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15KEK-TH-1788, TU-990, IPMU15-0003

Elliptic Inflation: Interpolating from natural inflation to R2-inflation

Tetsutaro Higaki1, ∗ and Fuminobu Takahashi2, 3, †

1Theory Center, KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan2Department of Physics, Tohoku University, Sendai 980-8578, Japan

3Kavli Institute for the Physics and Mathematics of the Universe (WPI),Todai Institutes for Advanced Study, University of Tokyo, Kashiwa 277-8583, Japan

We propose an extension of natural inflation, where the inflaton potential is a general periodicfunction. Specifically, we study elliptic inflation where the inflaton potential is given by Jacobielliptic functions, Jacobi theta functions or the Dedekind eta function, which appear in gauge andYukawa couplings in the string theories compactified on toroidal backgrounds. We show that in thefirst two cases the predicted values of the spectral index and the tensor-to-scalar ratio interpolatefrom natural inflation to exponential inflation such as R2- and Higgs inflation and brane inflation,where the spectral index asymptotes to ns = 1− 2/N ≃ 0.967 for the e-folding number N = 60. Wealso show that a model with the Dedekind eta function gives a sizable running of the spectral indexdue to modulations in the inflaton potential. Such elliptic inflation can be thought of as a specificrealization of multi-natural inflation, where the inflaton potential consists of multiple sinusoidalfunctions. We also discuss examples in string theory where Jacobi theta functions and the Dedekindeta function appear in the inflaton potential.

I. INTRODUCTION

The cosmic microwave background (CMB) is isotropicto roughly one part in 105, and its temperatureanisotropies have the correlations beyond the horizonsize at the last scattering. This strongly implies thatour Universe experienced an accelerated cosmic expan-sion, namely, inflation, at an early stage of the evolu-tion [1–5]. In the slow-roll inflation paradigm [6, 7],the quantum fluctuations of the inflaton become scalar-mode metric perturbations (i.e. density perturbations),while the quantum fluctuations of the graviton becometensor-mode metric perturbations [8], inducing primor-dial B-mode polarization of the CMB. If the primordialB-mode polarization is detected, it will provide a defini-tive proof of inflation, and determine the inflation scaleto be around the GUT scale.The inflaton field excursion during the last 50 or 60

e-foldings must be greater than or comparable to thePlanck scale, MP ≃ 2.4× 1018GeV, in order to generatelarge tensor mode perturbations within the reach of thecurrent and future CMB experiments. There have beenproposed many such large-field inflation models. Amongthem, there are two simple and therefore important infla-tion models: quadratic chaotic inflation [9] and naturalinflation [10]. The natural inflation consists of a sin-gle cosine function and it asymptotes to the quadraticchaotic inflation in the limit of a large decay constantF . The Planck constraint on the tensor-to-scalar ratio rreads [11]

r < 0.11 (95% C.L.), (1)

and the quadratic chaotic inflation model is on the verge

∗ thigaki@post.kek.jp† fumi@tuhep.phys.tohoku.ac.jp

of being excluded, while some of the parameter space forthe natural inflation (F . 5MP ) has been excluded.

1

There is one interesting class of inflation models wherethe predicted values of ns and r sits near the best-fit val-ues of the Planck observation: R2 inflation [3] or Higgsinflation [14] (see also Refs. [15–22] for related works).Of course this does not preclude any other models lyingsomewhere between the natural inflation and R2 infla-tion. In fact, there are various extensions of the quadraticchaotic and natural inflation models, where a variety ofvalues of ns and r is realized. For instance, if one con-siders a polynomial chaotic inflation with the potentialV (φ) = 12m

2φ2 + κφ3 + λφ4 · · · , the value of r can bereduced to give a better fit to the Planck data [23–26].Similarly, if there are multiple sinusoidal functions withdifferent potential heights and decay constants, a vari-ety of values of ns and r can be realized [27–29], andsuch extension of the natural inflation is called multi-natural inflation. It was shown in Ref. [27] that, evenwith two sinusoidal functions, the lower bound on thedecay constant disappears because a small-field inflation(axion hilltop inflation) is realized for a certain choice ofthe parameters. Note that, although multiple sinusoidalfunctions are generated in the extra-natural inflation [30],it does not significantly improve the bound on the decayconstant as the higher order terms are suppressed. Itwas recently pointed out that this problem can be cir-cumvented by adding appropriate matter fields in thebulk [31], and this provides another UV completion ofthe multi-natural inflation.In this paper we propose another extension of the nat-

ural inflation. The inflaton potential in natural inflation

1 The BICEP2 result [12] prefers a large r which is consistent withthe quadratic chaotic inflation. However, the subsequent Planckpolarization data [13] showed that the dust contamination in theBICEP2 region is larger than previously thought.

http://arxiv.org/abs/1501.02354v2

2

is given by a single sinusoidal function, which is a spe-cific type of periodic function. A periodic potential isattractive from the model building point of view becauseit often appears when the flatness of the inflaton poten-tial is protected by an approximate shift symmetry: theinflaton potential which arises from the breaking of acontinuos shift symmetry respects the residual discreteone, leading to such periodic potential. We therefore fo-cus on a class of inflation models with periodic potential.Specifically, we consider Jacobi elliptic functions, Jacobitheta functions, and the Dedekind eta function as the in-flaton potential, and we call such inflation models basedon elliptic functions and other related functions as ellipticinflation.

