Constraining ${\cal N}=1$ supergravity inflation with non-minimal K\"ahler operators using $\delta N$ formalism

In this paper I provide a general framework based on $\delta N$ formalism to study the features of unavoidable higher dimensional non-renormalizable K\"ahler operators for ${\cal N}=1$ supergravity (SUGRA) during primordial inflation from the combined constraint on non-Gaussianity, sound speed and CMB dipolar asymmetry as obtained from the recent Planck data. In particular I study the nonlinear evolution of cosmological perturbations on large scales which enables us to compute the curvature perturbation, $\zeta$, without solving the exact perturbed field equations. Further I compute the non-Gaussian parameters $f_{NL}$, $\tau_{NL}$ and $g_{NL}$ for local type of non-Gaussianities and CMB dipolar asymmetry parameter, $ A_{CMB}$, using the $\delta N$ formalism for a generic class of sub-Planckian models induced by the Hubble-induced corrections for a minimal supersymmetric D-flat direction where inflation occurs at the point of inflection within the visible sector. Hence by using multi parameter scan I constrain the non-minimal couplings appearing in non-renormalizable K\"ahler operators within, ${\cal O}(1)$, for the speed of sound, $0.02\leq c_s\leq 1$, and tensor to scalar, $10^{-22} \leq r_{\star} \leq 0.12$. Finally applying all of these constraints I will fix the lower as well as the upper bound of the non-Gaussian parameters within, ${\cal O}(1-5)\leq f_{NL}\leq 8.5$, ${\cal O}(75-150)\leq\tau_{NL}\leq 2800$ and ${\cal O}(17.4-34.7)\leq g_{NL}\leq 648.2$, and CMB dipolar asymmetry parameter within the range, $0.05\leq A_{CMB}\leq 0.09$.


Introduction
The primordial inflationary paradigm is a very rich idea to explain various aspects of the early universe, which creates the perturbations and the matter. For recent developments see Refs. [1,2]. Usually inflation prefers slow-rolling of a single scalar field on a flat potential, which has unique predictions for the Cosmic Microwave Background (CMB) observables. The induced cosmological perturbations are generically random Gaussian in nature with a small tilt and running in the primordial spectrum which indicates that inflation must come to an end in our patch of the universe. But a big issue may crop up in model discrimination and also in the removal of the degeneracy of cosmological parameters obtained from CMB observations [3][4][5][6]. Non-Gaussianity has emerged as a powerful observational tool to discriminate between different classes of inflationary models [7][8][9][10][11]. The Planck data show no significant evidence in favour of primordial non-Gaussianity, the current limits [6] are yet to achieve the high statistical accuracy expected from the single-field inflationary models and for this opportunities are galore for the detection of large non-Gaussianity from various types of inflationary models. To achieve this goal, apart from the huge success of cosmological linear perturbation theory, the general focus of the theoretical physicists has now shifted towards the study of nonlinear evolution of cosmological perturbations. Typically any types of nonlinearities are expected to be small; but, that can be estimated via non-Gaussian npoint correlations of cosmological perturbations. The so-called "δN formalism" (where N being the number of e-foldings) [12][13][14][15][16][17][18][19] is a well accepted tool for computing nonlinear evolution of cosmological perturbations on large scales (k << aH), which is derived using the "separate universe" approach [14,15,20,21]. Particularly, it provides a fruitful technique to compute the expression for the curvature perturbation ζ without explicitly solving the perturbed field equations from which the various local non-Gaussian parameters, f local N L , τ local N L , g local N L and CMB dipolar asymmetry parameter [22][23][24][25][26], A CM B are easily computable 1 .
We will be using the following constraints on the amplitude of the power spectrum, P s , spectral tilt, n s , tensor-to-scalar ratio, r, sound speed, c s , local type of non-Gaussianity, f local N L and τ local N L and CMB dipolar asymmetry from Planck data throughout the paper [4][5][6]27]: (1.7) In this paper I will concentrate our study for Hubble induced inflection point MSSM inflation derived from various higher dimensional Planck scale suppressed nonminimal Kähler operators in N = 1 supergravity (SUGRA) which satisfies the observable universe, and it is well motivated for providing an example of visible sector inflation.
In section 2, I will briefly review the setup with one heavy and one light superfield which are coupled via non-minimal interactions through Kähler potential. In section 3 I discuss very briefly the role of various types of Planck suppressed non-minimal Kähler corrections to model a Hubble induced MSSM inflation for any D-flat directions. Hence in section 4 I present a quantitaive analysis to compute the expression for the local types of non-Gaussianin parameters and CMB dipolar asymmetry parameter which characterize the bispectrum and trispectrum using the δN formalism. For the numerical estimations I analyze the results in the context of two D-flat direction, L L e and u d d within the framework of MSSM inflation [28,29]. 1 One can also compute all the non-gaussian parameters, f local N L , τ local N L , g local N L and CMB dipolar asymmetry parameter, A CMB using In-In formalism in the quntum regime. But the inflationary dynamics responsible for the interactions between the modes occurs at the super-horizon scales within the effective theory setup proposed in this paper. Here I use δN formalism as-(1) it perfectly holds good at the super-horizon scales and (2) are also independent of any kind of intrinsic non-Gaussianities generated at the scale of horizon crossing.

