  Little is known about the early evolution of Venus and a potential habitable
period during the first one billion years. In particular, it remains unclear
whether or not plate tectonics and an active carbonate-silicate cycle were
present. In the presence of liquid water but without plate tectonics,
weathering would have been limited to freshly produced basaltic crust, with an
early carbon cycle restricted to the crust and atmosphere. With the evaporation
of surface water, weathering would cease. With ongoing volcanism, carbonate
sediments would be buried and sink downwards. Thereby, carbonates would heat up
until they become unstable and the crust would become depleted in carbonates.
With CO$_2$ supply to the atmosphere the surface temperature rises further, the
depth below which decarbonation occurs decreases, causing the release of even
more CO$_2$.
  We assess the habitable period of an early stagnant-lid Venus by employing a
coupled interior-atmosphere evolution model accounting for CO$_2$ degassing,
weathering, carbonate burial, and crustal decarbonation. We find that if
initial surface conditions allow for liquid water, weathering can keep the
planet habitable for up to 900 Myr, followed by evaporation of water and rapid
crustal carbonate depletion. For the atmospheric CO$_2$ of stagnant-lid
exoplanets, we predict a bimodal distribution, depending on whether or not
these planets experienced a runaway greenhouse in their history. Planets with
high atmospheric CO$_2$ could be associated with crustal carbonate depletion as
a consequence of a runaway greenhouse, whereas planets with low atmospheric
CO$_2$ would indicate active silicate weathering and thereby a habitable
climate.
