Fractional chaotic inflation in the lights of PLANCK and BICEP2

In the lights of current BICEP2 observations accompanied with the PLANCK satellite results, it has been observed that the simple single field chaotic inflationary models provide a good agreement with their spectral index n_s and large tensor-to-scalar ratio r (0.15<r<0.26). To explore the other simple models, we consider the fractional-chaotic inflationary potentials of the form V_0 phi^(a/b) where a and b are relatively prime. We show that such kind of inflaton potentials can be realized elegantly in the supergravity framework with generalized shift symmetry and a nature bound a/b<4 for consistency. Especially, for the number of e-folding from 50 to 60 and some a/b from 2 to 3, our predictions are nicely within at least 1 $\sigma$ region in the r-n_s plane. We also present a systematic investigation of such chaotic inflationary models with fractional exponents to explore the possibilities for the enhancement in the magnitude of running of spectral index (\alpha_{n_s}) beyond the simplistic models.


Introduction
Among the plethora of inflationary models developed so far, the polynomial inflationary potentials have always been among the center of attraction since the very first proposal as chaotic inflation in Ref. [1]. The recent BICEP2 observations [2] interpreted as the discovery of inflationary gravitational waves have not only taken these models in the limelight but also supported these to be the better ones among many others. The BICEP2 observations fix the inflationary scale by ensuring a large tensor-to-scalar ratio r as follows [2]: r = 0.20 +0.07 −0.05 (68% CL) where H inf denotes the Hubble parameter during the inflation. Subtracting the various dust models and re-deriving the r constraint still results in high significance of detection and one has r = 0.16 +0.06 −0.05 . Thus, it suggests the inflationary process to be (a high scale process) near the scale of the Grand Unified Theory (GUT) and then can provide invaluable pieces of information on the UV completion proposal such as string theory, for example, in searching for a consistent supersymmetry (SUSY) breaking scale [3,4,5,6,7]. On these lines, some recent progresses on realizing chaotic as well as natural or axion-like inflationary models from string or supergravity framework have been made in Refs. [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. However, most of these works on natural as well as chaotic inflation can only produce integral power of inflaton in the polynomial potential. In this work, we will present a general fractional chaotic inflation which can be naturally generated from supergravity framework by utilizing the generalized shift symmetry.
For a given single field potential V (φ), the sufficient condition for ensuring the slow-roll inflation is encoded in a set of so-called slow-roll conditions defined as follows where ′ denotes the derivative of the potential w.r.t. the inflaton field φ. Also, the above expression are defined in the units of reduced Planck mass M Pl = 2.44 × 10 18 GeV. The various cosmological observables such as the number of e-foldings N e , scalar power spectrum P s , tensorial power spectrum P t , tensor-to-scalar ratio r, scalar spectral index n s , and runnings of spectral index α ns can be written in terms of the various derivatives of the inflationary potential via the slow-roll parameters as introduced above [24,25,26]. For example, the number of e-folding is given as, where φ end is determined by the end of the inflationary process when ǫ = 1 or η = 1. The other cosmological observables relevant for the present study are given by [24,25,26], where C E = −2 + 2 ln 2 + γ = −0.73, γ = 0.57721 being the Euler-Mascheroni constant. Therefore, it is natural to expect that the shape of the inflationary potential are tightly constrained by the experimental bounds on these cosmological observables coming from various experiments. These experimental constraints on the cosmological parameters are also useful for directly reconstructing the single field potential [27] by fixing the magnitude as well as the various derivatives of the potential. The experimental bounds on cosmological observables relevant in the present study are briefly summarized as follows, r = 0.16 +0.06 −0.05 , n s = 0.957 ± 0.015 , α ns = −0.022 +0.020 −0.021 .
Apart from the tensor-to-scalar ratio r and spectral index n s , the running of spectral index α ns has emerged as another crucial cosmological parameter on the lines of reconciling the results between the BICEP2 [2] and Planck satellite experiments [28]. To be more precise, without considering the running of spectral index, Planck + WMAP + highL data [28,29] result in n s = 0.9600 ± 0.0072 and r 0.002 < 0.0457 at 68 % CL for the ΛCDM model, and hence facing a direct incompatibility with recent BICEP2 results. However, with the inclusion of the running of spectral index, the Planck + WMAP + highL data result in n s = 0.957±0.015, α ns = −0.022 +0.020 −0.021 and r 0.002 < 0.263 at 95 % CL. The reconciliation of BICEP2 data along with those of Planck + WMAP + highL [28,29] demands a non-trivial running of spectral index, α ns < −0.002. Although the simplistic single field models are good enough to reproduce the desired values of tensor-toscalar ratio within 0.15 < r < 0.26 along with 50 − 60 e-foldings, most of them fail to generate large enough magnitude for α ns (which is needed to be of order 10 −2 ). On these lines, a confrontation, of realizing the desired values of n s , r and α ns within a set of reconciled experimental bounds of the BICEP2 and Planck experiments, has been observed for chaotic and natural inflation models [30].
In this letter, our aim is to consider the fractional chaotic inflation models with potentials V 0 φ a/b where a and b are relatively prime. We shall obtain such kind of inflaton potentials elegantly in the supergravity framework with generalized shift symmetry for a/b < 4 where the shift symmetry is broken only by superpotential 4 . We find that for the number of e-folding from 50 to 60 and some a/b from 2 to 3, our models are nicely within the 1σ region in the r − n s plane. Furthermore, we investigate the possibility of improvements for a better fit of the three confronting parameters n s , r and α ns in the lights of Planck and BICEP2 data. However, if the BICEP2 bounds on r is modified/diluted, and if r is found to be smaller than 0.11 in the upcoming Planck results, then the fractional chaotic inflation can still explain the data for exponents a/b < 2 with n s being a little bit larger than 0.96. On the other hand, if the BICEP2 claims would be confirmed in near future, the open question of interest, for the current model we propose, will be whether one can measure the exponent a/b.

