Chaotic inflation and unitarity problem

We consider a general chaotic inflation model with non-canonical kinetic term, resulting in attractor solutions for the inflation of quadratic or other monomial type. In particular, the form of the kinetic term and the potential is fixed due to the requirement that the inflation model is a quadratic form in the large field values of the inflaton. We show that a large coupling in the non-canonical kinetic term allows for the slow-roll inflation with sub-Planckian field values of the inflaton and the successful predictions of the quadratic or other monomial type chaotic inflation in light of BICEP2 results are maintained in our model. We find that due to the large rescaling of the inflaton field in the vacuum, there is no unitarity problem below the Planck scale.


Introduction
The BICEP2 collaboration [1] has recently announced the evidence for B-modes in the CMB polarization, which are presumably originated from the primordial gravitational waves of cosmic inflation. The reported value of the tensor-to-scalar ratio is r = 0.20 +0.07 −0.05 , which is quite larger than the previous upper bound, r < 0.11, given at 95% C.L. by Planck data combined with WMAP polarization and high-l [2]. In the case of single-field inflation models, such a large value of r implies that there was an excursion of the inflaton beyond the Planck-scale field values during inflation [3]. As a result, simple monomial chaotic inflation models have become favored 1 . On the other hand, many of the inflation models predicting a small value of r including Starobinsky model [4] and Higgs inflation [5] in a simple form 2 have been disfavored by a large r, on top of the problem of unitarity violation [9][10][11][12].
Among the monomial chaotic inflation models, the quadratic inflation has drawn a new attention due to the fact that the model predictions are consistent with Planck+ BICEP2 within 1σ. We consider a general quadratic inflation with a single scalar field containing the general kinetic term and potential. The condition that the quadratic inflation is reproduced at large field values fixes the form of the kinetic and potential terms. As a consequence, we show how the model parameters in the general quadratic inflation are constrained after BICEP2.
A large r suggests that the inflaton field with canonical kinetic term must have travelled to trans-Planckian field values during inflation, so there is a concern about how to address the quantum gravity effects from the pointview of the effective field theory. In our general quadratic inflation, the field value of the non-canonical inflaton remains sub-Planckian during inflation, thanks to a large coupling in the non-canonical kinetic term. Therefore, the higher order terms for the inflaton suppressed by the Planck scale can be safely ignored. Furthermore, we find that there is no unitarity violation coming from the large coupling below the Planck scale, because the wave function rescaling of the inflaton field needed in the vacuum eliminates any positive powers of the large coupling as in the induced inflation models [12]. We also mention the general power-like chaotic inflation in the model with a similar conclusion.
The paper is organized as follows. We begin with a model description of the general chaotic inflation focusing on the quadratic form and discuss the inflation constraints on the model in view of Planck and BICEP2. Then, we address the issue of unitarity violation in this model and conclusions are drawn.

General chaotic inflation
We introduce a real scalar inflaton minimally coupled to gravity, having a general kinetic term with two derivatives and a potential as follows, Without loss of generality, we choose the general kinetic term and the potential in the following discussion as where ξ, λ are constant parameters, f (φ), g(φ) are general functions of φ, and we have chosen the potential to vanish in the vacuum with φ = φ 0 . The generic feature of the above Lagrangian is that the inflaton field has a non-canonical kinetic term with f (φ 0 ) = 0 so the interaction terms get rescaled after the inflaton kinetic term is made canonical. This aspect leads to an interesting result for the unitarity scale as will be discussed in a later section. Here, we have used the units with M P = 1 but the Planck scale is introduced whenever necessary.
Even if the potential is not dominated by a mass term, the model can describe a quadratic inflation due to a non-canonical kinetic term during inflation. For simplicity, we take the functions to be power-like, Then, in the limit of ξf (φ) = ξφ n ≫ 1 during inflation 3 , we obtain the canonically normalized inflation as χ ≈ √ ξφ n/2+1 /(n/2 + 1). In this case, the potential function becomes g(φ(χ)) ∼ χ 2m/(n+2) . Therefore, choosing m = (n+2)/2, we obtain the Lagrangian during inflation as follows, where χ 0 ≡ √ ξφ 1+n/2 0 /(1 + n/2). Then, we can get rid of χ 0 by making a shift for χ without changing any physics, ending up with a quadratic inflation model. The mass of the canonical inflaton is given by Therefore, the predictions of the model for inflation are the same as for the usual quadratic inflation with canonical kinetic term.
The slow-roll parameters for the quadratic inflation at horizon exit are written in terms of the number of efoldings N as Then, we get the spectral index of the scalar perturbation and the tensor-to-scalar ratio, respectively, as Therefore, for N = 50(60), we find that n s = 0.941(0.950) and r = 0.158(0.132). As a result, the model leads to large primordial gravitational waves, and the model predictions can be consistent with n s = 0.9600 ± 0.0071 and r = 0.20 +0.07 −0.05 given by Planck+WP+high-l+BICEP2 within about 1σ. On the other hand, the running of the spectral index is negligibly small as Thus, there is a tension with the previous limit on the tensor-to-scalar ratio, r < 0.11, obtained at 95% C. L. for Planck+WP+high-l, in the case of a zero running of the spectral index. But, we do not discuss the solution to resolve the tension in this work.
As a consequence, we need a large ξ or a small λ to satisfy the COBE normalization. For a higher power of the polynomial, we would get a smaller ratio λ/ξ. The value of the inflaton mass is suggestive of solving the vacuum instability problem with a heavy scalar threshold in the SM with an inflaton coupling to the Higgs doublet [14].

