"Invisible" axion rolling through QCD phase transition

The origin of “invisible” axion in four dimensional effective beyond-standard models from string compactification is discussed and its refined passover through the QCD phase transition is presented toward a reliable estimate of the current axion energy density in terms of the initial misalignment angle θ̄1. The explicit examples are presented in a flipped SU(5) GUT model. This allows to introduce a flavor symmetry through string compactification, and hence we also comment on the source of flavor symmetries from string compactification and attempts to fit the resulting Yukawa couplings to the observed Cabibbo-Kobayashi-Maskawa and Pontecorvo-MakiNakagawa-Sakata matrices.