Early observations of supernova light curves are powerful tools for shedding light on the pre-explosion structures of their progenitors and their mass-loss histories just prior to explosion. Some core-collapse supernovae that are detected during the first days after the explosion prominently show two peaks in the optical bands, including the R and I bands, where the first peak appears to be powered by the cooling of shocked surface material and the second peak is clearly powered by radioactive

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... red by radioactive decay. Such light curves have been explored in detail theoretically for SN 1993J and 2011dh, where it was found that they may be explained by progenitors with extended, low-mass envelopes. Here, we generalize these results. We first explore whether any double-peaked light 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 a standard 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 luminosity as is seen after the first peak. We then explore the signature of a progenitor with a compact core surrounded by extended, low-mass material. This may be a hydrostatic low-mass envelope or material ejected just prior to the explosion. We show that it naturally produces both of these features. We use this result to provide simple formulae to estimate (1) the mass of the extended material from the time of the first peak, (2) the extended material radius from the luminosity of the first peak, and (3) an upper limit on the core radius from the luminosity minimum between the two peaks.