Superconformal Subcritical Hybrid Inflation

We consider D-term hybrid inflation in the framework of superconformal supergravity. In part of the parameter space, inflation continues for subcritical inflaton field value. Consequently, a new type of inflation emerges, which gives predictions for the scalar spectral index and the tensor-to-scalar ratio that are consistent with the Planck 2015 results. The potential in the subcritical regime is found to have a similar structure to one in the simplest class of superconformal alpha attractors.


I. INTRODUCTION
The observations of the cosmic microwave background (CMB) strongly support inflation as the paradigm of early universe. To discover the nature of inflation, intensive analysis of the CMB has been performed. The latest results by the Planck collaboration [1,2] provide the bounds on the scalar spectral index n s and the tensorto-scalar ratio r of the primordial density fluctuations, n s = 0.9655 ± 0.0062 (68% CL) , r < 0.10 (95% CL) . (1) In fact, some inflation models, such as canonical chaotic inflation [3] and hybrid inflation [4], are already disfavored due to the bounds. Although they are not supported by the current observations, the models are simple and still attractive in theoretical point of view. Recently Refs. [5,6] studied the hybrid inflation in the framework of superconformal supergravity [7][8][9][10]. It was found that the Starobinsky model [11] emerges in the supersymmetric D-term hybrid inflation [12][13][14], to give a good accordance with the Planck observations. On the other hand, the D-term hybrid inflation was considered in a different context. In a shift symmetric Kähler potential [15], a 'chaotic regime' was found in the subcritical value of the inflaton field [16]. In the framework, inflation lasts even after the critical point of the hybrid inflation to give rise to different predictions from chaotic inflation. The following study [17] showed that the energy scale of inflation coincides with the Grand Unification (GUT) scale using the Planck 2013 data [18]. However, there is a tension between the predictions and the observations, especially the Planck 2015 data [1,2].
In this letter we revisit D-term hybrid inflation in superconformal framework. It will be shown that there exists a single slow-rolling field in the subcritical value of the inflaton field. Since inflation continues for sufficiently long period, cosmic strings are unobservable as in Refs. [16,17]. The potential in the subcritical region turns out to be in a general class of superconformal α attractors [19,20], especially similar to the simplest version of the model. Consequently, non-trivial behavior and different predictions from the simplest ones are discovered.

II. SUBCRITICAL REGIME IN SUPERCONFORMAL D-TERM INFLATION
We consider D-term hybrid inflation in supergravity with superconformal matter [5,6]. In the model three chiral superfields S ± and Φ, which have local U(1) charge ±q (q > 0) and 0, respectively, are introduced. The superpotential and Kähler potential after fixing a gauge for the local conformal symmetry are respectively given by, with, where λ and χ are constants. 1 The term proportional to χ in the Kähler potential breaks superconformal symmetry explicitly. In the model the Fayet-Iliopoulos (FI) term can be accommodated. Then, the D-term potential in the Einstein frame is [5], where g is the gauge coupling and ξ is the FI term, which is taken as a constant. (See Refs. [21][22][23][24][25][26] for the subtleties of this issue in supergravity.) The F-term potential in the Einstein frame, on the other hand, is given in a simple form without exponentially growing terms [5,27], As in the canonical hybrid inflation, S − is stabilized to its origin meanwhile S + suffers from the tachyonic instability depending on the field value of Φ. The nature of Φ depends on the value of χ. In the Kähler potential there is a shift symmetry under Re Φ (Im Φ) → Re Φ (Im Φ) + const. for χ = −1 (+1), and Re Φ (Im Φ) can play a role of inflaton, as mentioned in Ref. [5]. We consider χ ≤ −1 in the later discussion without loss of generality. Then, the total potential is given by the waterfall field s ≡ √ 2|S + | and the inflaton field The waterfall field becomes tachyonic below the critical value φ c of the inflaton field, After the tachyonic growth, the waterfall field is expected to reach its local minimum, which is obtained by ∂V tot (φ, s)/∂s = 0, whereξ ≡ ξ/3q and, The expression for the local minimum given in Refs. [16,17] corresponds to the case for χ = −1 (and q = 1) from the facts that Ω(φ, 0)| χ=−1 = 1 andξ ∼ O(10 −4 ) in our targeted parameter space. Following Refs. [16,28] (see also Appendix), we have confirmed numerically that the waterfall field reaches to the local minimum after is the Hubble parameter at the critical point, and then it becomes a single field inflation. Since the inflation lasts well over O(10 2 /H c ), cosmic strings, which are produced during the tachyonic growth, are unobservable. After the waterfall field relaxed to the local minimum, the dynamics of the inflaton is described by the potential, As in Eq. (10), it is easily to see that the potential V with χ = −1 agrees with one given in Refs. [16,17] up to O(ξ). We note that non-zero λ explicitly breaks the shift symmetry for Re Φ as well as χ that deviates from −1 does. Thus, a parameter λ 1 and χ −1 is consistent with each other under the approximate shift symmetry. In addition, χ −1 is required for λ 1 otherwise φ 2 c gets negative. As it will be seen, the observational data indeed implies such a parameter space.