As an example of the elliptic inflation, we will first con-sider a Jacobi elliptic function as the inflaton potential:

V (φ) = 2Λ4(

1− cn2(

φ

2F, k

))

, (2)

where cn(x, k) is the Jacobi elliptic cosine function withelliptic modulus 0 ≤ k ≤ 1, and Λ and F are modelparameters. We call the inflation model with the abovepotential cnoidal inflation. Interestingly, the cnoidal in-flation is reduced to the natural inflation in the limit ofk → 0, while it is reduced to the exponential inflationincluding the brane inflation, R2 inflation, and the Higgsinflation, in the limit of k → 1.The Jacobi elliptic functions are closely related to the

Jacobi theta functions, which often appear in theorieswith periodic boundary conditions. Periodicity or mod-ular invariance play important roles in field theories inassociation with symmetries and dualities [32, 33]. Instring theories, periodicity appears through compactifi-cations of extra dimensions when a compact manifold hasperiodic properties. For instance, a holomorphic gaugecoupling at one loop level is given by the Jacobi thetafunction ϑ1(v, q) in toroidal compactification [34]. Here,v is a open string modulus (diagonal modes of the ad-joint representation on D-branes), and q = eiπτ is givenby a complex structure of a torus τ in type IIB mod-els (see also Ref. [35] for their interpretation and fur-ther generalization). In our later example, the inflaton isidentified with the real part of v. Another example is aYukawa coupling among matter-like fields, which can bedescribed by the Jacobi theta functions ϑ3(v, q) in mag-netized (or intersecting) D-brane models on tori [36, 37].2

The theta functions appear in the gauge and Yukawacouplings because of the periodicity of the tori. Simi-lar Yukawa couplings are obtained in Heterotic orbifoldsthrough world-sheet instantons [39].3 Therefore, a scalar

2 In both IIB and IIA models on tori, it is possible to realizemagnetized and intersecting D-brane system with any branes ifone does not require the preservation of supersymmetry [38].

3 In Ref.[40], they considered a case in which Yukawa couplingsare given by the Dedekind eta function of Kähler moduli withthe target space modular invariance.

potential which depends on the theta functions appearsnaturally, and we shall show that it takes the followingform,

V (φ) ∼[

ϑ1

(

φ

2πF, q

)]a[

ϑ3

(

φ

2πF, q

)]b

, (3)

where a and b denote rational numbers. We call the infla-tion model with the above potential form theta inflation.We find that the theta inflation exhibits similar behaviorto the cnoidal inflation.It is possible to consider inflation models based on

other special functions. In particular, inflation modelswith the Dedekind eta function was studied in Ref. [41].(See also Ref. [42] for inflation models with modular func-tions.) Interestingly, the inflaton potential can be ap-proximated by natural inflation with modulations, lead-ing to a sizable running of the spectral index. In fact,it was shown in Ref. [43] that a sizable running of thespectral index can be generated if there are small modu-lations on the inflaton potential. (See also Refs. [44–50]for related work.) Here we will study the inflaton poten-tial with the eta function in a greater detail.

The rest of this paper is organized as follows. In Sec. IIwe study phenomenological aspects of the elliptic infla-tion, where we calculate the predicted values of ns and rfor the above three elliptic inflation models. In Sec. IIIwe discuss possible origins of the Jacobi theta functionsand the Dedekind eta function in the string theories andimplications for the large-field inflation. The last sectionis devoted to discussion and conclusions.

II. ELLIPTIC INFLATION

In this section we first study two examples of the ellip-tic inflation. One is based on the Jacobi elliptic inflation(cnoidal inflation), and the other is based on the Jacobitheta function (theta inflation). As we shall see below,they provide a one-parameter extension of the naturalinflation, interpolating from natural inflation (approxi-mately) to the R2-inflation. Finally we study the infla-tion based on the Dedekind eta function, and show thata sizable running of the spectral index can be generatedbecause of the small modulations on the inflaton poten-tial.

A. Cnoidal inflation

Let us consider the cnoidal inflation model with thepotential Eq. (2). The Jacobi elliptic functions sn(x, k)and cn(x, k) can be thought of as an extension of trigono-metric functions, sin(x) and cos(x), respectively. In thelimit of k → 0, they become

sn(x, k) → sinx, (4)cn(x, k) → cosx, (5)

3

2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

0

0

k=0 0.6

0.9

0.999

FIG. 1. The inflaton potential V (φ) with the Jacobi ellipticfunction given by Eq. (2) for k = 0, 0.6, 0.9 and 0.999.

while, in the limit of k → 1−,

sn(x, k) → tanhx, (6)cn(x, k) → sechx. (7)

In these limits, the inflaton potential V (φ) becomes

V (φ) →

Λ4(

1− cos φF

)

for k → 0,

2Λ4 tanh2φ

2F≃ 2Λ4

(

1− 4e−φ/F)

for k → 1,

(8)

where we have assumed φ ≫ F in the second equality.Therefore, the elliptic inflaton potential Eq. (2) interpo-lates from natural inflation to the exponential inflation.See Fig. 1 for the inflaton potential V (φ) for several val-ues of k. One can see from the figure that the potentialbecomes flatter around the potential maximum as k ap-proaches unity.To see how the predicted values of the spectral index

ns and the tensor-to-scalar ratio r change as a function ofthe model parameters, we numerically solve the inflatonequation of motion,

φ̈+ 3Hφ̇+ V ′(φ) = 0, (9)

with V (φ) given by Eq. (2). The spectral index and thetensor-to-scalar ratio can be evaluated by the standardformulae,

ns ≃ 1 + 2η − 6ε, (10)r ≃ 16ε, (11)

where ε and η are the slow-roll parameters,

ε =1

2

(

V ′(φ)

V (φ)

)2

, (12)

η =V ′′(φ)

V (φ). (13)

Here and in what follows we have adopted the Planckunits, MP = 1, and the prime denotes the derivative

0.90 0.92 0.94 0.96 0.98 1.001 10

- 4

5 10- 4

0.001

0.005

0.01

0.05

0.1

x

x

r

ns

k=0 (na

tural in

flation

)

0.4 0.8

0.90.9

9

0.999

R inflation2

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00

0.00

0.05

0.10

0.15

0.20

0.25

r

ns

FIG. 2. The spectral index and the tensor-to-scalar ratioin the elliptic inflation Eq. (2) for k = 0, 0.4, 0.8, 0.99 and0.999 from top to bottom with the e-folding number N =60. The prediction of R2-inflation is shown by a red disc.We also overlay the Planck result (Planck+WP+BAO) [11]in the bottom panel for comparison. The elliptic inflationinterpolates from natural inflation to R2-inflation as k variesfrom 0 to 1.

with respect to the inflaton field φ. In the actual nu-merical calculation, we have used a refined version of theabove formulae including up to the second-order slow-rollparameters.In Fig. 2 we show the calculated values of ns and r for

several values of k by varying F . One can see that the(ns, r) predicted in the elliptic inflation indeed interpo-lates from natural inflation to exponential inflation. Thelatter includes the R2- or Higgs inflation, in which the po-

tential is given by V (φ) ∼ (1 − 2e−√

2

3φ). Here we have

fixed the e-folding number to be N = 60. We also showin Fig. 3 how the ns and r evolve as the decay constant Fchanges. For k sufficiently close to unity (k & 0.99), thespectral index ns stays around 1−2/N = 0.967 for a widerange of the decay constant including even sub-Planckianvalues.

