IPMU 12-0215

Chaotic Inflation with a Fractional Power-Law Potential
in Strongly Coupled Gauge Theories
Keisuke Harigaya,1 Masahiro Ibe,1, 2 Kai Schmitz,1 and Tsutomu T. Yanagida1
1

Kavli IPMU, TODIAS, University of Tokyo, Kashiwa, 277-8583, Japan
2
ICRR, University of Tokyo, Kashiwa, 277-8582, Japan(Dated: November 27, 2012)

Models of chaotic inflation with a fractional power-law potential are not only viable but also
testable in the foreseeable future. We show that such models can be realized in simple strongly
coupled supersymmetric gauge theories. In these models, the energy scale during inflation is dynamically generated by the dimensional transmutation due to the strong gauge dynamics. Therefore,such models not only explain the origin of the fractional power in the inflationary potential but also
provide a reason why the energy scale of inflation is much smaller than the Planck scale.

INTRODUCTION

Cosmic inflation [1] is a very successful paradigm
of modern cosmology which explains the origin of
the anisotropies of the Cosmic Microwave Background(CMB) as well as of the Large Scale Structure of the Universe [2, 3]. At present, a realistic and complete theory of
inflation is, however, still pending and hence, inflationarymodel building remains one of the most important tasksof particle physics and cosmology.
Among the various classes of inflation models proposedso far, the chaotic inflation scheme [4] is one of the most
attractive classes since it can realize an inflationary expansion even in the presence of large quantum fluctuations at the Planck time. Moreover, the large field values
typically encountered in models of chaotic inflation imply
a large contribution from gravitational waves to the CMB
power spectrum [5], rendering these models testable in
the foreseeable future. However, according to the precise
observations of the CMB anisotropies, the simplest versions of chaotic inflation, i.e. the models with a quadratic
potential or a quartic potential, are now somewhat disfavored [6]. With the forthcoming data provided by the
Planck satellite experiment [7], the constraints on those
simplest versions will be improved upon.
In light of this situation, a more general version of
chaotic inflation has been gathering attention, in which
the inflaton potential comes with a fractional power. As
analyzed in Refs. [8], the models of chaotic inflation witha fractional power-law potential are more favored than
the simplest versions of chaotic inflation. Despite suchsuccesses, these models, however, lack a firm field theoretical foundation, which apparently seems difficult to beachieved from regular field theories.1

In this paper, we show that such fractional power-law
chaotic inflation models can be realized in simple strongly
coupled supersymmetric gauge theories. There, the energy scale during inflation is generated by the dimensional transmutation due to the strong gauge dynamics.
Thus, these models not only explain the origin of the fractional power in the inflationary potential but also provide
a reason why the energy scale of inflation is much smaller
than the Planck scale.2
The organization of the paper is as follows. First, wederive the fractional power-law potential for the inflaton in strongly coupled gauge theories. Next, we discuss
distortions of the inflaton potential due to supergravity
contributions. Then, we outline the phenomenology ofchaotic inflation with the dynamically generated potential and summarize its observational consequences. The
final section is devoted to conclusions and discussion.

DYNAMICAL GENERATION OF THE
INFLATON POTENTIAL

Let us discuss how the fractional power-law potential of
the inflaton is generated dynamically. First, we consider
an SP (N ) supersymmetric gauge theory3 with 2(N + 2)
chiral superfields in the fundamental representation, QI(I = 1 · · · 2(N + 2)). Besides the fundamental representations, we also introduce (N + 2)(2N + 3) gauge-singlet
chiral superfields ZIJ (= −ZJI ) which couple to the fundamental representations in the superpotential via
W =1
λIJ ZIJ QI QJ ,2

(1)with coupling constants λIJ . It should be noted that all
the quantum moduli, QI QJ , are lifted by the couplings
to the gauge singlets Z IJ .

1 A recipe for embedding chaotic inflation with a fractional power-

law potential into supergravity is provided in Refs. [9, 10]; seealso Ref. [11], in which a fractional power-law potential is obtained from a running kinetic term for the inflaton. For fractional
power-law potentials derived in string theories, see Ref. [12].

