Polynomial Chaotic Inflation in Supergravity Revisited

We revisit a polynomial chaotic inflation model in supergravity which we proposed soon after the Planck first data release. Recently some issues have been raised in Ref.[12], concerning the validity of our polynomial chaotic inflation model. We study the inflaton dynamics in detail, and confirm that the inflaton potential is very well approximated by a polynomial potential for the parameters of our interest in any practical sense, and in particular, the spectral index and the tensor-to-scalar ratio can be estimated by single-field approximation. This justifies our analysis of the polynomial chaotic inflation in supergravity.

Inflation provides elegant solutions to several theoretical problems of the standard big bang cosmology such as the horizon and flatness problems [1,2], and the slow-roll inflation paradigm [3,4] successfully explains observations of cosmic microwave background (CMB) and large-scale structure. Particularly interesting is the so called large-field inflation that can generate a sizable tensor-to-scalar ratio, r, within the reach of the on-going and planned CMB experiments. Among various large-field inflation models, the simplest one is the quadratic chaotic inflation model proposed by Linde long time ago [5].
The Planck satellite observed the CMB temperature and polarization anisotropy with unprecedented accuracy. Planck released the first data with a series of papers in March 2013 [6], providing tight constraints on the scalar spectral index n s and the tensor-toscalar ratio r. Soon after the Planck first data release, we proposed a polynomial chaotic inflation in supergravity (SUGRA) [7] as an extension of Refs. [8,9]. We showed in Ref. [7] that the predicted values of n s and r can cover almost entire region allowed by the Planck data. The inflaton dynamics was further studied in a more general set-up in Ref. [10]. The polynomial chaotic inflation has gained momentum recently, especially after the BICEP2 collaboration claimed a detection of primordial B-mode polarization [11]. The dynamics of the polynomial chaotic inflation and its variation have been studied in Refs. [12,13,14,15]. In Ref. [12], several issues were raised concerning our inflation model: 1) the real component of the inflaton field acquires a non-zero vacuum expectation value which depends on the inflaton field, and so, we will have no longer the simple single-field inflation; 2) the potential is not exactly polynomial as a result of 1); 3) the kinetic term of the fields will be non-canonical and non-diagonal; 4) there is an extra minimum. The purpose of the present letter is to study our polynomial chaotic inflation model in detail, in answer to the above issues.
Our short answer is that, in any practical sense, the inflaton potential is very well approximated by a polynomial potential for the parameters of our interest, and in particular, the spectral index and the tensor-to-scalar ratio can be estimated by single-field approximation. The kinetic terms can be easily diagonalized and canonically normalized by only slightly rotating the field basis. There is an extra minimum which however is located outside of the validity region of our inflation model, and it does not affect the inflaton dynamics significantly. The typical change of the field basis is so small that the predicted values of (n s , r) remain almost unchanged. This justifies our analysis of the polynomial chaotic inflation in SUGRA in Ref. [7]. Therefore, our model is a concrete realization of the polynomial chaotic inflation in SUGRA 1 , for which the predicted values of the spectral index and the tensor-to-scalar ratio can cover almost entire region allowed by Planck.
The central issue in building successful chaotic inflation models in SUGRA is how to have a good control of the inflaton potential over super-Planckian field ranges. A simple prescription was given in the paper [8], where they introduce a shift symmetry on the inflaton field φ along its imaginary component: where C is a real transformation parameter. The Kähler potential takes the following which satisfies the shift symmetry, whereas it is explicitly broken by the superpotential of the form, where X is a singlet chiral superfield with R-charge 2. The introduction of X is crucial for avoiding a negative inflaton potential at large field values of φ. The scalar potential in SUGRA is given by Here and in what follows, we adopt the Planck units in which M P ≃ 2.4 × 10 18 GeV is set to be unity. The inflaton potential is generated by the small shift symmetry breaking superpotential, and it is given by where ϕ ≡ √ 2Im(φ), and both X and Re[φ] are stabilized at the origin. The approximate shift symmetry ensures the flatness of the potential along the imaginary component Im(φ) beyond the Planck scale.
