Natural inflation with and without modulations in type IIB string theory

We propose a mechanism for the natural inflation with and without modulation in the framework of type IIB string theory on toroidal orientifold or orbifold. We explicitly construct the stabilization potential of complex structure, dilaton and K\"ahler moduli, where one of the imaginary component of complex structure moduli becomes light which is identified as the inflaton. The inflaton potential is generated by the gaugino-condensation term which receives the one-loop threshold corrections determined by the field value of complex structure moduli and the axion decay constant of inflaton is enhanced by the inverse of one-loop factor. We also find the threshold corrections can also induce the modulations to the original scalar potential for the natural inflation. Depending on these modulations, we can predict several sizes of tensor-to-scalar ratio as well as the other cosmological observables reported by WMAP, Planck and/or BICEP2 collaborations.


Introduction
Cosmic inflation is the most successful scenario which not only explains the current cosmological observations but also solves the fine-tuning problems such as the horizon and flatness problems at the same time.
The inflation scenarios are mostly classified according to the size of the tensor-to-scalar ratio which measures the tensor perturbations of the metric in our universe. One is the small-field inflation scenario which gives the tiny tensor-to-scalar ratio due to the flat potential of the scalar field, called inflaton. The other scenario we consider is the large-field inflation model which gives a sizable and measurable tensor-to-scalar ratio. Recent data reported by BICEP2 collaboration [1] may be a signal of the gravitational wave, although such signal would be explained by the dust distributions reported by Planck collaboration [2]. In any case, it is interesting to propose the large-field inflation models which would be tested by future joint analysis of Planck, BICEP2 and the other cosmological observations.
When we consider the large-field inflation models, we always encounter the problems how to treat the trans-Planckian field values. For example, in the case of natural inflation [3] known as one of the large-field models, we need the corresponding trans-Planckian axion decay constant of the inflaton which is required by recent Planck [4] and/or BICEP2 data [1] (See Ref. [5] and references therein.) Especially, in the higher-dimensional theory, there are a lot of axions associated with the internal cycles of the internal manifold and then it would be natural to identify such axions as the inflaton. However, it is in general to be problematic that the scale of axion decay constant is severely constrained by the size of internal manifold and the cut-off scale of higherdimensional theory. To overcome such a problem, there are several approaches to realize trans-Planckian axion decay constant by employing Kim-Niles-Peloso alignment mechanism [6] in the case of multiple axions with sub-Planckian axion decay constant, for more details see Refs. [7,8,9,10,11,12,13,14,15,16]. In the case of single axion, the trans-Planckian axion decay constant can be realized based on the five-dimensional theory [17,18] with a small five-dimensional gauge coupling and the weakly-coupled heterotic string theory with certain loop-corrections to the gauge coupling [19].
In this paper, we propose the natural inflation scenario in the framework of type IIB string theory on toroidal orientifold or orbifold and the inflaton is identified as the imaginary part of the complex structure moduli, Im U 2 . The axion decay constant of inflaton is enhanced to the trans-Planckian field value due to the inverse of one-loop factor in the gauge threshold corrections which have a dependence on the complex structure moduli. The sections are organized as follows. We briefly review the gauge threshold corrections caused by the massive open strings between D-branes in the N = 2 sector of type II string theory in Sec. 2. In Sec. 3, we show the moduli stabilization procedure step by step and identify the lightest mode (Im U 2 ) as the inflaton. First, the dilaton and the complex structure moduli expect for the inflaton sector U 2 can be stabilized by three-form fluxes at the perturbative level. Second, we consider two classes of Kähler modulus stabilization by such non-perturbative effects as those employed in the racetrack scenario [20] in Secs. 3.1, 3.2 and as that adopted in the Kachru-Kallosh-Linde-Trivedi (KKLT) scenario [21] in Sec. 3.3. Then the real part of complex structure moduli Re U 2 can be also stabilized due to those nonvanishing superpotential terms at the same time. Finally, we extract the effective inflaton potential which is in a type of natural inflation with the trans-Planckian axion decay constant by identifying Im U 2 as the inflaton in the large complex structure moduli limit, Re U 2 > 1. On the other hand, in the case of Re U 2 ≃ 1, we find the modulations to the original scalar potential for the natural inflation to be discussed in Sec. 3.3. Sec. 4 is devoted to the conclusion. We show the mass-squared matrices of moduli in Appendix A.
