  Water is an important molecule in the chemical and thermal balance of dense
molecular gas, but knowing its history through-out the various stages of the
star formation is a fundamental problem. Its molecular deuteration provides us
with a crucial clue to its formation history. H$_2$O has recently been detected
for the first time towards the prestellar core L1544 with the Herschel Space
Observatory with a high spectral resolution (HIFI instrument). Prestellar cores
provide the original reservoir of material from which future planetary systems
are built, but few observational constraints exist on the formation of water
and none on its deuteration before the collapse starts and a protostar forms at
the centre. We report on new APEX observations of the ground state
1$_{0,1}$-0$_{0,0}$ HDO transition at 464 GHz towards the prestellar core
L1544. The line is undetected, and we present an extensive study of the
conditions for its detectability in cold and dense cloud cores. The water and
deuterated water abundances have been estimated using an advanced chemical
model simplified for the limited number of reactions or processes that are
active in cold regions (< 15 K). We use the LIME radiative transfer code to
compute the expected intensity and profile of both H$_2$O and HDO lines and
compare them with the observations. We present several ad hoc profiles that
best-fit the observations and compare the profiles with results from an
astrochemical modelling, coupling gas phase and grain surface chemistry. Our
comparison between observations, radiative transfer, and chemical modelling
shows the limits of detectability for singly deuterated water, through the
ground-state transitions 1$_{0,1}$-0$_{0,0}$ and 1$_{1,1}$-0$_{0,0}$ at 464.9
and 893.6 GHz, respectively, with both single-dish telescope and
interferometric observations.