The Superpotential
In this section I concentrate on two sectors; heavy hidden sector denoted by the superfield S, and the light visible sector denoted by Φ where they interact only via gravitation. Specifically the inflaton superfield Φ is made up of D-flat direction within MSSM and they are usually lifted by the F -term [30] of the non-renormalizable operators as appearing in the superpotential. In the present setup for the simplest situation I start with the following simplified expression for the superpotential made up of the superfields S and Φ as given by: where for MSSM D-flat directions, n ≥ 3 (In the present context n characterizes the dimension of the non-renormalizable operator) and the coupling, λ ∼ O(1). The scale M s characterizes the scale of heavy physics which belongs to the hidden sector of the effective theory. Furthermore, I will assume that the VEV, s = M s ≤ M p and φ = φ 0 ≤ M p , where both s and φ are fields corresponding to the super field S and Φ. We also concentrate on two MSSM flat directions, L L e and u d d, which can drive inflation with n = 6 via R-parity invariant ( L L e)( L L e)/M 3 p and ( u d d)( u d d)/M 3 p operators in the visible sector, which are lifted by themselves [31], where u, d denote the right handed squarks, and L denotes that left handed sleptons and e denotes the right handed slepton.

The Kähler potential
In this paper I consider the following simplest choice of the holomorphic Kähler potential which produces minimal kinetic term, and the Kähler correction of the form: where δK represent the higher order non-minimal Kähler corrections which are extremely hard to compute from the original string theory background. in a more generalized prescription such corrections allow the mixing between the hidden sector heavy fields and the soft SUSY breaking visible sector MSSM fields. Using Eq (2.2) the most general N = 1 SUGRA kinetic term for (s, φ) field can be written in presence of the non-minimal Kähler corrections through the Kähler metric as: In this paper I consider the following gauge invariant non-minimal Planck scale suppressed Kähler operators within N = 1 SUGRA [31][32][33]: (2.6) where a, b, c, d are dimensionless non-minimal coupling parameters. The · · · contain higher order non minimal terms which has been ignored in this paper. In Fig (1) let me have shown the schematic picture of the total effective field theory setup within N = 1 SUGRA in presence of non-minimal Kähler interaction.

