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A full renormalisation of the Inert Doublet Model (IDM) is presented andexploited for a precise calculation of the relic density of Dark Matter (DM) atone-loop. In this first paper, we study the case of a DM candidate with $m_{\rmDM} \sim 500$ GeV. In this regime, the co-annihilation channels are important.We therefore compute, for a wide range of relative velocities, the fullnext-to-leading order electroweak corrections to 7 annihilation/co-annihilationprocesses that contribute $\sim$ 70\% to the relic density of DM. Thesecorrected cross-sections are interfaced with {\tt micrOMEGAs} to obtain theone-loop correction to the freeze-out relic density. Due to the accuratemeasurement of this observable, the one-loop corrections are relevant. Wediscuss the one-loop renormalisation scheme dependence and point out theinfluence, at one-loop, of a parameter that solely describes the scattering inthe dark sector. A tree-level computation of the relic density is not sensitiveto this parameter.

Relic density of Dark Matter in the Inert Doublet Model beyond Leading Order. I) The Heavy Mass Case