Combining Universal and Odd RR Axions for Aligned Natural Inflation

We successfully embed the Kim-Nilles-Peloso (KNP) alignment mechanism for enhancing the axion decay constant in the context of large volume type IIB orientifolds. The flat direction is generated in the plane of ($C_0$-$C_2$) axions corresponding to the involutively even universal axion $C_0$ and odd axion $C_2$, respectively. The moduli stabilization with large volume scheme has been established as well.


Abstract
We successfully embed the Kim-Nilles-Peloso (KNP) alignment mechanism for enhancing the axion decay constant in the context of large volume type IIB orientifolds. The flat direction is generated in the plane of (C 0 − C 2 ) axions corresponding to the involutively even universal axion C 0 and odd axion C 2 , respectively. The moduli stabilization with large volume scheme has been established as well.

Introduction and Motivation
The recent BICEP2 results [1] have undoubtably shaken the status of inflationary model building in string cosmology. The discovery of the primordial B-mode polarization of the cosmic microwave background has been recently claimed by the BICEP2 Collaboration. It can be understood as the signature of gravitational wave being encoded in the so called tensor-to-scalar ratio (r). The BICEP2 observations fix the inflationary scale by ensuring a large tensor-to-scalar ratio r as follows [1] r = 0.20 +0.07 −0.05 (68% CL) , H inf ≃ 1.2 × 10 14 r 0. 16 GeV , (1.1) 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 . In order to reconcile the tension between BICEP2 [1] result and PLANCK [2], WMAP data [3], it demands the following windows for the cosmological observables ln 10 10 P s = 3.089 +0.024 −0.027 , n s = 0.957 ± 0.015, α ns = −0.022 +0.020 −0.021 , (1.2) where P s is the scalar power spectrum and α ns is the running of spectral index n s . All these cosmological observables can be written out in terms of the inflationary potential and its various derivatives. Thus, with the available experimental data from various sources so far, the shape of a single field inflationary potential is significantly constrained. As a reverse computation, writing out various derivatives of inflationary potential in terms of the aforementioned cosmological observables, the inflationary potential can be locally reconstructed [4,5,6]. In order to realize the required large value of tensor-to-scalar ratio r, the inflaton field needs to travel over trans-Planckian distance according to the famous Lyth bound [7]. Further, it also suggests the inflationary process to be (a high scale process) near the scale of the Grand Unified Theory (GUT). The UV sensitivity in chaotic inflation class of models is also addressed in [8]. As a result, embedding the inflationary models in a UV complete framework, such as string theory, is inevitable. It also provides invaluable pieces of information in searching for a consistent supersymmetry (SUSY) breaking scale [9,10].
With large field excursions, the other relevant issues from higher order corrections should also be taken care of for the viability of the model [11,12,13]. If the BICEP2 results are confirmed, it would serve as a huge discriminator filtering out many among the plethora of inflationary models developed so far. However, it is interesting that the three classes of inflationary models; namely the chaotic-type [14,15,16,17,18,19], natural-type [20,21,22,23] as well as Assisted/N/M-flation type [24,25,26,27,28,29,30,31] inflationary models are among the winners. In the context of models developed in a purely string framework prior to the BICEP2 results, the axion monodromy inflation [32,33] was found to be the most closer but still insufficient as per the BICEP2 claims. There has been very vibrant and speedy progress on these lines of developing chaotic-or (multi)natural-type of inflationary models utilizing axion monodromy in a very short post BICEP2 period so far [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53].
