PHYSICAL REVIEW D 72, 023505 (2005)

Implications of the curvaton on inflationary cosmology

Takeo Moroi1 and Tomo Takahashi21Department of Physics, Tohoku University, Sendai 980-8587, Japan

2Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan(Received 19 May 2005; published 8 July 2005)

1There is atuation such a[5,6]. In this

1550-7998=20

We study implications of the curvaton, a late-decaying light scalar field, on inflationary cosmology,paying particular attentions to modifications of observable quantities such as the scalar spectral index ofthe primordial power spectrum and the tensor-to-scalar ratio. We consider this issue from a generalviewpoint and discuss how the observable quantities are affected by the existence of the curvaton. It isshown that the modification owing to the curvaton depends on class of inflation models. We also study theeffects of the curvaton on inflation models generated by the inflationary flow equation.

DOI: 10.1103/PhysRevD.72.023505 PACS numbers: 98.80.Cq

I. INTRODUCTION

Recent precise observations of the cosmic microwavebackground (CMB) radiation [1] and the large scale struc-ture [2] have provided deep insights into the origin of thecosmic density fluctuations. In particular, it has becomeclear that the primordial density fluctuation is almost scaleinvariant. As a mechanism to generate the scale invariantdensity fluctuation, inflation [3] is the most prominentcandidate.

Even with inflation, however, it is not automatic to makethe present density fluctuations to be consistent with theobservational constraints. In the standard scenario, fluctua-tion of the inflaton field, whose potential energy is respon-sible for the energy density during the inflation, isgenerated during the inflation and it becomes the originof the present density fluctuations. In this case, propertiesof the present density fluctuations are determined once theinflaton potential is fixed, and we obtain constraints on theindividual inflation models.

If there exists some scalar field other than the inflaton,however, the mechanism of generating the cosmic densityfluctuations may become more complicated. In particular,with a scalar field whose mass is much smaller thanthe expansion rate during the inflation, fluctuation isimprinted in the amplitude of such scalar field, whichprovides another potential source of the present densityfluctuations. In this paper, we consider one of suchexamples, the curvaton [4].1 In the curvaton scenario,there exists a late-decaying scalar condensation (the cur-vaton) which acquires amplitude fluctuation during theinflation. When the curvaton decays, fluctuation of thecurvaton amplitude becomes the fluctuation of the radia-tion (and of other components in the universe). Thus, ifthe curvaton exists, properties of the present density fluc-tuations change compared to the standard scenario onlywith the inflaton. From the particle physics point of view,

nother mechanism producing the primordial fluc-s inhomogeneous reheating or modulated reheatingpaper, we do not consider this possibility.

05=72(2)=023505(10)$23.00 023505

there are various well-motivated candidates of the curvaton[7].

Since the recent results from Wilkinson microwavebackground probe (WMAP) [1] provided severe con-straints on the properties of the primordial density fluctua-tions, it is interesting to reconsider the observationalconstraints in the framework of the curvaton scenario.With the curvaton, it is expected that the constraints onthe inflation models are changed (and possibly relaxed)compared with the case without the curvaton.

Indeed, in the past, it was pointed out that the constraintson the scale of the inflation is drastically relaxed with thecurvaton [8]. Then, there have been subsequent worksfocusing on the low scale inflation in the curvaton scenario[9–13]. Furthermore, in Ref. [14], the authors has shownthat the quartic chaotic inflation model, which is margin-ally excluded by WMAP observations, becomes viablewhen the curvaton mechanism applies. In Ref. [15], severalother inflation models were investigated such as chaoticinflation with several monomials, the natural inflationmodel and the new inflation model. The discussionsmade in Refs. [14,15] are mainly based on the scalarspectral index and the tensor-to-scalar ratio. Importantly,although some inflation models can become viable with thecurvaton even if they are excluded by current observations,there exist other class of models which cannot be liberatedeven with the curvaton. In particular, in Ref. [15], it wasdiscussed how the liberation of inflation models dependson model parameters such as the initial amplitude, massand decay rate of the curvaton field.

