A More General Model for the Intrinsic Scatter in Type Ia Supernova
  Distance Moduli
  We describe a new formalism to fit the parameters $\alpha$ and $\beta$ that
are used in the SALT2 model to determine the standard magnitudes of Type Ia
supernovae. The new formalism describes the intrinsic scatter in Type Ia
supernovae by a covariance matrix in place of the single parameter normally
used. We have applied this formalism to the Sloan Digital Sky Survey Supernova
Survey (SDSS-II) data and conclude that the data are best described by
$\alpha=0.135_{-.017}^{+.033}$ and $\beta=3.19_{-0.24}^{+0.14}$, where the
error is dominated by the uncertainty in the form of the intrinsic scatter
matrix. Our result depends on the introduction of a more general form for the
intrinsic scatter of the distance moduli of Type Ia supernovae than is
conventional, resulting in a larger value of $\beta$ and a larger uncertainty
than the conventional approach. Although this analysis results in a larger
value of $\beta$ and a larger error, the SDSS data differ (at a 98% confidence
level) with $\beta=4.1$, the value expected for extinction by the type of dust
found in the Milky Way. We have modeled the distribution of supernovae Ia in
terms of their color and conclude that there is strong evidence that variation
in color is a significant contributor to the scatter of supernovae Ia around
their standard candle magnitude.
