Reducing the Spectral Index in Supernatural Inflation

where M P = 2.4 × 10 18 GeV is the reduced Planck mass. The spectral index can be expressed in terms of the slow-roll parameters as

The latest WMAP 5-year result prefers the spectral index around n s = 0.96 [7]. The spectrum is given by

With the slow-roll approximation the value of the inflaton field φ, in order to achieve N e-folds inflation, is

From observation [7] P

R ≃ 5 × 10 -5 at N ≃ 60 (we call this CMB normalization). In order to build a successful inflation model, we need a potential which is very flat when N ≃ 60 and becomes steep when inflation ends. It is difficult to achieve this form of potential by a single field without significant tuning of the coupling parameter [4]. The idea of hybrid inflation is more natural in achieve so, in which the jobs of ending the inflation and providing the scale of inflation is done by another scalar field (called waterfall field). The quadratic potential for the inflaton field φ is

This potential is used in the case of chaotic inflation [8] where φ > M P is required. It can be turned into a hybrid inflation [9] by adding a “false vacuum” to it [10]:

Consequently we can have an inflation model with φ < M P . In this case, the end of inflation is not due to the failing of slow-roll, but the tachyonic instability of the waterfall field. The idea of supernatural inflation [11] is that the inflaton field φ is a flat direction in supersymmetric field theory and the mass term is provided by a soft SUSY breaking term and V 0 by another field coupled to the inflaton field. Therefore, the model is basically a tree-level hybrid inflation model. However, this model predicts a spectral index n s > ∼ 1. A method to reduce the spectral index in this tree-level hybrid inflation is by converting the model into “hilltop inflation” [12] via introduction of a negative quartic term to the potential as shown in [13]. The only problem here is how to introduce this quartic term and whether the coupling parameter needs fine tuning. The answer relies on the fact that the flat direction is generically affected not only by the soft SUSY breaking term but also the nonrenormalizable term and A-term. In this paper, we show that by considering these additional terms the spectral index can be reduced to n s = 0.96 in a natural fashion.

This paper is organized as follows. In Sec. 2, we introduce the scalar potential of the model. In Sec. 3, the analytic solution base on the potential and the main result of this paper are provided. Finally, we present our conclusions in Sec. 4.

Suppose we want to build an inflation model based on a SUSY flat direction [14], we have to know that generically a flat direction is lifted by supersymmetry breaking terms and non-renormalizable superpotential terms [15,16], which has the form

where φ is the flat direction and 4 ≤ p ≤ 9 [14]. Therefore, the potential is [17,18,19,20] (after minimizing the potential along the angular direction)

By spirit of hybrid inflation we can add a “false vacuum” to this potential via a coupling to a waterfall field similar to the case of supernatural inflation. Therefore, the potential we consider is of the following form

If we just consider the first two terms, the result is the tree-level hybrid inflation, which is realized in supernatural inflation where the mass term comes from soft SUSY breaking. In this work, we focus on the case of p = 4 (the least of Planck-mass suppression), and neglect the last term (which is possible when φ ≪ M P and will be justified in the following section). Therefore, the potential is

with

This form of potential has been considered in [13]. The question is whether η 0 and λ here need fine-tuning. The natural value of soft SUSY breaking terms, m and A, are m ∼ A ∼ O(TeV) ∼ 10 -15 M P . The coupling λ 4 is of O(1), which makes λ ∼ O(10 -15 ). As in the case of supernatural inflation, we choose V 0 = M 4 I where M I ≃ 10 11 GeV ≃ 10 -7 M P is the intermediate scale, therefore η 0 = O(10 -2 ). In the following section, we will apply those natural values without fine-tuning to achieve our goal of reducing the spectral index of supernatural inflation.

From Eq. ( 12), by using Eq. (3-5) we can obtain

and

)

By imposing P 1/2 R ≃ 5 × 10 -5 and n s = 0.96, we plot φ 2 /M 2 P (at N = 60) and λ as functions of η 0 in Fig. (1). The reason why we plot φ 2 /M 2 P is to justify that we can ignore the last term in Eq. (10). By comparing the third and fourth terms in Eq. ( 10), λ p ≃ 1 is the requirement for φ 2 /M 2 P ≪ λ. As we can see from Fig. 1, when λ ≃ 5 × 10 -15 , φ 2 /M 2 P ≃ 5 × 10 -17 , which means the φ 6 term is at

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