In this paper, we discuss the implication of the curvatonon inflation models from some general points of view. Inthe previous works, the analysis have been rather depen-dent on inflaton potentials. Of course, it is important tostudy individual inflation models motivated from, in par-ticular, particle physics point of view. It is, however, alsopossible to adopt some other approach, parameterizing theinflation models by using the slow-roll parameters (at someepoch during the inflation). In this approach, one generatesinflation models with some stochastic method such as

-1 2005 The American Physical Society

TAKEO MOROI AND TOMO TAKAHASHI PHYSICAL REVIEW D 72, 023505 (2005)

using the inflationary flow equations [16,17]. Such amethod has been used in many studies including the oneby the WMAP team and constraints on inflation modelshave been discussed [18–23]. Although other approachessuch as stochastic approaches or implementations of theflow equation are proposed [24], we discuss the implica-tions of the curvaton on inflation models generated by theusual flow equation as an example.

The organization of this paper is as follows. In the nextsection, we start with briefly reviewing the formalism tostudy how observable quantities are affected in the systemcontaining the inflaton and the curvaton. We discuss howthe standard scenario is modified in terms of the back-ground evolution and the perturbation. We also review theclassification of single-field inflation models. After thepreparation, in Sec. III, we go into how observationalquantities are affected by introducing the curvaton anddiscuss it from some general viewpoints. In Sec. IV, weconsider the effects of the curvaton on inflation modelsgenerated by the inflationary flow equation. Then we con-clude this note by summarizing the results in the finalsection.

II. FORMALISM

A. Background evolution

First we present the background evolution in the curva-ton scenario comparing its counterpart of the standard case.

In the standard case, the potential energy of the inflatondrives the inflation. During inflation, the inflaton slowlyrolls down the potential. However, at some point, theinflaton starts to roll fast down the potential hill, then theinflaton begins to oscillate around the minimum of thepotential. When the potential of the inflaton can be ap-proximated by the quadratic form as V��� � �1=2�m2

��2,the energy density of the inflaton behaves as matter com-ponent. Then the universe is dominated by the oscillatinginflaton. We call this epoch ‘‘� dominated’’ or ‘‘�D.’’After some time, when the expansion rate of the universe,namely, the Hubble parameter, becomes as the same as thedecay rate of the inflaton, the inflaton decays into radiation.Then the universe becomes radiation dominated. This is,what we call, the standard thermal history of the universe.

When the curvaton exists, above picture is modified.First we consider the case where the initial amplitude ofthe curvaton is so small that the potential energy of thecurvaton does not drive the (second) inflation. Duringinflation, the curvaton field stays at somewhere on thepotential. If we adopt the potential of the curvaton to beof the form V��� � �1=2�m2

��2, the curvaton begins to

oscillate around the minimum of the potential when theHubble parameter becomes of the same order of the massof the curvaton. During this epoch, the energy density ofthe curvaton behaves as ordinary matter. This event usuallyhappens when the universe is dominated by radiation

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which comes from the decay of the inflaton. Since theenergy density of matter decreases slower than that ofradiation, at some point, the energy density of the curvatonbecomes to dominate the universe. We call this epoch as �dominated or �D. After some time, the curvaton alsodecays into radiation when the expansion rate of the uni-verse becomes of the same order as the decay rate of thecurvaton. Then the universe becomes radiation dominatedagain. We call this epoch as RD2 not to confuse with theradiation-dominated epoch from the inflaton decay. (Wecall the first radiation-dominated epoch as RD1.)

Next we discuss the case where the energy density of thecurvaton can drive the second inflation after the first oneinduced by the inflaton. When the initial amplitude of thecurvaton is large enough, the second inflation can happenafter the �D or RD1 epoch before � starts to oscillate. Inthis case, the universe experiences the inflation era drivenby the inflaton, �D era, RD1 era and the second inflationdriven by the curvaton potential energy, followed by the�D era and RD2 era. The second inflation can modify theobservable quantities drastically in some cases. We discussthis point later.

B. Density perturbations: Standard case

Now we briefly review the issue of density perturbationin the standard case.