2 Moduli-dependent threshold corrections in Type II string theory We briefly review the one-loop stringy threshold corrections to the gauge couplings on D-branes in the framework of type II string theory. (For more details, see Refs. [22,23], and references therein.) The running gauge coupling for scale µ below the string scale M s is written by where g a is the 4D gauge coupling at the string scale M s , b a is the beta-function coefficient of the gauge group G a and ∆ a represents the correction from stringy massive modes at the one-loop level. In type II string theory, in general, the charged open strings between two stacks of Dbranes or O-planes contribute to the gauge couplings on D-branes as the threshold corrections ∆ a which are mostly moduli-dependent [24]. In the case of type IIA string theory on toroidal orientifold or orbifold with O-planes and D6-branes wrapping on a supersymmetric three-cycle (special Lagrangian submanifold) of the internal tori, the gauge threshold corrections are explicitly computed by an exact CFT method via the cylinder and Möbius diagram [22]. (There are similar computations in type IIB string theory and F-theory on the local geometric cycle with fractional D-branes [25,26].) When we consider the T-dual picture, they correspond to the setup of D3/D7-branes or D5/D9-branes in type IIB orientifold or orbifold which depend on the choice of T-duality. For N = 2 SUSY sector in type IIB string with D3/D7-branes and O3/O7-planes, (which correspond to the stacks of parallel D6-and D6 ′ -branes or O-planes in type IIA string theory), one-loop gauge threshold corrections for the gauge theory living on D7-branes with the gauge group G a and the wrapping numbers (p k a , q k a ) on three two-tori labeled by k = 1, 2, 3 are expressed as where T k and U k are Kähler and complex structure moduli, respectively, κ is the IR regularization constant and η is the Dedekind eta-function. The beta-function coefficients b N =2 ac represent contributions from the charged massive modes in open strings stretched between the a-stack of D7-branes and the other c-stack of D-branes, and the summation over c implicitly extracts the all contributions from the other stacks of D-branes. Note that the imaginary part of T k is given by the Neveu-Schwarz field.
As pointed out in Ref. [27], only holomorphic threshold corrections contribute to the gauge kinetic function on D7-branes, which is extracted from the first term on the right-handed side of Eq. (2), Especially, in the large complex moduli limit, the logarithmic factor in Eq. (3) behaves as due to the asymptotic form of the Dedekind eta-function. In this limit, the gauge kinetic function on D7-branes receives the following threshold correction, where the summations of Kähler and complex structure moduli are only performed over the cycle wrapped by the D7-branes and b j represents the contribution from the massive open-string modes. Here we consider the D7-branes , otherwise the dilaton dependence also appears in the gauge kinetic function depending on the two-form fluxes, because such fluxes are irrelevant in our scenario of moduli stabilization and inflation. The case with a more general form of gauge kinetic function in terms of the Dedekind eta-function in Eq. (2) are discussed later. In the following, we propose the moduli stabilization and inflation scenario in the framework of type IIB string theory on toroidal orientifold or orbifold such as T 2 /Z 2 or T 2 /(Z 2 × Z 2 ) with D-branes.
3 Natural inflation in Type IIB string theory on toroidal orientifold or orbifold In this section, we propose the natural inflation in the framework of type IIB string theory on toroidal orientifold or orbifold such as T 2 /Z 2 or T 2 /(Z 2 × Z 2 ) with D-branes. The inflaton is considered as the axion paired with one of the complex structure moduli into N = 1 SUSY multiplet and the axion decay constant is enhanced to trans-Planckian value due to the inverse of a loop-factor accompanying the one-loop corrections to the gauge kinetic function which makes it possible to realize a successful natural inflation, as shown later. As pointed out in Ref. [28], in the type IIB string theory (unlike the heterotic string theory) on Calabi-Yau (CY) three-fold, three-form fluxes induce the superpotential W flux which depends on the dilaton S and complex structure moduli U k as where Ω is the holomorphic three-form of the CY manifold and G 3 = F 3 − i SH 3 is the threeform flux determined by Ramond-Ramond (RR) three-form flux F 3 and Neveu-Schwarz (NS) three-form flux H 3 . Such flux-induced superpotential can stabilize the dilaton and all complex structure moduli at the perturbative level [28].