Regarding the axionic inflation in Type IIB string framework, the models are preferred to be developed in the context of LARGE Volume Scenarios (LVS) [54]. The LVS provides a well-controlled moduli stabilization mechanism, exhibiting an exponentially large volume for the internal Calabi-Yau threefold. The exponentially large volume V is favored as it also provides a control against the (un-)known α ′ [55] as well as string loop corrections g s [56,57]. In fact, it has been observed that the known/conjectured forms of these corrections at the level of Kähler potential appear in terms of volume suppressed terms in the scalar potential [54,56,57], which makes large volume scenario more robust as well. Further due to the presence of (extended-)no-scale structure in the context of type IIB swiss-cheese compactification, various volume moduli directions orthogonal to the overall Calabi-Yau (CY) volume V remains flat, and the breaking of flatness via (non-)perturbative corrections leads to a flat enough inflationary potential; for example, see the models with inflaton being identified with divisor volume moduli [58,59,60,61,62,63,64,65].
In the context of LVS framework, embedding of axion monodromy type potential has been recently proposed in [35], in which the universal axion c 0 could drive the inflationary process. Based on certain assumptions on the background flux, the large volume expansion has been argued to be a useful for trusting the effective field theory (EFT) description even in a non-perturbative regime where the string coupling g s satisfies 1 < g s < 10 [35]. On the other hand, in the contex of axionic inflationary models of natural-type inflation [20], a large decay constant has been proposed to be realized in a Kim-Nilles-Peloso (KNP)-type two-field potential [22]. The main idea is to align two sub-planckian decay constants such that with a certain rotation of field basis, one could create a hierarchy in the decay constant of the two newly constructed axion basis. The best advantage of this type of axionic inflation is that unlike N-flation [27,28,31] which requires a large number of (O(10 3 − 10 4 )) axions assisting the inflationary process, this is a two field model. However, the standard KNP-model with two-field usually requires large anomaly coefficients or equivalently large gauge groups of the nonperturbative effects generating the potential, and this might be difficult to embed on the practical grounds. On these lines, the standard KNP-model has been generalized to N-fields (with N < 10) [47,66] to facilitate the axionic alignments (as well as keep the number of axions less as required in N-flation model).
Motivate by the KNP proposal for enhancing the decay constant, in the article, we propose a new class of inflationary potentials in the context of LVS framework. The inflationary direction lies in the plane of (C 0 − C 2 ) axions, where C 0 corresponds to the involutively even universal axion while C 2 is involutively odd axion. If we restrict the orientifold to be divisor exchange or reflection, in order to support large volume scenarios with the orientifold odd axion, the underline Calabi-Yau threefold should have h 1,1 (CY 3 ) ≥ 3 [67,68]. Using two such involutively odd axions and magnetized non-perturbative effects, recently a KNP-type scenario has been proposed in [45]. Unlike this proposal, we utilize the universal axion C 0 along with a single odd axion C 2 to get the required alignment for the natural inflation. This engineering solves one of the major challenges of [35] by taking the framework within perturbative regime as large decay constant is realized within g s < 1 in our model. Moreover, a combination of C 0 and C 2 axions provides a better decoupling in the kinetic sector unlike the case with two odd axions [45].
The article is organized as follows. In section 2 we provide a brief and relevant features of type IIB orientifolds. Section 3 summarizes the original KNP formalism [22] for enhancing the decay constant in a two-field potential. In section 4, we provide a successful embedding of KNP-type potential in large volume scenarios with the inclusion of odd axion along with universal axion. In section 6 we provide a summary with possible open challenges.

Relevant Ingredients of Type IIB Orientifolds
We consider type IIB superstring theory compactified on an orientifold Calabi-Yau threefold CY 3 with O3/O7−plane. The orientifold action is O = (−) F L Ω p σ, where the F L is the spacetime fermion number in the left-moving sector, worldsheet projective action Ω p and involution σ. By performing the detailed dimensional reduction from ten to four dimensions [69], the low energy effective action at the second order in derivatives is given by a supergravity theory, whose dynamics is encoded in three building blocks, namely the Kähler potential K, the holomorphic superpotential W , and the holomorphic gauge kinetic functions. These building blocks of the four-dimensional effective theory can be generically written in terms of appropriate N = 1 coordinates (S, G a , T α ) defined as where t α is the two-cycle volumes, and c 0 , c a , and ρ α correspond to RR axions C 0 , C 2 , and C 4 , respectively. Further, κ αβγ and κ αab are the intersections numbers of the two cycle supported these moduli. Here, the index α runs in even (1,1)cohomology of CY orientifold (h 1,1 + (CY 3 /σ)) while index a are counted via odd (1,1)-cohomology h 1,1 − (CY 3 /σ).