To discuss observational consequences, we have to setup the initial condition during radiation-dominated eraafter the decay of the inflaton. For this purpose, we repre-sent the primordial power spectrum with the (Bardeen’s)gravitational potential � which appears in the perturbedmetric in the conformal Newtonian (or longitudinal) gaugeas

ds2 � �a2�1� 2��d2 � a2�1� 2��dx2; (2.1)

with being the conformal time. The quantum fluctuationof the inflaton �� during inflation generates the curvatureperturbation as R � ��H= _����. Since �� ’ H=2� and,� and R are related as � � ��2=3�R during radiation-dominated era, the power spectrum of the curvature per-turbation from the inflaton fluctuation is [25]

P�inf�� � �1� 2�1� 3C��V � 2C�V

1

12�2M6pl

V3inf

V02inf

��������k�aH;

(2.2)

where the ‘‘prime’’ represents the derivative with respect to� and C � �2� ln2� �, with � ’ 0:577 being theEuler’s constant. Here, we used the ‘‘potential’’ slow-rollparameters. There are two ways to define the slow-rollparameters: one is the one using inflaton potential andthe other is the one using the Hubble parameter. Thepotential slow-roll parameters are defined as

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IMPLICATIONS OF THE CURVATON ON. . . PHYSICAL REVIEW D 72, 023505 (2005)

�V �1

2M2

pl

�V0inf

Vinf

�2; �V � M2

pl

V 00inf

Vinf;

�2V � M2

pl

V0infV

000inf

V2inf

:

(2.3)

where the third parameter �2V is considered to be the second

order in the slow-roll.The ‘‘Hubble’’ slow-roll parameters are defined as

�H � 2M2pl

�H0

H

�2� �V;

�H � 2M2pl

H00

H� ��V � �V;

�2H � 2M4

pl

H0H000

H2 � �2V � 3�V�V � 3�2V

(2.4)

where we also write the relation between the Hubble andpotential slow-roll parameters to the first order in slow-roll.

The scalar spectral index is defined as

ns � 1 �d lnP�

d lnk: (2.5)

Using the slow-roll parameters, the spectral index can bewritten as

n�inf�s � 1 � �6�V � 2�V � 2�7� 12C��2V

� 2�3� 8C��V�V � 2C�2V; (2.6)

where ‘‘(inf) ’’ represents that the expression of ns is forthe case where primordial fluctuation comes from thefluctuation of the inflaton.

In the second order, we have the running of the scalarspectral index which is

d lnn�inf�s

d lnk� �24�2V � 16�V�V � 2�2

V: (2.7)

During inflation, the gravity wave can also be generated.The primordial gravity wave (tensor) power spectrum isgiven by

P�inf�T �

2Vinf

3�2M4pl

: (2.8)

With this expression, the tensor-to-scalar ratio r is definedand given by

r�inf� �P�inf�T

P�inf�R

�4

9

P�inf�T

P�inf��

� 16�V�1� 4C�V � 2C�V�:

(2.9)

C. A classification of inflation models

Here we discuss one of possible classifications of single-field inflation models. In many literatures, the followingclassification has been used; single-field inflation modelscan be classified into the ‘‘small-field,’’ ‘‘ large-field’’ and

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‘‘hybrid-type’’ models [26]. In this classification, the mod-els are classified on the ns � r plane and distinguished bythe value of the slow-roll parameters or the first and secondderivatives of the inflaton potential. To the first order in theslow-roll parameters, the observational plane �ns; r� isuniquely divided into the three classes. In this section,we consider the slow-roll approximation in the first order.

The ‘‘small-field’’ models are identified with the condi-tion V00���< 0 and �V 0=V�2 >V 00=V, which also means2�V > 0>�V . This category includes, for example, thenew and natural inflations. The generic potential of thistype is

V��� � �v4

�1�

��v

�p�: (2.10)

The scalar spectral index and the tensor-to-scalar ratio ofthis type of model can be written with the slow-roll pa-rameter defined in Eq. (2.3) and the number of e-foldingsduring inflation which is defined as Ne � ln�aend=a��where aend and a� are the scale factor at the end of inflationand the time of horizon crossing. Using the slow-rollapproximation, Ne is given by

Ne �1

M2pl

Z ��

�end

Vinf

V 0inf

d�: (2.11)

Using the e-folding number Ne, the slow-roll parametersfor this type of model are given by, for p > 2,