Modeling MSSM inflation from light & heavy sector
In this section let me consider a situation where inflation occurs via the slow roll of φ field within an MSSM vacuum with a gauge enhanced symmetry, where the entire electroweak symmetry is completely restored. Let me imagine a physical situation where the heavy field is coherently oscillating around a VEV, s ∼ M s , during the initial phase of inflation, which arises quite naturally from the hidden sector string moduli field and is coherently oscillating before being damped away by the phase of inflation. The contribution to the potential due to the time dependent oscillating heavy field, with an effective mass M s ≫ H inf , is averaged over a full cycle (0 < t osc < H −1 inf ) is given by: Let me now concentrate on the Hubble induced potential when V (s) = 3H 2 M 2 p ∼ M 2 s |s| 2 , in which case the contributions from the Hubble-induced terms are important compared to the soft SUSY breaking mass, m φ , and the A term for all the cases mentioned in Eq (2.4-2.7). Consequently the potential is dominated by a single scale, i.e. H ∼ H inf , [34] where I have taken λ = 1 and, the Hubble-induced mass parameter c H and A-term a H 2 for s << M p can be expressed in terms of the non-minimal couplings appearing in Eq (2.4). Fortunately for this class of potential given by Eq (3.3), inflection point inflation can be characterized by a fine-tuning parameter, δ, defined as [35]: Here the tuning parameter, |δ| is small enough 3 , for which a point of inflection at the position of the VEV of the inflaton φ = φ 0 exists, so that the flatness condition V ′′ (φ 0 ) = 0 holds good in the present context, with For δ < 1, one can Taylor-expand the inflaton potential around an inflection point, φ = φ 0 , as [36][37][38][39][40][41]: where α denotes the height of the potential, and the coefficients β, γ, κ determine the shape of the potential in terms of the model parameters 4 . Note that once the numerical values of c H and H are specified, all the terms in the potential are determined. For c s = 1, the upper bound on the numerical value of the Hubble parameter (H), the scale of inflation ( 4 √ V * ) and the scale of the heavy string moduli field (M s ) are expressed as: where r * is the tensor-to-scalar ratio at the pivot scale of momentum k * ∼ 0.002Mpc −1 .
4 δN formalism in presence of non-minimal Kähler operators for c s = 1 In this section I have used the δN formalism [12][13][14][15][16][17][18][19] to compute the local type of non-Gaussianity, f local N L from the prescribed setup for c s = 1. In the non-attractor regime, the δN formalism shows various non trivial features which has to be taken into account during explicit calculations. Once the solution reaches the attractor behaviour, the dominant contribution comes from only on the perturbations of the scalar-field trajectories with respect to the inflaton field value at the initial hypersurface, φ, as the velocity,φ, is uniquely determined by φ. However, in the non-attractor regime of solution, both the information from the field value φ and alsoφ are required to determine the trajectory [42].
In order to compute the scalar-field trajectories explicitly, here I feel the need to solve the equation of motion of the scalar field, which is in general a second-order differential equation in a prescribed background. This can be solved by providing two initial conditions on φ andφ on the initial hypersurface. I have, therefore, integrated the equation of motion to the final time, t = t * . Here I have solved the equation of motion perturbatively by expanding it around a particular trajectory given by φ ∝ e ϑHt , where ϑ is time dependent function in the generalized physical prescription. So I have used these background solutions for the field trajectories to compute the perturbations of the number of e-folds with respect to the initial field value and its time derivative. During the computation of the trajectories let me assume here that the universe has already arrived at the adiabatic limit via attractor phase by this epoch, or equivalently it can be stated that a typical phase transition phenomenona appears to an attractor phase at the time t = t * . More specifically, in the present context, I have assumed that the evolution of the universe is unique after the value of the scalar field arrived at φ = φ * where it is mimicking the role of standard clock, irrespective of the value of its velocityφ * . Let me mention that only in this case δN is equal to the final value of the comoving curvature perturbation ζ which is conserved at t ≥ t * . In Fig (2) I have shown the schematic picture of the δN formalism.

General conventions
In the present context, further I have neglected the canonical kinetic term during the non-attractor phase for simplicity. The background equation of motion for the four physical situations are given by for Case IV.
(4.1) From the Eq (4.1), it is obvious that the determination of a general analytical solution is too much complicated. To simplify the task here I consider a particular solution, and further my prime objective is to obtain a more generalized solution for the background up to the second order in perturbations around this particular solution. Here I also assume that the non-attractor phase ends when the inflaton field value is achieved at φ = φ * . Let me define a perturbative parameter, which represents the difference between the true background solution and the reference solution to solve the background Eq (4.1) perturbatively. Here ∆ 1 and ∆ 2 are the general linearized and second order perturbative solution of the background field equations. The · · · contribution comes from the higher order perturbation scenario which I will neglect for further computation.

