Follow-up photometry in another band benefits reducing \emph{Kepler}'s false positive rates

Kepler Mission's single-band photometry suffers from astrophysical false positives, the most common of background eclipsing binaries (BEBs) and companion transiting planets (CTPs). Multi-color photometry can reveal the color-dependent depth feature of false positives and thus exclude them. In this work, we aim to estimate the fraction of false positives that are unable to be classified by Kepler alone but can be identified with their color-dependent depth feature if a reference band (z, Ks and TESS) were adopted in follow-up observation. We build up physics-based blend models to simulate multi-band signals of false positives. Nearly 65-95% of the BEBs and more than 80% of the CTPs that host a Jupiter-size planet will show detectable depth variations if the reference band can achieve a Kepler-like precision. Ks band is most effective in eliminating BEBs exhibiting any depth sizes, while z and TESS band prefer to identify giant candidates and their identification rates are more sensitive to photometric precision. Provided the radius distribution of planets transiting the secondary star in binary systems, we derive formalism to calculate the overall identification rate for CTPs. By comparing the likelihood distribution of the double-band depth ratio for BEB and planet models, we calculate the false positive probability (FPP) for typical Kepler candidates. Additionally, we show that the FPP calculation helps distinguish the planet candidate's host star in an unresolved binary system. The analysis framework of this paper can be easily adapted to predict the multi-color photometry yield for other transit surveys, especially for TESS.