�V �p2

2

�Mpl

v

�2�

1

p�p� 1�Ne

�vMpl

�2�2�p�1�=�p�2�

;

(2.12)

�V � �p� 1

p� 2

1

Ne(2.13)

For v <Mpl, the � parameter becomes very small, thus thespectral index is given by

n�inf�s � 1 � �2p� 1

p� 2

1

Ne(2.14)

The ‘‘large-field models’’ are identified with the condi-tion V00���> 0 and �V 0=V�2 >V 00=V, which also means2�V > �V > 0. This category includes, for example, thechaotic inflation. The generic potential of this type is

V��� � �M4pl

��Mpl

�": (2.15)

The ns and r are obtained as similarly as the case with thesmall-field models as

n�inf�s � 1 � �"� 2

2Ne; (2.16)

r�inf� �4"Ne

: (2.17)

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TAKEO MOROI AND TOMO TAKAHASHI PHYSICAL REVIEW D 72, 023505 (2005)

Thus inflation models in this category predict red-tiltedscalar primordial spectrum and relatively large tensor-to-scalar ratio.

The final one, ‘‘hybrid-type’’ models are identified withthe condition V 00���> 0 and �V 0=V�2 < V 00=V, which im-plies 2�V < �V . This category includes, the hybrid infla-tion model as the name tells. Only models of this categorycan give blue-tilted spectrum.

D. Effects of the curvaton fluctuation

Here we briefly review the effects of the fluctuation ofthe curvaton on the observable quantities such as ns and r.(For details, see [14,15,27].) For the case where the fluc-tuations of the inflaton and the curvaton both affect thecosmic density perturbation today, the gravitational poten-tial during RD2 era is given by

�RD2 � ��1� �1� 3C��V � C�V2

3M2pl

Vinf

V 0inf

��init

� f�X���init

Mpl; (2.18)

where ��init is the primordial fluctuation of the curvaton,and

X ��init

Mpl: (2.19)

The function f�X� represents the size of the contributionfrom the fluctuation of the curvaton. f�X� can be calculatedusing the linear perturbation theory [14]. Assuming that thecurvaton dominates the universe once in the course of thehistory of the universe, f�X� can be given, for small andlarge X, by

f�X� ’� 49X :�init Mpl13X:�init � Mpl

(2.20)

Using Eq. (2.20), the scalar spectral index can be writtenas

P� � �1� ~f2�X��V � 2�1� 3C��V � 2C�V

�Vinf

54�2M4pl�V

; (2.21)

where ~f � �3=���2

p�f. Thus the scalar spectral index is given

by using Eq. (2.6)2

2Here we assume that the mass of the curvaton is much smallerthan that of the inflaton. Thus we can neglect the term in theexpression of ns which comes from the curvature of the curvatonpotential as �2��� where the positive and negative signs are forthe case with X < 1 and X > 1, respectively and

��� � M2pl

m2�

Vtotal:

Here Vtotal � Vinf � V�.

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ns � 1 � �2�V �2�V � 4�V1� ~f2�V

� 2�7� 12C��2V

� 2�3� 8C��V�V � 2C�2V (2.22)

The running of the scalar spectral index is given by

d lnnsd lnk

� 4�V��V � 2�V� �16�2V � 12�V�V � 2�2

V

1� ~f2�V

�4~f2�V��V � 2�V�2

�1� ~f2�V�2

: (2.23)

The tensor power spectrum is not modified even with thecurvaton. However, since the scalar perturbation spectrumis modified, the tensor-to-scalar ratio becomes

r �16�

1� ~f2��1� 4C�V � 2C�V�: (2.24)

E. Effects of modification of the background evolution

Here we discuss effects of the modification of the back-ground evolution due to the curvaton. Generally, the ob-servable quantities such as the scalar spectral index and thetensor-to-scalar ratio depends on the field value of theinflaton at the time of horizon crossing which can usuallybe expressed with the number of e-foldings during infla-tion. The fluctuation which corresponds to some referencescale kref at present epoch can be approximately given, inthe standard case, as [28]

krefa0H0

�akHk

a0H0� e�Ne�k�

aendareh

areha0

Hk

H0(2.25)

where ‘‘end’’ and ‘‘reh’’ denote the time when the inflationends and the reheating epoch, i.e., the beginning of RDepoch. Thus the e-folding number during inflation is givenby