Linearized perturbation
Let me consider the contribution from the linear perturbation, ∆ 1 . Consequently in the leading order the background linearized perturbative equation of motion takes the following form: (4.3) where I have neglect the higher powers of ∆ 1 in the linearized approximation. The general solution is given by

Second-order perturbation
Next I have considered the contribution from the second-order perturbation, ∆ 2 . Consequently in the leading order the background Second-order perturbative equation of motion takes the following form: (4.5) where the source term, Π s , for the sub-Planckian Hubble induced inflection point inflation within N = 1 SUGRA is given by (4.6) Now to solve Eq (4.5) in presence of non-linear source term, let me assume that the contribution from φ L is sub-dominant. Consequently the general solution in presence of second-order perturbation is given by: for Case IV.

δN at the final hypersurface
In the present context my prime objective is to compute the perturbations of the number of e-folds, δN. The truncated background solution of φ up to the second-order perturbations around the reference trajectory, φ L ∝ e −ϑN in terms of N is given by, , (4.8) where the symbol is introduced to rescale the integration constants as well as the perturbative solutions by φ * . Here I have neglected the contribution from e −ϑN in ∆ 2 (N) to avoid over counting in the Eq (4.8). It is important to note that in the present context of all these sets of scaled integration constants parameterizes different trajectories, and I have set φ(0, W k ) = φ * for any value of W k ∀k = ( [1,2], [3,4], [5,6], [7,8]) in accordance with the assumption that the end of the non-attractor phase is determined only by the value of the scalar field, φ = φ * . Here W k = C k , G k represent collection of all integration constants.
Further inverting Eq (4.8) for a fixed set of W k , I have obtained N as a implicit function of φ and W k . Then the δN formula can be obtained by [43]: Here I have introduced the shift in the inflaton field φ → φ + δφ and the number of e-folds N → N + δN on both sides of Eq (4.8) to compute δN iteratively. In the present context I have obtained, perturbative solutions of the scalar-field trajectories around the particular reference solution, φ L = φ * e ϑHt , which are valid only when the perturbed trajectories are not far away from the reference solution. Additionally, since I have neglected the sub-dominant solution, ∆ 1 ∝ e ϑHt , my approximation holds good only at sufficiently late times. These imply that here I should choose the initial time as close as possible to the final time for which N 1. Then the simplest choice is to take the initial time to be infinitesimally close to t = t * . Now perturbing the number of e-folds N up to the second order at the epoch t = t * , I get [43]: Here · · · corresponds to the higher order contributions, which are negligibly small compared to the leading order contributions. By taking the derivatives of both sides of Eq. (4.8) and setting N = 0 = W k (= C k , G k )∀k at the end, my next task is to identify δφ * and W k (= C k , G k ) which are actually generated from quantum fluctuations on flat slicing, δφ. To serve this purpose let me consider the evolution of δφ on super-horizon scales. The shift in the inflaton field can be expressed here as: where the subscript "1" and "2" represent the solution at the linear and the second order respectively. It is important to note that both the solutions include the features of growing and decaying mode. Now imposing the boundary condition from the end of the non-attractor phase, where N = 0, I get: for Case IV.