2 For examples of models in which the scale of inflation is generated

dynamically, see Refs. [13].
3 We use the convention where SP (1) is SU (2).

2
For a later purpose, we decompose the above fields
into,
Zij = ZIJ=ij ,Qi = QI=i ,Ti = ZIJ=i(2N +3) ,P = QI=2N +3 ,T̄i = ZIJ=i(2N +4) ,P̄ = QI=2N +4 ,S = ZIJ=(2N +3)(2N +4) ,(2)where i, j = 1 · · · 2(N + 1). In terms of these fields, theabove superpotential is now rewritten as,
W =1
λZij Qi Qj + λT Ti Qi P + λT̄ T̄i Qi P̄ − λS SP P̄ . (3)2We have assumed λij = λ for simplicity and we also assume that λT,T̄ are larger than λ in the following. The
sign convention for λS is just for later convenience. As
we will show, the scalar potential for S generated by thestrong dynamics features a fractional power and eventually plays the role of the inflaton potential.
To see how the scalar potential is generated, let us remember that the SP (N ) gauge theory with 2(N + 2)
fundamental representations exhibits the so-called sconfinement [14, 15] at low energies below the dynamical
scale Λ. In this phase, the model is well described by thecomposite fields, M ∝ QQ, MP ∝ QP , MP̄ ∝ QP̄ andMP P̄ ∝ P P̄ [14], which may be assembled in the sameway in an antisymmetric matrix M as the gauge singlets
Z, T , T̄ , and S are assembled in the antisymmetric matrix Z. Their effective superpotential is given by4
Weff =P f (N +2) (M)Λ(N +2)−3(4)1+ λΛZij M ij + λT ΛTi MPi + λT̄ ΛT̄i MP̄i − λS ΛSMP P̄ .
2
The first term is the non-perturbative potential generated by the s-confinement.5 The other four terms in
Eq. (4) can be regarded as mass-mixing operators between the composite fields and the gauge singlets, inducing supersymmetric masses of O(λΛ, λT Λ, λT̄ Λ, λS Λ), respectively. The effective superpotential shows that themodel possesses a supersymmetric vacuum in which allof the M ’s and singlets vanish. That is, as expected,
there is a vacuum with vanishing vacuum energy.
Now, let us consider an effective potential for S 6= 0
and X(∝ J ij Zij ) 6= 0 around the vacuum. Notice thatthe mesons MP and MP̄ are still fixed at
MP = 0,MP̄ = 0 ,since we have assumed λT,T̄  λ. Thus, the effective
potential in Eq. (4) is reduced to(5)

Weff = MP P̄

P f (N +1) (M )− λS ΛS
Λ(N +2)−31+ λΛZij M ij .
2

The first term leads to the so-called deformed moduli
constraint on the moduli space of the SP (N ) gauge
theory with 2(N + 1) fundamental representations [14].
Therefore, for a given non-vanishing S, the above model
is nothing but the dynamical supersymmetry breaking
model of Ref. [16]. By solving the quantum deformed
constraint for S 6= 0, we obtain
Mij = ΛλS S
Λ N1+1× Jij ,

(6)

which leads to
Weff = λ (N + 1)

1/2

Λ

2



λS S
Λ

 N1+1X ,(7)

where X is defined by X = Zij J ij /(2(N + 1)1/2 ). Hence,for S 6= 0, supersymmetry is broken by the F -componentof X, which leads to a scalar potential for S,
2

V ' λ (N + 1)Λ

4

λS |S|
Λ N2+1.

(8)

As promised, we find that the scalar component of thesinglet S obtains a fractional power-law potential,pV ∝ |S| .

