The String Phases of Hawking Radiation, De Sitter Stage and De Broglie Type Duality

We explicitly describe the last stages of black hole evaporation in the context of string theory : the combined study of Quantum Field Theory (QFT) and String Theory (ST) in curved backgrounds allows us to go further in the understanding of quantum gravity effects. The string ``analogue model(or thermo-dynamical approach) is a well suited framework for this purpose.The results also apply to another physically relevant case: de Sitter background. Semiclassical (QFT) and quantum gravity (String) phases or regimes are properly determined (back reaction effects included). The Hawking-Gibbons temperature ${T_H}$ of the semiclassical regime becomes the intrinsic string temperature ${T_S}$ in the quantum gravity regime.The spectrum of black hole evaporation is an incomplete gamma function of $(T_S - T_H)$: the early evaporation is thermal (Hawking radiation), while at the end the black hole undergoes a phase transition to a string state decaying (as string decay) into pure (non mixed) particle states.Remarquably, explicit dynamical computations show that both gravity regimes: semiclassical (QFT) and quantum (string), are dual of each other, in the precise sense of the classical-quantum (de Broglie type) duality.