Embedding the Fractional Chaotic Inflation into N = 1 supergravity
In this section, we will provide a supergravity origin of the fractional-chaotic inflationary potential. The scalar potential in the supergravity theory with given Kähler potential K and superpotential W is where (K −1 ) ī j is the inverse of the Kähler metric Kj i = ∂ 2 K/∂Φ i ∂Φj, and D i W = W i + K i W . Further the kinetic term for the scalar field is given by To warm up, we consider a simple inflation model first in the N = 1 supergravity theory, where the Kähler potential and superpotential are as follows and Thus, the Kähler potential K is invariant under the shift symmetry [32,33,34,35,36,37,38,39,40,41] Φ → Φ + iC , with C being a dimensionless real parameter, i.e., the Kähler potential K is a function of Φ + Φ † , and so it is independent of the imaginary part of Φ. The scalar potential can be easily computed from the ansatz of Kähler potential K and the superpotential W using Eq.(6) Because superpotential is a holomorphic function of Φ and X, if we choose we get One might wonder whether we can select the superpotential in Eq. (13) due to the rational power of Φ. From a pure supersymmetric theory point of view, it is fine. In fact, we can derive the Kähler potential in Eq. (8) and superpotential in Eq. (9) or Eq. (13) from the Kähler potential and superpotential with positive integer powers of all the fields. Let us consider a superfield Φ ′ with the following generalized shify symmetry [42,43] The Kähler potential and superpotential are Here, m < 2n is required.
And then the kinetic term of Φ is Therefore, the Kähler potential in Eq. (16) is the same as that in Eq. (8), the shift symmetry in Eq. (15) is the same as that in Eq. (10), and the superpotential becomes Now, let us give the concrete models, which can realize the superpotential in Eq. (13). We consider U(1) R symmetry and introduce the Z 2n symmetry. The respective quantum numbers for Φ ′ and X are given in Table 1. Especially, we want to emphasize 0 < m < 2n for consistency. One can easily show that the Kähler potential, which is consistent with the U(1) R and Z 2n symmetries, is given by Eq. (16) up to the higher order terms, and the superpotential is With the canonical normalization of Φ ′ , we indeed obtain the superpotential in Eq. (13). In addition, for the inflation potential V 0 φ a/b which will be studied in the following, we get a nature bound as To fit the Planck and BICEP2 data, we need a/b from 2 to 3 on which we will elaborate later on while discussing the numerical results for the various cosmological observables. Thus, this consistency condition (a/b < 4) from supergravity model building can be obviously satisfied. However, if the BICEP2 claims about r is modified (or if the bounds on r are diluted), and r is found to be smaller than 0.11 from the upcoming Planck results, then our fractional chaotic inflation can explain the data only for a/b < 2 and n s being a little bit larger than 0.96.

Fractional Chaotic Inflation: Numerical Study
From last section, we end up with a single field inflationary potential of chaoticinflation type monomials with fractional exponents as where a and b are positive integers and a/b < 4. In the subsequent analysis, our main focus and the special attention would be to study the inflationary models in which a and b are coprime. The sufficient condition for the ensuring the inflationary process is encoded in terms of so-called slow-roll conditions ǫ ≪ 1, η ≪ 1, and ξ ≪ 1.
Now the various slow-roll parameters (as defined in Eq. (2)) simplify for the potential (23) as follows The number of e-folding generated between the phase of horizon exit and the end of inflation is calculated by Eq.(3) as N e = b 2 a (φ 2 * − φ 2 end ) , with φ end = a √ 2 b . The simplified expressions of the cosmological observables n s , r and α ns for the present class of models are From the above expressions, one naively observes the possibility of enhancing the magnitude of the running of spectral index α ns by a choice of ratio a/b ≫ 1. However, it is not as arbitrarily possible as it seems to be, since large ratio of a/b will lead the spectral index (n s ) values going outside the experimentally allowed window. With introducing a new parameter p = b a , all the cosmological parameters defined earlier can be equivalently written out in terms of the model dependent parameter p and one of the observables in {φ, N e , r, n s , α ns }. A few set of expressions are presented as follows.

