  H2S is thought to be the main sulphur reservoir in the ice, being therefore a
key molecule to understand sulphur chemistry in the star formation process and
to solve the missing sulphur problem. The H2S deuterium fraction can be used to
constrain its formation pathways. We investigate for the first time the H2S
deuteration in a large sample of starless cores (SC). We use observations of
the GEMS IRAM 30m Large Program and complementary IRAM 30m observations. We
consider a sample of 19 SC in Taurus, Perseus, and Orion, detecting HDS in 10
and D2S in five. The H2S single and double deuterium fractions are analysed
with regard to their relation with the cloud physical parameters, their
comparison with other interstellar sources, and their comparison with deuterium
fractions in early stage star-forming sources of c-C3H2, H2CS, H2O, H2CO, and
CH3OH. We obtain a range of X(HDS)/X(H2S)~0.025-0.2 and X(D2S)/X(HDS)~0.05-0.3.
H2S single deuteration shows an inverse relation with the cloud kinetic
temperature. H2S deuteration values in SC are similar to those observed in
Class 0. Comparison with other molecules in other sources reveals a general
trend of decreasing deuteration with increasing temperature. In SC and Class 0
objects H2CS and H2CO present higher deuteration fractions than c-C3H2, H2S,
H2O, and CH3OH. H2O shows single and double deuteration values one order of
magnitude lower than those of H2S and CH3OH. Differences between c-C3H2, H2CS
and H2CO deuterium fractions and those of H2S, H2O, and CH3OH are related to
deuteration processes produced in gas or solid phases, respectively. We
interpret the differences between H2S and CH3OH deuterations and that of H2O as
a consequence of differences on the formation routes in the solid phase,
particularly in terms of the different occurrence of the D-H and H-D
substitution reactions in the ice, together with the chemical desorption
processes.
