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The study of deuteration in pre-stellar cores is important to understand thephysical and chemical initial conditions in the process of star formation. Inparticular, observations toward pre-stellar cores of methanol and deuteratedmethanol, solely formed on the surface of dust grains, may provide usefulinsights on surface processes at low temperatures. Here we analyze maps of CO,methanol, formaldehyde and their deuterated isotopologues toward a well-knownpre-stellar core. This study allows us to test current gas-dust chemicalmodels. Single-dish observations of CH$_3$OH, CH$_2$DOH, H$_2$CO,H$_2\,^{13}$CO, HDCO, D$_2$CO and C$^{17}$O toward the prototypical pre-stellarcore L1544 were performed at the IRAM 30 m telescope. We analyze their columndensities, distributions, and compare these observations with gas-grainchemical models. The maximum deuterium fraction derived for methanol is[CH$_2$DOH]/[CH$_3$OH] $\sim$ 0.08$\pm$0.02, while the measured deuteriumfractions of formaldehyde at the dust peak are [HDCO]/[H$_2$CO] $\sim$0.03$\pm$0.02, [D$_2$CO]/[H$_2$CO] $\sim$ 0.04$\pm$0.03 and [D$_2$CO]/[HDCO]$\sim$ 1.2$\pm$0.3. Observations differ significantly from the predictions ofmodels, finding discrepancies between a factor of 10 and a factor of 100 inmost cases. It is clear though that to efficiently produce methanol on thesurface of dust grains, quantum tunneling diffusion of H atoms must be switchedon. It also appears that the currently adopted reactive desorption efficiencyof methanol is overestimated and/or that abstraction reactions play animportant role. More laboratory work is needed to shed light on the chemistryof methanol, an important precursor of complex organic molecules in space.

Mapping deuterated methanol toward L1544: I. Deuterium fraction and comparison with modeling