A Linear Relation Between the Color Stretch $s_{BV}$ and the Rising Color Slope $s_0^*(B-V)$ of Type Ia Supernovae

Using data from the Complete Nearby ($z_{host}<0.02$) sample of Type Ia Supernovae (CNIa0.02), we discover a linear relation between two parameters derived from the $B-V$ color curves of Type Ia supernovae: the "color stretch" $s_{BV}$ and the rising color slope $s_0^*(B-V)$ after the peak, and this relation applies to the full range of $s_{BV}$. The $s_{BV}$ parameter is known to be tightly correlated with the peak luminosity, and especially for "fast decliners" (dim Type Ia supernovae), and the luminosity correlation with $s_{BV}$ is markedly better than with the classic light-curve width parameters such as $\Delta{m_{15}(B)}$. Thus our new linear relation can be used to infer peak luminosity from $s_0^*$. Unlike $s_{BV}$ (or $\Delta{m_{15}}$), the measurement of $s_0^*(B-V)$ does not rely on the well-determined time of light-curve peak or color maximum, making it less demanding on the light-curve coverage than past approaches.