(9)

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

2
.
N +1

(10)

In the above analysis, we have tacitly assumed that thefield value of S is around or below the dynamical scale,i.e. λS S . Λ. The above scalar potential is, however,also obtained for λS S  Λ. For that purpose, let us
remember that P and P̄ are heavier than the dynamical
scale for λS S  Λ and decouple perturbatively. Thus,
the effective theory below λS S consists of the SP (N )gauge theory with 2(N + 1) fundamental representations
whose effective dynamical scale is given by [17]4 It should be noted that we have neglected O(1) differences be-

tween the λ’s in Eq. (3) and the ones in Eq. (4) due to nonperturbative effects. We also assume that the composite fields in
M are close to the canonically normalized ones.
5 In this paper, we define the Pfaffian of a 2n × 2n antisymmetricmatrix, P f (n) , so that the symplectic form J, where J = 1ln ⊗iσ2with 1ln being the n × n unit matrix and σ2 the second Paulimatrix, satisfies P f (n) (J) = 1.

Λeff = Λ ×

λS |S|Λ 2(N1+1).

(11)

Then, since the effective theory below λS S is again
the dynamical supersymmetry breaking model, we againreach the effective superpotential in Eq. (7). Therefore,

3
again, supersymmetry is broken by the F -component of
X at S 6= 0, which again leads to6
 2

λS |S| N +1
2
4
.
(12)
V ' λ (N + 1)Λ
Λ

One caveat to the above argument is that the shift
symmetry is explicitly broken by the interaction

In the following sections, we assume that S plays the roleof the inflaton. After inflation, S reaches its origin, whichleads to the restoration of supersymmetry.

in Eq. (3). With this explicit breaking, the Kähler potential in Eq. (15) obtains a radiative correction which
breaks the symmetry, 2 µ
2N λ2S 2|S| log
,
(17)
δK ∼
2
16π 2
MPl

SUPERGRAVITY CONTRIBUTIONS

In our model, we apply the above obtained fractional
potential to the chaotic inflation scenario, where the fieldvalue of the inflaton exceeds the Planck scale MPl . For
such a large field value, we need to carefully examine thesupergravity contributions to the scalar potential, which
could change the potential drastically from the fractional
power-law potential. In fact, if we assume, for example,
a minimal Kähler potential S † S for the inflaton S, theinflaton potential is modified in supergravity as follows, 22
λS |S| N +1
|S|2 /MPl24, (13)V 'e× λ (N + 1)Λ
Λ
which is too steep for chaotic inflation for S  MPl .7
The most attractive way to avoid such supergravity
contributions is to introduce a shift symmetry in the direction of S [18, 19],
S → S + ic ,

c∈R

(14)

(see also Ref. [9] for recent developments). With this shift
symmetry, the Kähler potential is a function of S + S † ,1
2
S + S† + · · · ,(15)2
and hence, it does not depend on the imaginary component of S, =(S). Therefore, the imaginary component of S has a fractional power-law potential even for=(S)  MPl , while the real part of S obtains a√Hubble
mass term around <(S) = 0. In the following, 2=(S)plays the role of the inflaton in the chaotic inflation scenario with the dynamically generated fractional powerlaw potential in Eq. (12).8K=

6 The agreement between the powers in the potential in the twodifferent field regimes is not a coincidence but can be understood by remembering that the effective superpotential consistent with (anomalous) R-symmetries, holomorphicity and dimensional analysis should have the form in Eq. (7), regardless of thesizes of X 6= 0 and S 6= 0.
7 Even for S < M , there is an eta problem. This problem is alsoPlavoided by the solution mentioned below.
8 One may assume R-symmetry with the following charge assignments: Q(0), P (1), P̄ (1), Z(2), T (1), T̄ (1), and S(0). With thesecharge assignments, R-symmetry allows the shift symmetry only
for the singlet S, which explains why only the imaginary part of
S can exceed the Planck scale.

