By EDMUND J COPELAND; SAMI M; TSUJIKAWA SHINJI; D V Ahluwalia-Khalilova
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Here λ and Γ are defined by λ≡− V,φ , κV 3/2 Γ≡ V V,φφ . V,φ2 The eigenvalues of the matrix M are (198) µ1,2 = 3 γ−2± 4 17γ 2 − 20γ + 4 + 48 2 γ λ2 The equation of state and the fraction of the energy density in the tachyon field are given by γφ = x2 , y2 . Ωφ = √ 1 − x2 (199) Then the allowed range of x and y in a phase plane is 0 ≤ x2 + y 4 ≤ 1 from the requirement: 0 ≤ Ωφ ≤ 1. 1. Constant λ From Eq. (196) we find that λ is constant for Γ = 3/2. This case corresponds to an inverse square potential V (φ) = M 2 φ−2 .
175) and (176). Although λ is a dynamically changing quantity, one can apply the discussion of constant λ to this case as well by considering “instantaneous” critical points [201, 203]. , x(N ) = λ(N )/ 6 and y(N ) = [1 − λ2 (N )/6]1/2 . When Γ > 1 this point eventually approaches x(N ) → 0 and y(N ) → 1 with an equation of state of a cosmological constant (γφ → 0) as λ(N ) → 0. See Refs. [201, 203] for more details. C. e, ǫ = −1 in Eq. (170). Let us first consider the exponential potential given by Eq.
They proposed the following potential V (φ) = λ(φ4 + M 4 ) λM 4 = 1 + (φ/M )α for φ < 0 , for φ ≥ 0 . (285) For φ < 0 we have ordinary chaotic inflation. Much later on, for φ > 0 the universe once again begins to inflate but this time at the lower energy scale associated with quintessence. Needless to say quintessential inflation also requires a degree of fine tuning, in fact perhaps even more than before as there are no tracker solutions we can rely on for the initial conditions. The initial period of inflation must produce the observed density perturbations, which constrains the coupling to be of order (0) λ ∼ 10−13 .