In order to show the essential idea of our scenario, as mentioned above, we consider the type IIB string on the simple toroidal orientifold or orbifold such as T 2 /Z 2 or T 2 /(Z 2 × Z 2 ) whose moduli are characterized by dilaton S, three complex structure moduli U 1 , U 2 , U 3 and single overall Kähler modulus T . 1 In order to obtain the desired inflation potential, we follow a similar step to the KKLT scenario [21] for stabilizing all the moduli other than Im U 2 which is identified as the inflaton field.

Moduli stabilization with three-form fluxes
First, let us focus on the stabilization of the dilaton and complex structure moduli by employing the three-form flux. We consider the following Kähler potential and superpotential of S, U 1 , U 2 and U 3 in the framework of 4D N = 1 supergravity, (7) in the Planck unit, M Pl = 1, where all the dimensionful quantities are measured by the reduced Plank mass 2 M Pl = 2.4 × 10 18 GeV and then the coefficients w m (m = 1, 2, . . . , 7) are integers determined by the RR-and NS-fluxes. 3 To brighten the outlook for analyzing the stabilization of dilaton S and complex structure moduli U 1 , U 2 and U 3 , we redefine one of the latter as In the field base S, U 2 , U 3 and U 4 , the superpotential and Kähler potential are rewritten as Then the vacuum expectation values of dilaton and complex structure moduli are determined by the supersymmetric condition, where D I W = W I + K I W , with W I = ∂W/∂Φ I and K I = ∂K/∂Φ I , is the covariant derivative with respect to the moduli fields Φ I , Φ I = S, U 2 , U 3 and U 4 .
The above stabilization condition (10) can be satisfied by For simplicity and concreteness, we further restrict the RR-and NS-fluxes to those satisfying with which the expectation value of S, U 3 and U 4 are given by at the minimum given by Eq. (11). Thus S, U 3 and U 4 can be stabilized at the supersymmetric Minkowski minimum and their mass-squared matrices are found as in the field basis (U 4 , U 3 , S), which becomes full rank with some appropriate choices of the integers w m . Note that since there is a Kähler mixing between U 2 and U 4 as can be seen from Eq. (9), we have to canonically normalize them as summarized in the Appendix A.
We remark that, with the above choice of RR-and NS-fluxes, the tadpole cancellation condition may not occur among themselves, however, our moduli stabilization and natural inflation scenario would not depend on the detail structure of tadpole condition. Only the facts that S, U 3 and U 4 are stabilized by three-form fluxes and there exists a Kähler mixing between U 2 and U 4 are the essential ingredients to realize a successful inflation scenario as discussed later.

Light moduli stabilization with non-perturbative effects
Next, we discuss the stabilization of remnant complex structure modulus U 2 and Kähler modulus T below the mass scale of the stabilized dilaton S and complex structure moduli U 3 and U 4 . As mentioned above, for simplicity, we focus on the case with a single overall Kähler modulus T and then its Kähler potential is expressed as in the large volume limit. As a source of stabilizing the Kähler modulus, we assume the nonperturbative effects such as the gaugino condensation on D7-branes, where f 1 and f 2 denote the gauge kinetic functions of pure SU(N) × SU(M) gauge theories on D7-branes, where we assume that both of them receive the threshold corrections depending on the complex structure moduli other than U 2 , that is, A(U) and B(U) are functions of only the heavy complex structure moduli stabilized by the flux-induced superpotential (6). Thus we can treat these functions A(U) and B(U) as constants, neglecting the fluctuations of these heavy moduli around the stabilized value. The stabilization of the Kähler modulus can be achieved by two gaugino-condensation terms in the same way as the racetrack scenario [20], i.e., which leads to the following value of the Kähler moduli at the racetrack minimum, where the explicit values of parameters are explored by evaluating the cosmological observables. In the following, let us discuss the stabilization of the remnant complex structure modulus U 2 . Since the Kähler modulus is stabilized at the minimum W = 0, the real part of U 2 is stabilized by the Kähler potential under the following condition: which determines the expectation value of Re U 2 as satisfying the extremal condition V U 2 = ∂V /∂U 2 = 0. We again ermark that U 3 , U 4 and S are fixed at a high-scale by the condition D U 3 W = D U 4 W = D S W = 0. Therefore, if the gaugino condensation scale is much smaller than the mass scale of U 3 , U 4 and S, their deviations from the minimum given by Eq. (13) are sufficiently small, and we can replace the heavy moduli U 3 , U 4 and S by their expectation values (13) in evaluating the stabilization of light moduli T and Re U 2 .
