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A Generalized $K$ correction for Type Ia Supernovae: Comparing R-band Photometry Beyond z=0.2 with B, V, and R-band Nearby Photometry

Photometric measurements show that as a group nearby type Ia supernovae follow similar lightcurves and reach similar peak magnitudes Thus, these supernovae can serve as standard candles or calibrated candles at cosmological distances. Magnitudes of local and distant supernovae, both in the same filter band, are compared using a $K$ correction to account for the different spectral regions incident on that filter. A generalized approach compares magnitudes in different bands for the nearby and distant supernova, bands that are selected to give sensitivity in corresponding regions of the redshifted and unredshifted spectra. Thus at a redshift of $z \approx 0.5$, local $B$ magnitudes are compared with distant $R$ magnitudes. We compute these generalized $K$ corrections over a range of redshifts and bandpass pairs and discuss their advantages over the traditional single-band $K$ correction. In particular, errors near maximum light can be kept below 0.05 mag out to at least $z=0.6$, whereas the traditional $K$ correction is difficult to use beyond $z > 0.2$.