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By Adnan Ghribi

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The thermal leptogenesis requires high reheating temperature, Treh ≥ 109 GeV (82), which is consistent with our eq. (187). The matter-coupled F (R) supergravity theory may contribute towards the origin and the mechanism of CP-violation and baryon asymmetry, because • complex coefficients of F (R)-function and the complex nature of the F (R) supergravity are the simple source of explicit CP-violation and complex Yukawa couplings; • the nonthermal leptogenesis is possible via decay of heavy sterile neutrinos (FY-mechanism) universally produced by (super)scalaron decays, or via neutrino oscillations in early universe (83); • the existence of the natural Cold Dark Matter candidates (gravitino, axion, inflatino) in F (R) supergravity; • F (R)-supergravity can naturally support hybrid (or two-field) inflationary models because it already has a pseudo-scalar superpartner (axion) of inflaton.

It is worth noticing that the corrections to the Einstein action in eqs. (128) and (134) are of ther same order (and small) at the borders of the intermediate region (130). The roots of the cubic equation (125) are given by the textbook (Cardano) formula (59), though that formula is not very illuminating in a generic case. The Cardano formula greatly simplifies in the most interesting (high curvature) regime where inflation takes place, and the Cardano discriminant is R 3 90 D≈ <0 (136) It implies that all three roots are real and unequal.

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