Physical conditions in Centaurus A's northern filaments II: Does the HCO$^+$ emission highlight the presence of shocks?

Abridged: We present the first observation of the HCO+(1-0) and HCN(1-0) emission in the northern filaments of Centaurus A with ALMA. HCO+(1-0) is detected in 9 clumps of the Horseshoe complex, with similar velocities as the CO(1-0) emission. Conversely, the HCN(1-0) is not detected and we derive upper limits on the flux. At a resolution of ~40 pc, the line ratio of the velocity-integrated intensities I_HCO+/I_CO varies between 0.03 and 0.08, while I_HCO+/I_HCN is higher than unity with an average lower limit of 1.51. These ratios are significantly higher than what is observed in nearby star-forming galaxies. Moreover, the ratio I_HCO+/I_CO decreases with increasing CO integrated intensity, contrary to what is observed in the star-forming galaxies. This indicates that the HCO+ emission is enhanced and may not arise from dense gas within the Horseshoe complex. This hypothesis is strengthened by the average line ratio I_HCN/I_CO<0.03 which suggests that the gas density is rather low. Using non-LTE, large velocity gradient modelling with RADEX, we explored two possible phases of the gas, that we call "diffuse" and "dense", and are characterised by a significant difference in the HCO+ relative abundance to CO, respectively N_HCO+/N_CO=10^-3 and 3x10^-5. The average CO(1-0) and HCO+(1-0) integrated intensities and the upper limit on HCN(1-0) are compatible with both "diffuse" and "dense" gas. The spectral setup of the present observations also covers the SiO(2-1). While undetected, the upper limit on SiO(2-1) is not compatible with the RADEX predictions for the "dense" gas. We conclude that the 9 molecular clouds detected in HCO+(1-0) are likely dominated by diffuse molecular gas. While the exact origin of the HCO+(1-0) emission remains to be investigated, it is likely related to the energy injection within the molecular gas that prevents gravitational collapse and star formation.