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The Peccei-Quinn symmetry is an automatic symmetry of the 3-3-1 gauge modelswhich, consequently are not plagued with the strong CP problem. Nevertheless,the axion that emerges from spontaneous breaking of Peccei-Quinn symmetrycannot be made invisible in the original versions of these models, unless weextend their scalar sector by an additional neutral scalar singlet. In thiscase we show that if, we also add heavy neutrinos in the singlet form, we getto solve three interesting open questions at once: the real component of theneutral scalar singlet driving inflation, the axion playing the role of thedark matter of the universe and standard neutrinos gaining masses throughseesaw mechanism.

Embedding cosmological inflation, axion dark matter and seesaw mechanism in a 3-3-1 gauge model