Deuteration of c-C$_3$H$_2$ towards the pre-stellar core L1544

Context: In the centre of pre-stellar cores, the deuterium fractionation is enhanced due to the cold temperatures and high densities. Therefore, the chemistry of deuterated molecules can be used to probe the evolution and the kinematics in the earliest stages of star formation. Aims: We analyse emission maps of cyclopropenylidene, c-C$_3$H$_2$, to study the distribution of the deuteration throughout the prototypical pre-stellar core L1544. Methods: We use single-dish observations of c-C$_3$H$_2$, c-H$^{13}$CC$_2$H, c-C$_3$HD, and c-C$_3$D$_2$ towards the pre-stellar core L1544, performed at the IRAM 30m telescope. We derive the column density and deuterium fraction maps, and compare these observations with non-LTE radiative transfer simulations. Results: The highest deuterium fractions are found close to the dust peak at the centre of L1544, where the increased abundance of H$_2$D$^+$ ions drives the deuteration process. The peak values are N(c-C$_3$HD)/N(c-C$_3$H$_2)=0.17\pm0.01$, N(c-C$_3$D$_2$)/N(c-C$_3$H$_2)=0.025\pm0.003$ and N(c-C$_3$D$_2$)/N(c-C$_3$HD$)=0.16\pm0.03$, which is consistent with previous single point observations. The distributions of c-C$_3$HD and c-C$_3$D$_2$ indicate that the deuterated forms of c-C$_3$H$_2$ in fact trace the dust peak and not the c-C$_3$H$_2$ peak. Conclusions: The N(c-C$_3$D$_2$)/N(c-C$_3$HD) map confirms that the process of deuteration is more efficient towards the centre of the core and demonstrates that carbon-chain molecules are still present at high densities. This is likely caused by an increased abundance of He$^+$ ions destroying CO, which increases the amount of carbon atoms in the gas phase.