The Type Ia Supernova Color-Magnitude Relation and Host Galaxy Dust: A Simple Hierarchical Bayesian Model

Conventional Type Ia supernova (SN Ia) cosmology analyses currently use a simplistic linear regression of magnitude versus color and light curve shape, which does not model intrinsic SN Ia variations and host galaxy dust as physically distinct effects, resulting in low color-magnitude slopes. We construct a probabilistic generative model for the dusty distribution of extinguished absolute magnitudes and apparent colors as the convolution of a intrinsic SN Ia color-magnitude distribution and a host galaxy dust reddening-extinction distribution. If the intrinsic color-magnitude ($M_B$ vs. $B-V$) slope $beta_{int}$ differs from the host galaxy dust law $R_B$, this convolution results in a specific curve of mean extinguished absolute magnitude vs. apparent color. The derivative of this curve smoothly transitions from $beta_{int}$ in the blue tail to $R_B$ in the red tail of the apparent color distribution. The conventional linear fit approximates this effective curve near the average apparent color, resulting in an apparent slope $beta_{app}$ between $beta_{int}$ and $R_B$. We incorporate these effects into a hierarchical Bayesian statistical model for SN Ia light curve measurements, and analyze a dataset of SALT2 optical light curve fits of 248 nearby SN Ia at z < 0.10. The conventional linear fit obtains $beta_{app} approx 3$. Our model finds a $beta_{int} = 2.3 pm 0.3$ and a distinct dust law of $R_B = 3.8 pm 0.3$, consistent with the average for Milky Way dust, while correcting a systematic distance bias of $sim 0.10$ mag in the tails of the apparent color distribution. Finally, we extend our model to examine the SN Ia luminosity-host mass dependence in terms of intrinsic and dust components.