B. Theta inflation

Next we consider another example of the elliptic in-flation, the theta inflation with the potential (3). It isknown that the Jacobi elliptic functions are closely re-

4

0.5 1.0 2.0 5.0 10.0 20.0

0.90

0.92

0.94

0.96

0.98

1.00

ns

F / Mp

k=0 (natural inflation)0.40.80.9

0.99

0.999

0.5 1.0 2.0 5.0 10.0 20.0

1 10- 4

5 10- 4

0.001

0.005

0.01

0.05

0.1

x

x

r

F / Mp

k=0 (natural inflation)0.4

0.80.90.99

0.999

FIG. 3. The dependence of ns and r on the decay constantF in the cnoidal inflation Eq. (2).

lated to the Jacobi theta functions, ϑ0,1,2,3(v, τ). Forinstance, there are following relations:

sn(u, k) =ϑ3(0, q)ϑ1(v, q)

ϑ2(0, q)ϑ0(v, q), (14)

cn(u, k) =ϑ0(0, q)ϑ2(v, q)

ϑ2(0, q)ϑ0(v, q), (15)

where

k =

(

ϑ2(0, q)

ϑ3(0, q)

)2

, u = πv(θ3(0, q))2, (16)

or equivalently,

v =u

2K(k), q = eiπτ , τ =

iK(√1− k2)

K(k), (17)

and K(k) is the elliptic integral of the first kind. TheJacobi theta functions are given by

ϑ0(v, q) =

∞∑

n=−∞

eπiτn2+2πin

(

v+ 12

)

= 1 + 2

∞∑

n=1

(−1)nqn2 cos 2nπv, (18)

ϑ1(v, q) =

∞∑

n=−∞

eπiτ(

n+ 12

)

2

+2πi(

n+ 12

)(

v+ 12

)

= 2

∞∑

n=0

(−1)nq(n+ 12 )2

sin (2n+ 1)πv, (19)

ϑ2(v, q) =

∞∑

n=−∞

eπiτ(

n+ 12

)

2

+2πi(

n+ 12

)

v

= 2

∞∑

n=0

q(n+1

2 )2

cos (2n+ 1)πv, (20)

ϑ3(v, q) =∞∑

n=−∞

eπiτn2+2πinv

= 1 + 2∞∑

n=1

qn2

cos 2nπv. (21)

Here, |q| < 1, i.e., Im(τ) > 0. The Fourier series con-verge very fast for |q| ≪ 1 as the higher order terms areexponentially suppressed.The Jacobi theta functions appear in gauge and

Yukawa couplings in string theories with toroidal com-pactifications. As we shall see in the next section, theJacobi theta functions appear naturally in the scalar po-tential.To be specific, let us consider the following inflaton

potential,

V (φ) =Λ4

2q

(

ϑ3(0, q)− ϑ3(

φ

2πF, q

))

(22)

which corresponds to Eq. (3) with a = 0 and b = 1.Note that a similar inflaton dynamics is obtained forthe potential with the other theta functions becauseof the identities between the theta functions: ϑ3(v +1/2, q) = ϑ0(v, q), ϑ3(v + τ/2, q) = e

−iπ(τ/4+v)ϑ2(v, q),ϑ2(v + 1/2, q) = −ϑ1(v, q) and so on. In the limit ofq → 0, the above potential is reduced to the natural in-flation since the higher order terms in the Fourier seriesare suppressed.

V (φ) → Λ4(

1− cos φF

)

for q → 0. (23)

On the other hand, for |q| → 1, many sinusoidal functionsadd up so as to make the inflaton potential extremelyflat around the potential maximum. This is essentiallysame as the axion hilltop inflation where the inflaton po-tential becomes flat due to cancellation among multiplesinusoidal functions [27–29]. In this sense, the theta in-flation can be thought of as a specific realization of themulti-natural inflation. (Also note that the Fourier se-ries converge fast unless |q| is extremely close to unity.)We show the inflaton potential for q = 0, 0.7, 0.8 and 0.9in Fig. 4. As one can see from the figure, the inflatonpotential around the local maximum becomes flatter asq approaches unity.We similarly solve the inflaton dynamics with inflaton

potential (22), and estimated ns and r. The results areshown in Figs. 5 and 6. The spectral index ns staysaround 0.967 for sub-Planckian values of F , if q & 0.7. Itis interesting to note that the predicted values of ns andr become closer to the R2-inflation as q becomes larger,but they do not exactly matches with the R2-inflation:the spectral index tends to be slightly smaller for thesame value of r.

5

-6 -4 -2 2 4 6

0.5

1.0

1.5

2.0

2.5

3.0

0

0.9

0.8

0.7

(natural inflation)

q=0

FIG. 4. The inflaton potential V (φ) with the Jacobi thetafunction given by Eq. (22) for q = 0, 0.7, 0.8 and 0.9 frombottom to top.

0.92 0.94 0.96 0.98 1.00

10-4

0.001

0.01

0.1

0.90

r

ns

q=0

0.4

0.6

0.7 0.8

R inflation2

FIG. 5. (ns, r) for the theta inflation with the potential (22).

