripped from a nondegenerate companion star, these
$L_{H\alpha}$ values correspond to hydrogen mass limits of
$M_H$$~\lesssim~$1-3$\times$10$^{-4}$ $M_{\odot}$, roughly three orders of
magnitude below that expected for the systems modeled, although we note that no
simulations of H$\alpha$ nebular emission in such weak explosions have yet been
performed. Despite the recent detection of strong H$\alpha$ in ASASSN-18tb (SN
2018fhw; $\Delta$m$_{15}(B)$ = 2.0 mag), we see no evidence that fast declining
systems are more likely to have late time H$\alpha$ emission, although a larger
sample is needed to confirm this result.
