Rapid Optical Variations Correlated with X-rays in the 2015 Second Outburst of V404 Cygni (GS 2023$+$338)

We present optical multi-colour photometry of V404 Cyg during the outburst from December, 2015 to January, 2016 together with the simultaneous X-ray data. This outburst occurred less than 6 months after the previous outburst in June-July, 2015. These two outbursts in 2015 were of a slow rise and rapid decay-type and showed large-amplitude ($\sim$2 mag) and short-term ($\sim$10 min-3 hours) optical variations even at low luminosity (0.01-0.1$L_{\rm Edd}$). We found correlated optical and X-ray variations in two $\sim$1 hour time intervals and performed Bayesian time delay estimations between them. In the previous version, the observation times of X-ray light curves were measured at the satellite and their system of times was Terrestrial Time (TT), while those of optical light curves were measured at the Earth and their system of times was Coordinated Universal Time (UTC). In this version, we have corrected the observation times and obtained a Bayesian estimate of an optical delay against the X-ray emission, which is $\sim$30 s, during those two intervals. In addition, the relationship between the optical and X-ray luminosity was $L_{\rm opt} \propto L_{\rm X}^{0.25-0.29}$ at that time. These features can be naturally explained by disc reprocessing.