C. Inflation with Dedekind eta function

Now we study an inflation model with the Dedekindeta function,

η(τ) = q1

12

∞∏

n=1

(1− q2n), (24)

where q = eiπτ and Im(τ) > 0. It is known that theeta function is given by the special values of the thetafunctions, for instance, η3(τ) = 12ϑ2(0, τ)ϑ3(0, τ)ϑ0(0, τ).To be specific, we will study the inflaton potential:

V (φ) = −Λ4[

η−2(

6

πFφ+ iC

)

+ c.c.

]

+ const.,(25)

∝ cos φF

+ e−2πC cos

(

11φ

F

)

+ · · · ,

where C is a real parameter, and the constant term isadded so as to make the inflaton potential vanish at theorigin, V (0) = 0. In the second line, we have expandedthe eta function in terms of |q| = e−πC ≪ 1. The inflaton

1 2 5 10 20 500.90

0.92

0.94

0.96

0.98

1.00

ns

F / Mp

q=0 (natural inflation)0.40.6

0.7

0.8

1 2 5 10 20 50

1 10- 4

5 10- 4

0.001

0.005

0.01

0.05

0.1

x

x

r

F / Mp

q=0

0.4

0.6

0.70.8

FIG. 6. Same as Fig. 3 but for the theta inflation (22).

potential (25) is shown in Fig. 7 for several values of C.One can see from the figure that the inflaton potentialhas a maximum at φ = πF and it receives modulationsfor C . 1. Here, (φ,C) = (0, 1) implies the self-dualpoint under the modular transformation of τ → −1/τ ,i.e., C → 1/C. For larger values of C, the above poten-tial asymptotes to the natural inflation with the decayconstant F , as one can see from the expansion of theinflaton potential. Note that the normalization of thepotential height is chosen for visualization purpose. Thisdoes not affect the predicted values of r, ns and its run-ning in the following analysis.

The predicted values of ns and r are shown in Fig. 8for C = 1.5, 1 and 0.85, where we have varied the valueof the decay constant F , and set the e-folding numberN to be 60. For C = 1 and C = 0.85, the predictedvalues of (ns, r) rotate counterclockwise as F decreases.In contrast to the previous two cases, this model does notasymptote to the exponential inflation. The dependenceof ns and r on the decay constant F is shown in Fig. 9,and we find that the super-Planckian decay constant isrequired to be consistent with the Planck data. We willreturn to the issue of super-Planckian decay constant atthe end of this section.

Finally let us evaluate the running of the spectral in-dex, dns/d ln k. Recently the inflation model with theDedekind eta function was shown to generate a sizablerunning of the spectral index [41]. The running can be

6

-1 1 2 3 4

1

2

3

4 C=1.3

1.1

0.9

0.65

FIG. 7. The inflaton potential with the Dedekind eta func-tion (25) for C = 1.3, 1.1, 0.9 and 0.65 from top to bottom.

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00

0

0.05

0.10

0.15

0.20

0.25

r

ns

C=1.5

C=1

C=0.85

FIG. 8. (ns, r) for the inflation potential with the Dedekindeta function (25).

expressed in terms of the slow-roll parameters as

dnsdlnk

≃ −24ε2 + 16εη − 2ξ, (26)

where

ξ ≡ V′(φ)V ′′′(φ)

V (φ)2. (27)

The current constraint on the running by Planck andWMAP polarization reads [51]

dnsd ln k

= −0.013± 0.009 (68%). (28)

For ε and η of O(0.01), the first two terms in Eq. (26)are of O(10−3), one order of magnitude smaller than thecenter value of the current bound. In order to generatea sizable running of order O(0.01), therefore, ξ must belarge. For a simple class of inflaton potentials expressedby Taylor series with the finite truncation, however, alarge negative running that is more or less constant overthe observed cosmological scales would quickly terminate

2 5 10 20 500.85

0.90

0.95

1.00

1.05

ns

F / Mp

C=1.5

C=1

C=0.85

2 5 10 20 50

F / Mp

1 10- 4

5 10- 4

0.001

0.005

0.01

0.05

0.1

x

x

r

C=1.5

C=1

C=0.85

FIG. 9. Same as Fig. 3 but for the inflaton potential (25).

-0.005

0.000

0.005

0.010

0.015

2 5 10 20 50

F / Mp

C=1.5

C=1

C=0.85

dn sdlnk

FIG. 10. The running of the spectral index dns/d ln k as afunction of the decay constant F in the inflation model (25).

inflation within the e-folding number N . 30 [52]. Thereis a loophole in this analysis, and it was pointed out inRef. [43] that a sizable and almost constant running overthe CMB scales ∆NCMB ∼ 8 can be generated in aninflaton potential with small modulations, because themodulations can give a dominant contribution to V ′′′,while V , V ′ and V ′′ are not significantly modified. Therunning of the spectral index as a function of the decayconstant is shown in Fig. 10. For slightly smaller valuesof C, |dns/d ln k| can be larger than 0.01, and it is moreor less constant over the CMB scales.

7

III. POSSIBLE UV COMPLETION

We have introduced the Jacobi elliptic functions andtheta functions in the scalar potential to extend the nat-ural inflation, motivated by the recent Planck data. Wehave shown that they predict the spectral index and thetensor-to-scalar ratio interpolating from natural inflationto exponential inflation including the R2- or Higgs infla-tion. Such functions indeed appear in the string the-ories with extra dimension containing tori. As notedalready, gauge and Yukawa interactions can be writtenin terms of the Jacobi theta functions in toroidal com-pactifications in intersecting or magnetized D-brane sys-tem [34, 36, 37]. Also on the magnetized orbifolds, e.g.,T 6/(Z2×Z2) and T 2×T 4/Z2, Yukawa couplings are sim-ilarly expressed with the theta functions [53]. This typeof function comes from the wave functions of the fields onthe torus, which respects the periodicity, in magnetizedbrane models (or summation of world-sheet instantonsin intersecting brane models).4 So, the scalar potentialwith the Jacobi theta function will appear naturally inthe low energy scales.

In passing, we note that the theta functions can alsoappear in scattering amplitudes at one-loop level of stringworld-sheet together with the Dedekind eta function (in-volving moduli integrals) [54]. This is similar in the fieldtheories [55]. Also the eta function can appear in one-loop threshold corrections to gauge couplings after inte-grating out heavy Kaluza-Klein modes in the extra di-mension [56]. Note that the eta function is given by thespecial value of the theta functions, as we have seen be-fore.