Computation of local type of non-Gaussianity and CMB dipolar asymmetry
The local type of non-Gausiianity is originally implemented as a position space expansion of non-Gaussian fluctuations around Gaussian pertabations [44]: where ζ g (x) satisfies the Gaussian statistics. Here f N L , g N L , · · · are the non-Gaussian estimators. However, the inflationary perturbation itself is implemented in the momentum space and thus the momentum space correlators provide a clear picture of non-Gaussianity compared to the isolated position space. Using Eq (4.16) in fourier space the three point and the four point correlator can be expressed as [44]: (4.18) My next job is to derive and to estimate the amount of the local type of non-Gaussianity from the prescribed setup. Further using the results obtained for δN in the earlier section, the non-Gaussian parameter corresponding to the local type of non-Gaussianity f local N L 5 , g local N L and τ local N L can be computed as: where the parameter ϑ, appearing in all the physical situations, can be expressed in terms of the sound speed (c s ), potential dependent slow roll parameter (ǫ V , η V ) and the model parameters (α, β, γ) as: (4.22) where the sound speed c s can be expressed in terms of non-canonical Kähler corrections, a, b, c, d and the scale of heavy field, M s , as: The dot denotes derivative w.r.t. physical time, t. Here V = V (φ) − V (S) and the symbol Σ = X, Y, Z, W , appearing for the four cases in Eq (4.23) are mentioned in the appendix. Additionally, it is important to note that the well-known Suyama-Yamaguchi consistency relation [45,46] between the three and four point non-Gaussian parameters, 5 In [43] the authors have shown that for a specific P (X, φ) theory with c s = 1 the non-Gaussian parameter, f local N L = 5 4 1 + 1 c 2 s . In this paper I have obtained different result as the non-minimal Kähler interactions within N = 1 SUGRA effective theory setup which is completely different compared to the case studied in [43]. f local N L , τ local N L and g local N L violates [47][48][49][50] in the present context due to the appearance of · · · terms in Eq (4.19,4.20,4.21). As the contributions form · · · terms are positive, the consistency relation is modified as: However, it is important to note that since · · · terms are small, the amount of violation is also small. Further using δN formalism, the CMB dipolar asymmetry parameter for single field inflationary framework can be expressed as [51]: where |N ,φ ∆φ| < 1 for which the perturbative expansion is valid here. In it the field excursion ∆φ = φ cmb − φ e ≈ φ * − φ e , can be expressed as [52,53]: √ c s r * 0.12 3 400 where φ cmb ≈ φ * and φ e be the value of the inflaton field at the horizon crossing and the end of inflation respectively. Here r * be the tensor-to-scalar ratio at the pivot scale of momentum, k * ∼ 0.002 Mpc −1 . Hence substituting Eq (4.19,4.20,4.21) in Eq (4.25) I derive the following simplified expression in terms of the tensor-to-scalar ratio, sound speed and : where |∆φ| φ * < 1 in the present sub-Planckian setup.