To confirm the stabilization of Re U 2 , we have to check that the rank of the full mass-squared matrices for U 2 , U 3 , U 4 , S and T . The explicit form of them and the canonical normalization of all moduli are summarized in Appendix A. From the mass matrices shown in Eq. (44), we find the squared mass of Re U 2 is positive, if and only if the mass scales of U 3 , U 4 and S are much heavier than the gaugino condensation scale determined by the superpotential (16), that is consistent with the above argument.
In the above analysis, the vacuum energy is negative at the minimum D I W = 0 for I = U 2 , U 3 , U 4 , S and T . Therefore we assume the existence of some uplifting sector with which the total scalar potential V is in KKLT-type [21], where V F is written by the usual N = 1 supergravity formula, with K IJ is the inverse of the Kähler metric K IJ = ∂ 2 K/∂Φ I ∂ΦJ for Φ I = S, T, U 2 , U 3 and U 4 . The uplifting potential V up may come from anti-D3-branes [21] and/or nonvanishing F-terms in some dynamical SUSY breaking sector [29,30,31,32], etc.. In the next section, we show the inflaton potential by identifying the light axion Im U 2 as the inflaton. Finally, we comment on the stabilization of the Kähler modulus T . In the above analysis, the Kähler modulus is stabilized in the same way as the racetrack scenario [20]. In the case of the KKLT scenario [21] which is achieved, instead of Eq. (16), by a single gaugino-condensation term and a tiny constant value of the nonvanishing flux-induced superpotential, we can also derive the similar inflaton potential with a trans-Planckian axion decay constant as seen in the next section. This is because our inflaton potential do not depend on the stabilization of T . However, in the latter case, we need to tune the RR-and NS-flux to obtain the tiny expectation value of superpotential W ≃ 10 −2 in order to realize the large volume limit required to ensure the form of Kähler potential shown in Eq. (15). When the three-form fluxes are turned on, we may have to consider a more general geometry than CY (locally) warped due to the energies of these fluxes as well as some sources for the tadpole cancellation [28]. Thus we further assume that the possible backreactions from the three-form fluxes are negligible in the relevant sector to our scenario of moduli stabilization and inflation.

Natural inflation with and without modulations
Now we are ready to write down the inflaton potential. The effective scalar potential for Im U 2 is generated from another SU(L) gaugino-condensation term, where we assume the gauge coupling on SU(L) gauge theory receives the threshold corrections which have U 2 -dependence. The factor C( U ) denotes possible threshold corrections from the heavy complex structure moduli, U 3 and U 4 , as in the previous step. We assume again that all the other moduli are strictly fixed at their minimum given by Eqs. After all, the effective scalar potential for Im U 2 is generated from V F in Eq. (23), which is written as where Re U 2 , λ 1 = bπ/6d L and φ = d Im U 2 is the canonically normalized axion field. The normalization factor d = 2 is determined by the canonical normalization of relevant complex structure moduli which is explicitly shown in Appendix A. Even though U 2 and U 4 have a kinetic-mixing from the structure of Kähler potential (9) as mentioned in Sec. 3.1, the effects from the mixing between φ and Im U 4 is negligible on the inflation mechanism discussed in the following. This is because Im U 4 is heavier enough than Im U 2 and already decoupled from the inflaton dynamics.
When we identify the inflaton as φ, the axion potential is considered as the type of natural inflation. As seen in the scalar potential (25), the axion decay constant is enhanced by the inverse of one-loop factor and is determined by the ratio b/L and the vacuum expectation value, Re U 2 . Since the trans-Planckian axion decay constant is required in order to explain the cosmological observations reported by WMAP, Planck [4] and/or BICEP2 collaborations [1], the ratio b/L and Re U 2 have to be properly chosen. Note that the beta-function coefficient b in N = 2 sector is not related with the sector of SU(L) gauge theory.