Observables/parameters in terms of p and
Observables/parameters in terms of p and N e Observables/parameters in terms of p and r where Y = ± √ 9 − 3p r + 9 p r C E . Imposing Y to be a real number will result in the following inequality and hence gives an important bound on choice of fractions. Similarly, the expressions for other combinations {p, n s } and {p, α ns } can also be written out. Now, looking for the solutions under the constraint Eq.(5), we find the constraint on α ns to be the hardest one to match. This is also consistent with the previous study in [30]. However, we manage to get α ns ∼ −0.00108 for N e = 60 with a/b ≤ 3, which is relatively better than the standard single field inflationary model. To illustrate this numerically, let us consider the set of Eq. (27) and fix the number of e-foldings to sixty, then solving the following inequalities sequentially, we get Thus, we observe that one can easily have n s ≃ 0.96, r ≃ 0.2 and N e ≃ 60, but with |α ns | < 0.001. There does not exist a solution for p if one considers larger value for |α ns | in the third inequality. In order to have larger magnitude of α ns , one has to compromise with the number of e-folding which is entangled with the spectral index n s . For a given a/b, the value of n s is lowered by lowering the number of e-foldings, and it will be outside the experimental bounds for a certain value of N e . The ratio a/b along with any one of {N e , φ, r, n s , α ns } generically fixes all the rest observables in this single field setup. Some of the possibilities are tabulated in Table 2. These numerical observations will be clearer in the graphical analysis as we discuss now.  Table 2: Some sampling values for cosmological observables and parameters.
The Figs. 1 and 2 represent the relations (n s -r) indicating that it is fairly possible to have well consistent n s and r values for chaotic inflationary models including integer as well as fractional exponents. In particular, for a/b = 5/2, 7/3, 8/3, 12/5, 14/5, 16/7 and 18/7, our predictions are within the 1σ region for the number of e-folding in the range from 50 to 60. However, as we have argued earlier that reconciling the Planck data with the recent BICEP2 observations demands a non-trivial running (α ns ) of the spectral index (n s ). It is in confrontation with desired n s and r values as can be well seen from the opposite slopes of lines plotted for different exponents in Fig. 3 and Fig. 4. Actually, this confrontation has been recently studied in detail in [30] in the context of polynomial-chaotic inflation [39,44], Natural inflation models [45] as well as S-dual inflation models [46].   Figure 4: The r − α ns plot for fractional chaotic potentials with n ≡ a/b = 2, 5/2, 7/3, 8/3 and 3. The colors are the same as these in Fig. 3. The number of e-folding is from 50 to 60 . The Fig. 3 and Fig. 4 in our analysis recover that for standard φ 2 chaotic inflation [44], α ns = −8 × 10 −4 is the best value corresponding to N e = 50. However, we can see that a larger value of a/b will result in a better situation for relaxing the tension between BICEP2 and PLANCK data. The magnitude of running of spectral index α ns increases together with the exponents up to α ns = −0.001 for N e = 50 with a φ 3 potential, but it takes the spectral index (n s ) values into the marginal regime. Also, the (α ns − n s ) relation for some more fractional exponents with intermediate values 2 < a/b < 3 are plotted in Fig[5]. From the plots in Fig. 3 -Fig. 5, we can see that the best fit with the BICEP2 experiment is with fractional powers such as 8/3 or 20/7 lying between 2 and 3, which is giving a reasonable (however not very large) enhancement in the negative running of spectral index α ns .

Conclusions
In this article, we have constructed the fractional-chaotic inflationary potentials from the supergravity framework utilizing the generalized shift symmetry. A nature bound on the fractional power is provided in our model for consistency. One of the motivations for this construction is to investigate the possibility of realizing a non-trivial and negative running (α ns ) of the spectral index n s . It has been well established by now that the simplistic polynomial chaotic inflation models successfully realize large tensor-to-scalar ratio r compatible with the allowed values of the spectral index n s . However, in order to reconcile the data from the Planck and recent BICEP2 observations, a non-trivial running of spectral index is needed which is usually suppressed at order 10 −4 in chaotic inflationary models. We have studied a generalization of polynomial chaotic inflation by including the fractional exponents in search for possible improvements along this direction. We found that, for the number of e-folding from 50 to 60 and some fractional exponents a/b from 2 to 3, our results are nicely within the 1σ region in the (r − n s ) plane along with an improvement (although insufficient) in the magnitude of the running of spectral index (α ns ). Such a class of fractional-chaotic inflationary potentials can also be interesting for facilitating large field excursions on the lines of [47], and also for studying other cosmological aspects on the lines of [48,49,50]. If the BICEP2 result will be confirmed in (near) future, the open question, relevant to our fractional-chaotic inflationary model, will be whether one can measure the exponent a/b at some future experiments.