In the following, we discuss several possibilities to gen-erate the scalar potential with the theta functions. Theinflaton candidate is an open string modulus on the D-branes, and hence it originates from the Wilson lines,positions of D-branes or their linear combination. Forinstance, if a four cycle S in a Calabi-Yau, wrapped byD7-branes, has the Hodge numbers with h1,0− (S) 6= 0 orh2,0− (S) 6= 0, there appear h1,0− (S) Wilson line modulior h2,0− (S) position moduli, respectively [57]. To discusswhether such models are really viable for inflation, how-ever, it is necessary to construct an explicit model of themoduli stabilization, supersymmetry (SUSY) breaking,as well as the embedding of the standard model sectoretc. The open string inflation and the related issues havebeen studied extensively in the literature. The purpose ofthis section is to present several possible scenarios wherethe Jacobi theta functions appear in the scalar potential,motivated by the recent Planck results on ns and r.

4 The wave functions are invariant under the periodic shift on thetorus up to gauge transformation, which is represented by doublepseudo periodicity of the Jacobi theta functions.

A. Field theoretic cases

As a toy model, we first consider a case where theYukawa coupling is given by the theta function ϑ3(v, q),where q = eiπτ , and τ is a torus complex structure mod-ulus τ in magnetized brane model 5 (or the volume mod-ulus and B-field in the intersecting brane system), and vis identified with the inflaton φ/2πF . In a more realisticscenario, one has to worry about the stabilization of theimaginary component of v, which we shall discuss in thenext subsection.6

Now the question is how to generate a scalar potentialwhich depends on the Yukawa coupling. One possibilityis to use a strong dynamics:

L = ϑ3(v, q)ϕψcψ

→ V = ϑ3(v, q)〈ϕ〉〈ψcψ〉 ≡ Λ4ϑ3(

φ

2πF, q

)

, (29)

where ϕ and ψ + ψc are scalar and vector-like fermionfields in the strongly coupled hidden sector respectively.Here, we have assumed that the scalar develops a non-zero vacuum expectation value (VEV) and the fermionpair condensates through the strong dynamics.In a supersymmetric case, we may consider the IYIT

model [58, 59], where there are four superfields ofΨj (i, j = 1, 2, 3, 4) with fundamental representationunder SU(2) gauge theory and six singlet superfieldsΦij = −Φji. The low energy superpotential becomes

W = ϑ3(v, q)Φij [ΨiΨj ] (30)

with Pf([ΨiΨj]) = Λ4SU(2). Here, [Ψ

ciΨj ] is a composite

operator and ΛSU(2) is dynamically generated scale underthe SU(2) theory. Thus, scalar potential is given by theF-component of Φ:

V = Λ4SU(2)|ϑ3(v, q)|2 +O(Φ). (31)Here, we have assumed that Φ = 0 is the minimum ow-ing to the Kähler potential K = |Φ|2 − c|Φ|4/Λ2SU(2),

5 A Yukawa coupling consists of a product of theta functions ifa compact extra dimension is given by a product of tori. Wewill focus on the contribution from a single torus and neglectcontributions from the other tori, assuming that other modulibecome heavy owing to flux compactification. One may considerT 4/Z2 × T 2, in which Wilson line moduli could be required toexplain the smallness of the observable Higgs mass, although themoduli are not necessary to explain Yukawa textures [53]. On theother hand, ϑ3 is often used for realizing a large (top) Yukawacoupling because it contains unity and ϑ2 is used for a smallertop Yukawas; see Eq. (21).

6 Note also that theta inflation with |q| ∼ 1 would require a smallcomplex structure, i.e., Im(τ) ≪ 1. Then we may have a diffi-culty to control the coupling expansion, because then a complexstructure on a torus can be translated into the volume under T-duality. Such a problem may be avoided, if q in the theta functiondepends on a linear combination of moduli or a large volume ex-tra dimension is realized with stabilized moduli, compensatingsmall values of complex structure moduli.

8

in which c = O(10−2) is obtained after integrating outheavy modes with the masses of order ΛSU(2). Alterna-tively, one may consider the trilinear scalar term of theYukawa interaction, assuming the scalars develop non-zero VEVs by some dynamics. In this case the scalar po-tential is approximately given by V ∼ m3/2ϑ3(v, q) 〈Φ〉3.

B. Supergravity models with moduli stabilization

In a more realistic stringy set-up one has to considerboth real and imaginary components of v. In order to re-alize an effective single-field inflation, one needs to ensurethat the other degrees of freedom including the imaginarycomponent of v are stabilized with a mass heavier than(or comparable to) the Hubble parameter during infla-tion. To see this, let us consider a supergravity model intype IIB orientifold with flux compactification [60–67].The following is the simplified effective action

K = K(S + S̄;T + T̄ )− 3 log[−i(τ − τ̄ )]+∑

α

F 2α|vα − v̄α|2 + · · · , (32)

W =

∫

G ∧ Ω(S, τ, vα) +A(S, τ, vα,Ψ)e−aT

+Whidden Yukawa(vα, τ,Ψ) + · · · , (33)where S is a dilaton-axion, T is a Kähler modulus, τ isa complex structure modulus, and vα is a open stringmodulus. Fα is a decay constant for open string mod-ulus, e.g., F 2W = 3/(T + T̄ ) and F

2P = 1/(S + S̄) for

D7 Wilson line moduli and D7 position moduli, respec-tively [57], and they are expected to be comparable tothe string/Planck scale. We shall return to the issue ofthe size of the decay constant later. G is the three formflux, Ω is holomorphic three form and the second termin the superpotential arises from instantons or gauginocondensations. We have included trilinear superpoten-tial Whidden Yukawa between matter-like fields Ψi in thehidden sector. In the Kähler potntial, the ellipsis includethe terms for Ψi and SUSY breaking sector and the quan-tum corrections [68] in addtion to the Standard Model(SM) fields. The ellipsis in the superpotential containscontributions from SUSY-breaking and the SM sector.With the above effective action, τ and S can be stabi-

lized by a choice of three form fluxesG:∫

G∧Ω ⊃ τ2+Sτ .We may obtain a light open string moduli v, a certaincombination of vα, if