The Kähler Potential K
Generically, the Kähler potential is given as where Y = 1 6 K ABC t A t B t C is the volume of the Calabi-Yau manifold expressed in terms of two-cycle volumes t A . The dots in (2.2) denote the potential appearance of other moduli like D3/D7-brane fluctuations (and hence complex structure moduli which get coupled after including brane-fluctuations) or Wilson line moduli. Unfortunately, Y is only implicitly given in terms of the chiral superfields. It is in general non-trivial to invert the last relation in (2.1), and so it is not possible to write K in terms of T α explicitly. Further, the most general Kähler potential can also depend on the derivatives of chiral superfield [11,12]. However, we ignore such higher order corrections in the present analysis.

The Superpotential W
The general schematic form of the superpotential W is given as where the first term is the Gukov-Vafa-Witten (GVW) three-form flux induced tree-level superpotential [70] (See [71,72] also for related work). The second term denotes a sum over non-perturbative corrections coming from the Euclidean D3brane instantons or gaugino condensation on D7-branes [73]. Again the dots indicate a further dependence on e.g. D3/D7-brane fluctuations or Wilson line moduli. Further, the prefactor contains not only the one-loop Pfaffian for fluctuations around the instanton background but also contributions from so-called (gauge-)fluxed instantons [74,75] and Euclidean D1-brane instantons [76]. The presence of gauge fluxes on the divisor contributing to the non-perturbative superpotential helps in alleviating [74] the chirality issue proposed in [77]. It also helps to stabilization all the odd moduli, with or without the help of poly-instanton effects 4 [68]. Also, in principle one has to sum over all the possible instanton or gaugino condensation effects, and in the presence of extra magnetic fluxes turned-on on the relevant odd two-cycles sitting inside the relevant divisor, this issue becomes more delicate in terms of satisfying tadpole/anomaly cancellation conditions, etc [79,80]; see also a related review in [81]. However in the present study, our approach would be more phenomenological with considering the most suitable ansatz of the superpotential without getting into these technicalities. We would consider the gauge flux effects on the orientifold invariant divisor D (having involutively odd two-cycles and) contributing to the non-perturbative superpotential. Usually, there are two kinds of non-perturbative corrections, one is induced through Euclidean D3-brane (E3-brane) instanton while the other one through gaugino condensation. For E3-brane instanton, we require that the gauge flux turned-on on the brane to be  [78,82], then a D-term will be generated by the U(1) subgroup with FI-term. Since O7 −/+ -plane carries −8/ + 8 times of the D7-brane charge, in the following, we always assume that we have O7 − -plane and turn-on only the odd gauge flux F D ∈ H 1,1 − (D) on the branes. Also, for the time being, we concentrate on the F-term dynamics and just consider the suitable form of superpotential W with multiple gaugino condensation configuration. The D-brane tadpole cancellation as well as the zero-modes condition are assumed can be done in a proper way. Nevertheless, one can extracts the following form of the odd moduli G a dependences from the holomorphic prefactor A D (zã, G a , F D , ...) of the expression as in eq.(2.3) where h i (F D )'s are gauge flux dependent constants turned-on along the odd twocycles of the divisor D supporting the non-perturbative superpotential contribution. This form of superpotential will be heavily utilized in the upcoming sections.
The Scalar Potential V From the Kähler potential and the superpotential one can compute the N = 1 scalar potential where the sum runs over all moduli. For studying the Kähler moduli dynamics, we will assume that the complex structure moduli and dilaton have already been stabilized supersymmetrically as D c.s. W = 0, D S W = 0. Although a delicate assumption, on the lines of [35], we assume that with the freedom available through the landscape of background fluxes, one can still keep universal axion c 0 massless or 'nearly' massless. We will quantify what we mean by 'nearly' and elaborate on this point later while considering the explicit computations in section 4.