Ne�k��standard� ’ � ln

ka0H0

�n� 2

6nln%end

%reh�

1

3ln

s0sreh

� lnHinf

H0(2.26)

where we assumed that the energy density of the oscillatinginflaton behaves as %� / a�6n=�n�2� which corresponds tothe case with inflaton potential Vinf / �n. sreh and s0 are theentropy density at present time and time of reheating,respectively. In the last term, we assumed that theHubble parameter is almost constant during inflation,thus we replaced Hk with Hinf . Notice that the decay rateof the inflaton �� and parameters in the potential areneeded to determine Ne exactly. Depending on �� andthe parameters in the potential, the number of e-foldingcan be changed as

Ne ’4� "6"

ln�� � � � � (2.27)

-4

X(=

φ ini

t/Mpl

)

0 -1

-2

-3

-4

-5

-6

1010 1012 1014 1016 1018 1020

mφ / Γφ

10-3

10-2

10-1

FIG. 1. Contours of constant $Ne in the m�=�� vs X plane.The region where the curvaton does not dominate the universe isrepresented with small circles.

IMPLICATIONS OF THE CURVATON ON. . . PHYSICAL REVIEW D 72, 023505 (2005)

Now we discuss the e-folding number in the curvatonscenario. When the curvaton is introduced, Eq. (2.25) ismodified as

krefa0H0

� e�Ne�k��curvaton�aendareh1

areh1ainf 2

ainf 2a�D

a�D

areh2

areh2a0

Hk

H0

(2.28)

where ‘‘reh1,’’ ‘‘inf2,’’ ‘‘�D2,’’ and ‘‘reh2’’ denote thetime when the first radiation-dominated epoch begins, thesecond inflation begins, the oscillating curvaton dominatedbegins and the second radiation-dominated epoch begins,respectively. When the initial amplitude of the curvaton issmall (�init Mpl), there is no second inflation. In thatcase, �areh1=ainf 2��ainf 2=a�D� should be replaced with�areh1=a�D�.

Assuming that the curvaton field does not dominate theenergy density of the universe when the curvaton begins tooscillate,3 namely the case with no second inflation drivenby the curvaton, we get [29].

Ne�k��curvaton� ’ Ne�k��standard� �1

12ln%�D

%reh2

’ Ne�k��standard� �1

6lnm�

���

2

3ln�init

Mpl(2.29)

In Fig. 1, we plot contours of constant $Ne � N�curvaton�e �

N�standard�e in the m�=�� vs X plane. Requiring that the

curvaton dominates the energy density of the universe afterthe first radiation-dominated epoch, i.e., %� > %rad at thetime when the curvaton decays into radiation, we have therelation among the mass, the decay rate and the initialamplitude of the curvaton as

m�

��>

1

9X4: (2.30)

In Fig. 1, the region where the above inequality is notsatisfied is represented with small circles.

As seen from Eqs. (2.14) and (2.16), the scalar spectralindex in the small-field and large-field models are inverselyproportional to Ne. Thus slight change of Ne affects n�inf�s asj$n�inf�s j � $Ne=N2

e . This implies that, requiringj$n�inf�s j< 0:001, $Ne < 4� 5 for Ne � 50� 60: Noticethat the uncertainty of the e-folding number Ne also comesfrom the inflaton sector through the decay rate of theinflaton and the form of the potential, in particular, aroundthe minimum (i.e., the potential which the inflaton feelswhen the inflaton oscillates.)

When the second inflation occurs, the number ofe-foldings can be written as

3This assumption almost corresponds to the condition �init �Mpl.

023505

Ne�k��curvaton� ’ Ne�k�

�standard� � N2 �1

12ln%�D

%reh2

’ Ne�k��standard� � N2 �

1

6lnm�

��

�1

6ln�init

Mpl(2.31)

where N2 represents the e-folding number during the sec-ond inflation which can be 20� 30 [15]. Thus, in this case,Ne drastically reduced due to the second inflation driven bythe curvaton, which significantly affects the observationalquantities.

