How did early Earth stay unfrozen under a fainter young Sun?
In plain words
Four billion years ago the Sun gave about 25 percent less light, enough to freeze today's Earth solid, yet rocks show liquid water. Something, such as extra greenhouse gases or a darker surface, must have made up the difference.
Precise statement
For Archean Earth (3.8 to 2.5 Gyr ago) with solar constant $S \sim 0.75 S_{0}$ at 3.8 Gyr ago rising to $\sim 0.82 S_{0}$ at 2.5 Gyr ago ($S_{0} = 1.36 \times 10^{6}\ \mathrm{erg}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}$), identify the combination of greenhouse gases ($\mathrm{CO_2}$, $\mathrm{CH_4}$, $\mathrm{N_2}$ pressure broadening, $\mathrm{H_2}$), cloud and surface albedo changes that yields global mean surface temperature above freezing while satisfying geochemical limits on pCO2 from paleosols and marine carbonates. An answer is a mechanism with an atmospheric composition consistent with all proxy bounds in a 3D climate model.
What would settle it
Proxy-consistent Archean pCO2 and pCH4 reconstructions that, in a 3D climate model with clouds, give above-freezing global temperatures without ad hoc assumptions.
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