Review of the KNP-Type Natural Inflation
Let us very briefly review the original KNP proposal for natural inflation [22]. We consider the following two-field inflationary potential where f i and g i 's can be sub-Planckian decay constants as the most natural choice. The determinant of the Hessian of this potential is simplified to Thus, it will have a flat direction if the following condition holds Therefore, a small enough deviation from this condition can create a mass hierarchy between the two axions rotated in a new basis. As we will see explicitly in a moment, one can not only create a mass hierarchy but also with appropriate axionic rotation, an alignment leading to the enhancement of decay constant of the lighter combination occurs. With the following rotation of axions we reformulate the expression eq.(3.1) as below where f ′ 1 , f ′ 2 and f eff take the form as below Thus, if the deviation from the flatness condition eq.(3.3) is small enough, one can generate an 'effectively' large decay constant for ψ 2 combination. Further, by engineering an appropriate hierarchy Λ 2 ≪ Λ 1 , one can make the field ψ 1 heavier than ψ 2 with the respective masses at the minimum given as . (3.7) Stabilizing ψ 1 at one of its minimum ψ 1 = 0 would result in a single axion potential with large decay constant Now we turn to the embedding of the KNP-type mechanism in large volume scenario setup in the next section. The main focus would be to utilize universal RR axion C 0 along with an involutively odd RR axion C 2 .

Realizing Natural Inflation in Large Volume Scenarios
Let us consider the following ansatz for the Kähler potential K motivated by the large volume scenarios 5 [68]. After introducing a single odd moduli G 1 via the appropriate choice of orientifold involution, the Kähler potential becomes and C α ′ = − χ(X)ξ(3) 4(2π) 3 gs 3/2 . This form of Kähler potential explicitly shows the shift symmetries in various RR axionic directions: namely the universal axion C 0 , the involutively even axion C 4 and the involutively odd axion C 2 . Although the presence of α ′ -corrections break the "no-scale structure", it still leaves the direction orthogonal to V (which is τ s ) flat. This flatness and axionic shift symmetries are broken via the non-perturbative effects appearing in the following racetrack form of the superpotential Such a form of superpotential eq.(4.3) could be thought of arising from different stacks of unfluxed and fluxed D7-branes wrapping the so-called small divisor in an orientifold invariant way. As a result, we can set the gaugino condensations with a 0 = 2π N 0 , a s = 2π N 1 , b s = 2π N 2 , with N 0 , N 1 and N 2 the ranks of the corresponding gauge groups. Further, W cs1 , W cs2 , A 0 , A s and B s are generically complex structure moduli and background flux dependent quantities. For the time being, these can be considered to be constants as in the standard moduli stabilization schemes. At the outset, let us clearly mention the following assumptions to be made before coming to the scalar potential computation • On the lines of [35], we assume that in the absence of non-perturbative corrections to the superpotential, the landscape of background fluxes can facilitate to keep the universal RR axion C 0 massless or at least nearly massless via creating a mass-hierarchy between dilaton mass and universal axion C 0 . Although the univesal axion appears as a linear term in the superpotential, by tuning the background flux dependent parameters in the tree-level superpotential, the c 0 axion shift symmetry does not get broken via the quadratic term induced in the scalar potential. In order to restore large volume scenarios as well as a decoupled KNP-type inflationary potential of c 0 − c 1 axion, the coefficient w 2 in W cs = w 1 + c 0 w 2 has to satisfy the following bound This is probably the strongest assumption in our model and should be examined to be realized in a concrete Calabi-Yau orientifold construction.
• Based on the aforementioned point, we assume the standard procedure to stabilize the complex structure moduli and dilation via the background flux superpotential. So we naively consider W cs = w 1 + c 0 w 2 such that w 2 ≪ w 1 and we will quantify how small w 2 should be to trustfully recover the large volume potential.