III. COSMOLOGICAL CONSEQUENCES
The slow roll parameters for the inflaton dynamics are given as, where V = dV /dφ and V = d 2 V /dφ 2 . Hereφ is canonically-normalized inflaton field that is related to φ as, where (Parametrically vant numerically. Although we will use λ and ξ in the following discussion, the results in terms ofλ andξ can be obtained by q → 1, g → 1, λ →λ, and ξ →ξ. s min 0 is a good approximation as discussed later.) Inflation ends at φ = φ f ≡ Max{φ , φ η } where (φ ) = 1 and |η(φ η )| = 1, and the last e-folds N * before the end of inflation is obtained by, The cosmological observables, i.e., the scalar amplitude A s , the spectral index, and the tensor-to-scalar ratio, are then determined by, We normalize the scalar amplitude by using the Planck 2015 data [2] A s = 2.198 +0.076 −0.085 × 10 −9 and compute n s and r for a given N * .
As we have stated before, our target is the parameter space λ 1. To search such a region, it is convenient to parametrize χ as, to satisfy φ 2 c = 2qg 2 ξ/λ 2 (1 − δχ) > 0. Now we are ready to discuss the cosmological consequences. Fig. 1 shows the predictions of n s and r in our model. Here q = g = 1 is taken (see footnote 2), and λ and ξ are determined for a δχ and N * by using the scalar amplitude observed by the Planck collaboration. In Fig. 2, the allowed regions due to the bounds on n s and r are shown for N * = 55-60. 3 The upper and lower bounds on ξ corresponds to the upper limit on r and lower limit on n s , respectively. In the n s -r plane, smaller values of n s and r are obtained for larger λ (and smaller ξ). In Fig. 1 the result in the previous work [17], i.e., the shift symmetric Kähler potential case, is also given as 'shift sym.'. 4 We have checked that the result for δχ = 0 agrees with it numerically and the similar behavior is seen around δχ 0. When δχ gets close to unity, on the contrary, a different behavior is observed. It is seen that r gets smaller meanwhile n s tends to stay in the same value, which is within the Planck bounds. As a result, a wider allowed parameter space is obtained, which is seen in Fig. 2.
In order to interpret the results, it is instructive to consider a canonically-normalized inflaton fieldφ. Although the r.h.s of Eq. (14) is complicated, it can be approximated in the parameter space we are considering as, Then it is solved analytically, where C is a constant and, We have found that C = 0 is appropriate choice. Then Ψ is simply given as, to express the potential in terms ofφ, 3 There is no allowed region for N * = 50 except for δχ = 0.9. 4 Do not confuse with the shift symmetric Kähler case with the present superconformal case where the shift symmetry is (weakly) broken in the Kähler potential.
This potential is valid inφ ≤φ c = 1 √ β sinh −1 √ βφ c . It is straightforward to check that the r.h.s is equal to g 2 ξ 2 /2 forφ =φ c , andφ c → ∞ for δχ → 1. We note that the potential coincides with a general class of superconformal α attractors [19]. It especially resembles to the simplest class of the model, Due to the additional term, however, it has a different asymptotic behavior as we will see below.
In the small λ (and large ξ) region, β gets small, then the potential reduces to, This is nothing but the potential for the shift symmetric Kähler case given in Refs. [16,17] in the limit δχ → 0, which leads toφ → φ. This feature is clearly seen in Fig. 1. We note that the quadratic term is rewritten as (λ 2 ξ/2q)φ 2 , which is independent of δχ. Therefore, n s and r approach to those in quadratic chaotic inflation in small λ limit (while λ 2 ξ constant), independent of δχ.