W ⊃ −λS SP P̄

(16)

where µ is a renormalization scale.9 This breaking termcauses a steep exponential potential of =(S) for =(S) 
MPl unless λS is small enough. Thus, to avoid a toolarge breaking of the shift symmetry, we assume in thefollowing analysis that λS is rather suppressed.10
Furthermore, we note that in general the mere introduction of a shift symmetry in the direction of the inflaton field S does not suffice to protect chaotic inflation from receiving disastrously large supergravity corrections. In addition we have to require that the superpotential be of the form W = Xf (S) [18, 9], where f isan arbitrary holomorphic function of S and X is a gauge
singlet that can be identified as the goldstino superfield
responsible for the spontaneous breaking of supersymmetry during inflation [10]. Evidently, the effective superpotential in Eq. (7), which we equally obtained in the smallS as well as in the large-S regime, is just of the required
form, with X ∝ J ij Zij playing the role of the goldstino
field. That is why, after supplementing our model with ashift symmetry in the direction of the inflaton field S, allnecessary conditions for the successful implementation of
chaotic inflation into supergravity are satisfied.

CHAOTIC INFLATION WITH A FRACTIONAL
POWER-LAW POTENTIAL

As we have shown, simple strongly coupled gauge dynamics are able to generate an inflationary potential featuring a fractional power. We have also discussed theshift symmetry of the model which suppresses the distortions of the inflaton potential due to the supergravity
contributions. Let us now outline the phenomenology ofthe model, summarize its predictions for the inflationary

9 Here, we have assumed that the Kähler potential in Eq. (15) withthe shift symmetry is defined around the Planck scale.
S to be small is thefact that the masses of the P and P̄ quarks, mP = mP̄ ' λS S,
must be smaller than the Planck scale at all times, even though
S might take huge values, S ∼ 10 · · · 100 MPl , during inflation.
That is, too large λS would entail mP , mP̄ & MPl at some point
during inflation, causing our approach based on ordinary quantum field theory to break down.

10 A second reason why we have to assume λ4observables encoded in the CMB power spectrum and, in
relation to that, discuss its testability.
Inflation starts out at an arbitrary initial value of theinflaton field above the Planck mass S  MPl . At its
early stages, i.e. as long as λS S  Λ, the SP (N ) gaugeinteractions are in the perturbative regime and inflationis characterized by the slow-roll motion of the inflaton
in the effective potential in Eq. (12). Similarly, we knowthat at small field values, i.e. when λS S  Λ, the system
is in the s-confinement phase, in which S and the composite mesons have masses of O (λIJ Λ). In this case theinflaton potential is given by Eq. (8). In the intermediate
regime, where λS S ' Λ, we however lack the ability to
precisely calculate the inflaton potential, which is why we
do not exactly know how the transition from the large-S
to the small-S regime takes place. For instance, it might
be that towards the end of inflation S becomes trapped ina metastable vacuum at a field value around Λ/λS suchit actually never reaches the small-S regime. Assumingthat the effective inflaton potential exhibits no such peculiar features around Λ/λS , we are led to the conclusionthat inflation continues without any hindrance until theslow-roll conditions become violated at small values of
the inflaton field.
The end of inflation marks the onset of preheating,which proceeds in a rather unconventional way in our
scenario due to the negative curvature of the inflaton potential. In fact, so far only small-field and hybrid models
of inflation have been studied in connection with a negatively curved scalar potential, where it was found that
preheating occurs via tachyonic oscillations [20] of the inflaton field or tachyonic preheating [21], respectively. As
for large-field, i.e. chaotic inflation only the case of a positively curved inflaton potential, for which preheating occurs via parametric resonance [22], has been considered
up to now. We presume that in our large-field modelfeaturing a negatively curved inflaton potential preheating ends up being a combination of both, tachyonic inflaton oscillations as well as parametric resonance. The
verification of this conjecture certainly requires a morecomprehensive and ultimately numerical study.
After inflation the inflaton decays through its coupling to the Higgs fields Hu and Hd of the supersymmetric standard model in the Kähler potential, K ⊃(S + S † )Hu Hd , at a rate
ΓS ∼

MS3
2 ,
MPl

(18)

with MS ' λS Λ denoting the inflaton mass.11 Hence,

the inflaton eventually reaches the supersymmetric vacuum, in which S = M = 0. The rate of the perturbative
inflaton decays√ directly determines the reheating temperature, TR ∼ ΓS MPl , or, using Eq. (18),3/2 