∂

∂v

∫

G ∧ Ω(S, τ, vα) = 0. (34)

The other degrees of freedom in {vα} are assumed to bestabilized with a heavy mass by

∫

G∧Ω ⊃ v2α, and there-fore can be integrated out in the low energy. For instance,the Wilson line moduli are not included in flux-inducedsuperpotential because of the internal gauge symmetry[67]. Alternatively, only a part of the position modulion D-branes may be fixed by a selection of the closed

string fluxes, magnetic (worldvolume) fluxes and geom-etry through the tadpole condition [69]. (Similar argu-ment can be applied also to the complex structure mod-uli [41, 70] and to type IIA moduli stabilization with-out metric fluxes [71].) Finally, through instantons orgaugino condensations on the stack of branes, the Kählermoduli T can be stabilized taking account of the SUSYbreaking. The SUSY breaking also stabilizes the remain-ing moduli and produces the scalar potential of the Ja-cobi theta function through a gaugino mass or a trilinearscalar term. Here, we expect that Im(v) obtains the grav-itino mass m3/2 ∝

∫

G ∧Ω via the SUSY-breaking effectbecause the Kähler potential depends only on the imag-inary part of v as in Eq. (33) at the tree level. Thus,Re(v) ≡ φ/2πF will be the lightest degrees of freedom, agood candidate for the inflaton.

In the following we consider various ways to generatethe inflaton potential, which depends on the Jacobi thetafunctions, ϑi(v, q). We will discuss also an issue of one-loop breakdown of the shift symmetry along Re(v).

Case 1: Gauge coupling depending on the theta function

Let us first focus on the dependence on v through agauge coupling in the strongly coupled hidden sector.This v-dependence appears in the non-perturbative termas we shall see below.

On the stack of N D7-branes, the holomorphic gaugecoupling at one-loop level contains the Jacobi theta func-tion in the presence of e.g. a stack of n D3-branes [34]:

fone−loop ⊃ −n

2πlog(ϑ1(v, q)), (35)

where q = eiπτ is given by a torus complex structuremodulus τ and we have assumed that the gauge couplingdepends on v, and the tree level coupling is given byftree = T . This is regarded as a backreaction from n D3-branes to the worldvolume of N D7-branes [35]. In thiscase v denotes the position of D3-branes on the torus.When the N D7-branes undergo gaugino condensation7,

7 Similar type of the superpotential has been observed throughD-instanton contributions in Ref. [72, 73] with SL(2,Z) dual-ity in type IIB supergravity compactified on Calabi-Yau orien-tifold with odd parity Kähler moduli G: For instance, W ∼θm(G, iS)η−10(iS) may be generated by D-instantons. Here,

θm(G, iS) =∑

n∈L+r e−Sn2/2eimGn is the theta function of

weight s/2 with index m. L is some positive definite rationallattice of dimension s, and r is some vector which admits anexpansion in a basis of L with rational coefficients. Then, wehave ∂G

∫G ∧ Ω = 0, hence Re(G) can be an inflaton candi-

date if exists, because there will be just Im(G) dependence inthe Kähler potential at a perturbative level. This looks similarto the superpotential discussed in inflation model with Dedekindeta function.

9

the second non-perturbative term in Eq. (33) becomes

Wnon−pert ∝ exp[

− 2πN

(

T − n2π

log(ϑ1(v, q)

)]

(36)

≡ Λ3SU(N)(

ϑ1(v, q)

)nN

. (37)

Here, ΛSU(N) is the dynamically generated scale underthe pure SU(N) super Yang-Mills theory. Now we havethe gaugino mass term m3/2λλ in the SU(N) sector, thescalar potential is given by

V ∼ m3/2Λ3SU(N)(

ϑ1(v, q)

)nN

+O(Λ6SU(N)), (38)

where we neglected the presence of matter Ψi and higherorder terms in ΛSU(N). The inflaton is identified with thereal component of v, the position of D3-branes: Re(v) ≡φ/2πF . Thus, the inflaton potential depending on ϑ1arises from the interactions between n D3 branes and ND7 branes through gaugino condensation in this case.If the inflation is driven by the above inflaton poten-

tial, the Hubble scale during the inflation will be givenby H2inf ∼ m3/2Λ3SU(N). In the KKLT case [60], we ex-pect Λ3SU(N) . m3/2 . ΛSU(N), and so, Hinf . m3/2.

Therefore we may be able to avoid the decompactifica-tion problem [74]. The potential with the double gaug-ino condensations can ameliorate this problem. (See alsodiscussions in Ref. [75].) The scalar potential inducedby a similar mechanism was used to drive slow-roll infla-tion in the context of the warped brane inflation [35, 76–79], where inflation is driven by moving D3-branes in thewarped geometry with anti D3-branes at the tip. Onthe other hand, here we consider the inflation with aD-brane modulus in the bulk extra dimension, and theinflaton potential arises from the interactions betweenD3-branes and D7-branes. We may use anti D3-branesat the tip of a warped throat to uplift the AdS vacuumto the dS one with a small and positive cosmological con-stant. The interaction between branes and anti branescan be neglected if it is a sufficiently strongly warpedthroat. Alternatively, we may consider a field-theoreticspontaneous SUSY breaking for uplifting [80–83]. In ei-ther case the uplifting sector is required to preserve anapproximate shift symmetry along inflaton direction [76]for successful slow roll inflation.Lastly, we note that there are v-dependent one-loop

corrections in the Kähler potential [68], which inducethe mass term for the inflaton V ∼ cm23/2φ2 with anexpectation of c = O(10−2) or so. For successful slow-roll inflation, such corrections must be sufficiently small,c . 10−2(Λ3SU(N)/m3/2).