Utilizing these pieces of information, the F-term scalar potential can be computed from eq.(2.5) for the given ansatz of K and W , and various terms can be categorically collected as follows where V LVS (V, τ s ; ρ s , b 1 ) is the large volume potential contributing at the leading order O 1 V 3 . With stabilizing the axions at one of their minima ρ s = 0 = b 1 , the stabilized values of volume moduli is obtained by the solutions of following coupled expressions Let us mention an important point that in our approach of stepwise moduli stabilization, one has to maintain the hierarchy |V LVS | ≪ |V rest | throughout and so one has to be careful while sampling of the model dependent parameters. As we will see later, relatively larger gauge groups N 1 and N 2 are needed for realizing large decay constant, and in order to stabilize the overall volume of the CY to order O(10 3 ), we need N 0 < N 1,2 and then to maintain the mass hierarchy between standard Kähler moduli and universal axion together with odd moduli, one has to appropriately choose the gauge flux parameters h 1 and h 3 large enough. At the sub-leading order, the shift symmetry for the odd axion c 1 is broken and after stabilizing the heavier moduli and orientifold even axion C 4 , the potential reduces to the form as below where in the aforementioned simplification, the coefficients ∆ 1 and ∆ 2 are suppressed by factors e −as h 1 /gs and e −as h 3 /gs respectively as compared to |V LVS | while ∆ 0 is the collection of all the terms independent of c 1 and c 0 axions given our assumption that the coefficient of quadratic potential for universal axion c 0 generated at tree level can be fairly negligible by utilizing the flux freedom. Note that these model dependent parameters h 1 and h 3 depend on the gauge flux F supported on the divisor with odd two-cycles contributing to the non-perturbative superpotential. Further, the dots denote those terms which are doubly suppressed by flux dependent exponentials and hence are subleading for small string coupling regime. Now after using an appropriate uplifting mechanism, one can rearrange the terms to result in the desired KNP-type potential [22] V where Λ i 's can be collected in expressions with model dependent parameters as below Further, in expression eq.(4.9) of the potential, n i = a s h i and m i = b s h i+2 for i = 1, 2. Subsequently, the canonically normalized fields φ 1 and φ 2 are defined as follows With the following redefinitions of the two-fields (similar to the original KNPformalism reviewed in the last section) the expression of axionic potential eq.(4.9) can be adjusted into the form as below , and f eff = n 2 1 f 2 2 + n 2 2 f 2 1 |n 1 m 2 − n 2 m 1 | . (4.14) Assuming a reasonable hierarchy Λ 2 ≪ Λ 1 , a justified diagonalization follows with the splitting into a heavy (ψ 1 ) and a light (ψ 2 ) axionic combinations. Further, stabilizing the heavier axion at its minimum ψ 1 = 0 leads to a single-field natural inflation driven by a trans-Planckian axion as below Before the detailed numerical analysis towards inflationary aspects, let us exemplify the moduli stabilization part by providing a benchmark sampling as below Using these samplings in eq.(4.7), eq.(4.10), eq.(4.11) and eq.(4.14), one gets Here, the parameters are chosen to get Λ 2 ≃ 10 −9 , which is needed to have a high inflationary Hubble scale H inf ≃ 10 14 GeV as per the requirement of constraints eq.(1.1) or equivalently power spectrum P s ≃ 2.2 × 10 −9 given in eq.(1.2). The parametric setting has been done such that the effective decay constant f eff > 7. It is the minimal values to fit the PLANCK and BICEP2 data as we will systematically explore in the numerical section. For the sampling eq.(4.16) and eq.(4.17), the enhancement of decay constant can be seen from Fig.1.

