= Standardization of Type Ia supernova light curves
{title2=$m_{B,\mathrm{corr}}=m_B+\alpha(s-1)$}
The stretch method rescales the <light curve>'s time axis by $s(1+z)$ and corrects peak <apparent magnitude> by $\alpha(s-1)$, fitting $\alpha$ jointly with cosmology.<Measurements of Omega and Lambda from 42 High-Redshift Supernovae by Saul Perlmutter and others>[Section 2, equations (1)–(2): stretch and time dilation]{ref}
Here is a deliberately stipulated local model that exposes the signs. Let $t_{\rm obs}$ be observer time in <days>, $t_{\max}$ the observed peak time and $T(u)$ a dimensionless template with $T(0)=1$. In a single <photometric passband>, write
$$
u=\frac{t_{\rm obs}-t_{\max}}{s(1+z)},\qquad F(t_{\rm obs})=F_{\max}T(u).
$$
Here $u$ is rest-frame time in <days>; the template's argument carries those units. For illustration, assign the intrinsic relation $M_B(s)=M_{B,0}-\alpha(s-1)$ with $\alpha>0$: broader events are intrinsically brighter. Define the sign of the <K correction> by $m_{\rm obs}=M_B(s)+\mu+A_B+K$, where $A_B$ is <interstellar extinction> in magnitudes. Then
$$
m_{B,\rm corr}=m_{\rm obs}-K-A_B+\alpha(s-1)=M_{B,0}+\mu.
$$
Removing attenuation makes the magnitude smaller; standardizing a broad, intrinsically bright event makes it larger. The fitted <cosmological time dilation> must be removed before comparing intrinsic durations. This schematic factorization explains the bookkeeping; an actual multicolour fit must also model changing intrinsic colours and correlated uncertainties.
For two equally standardized sources, cancellation of their common <absolute magnitude> gives an especially clean result:
$$
\mu_2-\mu_1=m_{B,\rm corr,2}-m_{B,\rm corr,1},\qquad
\frac{d_{L,2}}{d_{L,1}}=10^{(\mu_2-\mu_1)/5}.
$$
A stipulated residual of $+0.20$ mag therefore means a distance ratio $10^{0.04}=1.09648$ and a <radiative flux> ratio $10^{-0.08}=0.83176$ relative to the comparison prediction, provided <luminosity> and attenuation have been correctly standardized. This is a conversion of a hypothetical residual, not a measured survey value.
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