3/2
Λ
λS
7
.
(19)
TR ∼ 10 GeV
10−41015 GeV
It is interesting to note that non-perturbative effects,
i.e. the formation and evaporation of Q-balls near the endof inflation [23], could speed up the decay of the inflaton field, thus leading to a reheating temperature muchhigher than in the mere perturbative picture. In principle, the formation of Q-balls is feasible in our model
since our effective inflaton potential is shallower than aquadratic one. Nonetheless, we suppose that no Q-balls
emerge towards the end of inflation because, owing to its
Hubble induced mass term, the real part of S, <(S), isstabilized at zero. This presumably renders it impossible
to induce inspiraling orbits in S field space continuously
connected to the inflationary trajectory, which would be
a necessary prerequisite for Q-balls to occur [24]. A further study of this issue is beyond the scope of this paperand shall be carried out elsewhere.
Finally, let us summarize the implications of our fractional power-law inflaton potential for the CMB observables and discuss the testability of our model. Given appotential V (S) ∝ Λ4 (|S| /Λ) , with p = 2/(N + 1), one
finds for the power spectrum Pζ of the curvature perturbations ζ [3]

4−p
Λ
1
1+p/2
(2pNe ),(20)Pζ =12π 2 p2 MPl
where Ne is the number of e-foldings. The observational
result Pζ = 2.42 × 10−9 [6] then requires the dynamical
scale Λ to be shortly below the GUT scale,Λ ' 1015 GeV .

(21)

The spectral index ns and the tensor-to-scalar ratio r of
the curvature perturbations are respectively given byp+2
4p
ns = 1 −
, r=
.
(22)
2Ne
Ne
For Ne = 50 and N ≥ 1, such that 0 < p ≤ 1, we obtainns = 0.97 · · · 0.98 and r = 0.16/(N + 1), which is consistent with the recent CMB observations [6]. The Planck
experiment is expected to detect the presence of tensor
modes if r > 0.05 [7], that is, if the inflaton potential isgenerated by the strong dynamics of an SP (1) ∼= SU (2)gauge theory. Future experiment such as CMBPol [25]and LiteBIRD [26], which are expected to reach sensitivities to r of O(10−3 ), will detect tensor modes unless theunderlying gauge group is very large, N > O(100).

11 The same coupling leads to the non-adiabatic production of ra-

diation during preheating. As it is strongly suppressed, we assume that during preheating most of the initial vacuum energy
is transferred into non-relativistic inflaton particles and only asmall fraction into radiation. This implies in particular that thestandard definition of the reheating temperature is applicable.

5
CONCLUSIONS AND DISCUSSION

In this paper, we have shown that fractional power-law
chaotic inflation models can be realized in simple supersymmetric gauge theories. In this class of models, theenergy scale during inflation is dynamically generated by
the dimensional transmutation due to the strong gauge
dynamics. Therefore, the models not only explain theorigin of the fractional power in the inflationary potential but also provide a reason why the energy scale ofinflation is much smaller than the Planck scale. We also
discussed how well the model fits together with the current data on the inflationary observables.
Several comments are in order. In our analysis, we haveconfined ourselves to models with 2(N + 2) fundamental
representations. One of the reasons for this assumption isthat for models with more fundamental representations
the system is in the so-called conformal window [27], andhence, exhibits conformal symmetry after inflation. In
such cases, the inflaton becomes an unparticle [28] after
inflation, which may change the evolution of the universe
drastically compared to conventional inflaton scenarios.
Although such a possibility is intriguing, we do not pursue it further since it is beyond the scope of this paper.
In our scenario, we have considered models in which
supersymmetry breaking in the inflaton sector vanishes
after inflation. It is, however, possible that some portionof supersymmetry breaking in the inflaton sector remains
non-vanishing even after inflation, providing the dominant source of supersymmetry breaking in the true vacuum. In this case, we can consolidate two well motivated
new physics, supersymmetry breaking and inflation into
one model. We will discuss this possibility elsewhere.

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