Period--Luminosity Relations for Double-mode $\delta$ Sct Stars

Previous studies of the Period--Luminosity relations (PLRs) of Delta Scuti ($\delta$ Sct) stars have focused on those with a single pulsation mode. However, for $\delta$ Sct stars with many different pulsation modes, classifying a single mode is difficult. In this study, an all-sky dataset is constructed using double-mode $\delta$ Sct stars from ZTF and OGLE, and is used to determine F-mode and 1O-mode PLRs for eight single bands and six Wesenheit bands. In the $W1$ band, the PLR dispersion is about 0.171 mag and the total zero point error is 1\%. Our results show that to accurately classify the 1O modes of $\delta$ Sct stars requires authentication based on multiple modes. Classification based on amplitude alone leads to impure 1O-mode $\delta$ Sct stars and significant deviations in the PLRs. We compare the PLRs of the different sequences in the Petersen diagram and find that they are consistent after a strict criterion filtering, suggesting that their evolutionary state is similar. In addition, we find a weak period--metallicity relation for double-mode $\delta$ Sct stars, unlike double-mode RR Lyrae stars. As distance tracers, large-amplitude F-mode $\delta$ Sct and double-mode $\delta$ Sct stars are the most suitable of the $\delta$ Sct family.