Case 2: Yukawa coupling depending on the theta function

As noted already, Yukawa couplings are given by theJacobi theta function in the toroidal compactification of

extra dimension with magnetic fluxes in the string theory.The superpotential is given by

W = ϑ3(v, q)ΨiΨjΨk, (39)

where Ψi,j,k are hidden matter-like superfields localizedby the magnetic fluxes on the extra dimension, v is a openstring modulus and q is a torus complex structure mod-uli. All the moduli other than v are assumed to be sta-bilized with a heavy mass. Taking account of the SUSYbreaking effects, there will be a SUSY-breaking trilinearscalar term: VSUSY−breaking = m3/2ϑ3(v, q)ΨiΨjΨk+c.c..(Here, note that Ψi,j,k are scalar component of the re-spective superfields.) The scalars can develop non-zeroVEVs, for instance, via D-term potentials involving withFayet-Iliopoulos term ξ generated by the magnetic fluxesand stabilized T : D ∝ ξ + ∑i qi|Ψi|2, where qi is anU(1) charge of Ψi. See, e.g., Refs.[64, 84–88] for modulistabilization with D-term potential. Then the potentialbecomes:

V ∼ aϑ3(v, q) + b(

ϑ3(v, q)

)2

+O(Ψ5). (40)

Here, Re(v) ≡ φ/2πF is a candidate for the inflaton,a = O(m3/2Ψ3), b = O(Ψ4), and we have added SUSYF-term contribution VSUSY ∝ |∂ΨW |2, and neglected su-pergravity contributions of O(Ψ5), which contain deriva-tives of the theta function. Whether decompactificationoccurs depends the VEVs of {Ψi}, and a, b . m23/2 is re-quird. As discussed in the previous example, we needto suppress one-loop corrections to the inflaton massfor successful slow-roll inflation: c . 10−2a/m23/2 or

c . 10−2b/m23/2. Here c is a coefficient of the one-loop

induced inflaton mass term normalized by the gravitinomass squared.

Case 3: Yukawa coupling depending on the theta functionand strong dynamics

Now we consider a scenario similar to the case 2, butΨi are stabilized by a strong dynamics as in Refs. [64, 89]:

Wnon−pert = (Nc −Nf )(

Ae−2πT

det(ΨcΨ)

)1

Nc−Nf

,(41)

Whidden Yukawa = ϑ3(v, q)ΦΨcΨ, (42)

where A includes ϑ1-dependent term in the non-perturbative effect as discussed in the case 1, Ψc + Ψare vector-like superfields charged under SU(Nc) gaugegroup with the number of Nf(< Nc) flavors, and Φ is asinglet superfields. Note that there are no open stringmoduli (adjoint modes) in the strong coupling brane torealize an asymptotically freedom. Here, let us considerthree stacks of D-branes labeled by (α, β, γ) for simplic-ity;

v = Iαvα + Iβvβ + Iγvγ (43)

10

is a linear combination of open string moduli (vα, vβ , vγ)on the respective branes [36, 37], and Re(v) ≡ φ/2πF isthe inflaton. Here, Iα = Ind(D)βγ , Iβ = Ind(D)γα, andIγ = Ind(D)αβ , where Ind(D)αβ ∈ Z is the index of Diracoperator, which depends on magnetic flux between theα- and β-th D-brane and so on8. Hence Iα = −Iβ = Nfand Iγ counts the number of singlet Φ. Note that vγ ≡ 0when there exists the strongly coupled gauge theory onthe γ-th D-brane, because we have assumed there is noopen string moduli on γ-th D-brane. Note that q is givenby q = exp(iπτ |IαIβIγ |), where τ is a complex structuremoduli on the torus.Here, we shall consider stabilization of matter-like

fields. Φ can be fixed by D-term potential in the presenceof Fayet-Iliopoulos term as discussed in the case 2:9

D ∝ ξ − |Φ|2 ∼ 0. (44)This implies that Φ multiplet is eaten by the U(1) gaugemultiplet on which we are focusing. Here, we have usedthe canonical wave function for simplicity and ξ is com-parable to the cutoff scale. Thus 〈Φ〉 plays a role ofheavy mass against vector-like pairs, and they condense

as 〈ΨcΨ〉 = A 1Nc e− 2πNc T(

ϑ3(v, q))

NfNc

−1〈Φ〉NfNc

−1. We inte-grate out them and obtains low energy effective potential

Weff = NcA1

Nc e−2πNc

T(

ϑ3(v, q))

NfNc 〈Φ〉

NfNc (45)

≡ Λ3SU(Nc)A1

Nc

(

ϑ3(v, q))

NfNc . (46)

Here, ΛSU(Nc) is defined as the dynamically generatedscale in the pure SU(Nc) Yang-Mills theory in the lowenergy. Thus, low energy potential can be rewritten as

V ∼ m3/2Λ3SU(Nc)(

ϑ3(v, q))

NfNc through the gaugino mass

and the trilinear scalar term between Φ and Ψc + Ψ,if A does not depend on v. We expect, however, thattwo stacks of flavor branes can give an effect on theworldvolume of strong coupling brane, hence we will findA ∼ [ϑ1(vα, q̃)]nα [ϑ1(vβ , q̃)]nβ , where q̃ = eiπτ , nα and nβdenote the number of α-th and β-th D-branes for flavors:Two ϑ1 terms come from backreaction of α-th and β-thD-branes to the γ-th brane. Thus the scalar potential is

V ∼ m3/2Λ3SU(Nc)[

ϑ1

(

v

2Nf, q̃

)]2

Nc[

ϑ3(v, q)

]

NfNc

+O(Λ6SU(Nc)). (47)Here we have assumed that vα + vβ is stabilized byclosed string fluxes around the origin, v/Nf = vα − vβ ,ϑ1(−v, q) = −ϑ1(v, q), and we took nα = nβ = 1. Thispotential is similar to the case 1, but now it is accompa-nied with the ϑ3-dependent term. Note that the decayconstant is enhanced by 2Nf in the ϑ1 term compared tothat in the ϑ3 term. .

8 Dirac operator is given by Dαβ = ∂ + iAα − iAβ , where Aαdenotes the background of gauge potential on the α-th D-braneand so on.

