IPMU 14-0061

Dynamical Chaotic Inflation in the Light of BICEP2
Keisuke Harigaya,1 Masahiro Ibe,1, 2 Kai Schmitz,1 and Tsutomu T. Yanagida1

arXiv:1403.4536v1 [hep-ph] 18 Mar 2014

1

Kavli IPMU (WPI), TODIAS, University of Tokyo, Kashiwa, 277-8583, Japan
2
ICRR, University of Tokyo, Kashiwa, 277-8582, Japan(Dated: March 19, 2014)

The measurement of a large tensor-to-scalar ratio by the BICEP2 experiment, r = 0.20+0.07−0.05 ,
severely restricts the landscape of viable inflationary models and shifts attention once more towards
models featuring large inflaton field values. In this context, chaotic inflation based on a fractional
power-law potential that is dynamically generated by the dynamics of a strongly coupled supersymmetric gauge theory appears to be particularly attractive. We revisit this class of inflation models
and find that, in the light of the BICEP2 measurement, models with a non-minimal gauge group
behind the dynamical model seem to be disfavored, while the model with the simplest group, i.e.
SU (2), is consistent with all results. We also discuss how the dynamical model can be distinguishedfrom the standard chaotic inflation model based on a quadratic inflaton potential.

INTRODUCTION

Cosmic inflation [1] is an enormously successful
paradigm of modern cosmology, which not only explains why the universe is almost homogeneous andspatially flat but which also accounts for the origin of
the anisotropies in the Cosmic Microwave Background(CMB) radiation as well as for the origin of the Large
Scale Structure of the Universe [2, 3]. Among the various models of inflation, chaotic inflation [4] is particularly
attractive since it is free from the initial condition problem at the Planck time. Moreover, the large field values typically encountered in models of chaotic inflation
predict a large contribution from gravitational waves to
the CMB power spectrum [5], which can be tested bymeasuring the so-called B-mode of the CMB polarization spectrum. Interestingly, the BICEP2 collaboration
recently announced the first measurement of just sucha B-mode signal, corresponding to a tensor-to-scalar ratio of r = 0.20+0.07−0.05 at 1 σ [6], which strongly suggests thepresence of primordial B-mode polarization in the CMB.1
This recent progress motivates us to revisit dynamical chaotic inflation, which was proposed in [8] and inwhich the inflaton potential is generated by the dynamics of a simple strongly coupled supersymmetric gauge
theory. One prominent feature of this class of models isthat it predicts a fractional power-law potential for the
inflaton with the fractional power being 1 or smaller.2
Chaotic inflation of this type can be distinguished fromthe simplest versions of chaotic inflation, i.e. models witha quadratic or quartic potential, by precise measurementsof the inflationary CMB observables. Furthermore, dynamical chaotic inflation is also attractive since it entails
that the energy scale of inflation is generated via dimensional transmutation due to the strong gauge dynamics.
This provides an explanation for why inflation takes placeat energies much below the Planck scale.3
The organization of the paper is as follows. First, we
review chaotic inflation emerging from a strongly coupled
supersymmetric gauge theory, which eventually leads us
to an inflaton potential proportional to some fractional
power of the inflaton field. Then, we discuss how thevalue of the tensor-to-scalar ratio observed by BICEP2
can be explained in this class of models.
DYNAMICAL GENERATION OF THE
INFLATON POTENTIAL

Let us briefly review the scenario of dynamical chaotic
inflation, in which strong gauge interactions such as those
proposed in [8] are responsible for the dynamical generation of the inflaton potential. For that purpose, let us
consider an SP (N ) supersymmetric gauge theory4 with2(N +2) chiral superfields in the fundamental representation, QI (I = 1 · · · 2(N + 2)). Besides these fundamental
representations, we also introduce (N +2)(2N +3) gaugesinglet chiral superfields ZIJ (= −ZJI ), which couple tothe fundamental representations in the superpotential via
W =1 As pointed out in [6], the observed ratio, r = 0.20+0.07 , is in ten−0.05

sion with the upper limit on this ratio, r < 0.11 (at 95%C.L.)[7],
which is deduced from a combination of Planck, SPT and ACTdata with polarization data from WMAP. In the following discussion, we shall keep this tension in mind when applying the
BICEP2 result to our dynamical model of chaotic inflation.
2 A fractional power-law potential can also be realized by intro-

ducing a running kinetic term for the inflaton [9]. For fractional
power-law potentials derived in string theory, cf. [10].

1λIJ ZIJ QI QJ ,2

(1)with coupling constants λIJ , which we shall assume to
be close to each other in the following, i.e. λIJ ' λ, for

3 For other examples of models in which the scale of inflation is

generated dynamically, cf. Refs. [11].
4 In our convention SP (1) is equivalent to SU (2).