III. EFFECTS ON THE OBSERVABLE QUANTITIES

Now we discuss how the curvaton affects the quantitiessuch as ns and r. In this section, we assume that the initialamplitude of the curvaton is small compared with thePlanck mass and the change of the number of e-foldingsis also small. Furthermore, we denote the scalar spectralindex for the standard case (i.e., the case with the inflatonfluctuation only) as n�inf�s and that for the case with theinflaton and the curvaton as ns. In this section, we considerthe slow-roll in the first order.

First, we discuss to what extent the scalar spectral indexis modified when the curvaton comes into play. For thispurpose, we show contours of constant ns � n�inf�s in then�inf�s � 1 vs r�inf� plane in Fig. 2. In the figure, we fixcontribution from the fluctuation of the curvaton as ~f �5 which corresponds to the case with �init � 0:1Mpl.Interestingly, in the ‘‘hybrid-type’’ models, the scalar spec-tral index always decreases, on the other hand, in the ‘‘-small-filed’’ and ‘‘large-field’’ models, ns increases. This iseasily understood from Eq. (2.22). When 2�V � 4�V ispositive, which is the boundary of the ‘‘hybrid-type’’ and

-5

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0 2 4 6 8 10

n s −

1

f~

r(inf)= 0.4r(inf)= 0.2r(inf)=0.01

FIG. 3. ns � 1 as a function of ~f for n�inf�s � 1 � �0:06. Here,we take r�inf� � 0:4 (solid line), 0:2 (dashed line) and 0:01(dotted line).

-0.03

-0.02

-0.01

0

0.01

0.02

0 2 4 6 8 10

n s −

1

f~

r(inf)= 0.4r(inf)= 0.2r(inf)=0.01

FIG. 4. Same as Fig. 3, except for n�inf�s � 1 � 0:02.

r(inf

)

0.04

0.02

0.0

-0.02

-0.06

-0.1

-0.14

-0.18

-0.2 -0.1 0 0.1 0.2

ns(inf) − 1

0

0.2

0.4

0.6

0.8

1

FIG. 2. Contours of constant ns � n�inf�s in the n�inf :�s � 1 vsr�inf� plane. For reference, boundaries of ‘‘small-field,’’ ‘‘large-field’’ and ‘‘hybrid-type’’ models are also shown. Notice that theboundary between the ‘‘large-field’’ model and ‘‘hybrid-type’’model coincides with the contour of ns � n�inf�s � 0. We fixedthe size of the contribution from the curvaton fluctuation as ~f �5.

TAKEO MOROI AND TOMO TAKAHASHI PHYSICAL REVIEW D 72, 023505 (2005)

‘‘large-field’’ models, the contribution from the curvatonalways suppresses the second term in Eq. (2.22). Thus nsbecomes smaller. When 2�V � 4�V is negative, which isthe case for ‘‘small-field’’ and ‘‘large-field’’ models, theopposite happens, namely ns becomes larger. In addition,as seen from Eq. (2.22), the contribution from the curvatonfluctuation comes in the combination of �V ~f

2. Thus modelswith small �V cannot be affected by the curvaton fluctua-tion much. The ‘‘small-field’’ model mostly covers a re-gion where the �V parameter is small. Thus, as for thescalar spectral index, we can observe the following featurefor each class of inflation models: In the hybrid-typeinflation models, the spectral index always decreases. Inthe large-field inflation models, the scalar spectral indexalways increases. In the small-field inflation models, thescalar spectral index increases very slightly, and this classof models are not significantly affected by the existence ofthe curvaton much.

Next we discuss ~f dependence of the scalar spectralindex. In Figs. 3 and 4, the changes of ns � 1 are shownas a function of ~f. In Fig. 3, the case with the ‘‘small-field’’model (r�inf� � 0:01) and the ‘‘large-field’’ case (r�inf� �0:2 and 0:4) are shown. As seen from the figure, for ‘‘small-field’’ model, (i.e., the case with small r�inf�), the change ofns is small compared to the case with the ‘‘large-field’’models. In Fig. 4, the case with the ‘‘hybrid-type’’ areshown. We can see from these figure that cases with smallr�inf� (or �) shows slow response to the increase in f asmentioned above.