Revisiting the Standard Natural Inflation
Let us recall the relevant features of standard natural inflation by checking the consistency requirements of cosmological observables from the PLANCK and BI-CEP2 data. Usually it is qualitatively mentioned that the decay constant for axion utilized in the natural inflation must be trans-Planckian, i.e., much larger than the reduced Planck scale M Pl . As the realization of large decay constant in string models has always been a challenge, and in one way or the other, the choice of model dependent parameters are crucially affected (and in confrontation within) to accommodate the observables in best possible manner. One of the reasons for this revisit is to quantify the decay constant window to fulfill the minimal experimental bounds. For a given single field potential V (φ), the sufficient conditions for ensuring the slow-roll inflation is encoded in a set of so-called slow-roll conditions defined as below where ′ denotes the derivative of the potential w.r.t. the inflaton field φ. Also, the above expressions are defined in the units of reduced Planck mass M Pl with 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 running of spectral index α ns can be written as the various derivative of the inflationary potential via introducing the aforementioned slow-roll parameters as follows where C E = −2 + 2 ln 2 + γ ≃ −0.73, γ = 0.57721 being the Euler-Mascheroni constant.
For the standard singe filed natural inflation potential the slow-roll parameters as well as the three main cosmological observables (n s , r and α ns ) to be constrained as per the relations in eq.(1.1-1.2) are simplified as below where N end e = f 2 ln 1 − 1 2 f 2 is evaluated by using ǫ = 1 where inflation ends, and these N end Number of e-foldings N e Because the natural inflation potential has a maximum at φ = π f , depending on the decay constant, there is an upper limit on N e which can be realized for a given f . It can be shown that even f = 1 can generate around 20 e-foldings as shown in Fig. 2, while f = 2 can result in a maximal value of N e around 80. However, in order to have |n s − 1| < 0.05 one needs larger decay constant. The Spectral Index n s and Tensor-to-Scalar ratio r Although more than sixty number of e-foldings can be generated even with the decay constant in the range 1 < f < 2, the fitting of the spectral index n s and tensor-to-scalar ratio r pushes the f window towards f > 4. To be more precise, one finds that for 1 < f < 4, the spectral index lies in the range 0.1 < n s < 0.9 while increasing the decay constant values enhances the spectral index. As can be seen from the Fig. 3, for e-foldings 50 < N e < 60, the decay constant f should be in the range of 4 < f < 12 in order to be consistent with PLANCK result, and larger than f > 7 in order to fall in the 2σ regions of r and n s for the BICEP2 data.
The running of spectral index α ns is small. It needs to be the same order (10 −2 ) to reconcile the PLANCK and BICEP2 data eq.(1.2). This confrontation has been investigated recently in [83] as can be also seen from Fig. 4. Again, it shows that in order to be consistent with both the PLANCK and BICEP2 data, one needs larger decay constant. However, we will show later that larger decay constant results in a larger rank of gauge group for gaugino condensation. Of course, f should not be too large. In our case, we constrain the decay constant f to be less than 20 for a natural choice.

Benchmark Points in our Aligned Natural Inflation Models
As we conclude from the Fig. 3 of the analyses done in the previous subsection, the best fit requirement for the n s and r values from the PLANCK and BICEP2 observations demands the decay constant to lie within 5 < f eff < 20. Further, as we have already matched our aligned natural inflation eq.(4.15) with the standard form eq.(5.2), and having all the cosmological observable analyses been revisited already, now all we need to do is to realize a large decay constant; for our samplings we would focus in the range 7 < f eff < 12.
Before the explicit numerical analysis and sampling of model dependent parameters, let us make the following important points • Although the form of scalar potential suggests that the two fields φ 1 and φ 2 in eq.(4.9) or ψ 1 and ψ 2 in eq.(4.13) should be arbitrarily interchangable, the canonical normalizations fix the choice for a given sampling. This argument is in the sense that the decay constants of the two axions are different In the large volume limit, one naturally expects f 1 > f 2 , and this hierarchy restricts the interchangeability of the two fields in eq.(4.9). The reason for considering this constraint f 1 > f 2 is to put a lower bound on the volume of the CY such that V > g −3/2 s for natural orientifold constructions. The positive definiteness of kinetic sector demands κ B11 < 0, and some explicit examples of swiss-cheese Calabi-Yau orientifolds with such intersections can be found in [67]. For orientifold examples with κ B11 = 0, the leading order contribution to the decay constant f c 1 scales as V −1/2 in the CY volume.