Now we move on to the polynomial chaotic inflation model [7,10]. We consider the following Kähler potential satisfying the shift symmetry (1), where c φ and c X are constants of order unity, the dots represent higher order terms, and a linear term of φ + φ † is dropped, since it does not affect the inflaton dynamics [10]. We introduce shift symmetry breaking terms in the superpotential as 2 where k i represents the numerical coefficient of higher order terms. See Ref. [16] for the case with shift symmetry breaking terms in the Kähler potential. To be concrete, we focus on the case where the first two terms in the superpotential make the dominant contribution to the inflaton dynamics: where we have defined λ ≡ |k 2 | and θ ≡ arg[k 2 ], and we assume λ = O(0.1) [7, 10] 3 . See also Appendix of this letter for another case.
First let us see that X is stabilized at the origin X = 0 during inflation. This is because X obtains an inflaton-dependent mass term as where H denotes the Hubble parameter during inflation. Therefore, for c X O(0.1), X obtains a mass of order of the Hubble scale and it is stabilized at the origin during 2 In Ref. [15], we proposed an extension to include multiple X fields. For instance we can consider By taking e.g. f (φ) ∝ φ and g(φ) ∝ φ 2 , one can realize a polynomial chaotic inflation in supergravity without a cross term. 3 The cut-off scale one order of magnitude larger than the Planck scale can be understood as follows. Suppose that the shift symmetry is broken by various Planck-suppressed shift symmetry breaking terms. Then, if the kinetic term coefficient happens to be enhanced by by a factor of O(10 − 100), all the higher order terms are suppressed when they are expressed in terms of the canonically normalized field. Such an enhancement may be realized if there are many singlet scalars whose kinetic term coefficients are subject to a certain random distribution [15]. inflation. 4 Thus in the following analysis we take X = 0. Note that the inflaton field is canonically normalized for X = 0.
Let us decompose the scalar field φ as where χ and ϕ are real and imaginary components, respectively. As noted above, ϕ can develop a field value much larger than the Planck scale because of the shift symmetry.
On the other hand, χ obtains a Hubble-induced mass and stabilized at sub-Planckian field values |χ| ≪ 1. In Refs. [7,10] we approximated χ ≈ 0 and obtained the inflaton potential as The potential shape is shown in Fig. 1. One can see that the inflaton potential changes its form as one varies θ. Therefore, as long as the approximation is valid, the polynomial chaotic inflation can be realized by the first two terms in (7).
In Ref. [9] the superpotential was extended to be the form of is an arbitrary holomorphic function, and it was shown that the real component of φ can be stabilized at the origin for a certain class of f (φ), where the coefficients are either purely real or imaginary depending on the definition of the shift symmetry. In this case, the polynomial chaotic inflation can be realized for a certain combination of three terms in the superpotential [14].
The above inflaton potential (11) is slightly modified once one takes account of the fact that the real component χ acquires a vacuum expectation value which depends on ϕ, as pointed out in Ref. [12]. To see this, we expand the full SUGRA potential in χ. Then we obtain Thus χ obtains a mass of order Hubble scale and it is stabilized at during inflation. It is the ϕ-dependence of χ min that modifies the inflationary path and the inflaton potential, because the constant part of χ min simply modifies the coefficients of the polynomial potentials as can be seen from (12). In fact, it is easy to see that the ϕ-dependence of χ min is rather suppressed: which becomes even smaller for ϕ λ −1 . For λ = O(0.1), therefore, the modification of the inflationary path as well as to the inflaton potential is at most of order O(1)% level.
The corrections to n s and r are expected to be of a similar order.
