Cosmic rays cannot explain the high ionisation rates in the Galactic centre

The H$_2$ ionisation rate in the central molecular zone, located in the Galactic centre, is estimated to be $\zeta\sim2\times10^{-14}~\mathrm{s}^{-1}$, based on observations of H$_3^+$ lines. This value is two to three orders of magnitude larger than that measured anywhere else in the Galaxy. Due to the high density of the gas in the central molecular zone, UV and X-ray photons do not penetrate this region. Hence, cosmic rays are expected to be the exclusive agents of ionisation. A high cosmic-ray density has been invoked to explain the unusually high ionisation rate. However, this excess is not seen in the $\gamma$-ray emission from this region, which is produced by high-energy cosmic rays. Therefore, an excess is expected only in the low-energy cosmic-ray spectrum. Here, we derive constraints on this hypothetical low-energy component in the cosmic-ray spectra, and we question its plausibility. To do so, we numerically solved the cosmic-ray transport equation in the central molecular zone. We derived stationary solutions under the assumption that cosmic rays are continuously injected by a source located in the Galactic centre. The high-energy component in the cosmic-ray spectrum was then fitted to available $\gamma$-ray and radio data, and a steep low-energy component was added to the cosmic-ray spectrum to explain the large ionisation rates. We find that injection spectra of $p^{-7}$ for protons below $p_\mathrm{enh,p}c\simeq780~\mathrm{MeV}$ and $p^{-5.2}$ for electrons below $p_\mathrm{enh,e}c=1.5~\mathrm{GeV}$ are needed to reach the observed ionisation rates. This corresponds to a cosmic-ray power of the order of $\sim10^{40-41}~\mathrm{erg}\,\mathrm{s}^{-1}$ injected at the Galactic centre. We conclude that cosmic rays alone cannot explain the high ionisation rates in the Galactic centre.