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We show that the luminous supernovae (SNe) associated with ultra-longgamma-ray bursts (GRBs) can be related to the slow cooling from the explosionsof hydrogen-free progenitors extended by pulsational pair-instability. In theaccompanying paper (Marchant & Moriya 2020), we have shown that somerapidly-rotating hydrogen-free GRB progenitors that experience pulsationalpair-instability can keep an extended structure caused by pulsationalpair-instability until the core collapse. Such progenitors have large radiiexceeding 10 Rsun and they sometimes reach beyond 1000 Rsun at the time of thecore collapse. They are, therefore, promising progenitors of ultra-long GRBs.We here perform the light-curve modeling of the explosions of one extendedhydrogen-free progenitor with a radius of 1962 Rsun. Thanks to the largeprogenitor radius, the ejecta experience slow cooling after the shock breakoutand they become rapidly evolving (<~ 10 days) luminous (>~ 1e43 erg/s) SNe inoptical even without the energy input from the 56Ni nuclear decay when theexplosion energy is more than 1e52 erg. The 56Ni decay energy input can affectthe light curves after the optical light-curve peak and make the light-curvedecay slow when the 56Ni mass is around 1 Msun. They also have fastphotospheric velocity above 10,000 km/s and hot photospheric temperature above10,000 K at around the peak luminosity. We find that the rapid rise andluminous peak found in the optical light curve of SN 2011kl, which isassociated with the ultra-long GRB 111209A, can be explained as the coolingphase of the extended progenitor. The ultra-long GRB progenitors proposed inMarchant & Moriya (2020) can explain both the ultra-long GRB duration and theaccompanying SN properties. When the GRB jet is off-axis or choked, theluminous SNe could be observed as fast blue optical transients withoutaccompanying GRBs. (abridged)

Luminous supernovae associated with ultra-long gamma-ray bursts from hydrogen-free progenitors extended by pulsational pair-instability