To evaluate the cosmological observables, we define the slow-roll parameters for the inflaton φ, and then the cosmological observables such as the power spectrum of the scalar density perturbation P ζ , the spectral index n s , its running dn s /d ln k and the tensor-to-scalar ratio r are written as by employing the slow-roll approximation at the leading order. The e-folding number is also evaluated as where φ end denotes the field value at the end of inflation with which the slow-roll condition, max{ǫ, η, ξ} = 1, is violated.
In order to explain the power spectrum of the scalar density perturbation, P ζ ≃ 2.2 × 10 −9 reported by Planck [4], we set the parameters in the superpotential given by Eqs. (9), (19) and (24), and the other parameters in Eq. (9) are fixed in a such way that the conditions given by Eq. (12) are satisfied, those lead to the vacuum expectation values of moduli, With the above set of parameters, the above cosmological observables and the e-folding number are evaluated as n s ≃ 0.96, r ≃ 0.11, dn s /d ln k ≃ −10 −3 , N e = 54, (31) which are consistent with WMAP, Planck data [4], at the pivot scale k * = 0.05Mpc −1 and/or BICEP2 data [1], when we properly choose the initial condition for inflaton φ.
We remark that, in our model, the axion decay constant is enhanced by the inverse of loop factor through the stringy threshold corrections which are characterized by the Dedekind etafunction and the beta-function coefficients b induced by the massive open-string modes between D-branes in Eq. (17). Thus, we can realize several values of axion decay constant depending on the brane configurations, which means that the tensor-to-scalar ratio can be of O(0.01 − 0.1) in our framework.
Our estimation is valid only if the Dedekind eta-function is approximated by the leading term as shown in Eq. (4) in the large field limit of complex structure moduli. We can estimate the deviations from the large complex-structure limit, by introducing the next leading term in the Dedekind function, which induces the following correction to the inflaton potential given by Eq. (25), where with Λ 2 = Λ 1 2b L e −(2π+ b π 6L ) Re U 2 , λ 2 = (2π + b π/6L)/d. It is remarkable that the correction V mod would in general yield the modulations [33,34,35] to the leading inflaton potential V eff in the case of Re U 2 ≃ 1, though it is not the case in the above analysis with the numerical values of parameters (29) resulting Eqs. (30)- (33).
In the following analysis, we also take care of the vacuum expectation value of Re U 2 and the ratio b/L in order to avoid the tachyonic scalar potential around the origin, φ = 0. Actually we can avoid the nonvanishing field value of the axion φ at the minimum, which would lead to the strong CP problem if it couples to the QCD sector, that is, the physicalθ term is severely constrained by the non-observation of electric dipole moment of the neutron [36,37]. The axion mass squared at the origin is described by and its positivity is ensured by the following condition: For general cases, it is interesting to discuss the contributions from the additional scalar potential V mod to the inflaton dynamics. By the inclusion of V mod , the slow-roll parameters of the inflaton potential V inf for the inflaton φ are written as while the spectral index n s including the higher-order corrections is found as where C = −2 + ln 2 + γ with γ ≃ 0.577 is the Euler-Mascheroni constant and the ellipsis stands for more higher corrections which are given by the fourth derivative with respect to the inflaton. (See Ref. [38] and references therein.) As discussed later, in models which have ξ 2 = O(0.01), the higher-order terms contribute to the numerical value of n s in Eq. (40), while the higher-order corrections to P ζ do not give sizable effects. Note that our inflaton effective potential is controlled by Re U 2 and b/L in the superpotential given by Eq. (9). In the following analysis, we focus on particular value Re U 2 ≃ 1, with which V mod does affect the inflaton dynamics. Then we numerically evaluate the cosmological observables r, n s , dn s /d ln k by putting several values of b/L. The scalar density perturbation P ζ can be obtained as 2.2 × 10 −9 also in this case by suitably choosing the gaugino-condensation terms in Eq. (19) which stabilize the Kähler modulus at the racetrack minimum. Fig. 1 shows the prediction of the spectral index n s and the tensor-to-scalar ratio r in the range of e-folding number, 50 ≤ N e ≤ 60. Several oscillating curves are drawn by varying N e with the corresponding fixed values of the ratio b/L in Fig. 1. This is because the slow-roll parameters oscillate due to the inclusion of the deviations from the large complex-structure limit as can be seen in Fig. 2 which shows the behavior of the slow-roll parameters by setting b/L = 1/5 (1/10) and Re U 2 = 1.2 (2.4) in the left (right) panel. Although, in the both left and right panels in Fig. 2, the leading scalar potential V eff has the same structure, the next-leading scalar potential V mod gives sizable corrections in the left rather than the right panel. The scalar potential with and without such modulations is shown in the Fig. 3. As mentioned above, the detectability of such modulations is governed by the expectation value of Re U 2 and then the next-to-next leading scalar potential which comes from the expansion of the Dedekind functions would be important in the case of Re U 2 < 1. We summarize our predictions for the cosmological observables in Table 1.   The input values of b/L, Re U 2 and the output values of the e-folding number N e , spectral index n s , tensor-to-scalar ratio r and the running of spectral index dn s /d ln k.
