What kept early Mars warm enough for rivers and lakes?
In plain words
About 3.7 billion years ago Mars had river valleys and lakes even though it is farther from a then fainter Sun. Carbon dioxide and water vapour alone do not seem able to warm it enough, so the question is what else did.
Precise statement
For Noachian to early Hesperian Mars (about $3.8\ \text{to}\ 3.5\,\mathrm{Gyr}$ ago, insolation $\sim 0.32\,S_0$ relative to modern Earth) determine whether surface runoff required sustained warm conditions (mean $T$ above $273\,\mathrm{K}$) or episodic warming, and identify the forcing: $\mathrm{CO_2}$-$\mathrm{H_2}$ or $\mathrm{CO_2}$-$\mathrm{CH_4}$ collision-induced absorption, cirrus clouds, impacts or volcanic outgassing. An answer specifies atmospheric composition, pressure and duration consistent with valley-network and lake-sediment volumes and with surviving carbonate inventories.
What would settle it
A climate scenario whose required $\mathrm{H_2}$ or $\mathrm{CH_4}$ sources, surface pressure and warm-period duration match sediment volumes and the measured carbonate and atmospheric-loss record.
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