The case with quartic potential
In particular, for f (φ) = φ 2 and g(φ) = φ 2 , the Lagrangian (1) with eq.(2) becomes 4 In this case, we can get the analytic expression for the canonical inflation field as follows, For ξφ 2 ≫ 1 during inflation, we obtain χ ≈ √ ξφ 2 /2. The slow-roll condition with χ ≫ 1, however, requires a stronger condition on the φ field value as φ ≫ 1/ξ 1/4 . But, for a large ξ, the φ field value can be sub-Planckian during inflation so the higher order quantum gravity corrections suppressed by the Planck scale can be ignored.
We remark that as noted before, for an arbitrary choice of powers, n and m, in eq. (3), the power of the potential for the canonically-normalized inflaton takes a general form of chaotic inflation as V (φ(χ)) ∼ χ 4m/(n+2) for large filed values as follows, Therefore, keeping the non-canonical inflaton field sub-Planckian during inflation due to the large coupling ξ, we can accommodate various power-like chaotic inflation models containing the canonical kinetic term.

Quadratic inflation and unitarity scale
It is remarkable that the unitarity scale depends on the background field values and the vacuum expectation value of the inflaton field plays an important role in determining the unitarity scale [12,13]. In this section, focusing on the general polynomial functions leading to the quadratic inflation, we consider the effective Lagrangian in the vacuum and discuss the unitarity problem. A similar result applies to the general power-like chaotic inflation.
By expanding the φ field around the vacuum with φ = φ 0 +φ, we obtain the Lagrangian as Then, after the scalar perturbation is canonically normalized byφ = 1 + ξφ n 0φ , the above Lagrangian is rewritten as Therefore, for ξφ n 0 1, from higher order interactions such asφ(∂φ) 2 ,φ 2 (∂φ) 2 and/orφ l+4 with l ≥ 1, we identify the unitarity cutoff Λ U V as follows, Consequently, for Λ U V ∼ M P , we need the vacuum expectation value of the φ field to be For instance, for ξ ∼ 10 12 and n = 2, we would need φ 0 ∼ 10 −3 M P ∼ 10 15 GeV. The higher n, the larger the necessary vacuum value for the Planck-scale cutoff.
In contrast to the above conclusion, if the vacuum expectation value of the inflaton field is negligible such that ξφ n 0 ≪ 1, theφ perturbation can be regarded as being canonical so the unitarity cutoff is identified fromφ n (∂φ) 2 as Λ U V = ξ −1/n M P , which is smaller than the Planck scale. On the other hand, during inflation, the inflaton φ runs over the field values, φ ≫ ξ −1/n M P = Λ U V , which are beyond the unitarity cutoff.
We note that the inflaton mass in the vacuum is the same value as during inflation, which is given by eq. (5).

Conclusions
We have proposed the general chaotic inflation models that share the predictions of the usual chaotic inflation with canonically-normalized inflaton field and are favored by the recent BICEP2 observation of a large tensor-to-scalar ratio. Focusing on the general quadratic inflation which is favored by Planck and BICEP2, we have shown that the non-canonical inflaton remains sub-Planckian during inflation while the unitarity scale identified in the vacuum is of order the Planck scale, after the non-canonical inflaton field obtains a large vacuum expectation value and it is canonically normalized. Our result indicates that the well-known chaotic inflation models such as quadratic inflation can be self-complete in the presence of the non-canonical kinetic term after the inflaton gets a large vacuum expectation value.