2simplicity. Note that the couplings to the gauge singlets
ZIJ in Eq. (1) lifts all of the quantum moduli, QI QJ .
To see how the inflaton potential is generated, imagine that one of the singlet fields, S = Z(2N +3)(2N +4) for
instance, has a very large field value, so that the effective mass of QI=2N +3 and QJ=2N +4 becomes much larger
than the dynamical scale of the SP (N ) gauge interactions Λ. Then, QI=2N +3 and QJ=2N +4 decouple perturbatively and the model reduces to an SP (N ) gauge
theory with 2(N + 1) fundamentals and (N + 1)(2N + 1)
singlets. This low energy effective theory is nothing but
the dynamical supersymmetry breaking model proposedin [12]. Therefore, for a given non-vanishing S, the model
breaks supersymmetry dynamically and results in a “vacuum energy” that depends on the field value of S,V2' λ (N + 1)Λ4eff (S) ' λ2 (N + 1)Λ4



λ |S|
Λ

 N2+1. (2)where we have substituted the effective dynamical scaleΛeff = Λ ×

λ |S|
Λ

 2(N1+1),(3)

for field values λS  Λ.
As a result, we find that the scalar component of thesinglet S obtains a fractional power-law potential,pV ∝ |S| ,(4)

in which the power is solely determined by the size of the
SP (N ) gauge group,5
p=

2
.
N +1

(5)

For example, for SU (2) = SP (1), we obtain a linear
potential, while a much flatter potential is generated forN  1. In everything what follows, we will assume thatthe field S plays the role of the inflaton, although any of
the other singlets could be equally used as well.
During chaotic inflation, the field value of the inflaton
exceeds the Planck scale MPl . Before we can be surethat the above model allows for a successful implementation of chaotic inflation, we thus have to carefully examine the supergravity contributions to the scalar potential.
For example, naively coupling the above model to supergravity by simply assuming a minimal Kähler potential,
K = S † S, leads to a very steep scalar potential22

V ' e|S| /MPl × λ2 (N + 1)Λ4



λ |S|
Λ

 N2+1,

above the Planck scale. To avoid such a steep potential,
we assume a shift symmetry in the direction of S [13, 14],
S → S + ic ,

c∈R,

(7)

(cf. also [15] for recent developments) which renders theKähler potential a function of S + S † only,K=2
1
S + S† + · · · ,2(8)such that it no longer depends on =(S), the imaginary
component of S. Consequently, the imaginary component of S merely has a fractional power-law potential√
even for =(S)  MPl . In the following, φ = 2=(S)is identified as the inflaton in the scenario of chaotic
inflation based on the dynamically generated fractional
power-law potential in Eq. (2).
It should be noted that the shift symmetry introducedin Eq. (7) is explicitly broken by the Yukawa interactions
in Eq. (1), which induce the Kähler potential 2 µ
2N λ2 2
|S|
log
,
(9)
δK ∼
2
2
16π
MPl
where µ is a renormalization scale.6 This breaking term
leads again to a steep exponential potential for =(S) once=(S)  MPl . To avoid such dangerous effects, we therefore assume that λ is rather suppressed, λ  O(10−1 ).7

TESTING DYNAMICAL CHAOTIC INFLATIONAs we have demonstrated, simple strongly coupledgauge dynamics are able to generate an inflationary potential featuring a fractional power. In this section, we
now outline how chaotic inflation proceeds in these models. We also summarize the predictions for the inflationary observables encoded in the CMB power spectrum.
Inflation starts out at an√ arbitrary initial value ofthe inflaton field S = iφ/ 2 above the Planck scale,φ  MPl . There, the SP (N ) gauge interactions are perturbative and inflation is characterized by the slow-rollmotion of the inflaton in the effective potential in Eq. (2)towards smaller field values. We assume that, during inflation, the strong gauge dynamics are negligible, which
requires λpMPl  Λ. Inflation finally ends once theslow-roll conditions are no longer satisfied, which happens when the inflaton field reaches φ ' pMPl . Well after inflation, the inflaton oscillates around its origin with

(6)
6 Here, we have assumed that the shift-symmetric Kähler potential

in Eq. (8) is defined around the Planck scale.
5 In this letter, we only discuss the scalar potential for large field

7 Small λ is also required in order to keep the effective Q mass

values, λS  Λ. The vacuum structure for λS  Λ has beenaddressed in [8].

below the Planck scale even during inflation, S ∼ 10 · · · 100 MPl ,i.e. λS  MPl .

