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

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