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 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.