The contour of the inflaton potential in the complex φ plane is shown in Fig. 2. It is seen that there are two global minima at and the potential is deformed asymmetrically. Note that the second minimum is super-Planckian along the real component for λ = O(0.1) and a general value of θ, and therefore it is outside the validity region of our inflation model. We draw the contour only for visualization purpose, simply by extrapolating the SUGRA potential to |χ| ≫ 1. In any case, there is an exponential potential barrier between these two minima, and the effect of deformation is not significant. Also for c φ > 0, χ becomes heavier and the inflationary trajectory becomes closer to χ = 0. For the reasons stated above, we expect that we can approximately set χ ≃ 0 during inflation as in our previous study [7,10]. Next we study the inflaton dynamics numerically to show this explicitly.
In order to see how large is the effect of the deformation of the inflaton potential on the predicted values of n s and r, we have performed numerical calculation using the full SUGRA potential. We have solved the two field inflaton dynamics χ and ϕ and identify the inflaton directionφ as a mixture of χ and ϕ asφ = cϕ + sχ where [17] with subscript ϕ and χ being the derivative with respect to it. The scalar spectral index and the tensor-to-scalar ratio is obtained from n s = 1 − 6ǫ + 2η and r = 16ǫ where Figure 3: Comparison of (n s , r) between full SUGRA result and approximate result for θ = π/3, 3π/8 and π/2 for the model (8). We have taken c φ = 0 (upper panel) and c φ = 1 (lower panel).
with prime denoting the derivative with respect toφ: they are given by V ′ = cV ϕ + sV χ , V ′′ = c 2 V ϕϕ + 2scV ϕχ + s 2 V χχ . They are evaluated at the point where the e-folding number is N e . In the numerical analysis, we take N e = 60.
The Planck normalization on the density perturbation is imposed. For comparison, we have also plotted the result for the approximate case where χ is set to be zero ((black) dashed lines). For c φ = 0 (upper panel), the results based on the full SUGRA potential agree well with the one based on the single-field approximation. The discrepancy between these two results are actually small: the change of (n s , r) can be absorbed by small change of θ. This is because the ϕ-dependence of χ min stretches the inflaton potential by a small amount, which effectively amounts to shifting the parameters λ and θ slightly. The lower panel is the same plot but for c φ = 1. As can be clearly seen, the full SUGRA results almost coincide with those of polynomial potential (11). In any case, even if we take account of the full SUGRA potential, the predicted values of (n s , r) of our model cover the almost entire region allowed by Planck, as in the case where the inflaton potential is approximated by a polynomial. Considering the uncertainties on N e and observational errors of n s and r, the single-field approximation with a polynomial potential is sufficient to estimate the prediction of n s and r. We have also estimated n s and r based on δNformalism solving the two field dynamics numerically, and obtained consistent results.

Acknowledgments
We thank Andrei Linde for pointing out the issues of the polynomial chaotic inflation A The case of W = X(φ + φ 3 ) In this Appendix we similarly study the case of different choice of the superpotential: where ξ is a real constant. This form of the superpotential is of particular interest because it is ensured by imposing a Z 2 symmetry on φ and X and, because of the Z 2 symmetry, this model is free from the gravitino overproduction from the inflaton decay [18,19,20,21,22].
It is seen that the potential is slightly deformed in the direction of χ and there exist three golobal minima: We obtain the following approximate inflaton potential by setting χ ≃ 0: A schematic picture for the scalar potential (20) is shown in Fig. 5. We focus on the case of ξ = O(0.01) where the second term affects the inflaton dynamics during the last 50 − 60 e-foldings. We have numerically solved the inflaton dynamics under the full SUGRA potential and the result is plotted in Fig. 6 for θ = 0, π/5 and π/3. We have taken c φ = 0 (upper panel) and c φ = 1 (lower panel). Similarly to the case studied in the main text, it is seen that the difference between approximate results and the full results are very small.   : Comparison of (n s , r) between full SUGRA result and approximate result for θ = 0, π/5 and π/3 for the model (18). We have taken c φ = 0 (upper panel) and c φ = 1 (lower panel).