Constraining local type of non-Gaussianity and CMB dipolar asymmetry via multi parameter scanning
My present job is to now scan the parameter space for c H , a H with the help of, by fixing λ = O(1) and δ ∼ 10 −4 . In order to satisfy the inflationary paradigm, the Planck observational constraints, as stated in the introduction of the paper, I obtain the following constraints on our parameters for H inf ≥ m φ ∼ O(TeV):  Inflation would not occur outside the scanning region since, at least, one of the constraints would be violated. Note that for the above ranges, the VEV of the inflaton, setup. This analysis will further constrain the non-minimal Kähler coupling parameters a, b, c, d 6 appearing in the higher dimensional Planck scale suppressed opeartors within the following range: (4.35) In Fig (3) and Fig (4) I have shown the behaviour of the local type of non-Gaussian parameter f local N L and τ local N L with respect to the sound speed c s in the Hubble induced inflationary regime (H >> m φ ). In Fig (3), the shaded yellow region represent the allowed parameter space for Hubble induced inflation which satisfies the combined Planck constraints on the f local N L and c s . For all the four cases, the region above the f local N L = 8.5 line is observationally excluded by the Planck data. The four distinctive features are obtained by varying the model parameters of the effective theory of N = 1 SUGRA, c H , a H and M s subject to the constraint as stated in Eq (4.28-4.31). As Planck puts an upper bound, τ local N L ≤ 2800, the rest of the region above the τ local N L = 2800 line in Fig (4) is excluded. In the present setup I have obtained the following stringent bound on the f local N L , τ local N L and g local N L within the following range:  Here the theoretical upper and lower bound on f local N L 7 satisfy both the constraints on the f local N L and c s observed by Planck data. Also it is important to note that, within this prescribed framework, τ local N L is bounded by below for all the four cases and consequently it is possible to put a stringent lower bound on τ local N L which satisfies the constraints on τ local N L and c s both. Till date the observational results obtained from Planck do not give any significant constraint on g local N L . However in this paper I have provided a theoretical lower and upper bound of g local N L using the consistency relation between τ local N L and g local N L as stated in Eq (4.24). Finally, in Fig (5) I have shown the behaviour of the CMB dipolar asymmetry parameter A CM B with respect to the tensor-to-scalar ratio r * within, 10 −22 ≤ r * ≤ 0.12, at the pivot scale, k * ∼ 0.002 Mpc −1 for the Hubble induced inflation. Here the red and blue coloured boundaries are obtained by fixing the sound speed at c S = 0.02 and c S = 1. The orange dark coloured region satisfied the Planck constraint on the A CM B 7 In the prescribed setup the consistency relation between the non-Gaussian parameter f local N L and the spectral tilt n s [7], f local N L ∼ 5 12 (1 − n s ), does not hold as in the present setup sound speed, c s = 1 and for such non-minimal N = 1 SUGRA setup, Planck data favours lower values of the sound speed (within 0.02 < c s < 1).
i.e. 0.05 ≤ A CM B ≤ 0.09 8 for 10 −22 ≤ r * ≤ 0.12 within our proposed framework. In Fig (5) only the region bounded by the red, blue and brown colour is the allowed one and the rest of the region (A CM B < 0.02 and A CM B > 0.09) is excluded by the Planck data.

Conclusion
In this paper, I have shown that in any general class of N = 1 SUGRA inflationary framework, the behaviour of Kähler potential in presence of non-minimal Kähler corrections in effective theory setup are constrained via the non-minimal couplings of the non-renormalizable gauge invariant Kähler higher dimensional Planck scale suppressed operators from the observational constraint on non-Gaussianity, sound speed and CMB dipolar asymmetry as obtained from the Planck data. In the present setup the hidden sector based heavy field is settled down in its potential via its Hubble induced vacuum energy density. In particular, for the numerical estimations in this paper I have used a very particular kind of (inflection point) inflationary model, which is fully embedded within MSSM, where the inflaton is made up of L L e and u d d gauge invariant D-flat directions. However the prescribed methodology holds good for other kinds of inflationary models too.
Further I have scanned the multiparameter region characterized by the Hubble induced mass parameter, c H , A-term, a H and the scale of the heavy field M s , where I have satisfied the current Planck observational constraints on the, inflationary parameters: P S , n S , c s , r * (within 2σ CL), non-Gaussian parameters: f local N L , τ local N L (within 1σ − 1.5σ CL) and CMB dipolar asymmetry parameter A CM B (within 2σ CL). Consequently the non-minimal Kähler couplings, a, b, c, d are fixed within ∼ O(1) in the present effective theory setup. Finally, using this methodology, I have obtained the theoretical upper and lower bound on the non-Gaussian parameters within the range, O(1 − 5) ≤ f N L ≤ 8.5, O(75 − 150) ≤ τ N L < 2800 and O(17.4 − 34.7) ≤ g N L ≤ 648.2, and the CMB dipolar asymmetry parameter within, 0.05 ≤ A CM B ≤ 0.09, which satisfy the observational constraints stated in Eq (1.1-1.7), as obtained from Planck data.
There is also a future prospect of upgrading the present methodology proposed in this paper by studying the further stringent phenomenological constraints on the nonminimal couplings a, b, c, d, as appearing in the context of higher dimensional Planck scale suppressed Kähler operators within N = 1 SUGRA by imposing the constraint on Higgs mass [55,56] and the dark matter abundance [5,57] via WIMPy baryogenesis scenario (see also Refs. [58][59][60] for the various theoretical issues). A detailed analysis on these aspects have been reported shortly as a separate paper [61].