Natural Multi-Higgs Model with Dark Matter and CP Violation

We explore an extension of the Inert Doublet Model which allows also for CP violation in the Higgs sector. This necessitates two non-inert doublets. The lightest neutral scalar of the inert doublet is a candidate for dark matter. Scanning over parameters we preserve the abundance of the dark matter in agreement with the WMAP data. We also impose all relevant collider and theoretical constraints to determine the allowed parameter space for which both the dark matter is appropriate and CP is violated. In addition we find regions where the cut-off of the model originating from naturality arguments can be substantially lifted compared to its Standard Model value, reaching 2-3 TeV.