3

,

(10)

TR,dim6 ∼ 108−9 GeV ×mφ 5/2,
1015 GeV

m

ΛΦ

=

16.5

=

eV

10 1

5

m

G

Φ

Φ

15.5

16

G

Φ

16.0

10

m

=

eV

10

m

14

G
=

eV

10

(11)

13

G
eV

15.0
- 3.0

where we have assumed that the coefficients of the operators responsible for the decay of the inflation are ofO(1). In Eqs. (10) and (11), we have worked with an
inflaton mass of mφ = O(1015 ) GeV, which turns out tobe a typical value (cf. below).9 As a result, we find thathigh reheating temperatures can be realized rather easily,
which is quite favorable for successful leptogenesis [19].

Φ

en
d

17.0

ΛΦ Ne > M PL

when the inflation decays into Hu and Hd Higgs fields
through a dimension five operator, K ∼ (S + S † )Hu Hd .
When, instead, the inflaton decays through dimension six
operators, the reheating temperature is roughly

17.5

>

mφ
1015 GeV

3/2

N = 1

L

TR,dim5 ∼ 1012−13 GeV ×



18.0

Log 10 @ L  GeV D

a mass of mφ ' λΛ. At small field values, the stronglyinteracting theory is in the s-confinement phase [16, 17],
which is well-behaved and free of singularities at the origin in field space.8
After inflation, the inflaton finally decays into radiation through appropriate operators. For example, thereheating temperature can be estimated as

- 2.5

- 2.0

- 1.5

- 1.0

Log 10 @ Λ D

- 0.5

18.0

N = 5
17.5

1
6π 2 p3



Λ
MPl

4−p

λ2 pNe1+p/2,(12)

where Ne is the number of e-foldings before the end of
inflation when the CMB scales leave the Hubble horizon.
In Fig. 1, the red lines mark the region in the (λ, Λ) parameter space which is consistent with the observed curvature power spectrum, ln 1010 Pζ = 3.080 ± 0.025 [21]for N = 1 and N = 5, respectively. The blue dot-dashedlines represent contour lines indicating the values of theinflaton mass. In the blue-shaded regions, the dynamical scale is rather large, so that is also important during
inflation, i.e. Λ > λφend , where φend ' pMP L . In thissituation, we loose control over the inflaton potential, as

d

ΛΦ

en

=

>

m

16.5

=

eV

10 1

5

m

G

Φ

=

m

15.5
15.0
- 3.0

Φ

16

G

Φ

16.0

10

eV

10

ΛΦ Ne > M PL

Pζ =

Φ

L

Now, let us discuss the predictions of our fractional
power-law inflaton potential for the CMB observables (cf.
also [20]). Given the potential in Eq. (2), one finds for the
power spectrum Pζ of the curvature perturbations ζ [3]Log 10 @ L  GeV D

m

17.0

14

G
=

eV

10

13

G
eV

- 2.5

- 2.0

- 1.5

Log 10 @ Λ D

- 1.0

- 0.5

FIG. 1: (λ, Λ) plane for N = 1 (upper panel) and N = 5(lower panel). The red lines indicate where the curvature
power spectrum is consistent with the observed value within
2 σ. Solid and dashed lines correspond to Ne = 50 and 60,
respectively. We also show contour lines for the inflaton mass
as blue dot-dashed lines. The shaded regions are theoreticallyinaccessible because either the dynamical scale is too large,
i.e. Λ > λφend , or the coupling constant λ is too large, i.e.
λφN e > MPl , as denoted in the figure.

8 At intermediate field values, λφ ' Λ, where the gauge dynamics

transit from the perturbative to the strongly coupled picture, welack the ability to precisely calculate the inflaton potential. In
our discussion, we assume that the effective inflaton potential
exhibits no peculiar features around Λ/λ, but that it is insteadsmoothly connected from one regime to the other.
9 Even if the mass of the inflaton is as large as 1015 GeV, such thatits decay products have extremely large momenta, the inflaton
decay products thermalize soon after their production [18].

it becomes distorted by incalculable strong coupling effects. On the other hand, in the gray-shaded regions, thecoupling λ is too large, such that the effective mass of theheavy Q’s exceeds MPl during inflation, i.e. λφNe > MPl
with φNe ' (2pNe )1/2 MPl . In conclusion, Fig. 1 illustrates that the observed curvature power spectrum can

4
Planck HarXiv:1312.3313L

0.4

r0.002

0.3

Φ2

Φ

Φ 23

Φ 12

0.96

0.97

0.98

0.99

0.2

BICEP2 1Σ
BICEP2 2Σ
0.1 BICEP2 3Σ

0.0
0.93

0.94

0.95

1.00

ns
FIG. 2: Predicted values for ns and r for p = 1 (N = 1), p =2/3 (N = 2), and p = 1/2 (N = 3). Here, r0.002 denotes thetensor-to-scalar ratio at the pivot scale k = 0.002 Mpc−1 . The
predictions according to chaotic inflation with a quadratic potential are also shown for comparison. Small and big dots
stand for Ne = 50 and 60, respectively. The green contours
are the constraints extracted from [21]. The orange bands correspond to the 1, 2 and 3 σ ranges for r measured by BICEP2.

