Mitigating Random-Phase Sampling Noise in the Cepheid Period-Luminosity Relation: A Cross-Filter Consistency Approach

The Period-Luminosity (PL) relation is usually derived using time-averaged magnitudes, which require multiple-epoch observations to determine periods and adequately sample the light curves. Although single-epoch observations are more practical and require significantly less observational effort, they inherently introduce greater photometric scatter, leading to an increased dispersion in the derived Period-Luminosity relations. In this paper, we explore, in detail, a method that transforms single random-phase data to their mean-light values, using information obtained in other bands for the same Cepheid. This approach enables the accurate re-construction of mean-light PL relations for wavelengths observed with space-based facilities, for instance, where the number of epochs per star makes simple averaging or template fitting less than optimal, with the latter requiring very high-precision periods for predictive phasing. While applying this technique across multiple bands, from optical to mid-IR, we focus particularly on widely separated bands covering the mid-IR to the optical. We showcase this method using the J band (as being observed by JWST) as the random-phase component. Our results show that this correction reduces the scatter of the PL relation in the J band by a factor of approximately $0.7\times$, equivalent to increasing the number of random-phase observations by a factor of 10, needed to obtain the same increase in precision as delivered here.