9 D will develop the VEV as D ∼ m23/2

.

C. Potential with a gauge coupling depending on

the Dedekind eta function

Now let us consider inflation models involving theDedekind eta function, in which a complex structuremodulus τ is an inflaton candidate [41] and a run-ning spectral index is generated naturally. See alsoRefs.[90, 91] and [92] in Heterotic cases. We choose closedsting fluxes such that ∂τ

∫

G ∧ Ω = 0 in Eq. (33) withassumption that all open string moduli are stabilized. Itis known that the a holomorphic gauge coupling includesthe Dedekind eta function [34, 56]:

fone−loop ⊃ −b

2πlog(η(τ)). (48)

Here this is obtained after integrating out open stringmodes stretched among D-branes which are propagatingon a part of the bulk extra dimension and hence b denotes1-loop coefficient of beta function for the gauge couplingin N = 2 SUSY sector. We expect that, as in the case1, the gauge coupling can depend on ϑ1(〈v〉, q), whereq = eiπτ , when there is another stack of n D-branes.Including these contributions to the gauge coupling withftree = T , the gaugino condensation in the pure SU(N)theory is given as

Wnon−pert ∝ exp[

− 2πN

(

T − b2π

log(η(τ))

)]

enN

log(ϑ1(〈v〉,q))

≡ Λ3SU(N)[

η(τ)]

bN(

ϑ1(〈v〉, q))

nN . (49)

Here, ΛSU(N) is the dynamically generated scale in theSU(N). Though the gaugino mass term depending onthe gravitino mass, the scalar potential includes

V ∼ m3/2Λ3SU(N)[

η(τ)]

bN(

ϑ1(〈v〉, q))

nN + c.c.

+O(Λ6SU(N)). (50)In Refs. [90, 91] on Heterotic otbifolds, b/N = −2 can beobtained with a requirement of the modular invariancealong τ direction. If τ is a kind of overall modulus, wefind b/N = −6.

D. Super-Planckian decay constant

The decay constant for the inflaton is expected to beof order the string/Planck scale. It is, however, possi-ble to realize a decay constant larger than the Planckscale by the alignment mechanism in the presence ofmultiple light moduli [93]. For instance, when thereare two Yukawa couplings and two light brane moduliv1 and v2, i.e., ∂v1,2

∫

G ∧ Ω = 0 and Whidden Yukawa =ϑ3(I1v1+I2v2, q)Ψ

3+ϑ3(J1v1+J2v2, q)Ψ′3, an enhanced

decay constant can be obtained with |J1 − J2|/I < 1,I1 = I2 ≡ I and 〈Ψ′〉 < 〈Ψ〉. The inflaton becomesRe(v1− v2) and the decay constant is enhanced by a fac-tor of I/|J1−J2|. Indeed, we have seen the enhancementof the decay constant by 2Nf in the scalar potential inthe above case 3.

11

IV. DISCUSSION AND CONCLUSIONS

The Universe must be reheated by the inflaton decaysome time after inflation. The inflaton appears in thegauge and Yukawa couplings, and so, it may decay intothe SM gauge bosons (or gauginios if kinematically acces-sible) through the gauge couplings or into the SM matterand the Higgs or right-handed neutrinos and B−L Higgsthrough Yukawa couplings. For instance, the followingoperators will be relevant to the reheating process:

φ

Fv(Fµν)

2,φ

Fvytt̄tH,

φ

FvyNϕB−LNN. (51)

Here, we have taken 〈φ〉 6= 0 because the modulus devel-ops a non-zero VEV in general, and the decay constantFv is comparable to the string scale, which is not nec-essarily equal to the decay constant in the inflaton po-tential. Fµν , t, H , ϕB−L and N denote the SM gaugebosons, top quark, B − L Higgs and right-handed neu-trino respectively. The thermal leptogenesis is viable ifthe reheating temperature is enough high as 109 GeV[94]. The last operator will be useful to implement non-thermal leptogenesis, even if the reheating temperaturebecomes lower [95, 96].

In this paper we have studied an extension of the nat-ural inflation, where the inflaton potential has a peri-odic property. We have proposed elliptic inflation wherethe inflaton potential is based on the elliptic functionsand related functions. Focusing on the elliptic and thetainflation, we have shown that the predicted ns and rinterpolate from natural inflation to the exponential in-flation including R2-inflation, where ns asymptotes tons ∼ 1− 2/N ≃ 0.967 for N = 60. We have also studiedthe inflation model with the Dedekind eta function and

shown that a sizable running of the spectral index canbe generated owing to the small modulations on the in-flaton potential. For a certain value of the parameter, itis possible to realize dns/d lnk ∼ −0.01 which is more orless constant over the CMB scales.Periodicity often appears in the field theories and the

string theories through compactifications. Interestingly,such periodic backgrounds restrict the allowed form ofthe potential, and as a result, the inflaton potential takesa specific form rather than simple cosine function(s). Forinstance, toroidal background forces infinite series of co-sine functions to add up to give the Jacobi theta functions(cf. Eqs. (18) - (21)). Indeed, the Jacobi theta functionsare known to appear in gauge and Yukawa couplings intoroidal compactifications, where the variables are openstring and complex structure moduli. We have exploredvarious ways to generate the scalar potential which de-pends on the Jacobi theta functions. The inflaton is iden-tified with the real component of the lightest open stringmodulus, and we have discussed the stabilization of theother moduli fields based on a supergravity model in typeIIB orientifold with flux compactification.

ACKNOWLEDGMENTS

The authors would like to thank to Tatsuo Kobayashifor discussions on Yukawa interactions in string mod-els. This work was supported by JSPS Grant-in-Aidfor Young Scientists (B) (No.24740135 [FT] and No.25800169 [TH]), Scientific Research (A) (No.26247042[TH, FT]), Scientific Research (B) (No.26287039 [FT]),the Grant-in-Aid for Scientific Research on InnovativeAreas (No.23104008 [FT]), and Inoue Foundation forScience [FT]. This work was also supported by WorldPremier International Center Initiative (WPI Program),MEXT, Japan [FT].

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