• As we have neglected the subleading corrections which are doubly suppressed in e −as h 1 /gs or e −bs h 3 /gs (or a product of the two factors) with an inherent assumption that e −as h 1 /gs ≃ e −bs h 3 /gs , we have to choose a s h 1 ≃ b s h 3 for consistancy, or equivalently where N 1 and N 2 are the ranks of the gauge groups corresponding to the gaugino condensations.
Several benchmark points in Table 1 and 2 have been presented for various model dependent parameters to realize a consistent r and n s value in Fig. 3 for different f eff .

Open Challenges and Conclusion
In this paper, we have successfully embedded the idea of KNP [22] for the enhancement of axion decay constant relevant for realizing the natural inflation requirement. The inflaton is identified with a linear combination of the universal axion c 0 and an involutively odd axion c 1 . The expressions of decay constants for these two axions enjoy appearance of string coupling g s and the Calabi-Yau volume V with a less suppressed factor as compared to the C 4 axions. Moreover, their decouplings in the kinetic sector via the Käehler potential are more natural in large volume limit as compared to considering two C 2 axionic setup as then, one has to diagonalize the intersection matrix κ Sab along the odd directions a and b. Despite of the several nice features of our model, there are certain assumptions to be consistently realized in concrete setups, especially on the technical grounds. On these lines, let us recall that the original universal axion monodromy inflation [35] has two delicate issues as below • The decay constant for universal axion c 0 is given as f c 0 = gs √ 2 , and so natural inflation embedding demands string coupling to lie in the window 1 < g s < 10 and thus pushing the whole description into the nonperturbative regime. Our approach of realizing the KNP-type inflation with inflaton being a combination of the universal axion c 0 and the odd axion c 1 provides a natural way of enhancing the decay constant in the regime where the perturbative description remains trustfully valid along with the support of large volume scenarios.
• The second delicate assumption of inflationary model in [35] is related to facilitate a hierarchy in the dilaton and universal axion at the tree-level superpotential. This flux superpotential depends on the landscape of background fluxes and it would be interesting to construct the explicit models in which this requirement could be satisfied.
In addition to the second point, it would be interesting to address more technical issues like the tadpole/anomaly cancellations in concrete Calabi-Yau orientifold examples with all the suitable gauge fluxes arranged through the incorporation of relevant involutively odd two-cycles to contribute the non-perturbative effects. Further, the trans-Planckian nature of the inflaton opens up some more challenges and hence there are some cautionary concerns on the lines of [11,12,13,84,85,86]. One of such concerns could be the inflaton coupling to the gauge degrees of freedom living on the two stacks of D7-brane wrappings with magnetic-fluxes turned-on, and those could be of the following kind where a s = 2π N 1 and b s = 2π N 2 with N i 's being the ranks of gauge groups. In order not to make gauge degrees of freedom supermassive with acquiring the trans-Planckian masses out of the axion vacuum expectation values, one has to ensure that the overall coupling still remains under control. For that one has to have large rank of gauge group which comes out to be an unnatural requirement beyond a certain value. Also in our setup, as we have required some of gauge flux parameters (h 1 and h 3 ) to be relatively large (of order 10) to sustain the masshierarchy, and this has to done in a consistent manner by not letting these fluxes very large, via keeping a s h 1 or b s h 3 less than unity. Although in our samplings we have successfully realized large effective decay constant with the rank of the gauge groups being not too large (N 1 = 30 and N 2 = 32), it would be interesting to increase the number of the odd axions and take the requirement for the rank of gauge group to be below ten on the lines of [47]. It suggests an exponential enhancement of the decay constant with increasing the number of axions in the KNP formalism.