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Eigenvalues of the Stewart–Lyth equation for inflation with a blue spectrum

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Abstract

By using the rather stringent nonlinear second order slow-roll approximation, we reconsider the nonlinear second order Abel equation of Stewart and Lyth. We determine a new blue eigenvalue spectrum. Some of the discrete values of the spectral index ns have consistent fits to the cumulative COBE data as well as to recent ground-base CMB experiments.

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... (As an example, f = (1 + cφ 2 ) −1 , where c is a constant, does not have a convergent Maclaurin series expansion for |c|φ 2 > 1, but g = f −1 does, and therefore both g and f satisfy the condition (14) whenever the condition (12) is satisfied.) To obtain our generalized slow roll EOM, it will be assumed that we are dealing with functions F , F ,φ , k, and U (or V ) that satisfy (14). Applying the above slow roll conditions to (10) and (11) (see the appendix for details) gives the slow roll EOM ...
... Next, we use the fact that an application of (14) to the functions F (φ) and U(φ) implies that |Ḣ/H| ≪ 1. Then taking the time derivative ofφ, using (16), and applying (15), the condition that |φ/(Hφ)| ≪ 1 translates into the condition |η SR | ≪ 1. ...
... For the case of a minimally coupled inflaton, it has been shown how the inflaton potential can be reconstructed from slow roll parameters (see, e.g., [13,14]). This type of approach of reconstructing the form of the theory has been extended to scalartensor theory by Boisseau, Esposito-Farese, Polarski, and Starobinsky [15], who have shown how both of the functions F (φ) and V (φ) can be determined (with a rescaling of φ to set k = 1) from future observations. ...
Article
Generalized slow roll conditions and parameters are obtained for a general form of scalar-tensor theory (with no external sources), having arbitrary functions describing a nonminimal gravitational coupling F(\phi), a Kahler-like kinetic function k(\phi), and a scalar potential V(\phi). These results are then used to analyze a simple toy model example of chaotic inflation with a single scalar field \phi and a standard Higgs potential and a simple gravitational coupling function. In this type of model inflation can occur with inflaton field values at an intermediate scale of roughly 10^{11} GeV when the particle physics symmetry breaking scale is approximately 1 TeV, provided that the theory is realized within the Jordan frame. If the theory is realized in the Einstein frame, however, the intermediate scale inflation does not occur. Comment: 14 pages, no figs. Accepted to Classical and Quantum Gravity
... We have applied the H-formalism 3 to the rather accurate first order corrections to the slow-roll approximation of Stewart and Lyth, and transform their nonlinear equation into the Abel equation (1). The spectral index of our new class 1 of solutions can only adopt the following discrete eigenvalues ...
Article
We reconsider the nonlinear second order Abel equation of Stewart and Lyth, which follows from a nonlinear second order slow-roll approximation. We find a new eigenvalue spectrum in the blue regime. Some of the discrete values of the spectral index n_s have consistent fits to the cumulative COBE data as well as to recent ground-base CMB experiments.
... In the next order slow-roll approximation arises a nonlinear Abel equation 13 with a continuous spectrum for n < 1 and a discrete 'blue' one for n > 1, see Ref. 57 for more details. This division of the spectrum persists in the next to second order slow-roll approximation. ...
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Chapter
We reconsider the second order slow-roll approximation of Stewart and Lyth. For the resulting nonlinear Abel equation, we find a new eigenvalue spectrum of the spectral index in the ‘blue’ regime n 5〉1. Some of the discrete values have consistent fits to the cumulative COBE data as well as to recent ground-base CMB experiments.
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