Gas phase Elemental abundances in Molecular cloudS (GEMS). II. On the
  quest for the sulphur reservoir in molecular clouds: the $H_{2}S$ case
  Sulphur is one of the most abundant elements in the Universe. Surprisingly,
sulphuretted molecules are not as abundant as expected in the interstellar
medium, and the identity of the main sulphur reservoir is still an open
question. Our goal is to investigate the H$_{2}$S chemistry in dark clouds, as
this stable molecule is a potential sulphur reservoir. Using millimeter
observations of CS, SO, H$_{2}$S, and their isotopologues, we determine the
physical conditions and H$_{2}$S abundances along the cores TMC 1-C, TMC 1-CP,
and Barnard 1b. The gas-grain model Nautilus is then used to model the sulphur
chemistry and explore the impact of photo-desorption and chemical desorption on
the H$_2$S abundance. Our model shows that chemical desorption is the main
source of gas-phase H$_2$S in dark cores. The measured H$_{2}$S abundance can
only be fitted if we assume that the chemical desorption rate decreases by more
than a factor of 10 when $n_{\rm H}>2\times10^{4}$. This change in the
desorption rate is consistent with the formation of thick H$_2$O and CO ice
mantles on grain surfaces. The observed SO and H$_2$S abundances are in good
agreement with our predictions adopting an undepleted value of the sulphur
abundance. However, the CS abundance is overestimated by a factor of $5-10$.
Along the three cores, atomic S is predicted to be the main sulphur reservoir.
We conclude that the gaseous H$_2$S abundance is well reproduced, assuming
undepleted sulphur abundance and chemical desorption as the main source of
H$_2$S. The behavior of the observed H$_{2}$S abundance suggests a changing
desorption efficiency, which would probe the snowline in these cores. Our
model, however, overestimates the observed gas-phase CS abundance. Given the
uncertainty in the sulphur chemistry, our data are consistent with a cosmic
elemental S abundance with an uncertainty of a factor of 10.
