{Abridged} We investigate the observability of cold accretion streams at redshift 3 via Lyman-alpha (Lya) emission and the feasibility of cold accretion as the main driver of Lya blobs (LABs). We run cosmological zoom simulations focusing on 3 halos spanning two orders of magnitude in mass, roughly from 10^11 to 10^13 solar masses. We use a version of the Ramses code that includes radiative transfer of UV photons, and we employ a refinement strategy that allows us to resolve accretion streams in their natural environment to an unprecedented level. For the first time, we self-consistently model self-shielding in the cold streams from the cosmological UV background, which enables us to predict their temperatures, ionization states and Lya luminosities with improved accuracy. We find the efficiency of gravitational heating in cold streams in a ~10^11 solar mass halo is around 10-20% throughout most of the halo but reaching much higher values close to the center. As a result most of the Lya luminosity comes from gas which is concentrated at the central 20% of the halo radius, leading to Lya emission which is not extended. In more massive halos, of >10^12 solar masses, cold accretion is complex and disrupted, and gravitational heating does not happen as a steady process. Ignoring the factors of Lya scattering, local UV enhancement, and SNe feedback, cold accretion alone in these massive halos can produce LABs that largely agree with observations in terms of morphology, extent, and luminosity. Our simulations slightly and systematically over-predict LAB abundances, perhaps hinting that the interplay of these ignored factors may have a negative net effect on extent and luminosity. We predict that a factor of a few increase in sensitivity from current observational limits should unambiguously reveal continuum-free accretion streams around massive galaxies at z=3.
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