Moreover, as far as our theoretical prediction is concerned, we also remark that, if the shift symmetry is broken not only in the superpotential but also in the Kähler
potential, the prediction for r can be still be raised tolarger values [15, 22, 23]. Therefore, it is certainly premature to declare dynamical chaotic inflation ruled out
by the data at this point, in particular, the model withthe simplest gauge group SU (2). Quite the contrary, as
further measurements of the CMB B-mode polarization
are being performed, dynamical chaotic inflation basedon strong SU (2) dynamics might eventually develop into
one of the most promising models correctly describing
the data.
Another important key feature of dynamical chaotic
inflation is that it predicts a slightly larger value for the
spectral index compared with the simplest chaotic inflation model. Therefore, by further observational investigation of ns and r, dynamical chaotic inflation can be
distinguished from the simplest model of chaotic inflation
based on a quadratic potential.10

CONCLUSIONS

be reproduced for λ ' 10−(2−1) and Λ ' 1016 GeV, where
the inflaton mass is typically of O(1015 ) GeV.

The spectral index ns , the running of the spectral index dns /d ln k, and the tensor-to-scalar ratio r of thecurvature perturbations are predicted to be,
ns = 1 −

p+2
,
2Ne

2+p
dns
=−
,
d ln k2Ne2r=

4p. (13)Ne

In Fig. 2, we show the predicted values for ns and r
for p = 1 (N = 1), p = 2/3 (N = 2), and p = 1/2(N = 3). At the same time, dynamical chaotic inflation
predicts the running of the spectral index to be negligibly small. In Fig. 2, we also reproduce the constraints on
ns and r presented in [21], in which the Planck data has
been re-analyzed taking particular care of possible systematics in the 217 GHz map. Furthermore, we include
constraints on the tensor-to-scalar ratio as observed by
BICEP2, r = 0.20+0.07−0.05 . In summary, the figure showsthat models with N > 1 are excluded at the 3 σ levelby the BICEP2 results. By contrast, the simplest case,i.e. N = 1, is consistent with the BICEP2 result within
3 σ. In fact, for N = 1, dynamical chaotic inflation predicts r ' 0.08, which deviates from the BICEP2 best-fit
value, r = 0.20, by 2.9 σ. It should be noted, however,that there is a tension between the constraints deduced
from the Planck data and the value measured by BICEP2, where the Planck data favors a smaller value ofthe tensor-to-scalar ratio. Furthermore, the best fit rangeof the tensor-to-scalar ratio shifts down to r = 0.16+0.06−0.05
at 1 σ level when a different foreground model is used [6].

In this paper, we revisited the class of models of chaotic
inflation the potential of which is generated by the dynamics of a strongly coupled supersymmetric gauge theory. A prominent feature of this scenario of dynamical
chaotic inflation is that the inflaton potential featuresa fractional power. Contrasting dynamical chaotic inflation with the tensor-to-scalar ratio recently observedby the BICEP2 experiment, we find that models with
non-minimal gauge group seem to be disfavored, while
the model based on the simplest gauge group, SU (2),is barely consistent with observations. However, since
there is a tension between the Planck constraints andthe BICEP2 measurement, we need to wait for further
confirmation/refutation by other observations such as
Planck, ACTpole, SPT, and POLARBEAR. Only withadditional data at hand, it will be become clear whether
dynamical chaotic inflation is excluded or in fact a good
description of the CMB data. Likewise, improved constraints on ns will also help to distinguish the dynamical
chaotic inflation model from chaotic inflation based on aquadratic potential.

10 Fig. 2 slightly suggests that a smaller number of e-foldings Ne(lower TR ) is preferred in the case of the dynamical model, so
as to raise the tensor-to-scalar ratio towards the BICEP2 bestfit value, while a larger number of e-foldings Ne (higher TR ) ispreferred in the case of the quadratic potential model, so as tomake the spectral index larger.

5
Acknowledgements This work is supported by
Grant-in-Aid for Scientific Research from the Ministryof Education, Science, Sports, and Culture (MEXT),
Japan, No. 22244021 (T.T.Y.), No. 24740151 (M.I.), andby the World Premier International Research Center Initiative, MEXT, Japan. The work of K.H. is supported inpart by a JSPS Research Fellowship for Young Scientists.

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