Our results suggest that we can realize several values of the tensor-to-scalar ratio and spectral index independently to each other when we consider the particular value of complexstructure modulus, Re U 2 ≃ 1. This nature is different from the original natural inflation model [3] and is also seen in the multi-natural inflation scenario [8]. However, up to now, we do not know which amount of gravitational waves are observed reported by BICEP2 collaborations [1]. We expect that future cosmological observations select more precisely certain values of cosmological observables.
In summary, in our framework of type IIB string theory on toroidal orientifold or orbifold, the deviation from the natural inflation depends on the expectation value of the real part of complex structure modulus, Re U 2 . In the large field limit of complex structure moduli, our inflaton potential is considered as the original natural inflation scenario [3]. When we construct the standard model sector on Dp-branes (p > 3), the matter fields in the standard model generically couple to the complex structure moduli. Such couplings affect the inflaton dynamics after the end of inflation which are related to the reheating processes. Thus, it is interesting to study toward such a direction in a future work.

Conclusion
We proposed a mechanism for the natural inflation with and without modulations in the framework of type IIB string theory on toroidal orientifold or orbifold. The essential ingredient to obtain the trans-Planckian decay constant which is required in the natural inflation is the holomorphic gauge threshold corrections to the gauge kinetic function. Such threshold corrections are exactly computed in type II string theory on toroidal orientifold or orbifold by employing the CFT method (see Ref. [22,23], and references therein) which suggests the gauge threshold corrections have moduli dependences. Note that when one of the moduli are identified as the inflaton, the moduli-dependent threshold corrections are important not only to discuss about the gauge coupling unification, but also to enhance the axion decay constant of the inflaton by the inverse of one-loop factor accompanying the correction.
In our model, the inflaton is considered as Im U 2 which is the imaginary part of the complex structure modulus and the inflaton potential is extracted from the gaugino-condensation term whose gauge coupling receives the complex structure moduli-dependent terms characterized by the Dedekind function. We presented that in the large complex-structure limit, Re U 2 > 1, the Dedekind function is approximated as the single exponential term and then the inflaton potential is close to that of the natural inflation which is consistent with cosmological observations such as WMAP, Planck [4] and/or BICEP2 [1]. On the other hand, in the regime with Re U 2 ≃ 1, we have to take account of the explicit Dedekind function, which leads to the modulations to the original natural inflation [3]. The modulations give a sizable modification to the predictions [33,34,35] of the original natural inflation in the same way as the multi-natural inflation scenario [8]. The natural inflation with modulations predict the different predictions unlike the original natural inflation without modulations. In fact, we can achieve the small and large tensor-to-scalar ratio without changing the value of spectral index so much.
In both inflation scenarios, we stabilize the complex structure moduli U 1 , U 3 and dilaton S by employing three-form fluxes in the usual manner. The overall Kähler modulus T and Re U 2 are stabilized at the racetrack (KKLT) minimum by double (single) gaugino-condensation terms (term) above the inflation scale. In general, although it seems to be difficult to obtain the mass difference between Re U 2 and Im U 2 , it can be achieved by the Kähler mixing between U 2 and the other complex structure moduli in our model.
We have not discussed the reheating process. When we construct the standard model sector on Dp-branes (p > 3), the matter fields in the standard model generically couple to the complex structure modulus (inflaton). Since such couplings affect the inflaton dynamics after the end of inflation, it is interesting to study in such a direction for the future work.
where (K eig ) IK = δ IK / (K eig ) I , and Note that here the mass-squared matrix is evaluated in the canonically normalized field basis