By Adnan Ghribi

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155). Therefore, pre-heating (ie. the nonperturbative enhancement of particle production due to parametric resonance (73)) is a generic feature of supergravity models. The axion χ and fermion ψ are both requred by supersymmetry, being in the same chiral supermultiplet with the inﬂaton φ. The scalar interactions are Vint (φ, χ) = mλφ(φ2 + χ2 ) + λ2 2 ( φ + χ2 )2 4 (178) whereas the Yukawa couplings are given by LYu = 1 1 ¯ ) + λχ(ψiγ ¯ 5 ψ) λφ(ψψ 2 2 (179) Supersymmetry implies the uniﬁcation of couplings since h = − 12 λ and g2 = λ2 in terms of the single coupling constant λ.

F. R. H. S. A. Starobinsky, Phys. Lett. A. Starobinsky, Nonsingular model of the Universe with the quantum-gravitational de Sitter stage and its observational consequences, in the Proceedings of the 2nd Intern. Seminar “Quantum Theory of Gravity” (Moscow, 13–15 October, 1981); INR Press, Moscow 1982, p. A. C. , Plemum Publ. , New York, 1984, p. P. Woodard, Lect. Notes Phys. D. Linde, Phys. Lett. B 129 (1983) 177 D. Wands, Class. and Quantum Grav. P. Sotiriou and V. Varaoni, Rev. Mod. Phys. 82 (2010) 451 A.

112) to be “small” and positive. We ﬁnd that it amounts to 4 34 · 72 · 11 MPl Rmax ≈ ≡R (115) Λ =0 25 · 192 f 22 cb = . It is also with the actual value of Rmax to be “slightly” above of that bound, Rmax > R Λ =0 . posssible to have the vanishing cosmological constant, Λl = 0, when choosing Rmax = R Λ =0 It is worth mentioning that it relates the values of Rmax and f 2 . The particular R2 -supergravity model (with f 0 = 0) was introduced in ref. (5) in an attempt to get viable embedding of the Starobinsky model into F (R)-supergravity.