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Early observations of supernova light curves are powerful tools for sheddinglight on the pre-explosion structures of their progenitors and their mass-losshistories just prior to explosion. Some core-collapse supernovae that aredetected during the first days after the explosion prominently show two peaksin the optical bands, including the $R$ and $I$ bands, where the first peakappears to be powered by the cooling of shocked surface material and the secondpeak is clearly powered by radioactive decay. Such light curves have beenexplored in detail theoretically for SN 1993J and 2011dh, where it was foundthat they may be explained by progenitors with extended, low-mass envelopes.Here we generalize these results. We explore, first, whether any double-peakedlight curve of this type can be generated by a progenitor with a "standard"density profile, such as a red supergiant or a Wolf-Rayet star. We show that astandard progenitor (1) cannot produce a double-peaked light curve in the $R$and $I$ bands, and (2) cannot exhibit a fast drop in the bolometric luminosityas is seen after the first peak. We then explore the signature of a progenitorwith a compact core surrounded by extended, low-mass material. This may be ahydrostatic low-mass envelope or material ejected just prior to the explosion.We show that it naturally produces both of these features. We use this resultto provide simple formulae to estimate (1) the mass of the extended materialfrom the time of the first peak, (2) the extended material radius from theluminosity of the first peak, and (3) an upper limit on the core radius fromthe luminosity minimum between the two peaks.

Supernovae with Two Peaks in the Optical Light Curve and the Signature of Progenitors with Low-mass Extended Envelopes