  Hycean worlds are a proposed subset of sub-Neptune exoplanets with
substantial water inventories, liquid surface oceans and extended
hydrogen-dominated atmospheres that could be favourable for habitability. In
this work, we aim to quantitatively define the inner edge of the Hycean
habitable zone using a 1D radiative-convective model. As a limiting case, we
model a dry hydrogen-helium envelope above a surface ocean. We find that 10 to
20 bars of atmosphere produces enough greenhouse effect to drive a liquid
surface ocean supercritical when forced with current Earth-like instellation.
Introducing water vapour into the atmosphere, we show the runaway greenhouse
instellation limit is greatly reduced due to the presence of superadiabatic
layers where convection is inhibited. This moves the inner edge of the
habitable zone from $\approx$ 1 AU for a G-star to 1.6 AU (3.85 AU) for a
Hycean world with a H$_2$-He inventory of 1 bar (10 bar). For an M-star, the
inner edge is equivalently moved from 0.17 AU to 0.28 AU (0.54 AU). Our results
suggest that most of the current Hycean world observational targets are not
likely to sustain a liquid water ocean. We present an analytical framework for
interpreting our results, finding that the maximum possible OLR scales
approximately inversely with the dry mass inventory of the atmosphere. We
discuss the possible limitations of our 1D modelling and recommend the use of
3D convection-resolving models to explore the robustness of superadiabatic
layers.
