Evaluating K bands for Nancy Grace Roman Space Telescope Rest-Frame NIR SN Ia Distances

Recently, the Nancy Grace Roman Space Telescope (Roman) Project raised the possibility of adding another filter to Roman. Based on the Filter Working Group's recommendations, this filter may be a K-band filter, extending significantly redder than the current-reddest F184. Among other scientific possibilities, this K filter raises the possibility of measuring SNe Ia in the rest-frame NIR out to higher redshifts than is possible with the current filter complement. I perform a simple survey optimization for NIR SN Ia distances with Roman, simultaneously optimizing both filter cutoffs and survey strategy. I find that the roughly optimal K band extends from 19,000A--23,000A (giving exposure times roughly half that of a 20,000A--23,000A Ks filter). Moving the K much redder than this range dramatically increases the thermal background, while moving the K band much bluer limits the redshift reach. Thus I find any large modification reduces or eliminates the gain over the current F184. I consider both rest-frame Y band and rest-frame J band surveys. Although the proposed K band is too expensive for a large rest-frame Y band survey, it increases the rest-frame J Figure of Merit by 59%.