We use GOODS and CANDELS images to identify progenitors of massive (log M > 10 Msun) compact early-type galaxies (ETGs) at z~1.6. Since merging and accretion increase the size of the stellar component of galaxies, if the progenitors are among known star-forming galaxies, these must be compact themselves. We select candidate progenitors among compact Lyman-break galaxies at z~3 based on their mass, SFR and central stellar density and find that these account for a large fraction of, and possibly all, compact ETGs at z~1.6. We find that the average far-UV SED of the candidates is redder than that of the non-candidates, but the optical and mid-IR SED are the same, implying that the redder UV of the candidates is inconsistent with larger dust obscuration, and consistent with more evolved (aging) star-formation. This is in line with other evidence that compactness is a sensitive predictor of passivity among high-redshift massive galaxies. We also find that the light distribution of both the compact ETGs and their candidate progenitors does not show any extended halos surrounding the compact core, both in individual images and in stacks. We argue that this is generally inconsistent with the morphology of merger remnants, even if gas-rich, as predicted by N-body simulations. This suggests that the compact ETGs formed via highly dissipative, mostly gaseous accretion of units whose stellar components are very small and undetected in the HST images, with their stellar mass assembling in-situ, and that they have not experienced any major merging until the epoch of observations at z~1.6.
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