In this Appendix we provide the mathematical details needed to understand the relation between the ΣHα group magnitudes and the observed mass step. This derivation provides the relation between the numerical values of the Hα bias and the mass step in the SNfactory dataset.
To begin with, the dataset is partioned into four subsets consisting of SNe Iaα and SNe Iaϵ above and below the canonical mass division point of log (M/M⊙) = 10. We denote the number in each subset as Nα,L, Nα,H, Nϵ,L, and Nϵ,H, with L and H symbolizing SNe Ia in low- and high-mass hosts, respectively. Likewise, the mean Hubble residuals for each subset are denoted as , , , and .
To make the equations more readable we introduce the following notation for fractions: means “the fraction of SNe from subset A that have property B”. For instance, means “the fraction of SNe Ia from high-mass hosts that are Iaϵ”, i.e., Nϵ,H/NH. Mathematically, B is the numerator and A the denominator. Naturally, the partitioning imposes the requirement that , e.g., .
The Hα bias, , is defined as the weighted mean difference of the between the Iaϵ and Iaα subsamples: (A.1)The mass step, , is defined as the weighted mean difference of the between SNe Ia in low-mass and the high-mass hosts: (A.2)The individual weighted mean magnitudes entering Eqs. (A.1) and (A.2) can be written in terms of the weighted mean magnitudes for each portion. Introduction of our notation then gives Combining these we obtain the equations for the Hα bias and mass step in terms of the partitioned data: With this development it is now possible to write the mass step directly in terms of the Hα bias, as follows: (A.9)In this expression we have simply rearranged terms to demonstrate that the same weighted mean magnitudes, just having different signs and fractions, appear in both the numerator and denominator.
Equation (A.9) is exact when all of the quantities can be directly measured, as they are in our dataset. These variables can be generalized to any similar dataset. In order to be exact, though, observations of the local environment would be required. However, in the main text we presented two observational constraints that can be used to greatly simply Eq. (A.9). First, we have shown in Sect. 5.2.2 that the averaged of SNe Iaα does not depend on the masses of their hosts: (A.10)Second, we observed in Fig. 10 that SNe Iaα and Iaϵ from low-mass hosts share the same mean magnitude (within error bars): (A.11)As a consequence of these two observations, the righthand side of Eq. (A.7) becomes (A.12)Similarly, the righthand side of Eq. (A.8) becomes (A.13)Finally, using the relation , where Fϵ is the fraction of SNe Iaϵ, and FH the fraction of SNe Ia from high-mass hosts, Eq. (A.9) can simply be expressed as (A.14)In the main text, the dependence of Fϵ with redshift is denoted ψ(z), and is assumed to be constant. Furthermore, based on the dataset we compiled in Childress et al. (2013a), FH appears to be constant within 10% up to z ~ 1. Subject to these approximations, we conclude that the mass-step evolution is simply proportional to ψ(z), as given in Eq. (6).
© ESO, 2013