(Such a region is excluded, thus it is not shown in Fig. 2.) The potential V α-attr , on the other hand, has a similar structure, Although it coincides with V in the limitφ → 0 for m = 1 and Λ 4 /6α = λ 2 ξ/2q, it is not possible to get the same factor for the quartic term. In large λ (and small ξ) region, on the contrary, β increases, which leads us to expand Ψ in large √ βφ limit to obtain, with a 1 = 8(1−δχ)/(2δχ−1) and a 2 = 16(2−δχ)/(2δχ− 1). This expression should be compared with Eq. (24) in the α 1 limit. As shown in Ref. [19], it reduces to the potential in R 2 inflation [29] at large field value 5 , Now it is clear that the form of the potential with δχ = 1 (in large λ region) reduces to R 2 inflation, or the simplest 5 To be precise, α = 1 gives the original R 2 inflation. The factor 4m(> 0) is quantitatively irrelevant for the slow-roll predictions.  class of superconformal α attractors in α 1 limit. To be specific, a choice of Λ 4 = g 2 ξ 2 /2 and α = 1/24β leads to the same asymptotic form. Then, we get n s 1 − 2/(N * + 1) − 3qg 2 ξ/8λ 2 (N * + 1) 2 , r qg 2 ξ/λ 2 (N * + 1) 2 , while satisfying λ 2 ξ constant. Namely, when λ increases n s approaches to 1 − 2/(N * + 1) and r gets smaller and smaller. We have confirmed this behavior using Eq. (23) with δχ = 1. Recall that, however, the critical value becomes infinity, which is unphysical.
Such a behavior, on the contrary, can not be seen for δχ = 1 case shown in Fig. 1. This arises from non-zero a 1 in Eq. (27). This is why we have seen the different cosmological consequences.
In Fig. 3, the potential as function of canonicallynormalized inflaton field is plotted for δχ = 0.9 and 1. Here λ = 9.4 × 10 −4 (1.9 × 10 −3 ), √ ξ = 1.3 × 10 16 (5.7 × 10 15 ) GeV for δχ = 0.9 (1) to give n s = 0.966 and r = 0.051 (0.00052) for N * = 60. In the plot the potentials in superconformal α attractors V α-attr , R 2 inflation, and the shift symmetric Kähler case, are also shown for comparison. Each potential is normalized to unity when φ reaches to the critical point (δχ = 0.9 case and the shift symmetric Kähler case) or infinity (δχ = 1 case, α attractors, and R 2 inflation). For the shift symmetric Kähler case, the critical point is taken to the same value as δχ = 0.9 case. We take the parameters for α attractors and R 2 inflation to have the same asymptotic form as δχ = 1 case in large field limit. It is seen that δχ = 1 case is similar to α attractors, but not exactly the same. As a result, the predictions for the slow-roll quantities are different, i.e., n s = 0.967 and r = 0.00044. It is clear, on the other hand, that δχ = 0.9 case shows a different behavior from the others. To summarize, the model has a nature of both the shift symmetric Kähler case and the simplest superconformal α attractors, and the slow-roll predictions change accordingly.

IV. CONCLUSION
We have revisited superconformal D-term hybrid inflation. After reaching its critical value, the inflaton field is slowly rolling thus inflation continues for a small coupling λ of inflaton to the other fields. Because of a sufficiently long period of slow-roll regime, cosmic strings, which are formed during the tachyonic growth of the waterfall field, are unobservable. The potential which determines the dynamics of the canonically-normalized inflaton in the subcritical regime has been found to resemble to the simplest version of superconformal α attractors but with an additional term. Consequently, different predictions for the slow-roll parameters are obtained. For λ ∼ 10 −4 -10 −3 and √ ξ ∼ 10 16 GeV, n s and r are consistent with the Planck data.
The predictions depend on a parameter χ that explicitly breaks superconformal symmetry in the Kähler potential. In addition, the Kähler potential with |χ| = 1 has a shift symmetry for the inflaton field, which is explicitly broken by non-zero λ in the superpotential. On the other hand, |χ| 1 is required from the consistency of the model setting, thus λ 1 is parametrically natural. It has been found that the observational bounds indeed prefer such a parameter space.