Next we consider how the tensor-to-scalar ratio is af-fected by the existence of the curvaton. For this purpose, in

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Fig. 5, we plot r=r�inf� as a function of r�inf� for severalvalues of ~f. As easily seen from Eq. (2.24), the effects ofthe curvaton always suppress the tensor-to-scalar ratio.Furthermore, its effect is larger when the �V parameter(i.e., r�inf�) is large since the contribution from the curvatoncomes in the form of �V ~f

2 as discussed above.Here we comment on the consistency relation of the

inflationary quantities. In the standard single inflatoncase, we have the consistency relation which is

r�inf� � �8nT: (3.1)

In the curvaton scenario, this equation is modified as [14]

r ��8nT

1� ~f2nT=2: (3.2)

Thus the consistency relation is also modified. This can beused to differentiate between the standard and curvaton

-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

r/r(i

nf)

r(inf)

f~= 5f~=10f~=50

FIG. 5. The ratio r=r�inf� as a function of r�inf�. Here, we take~f � 5, 10, and 50.

IMPLICATIONS OF THE CURVATON ON. . . PHYSICAL REVIEW D 72, 023505 (2005)

scenarios. It is also important to notice that Eq. (3.2) can beused to obtain ~f, in other words, the initial amplitude of thecurvaton since ~f is determined once nT and r are observa-tionally determined.

IV. ANALYSIS WITH THE FLOW EQUATION

In this section, we discuss the effects of the curvaton oninflation models using the flow equation approach [16],which is widely accepted to study some aspects of genericinflation models [17–24]. Before we consider the effects ofthe curvaton, first we briefly review this approach, follow-ing Ref. [17].

A. Flow equation

In this section, we make use of the Hubble slow-rollparameters which are defined in Eqs. (2.4). We can extendthe Hubble slow-roll parameters up to an arbitrary order as

l�H � �2M2pl�

l 1

Hl

�dHd�

�l�1 d�l�1�H

d��l�1�; l � 1: (4.1)

In fact, 1�H and 2�H correspond to �H and �H, respec-tively. Using the definitions of the Hubble slow-roll pa-rameters and

ddN

����2

pMpl

�������H

p dd�

; (4.2)

we can obtain differential equations which are called ‘‘in-flationary flow’’ equation as

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d�HdN

� 2�H��H � �H�;

d'H

dN� �5�H'H � 12�2H � �2�H�

2;

d�l�H�

dN�

�l� 1

2'H � �l� 2��H

�l�H �l�1 �H; l � 2:

(4.3)

Here we defined 'H � 2�H � 4�H for convenience. It isknown that there are two classes of fixed points in thesystem of the Eqs. (4.3) [17]. The first class is the casewith �H � l�H � 0 and 'H � constant. The second one isthe case with 'H � �2�H � constant and 2�H � �2H,which corresponds to the case with the power-law inflation.

Using the flow equation, we computed 50000 realiza-tions of inflation models following the method of Ref. [17].We truncated the hierarchy at the fifth order to have a finiteset of equations for numerical calculations by settingl�H � 0 for l � 6. To generate inflation models, we ran-domly choose the initial conditions for the slow-roll pa-rameters as

0 � �H � 0:8; �0:5 � 'H � 0:5; (4.4)

and

�5� 10�l � l�H � 5� 10�l�l � 2; 3; 4; 5�; (4.5)

reducing the width of the range of the parameters by factorof 10 for each higher order in slow roll. For l � 6, weapproximate l�H � 0.

We also have to set the range of the number of e-foldingduring inflation to generate inflation models. In mostanalysis done so far [17,19–21], the following range isused 40 � Ne � 70. However, since we are going to con-sider the effects of the curvaton on the models generated bythis method and we have discussed that the e-foldingnumber can be changed due to the existence of the curvatonin the previous section, we need to set this quantity takingthe modification by the curvaton into account.

B. Effects of the curvaton

Now we discuss the effects of the curvaton on inflationmodels generated by this method. Since the fluctuation ofthe inflaton and the curvaton are independent, we cancalculate the observational quantities according toEqs. (2.22) and (2.24). As mentioned above, we have toset the range of the e-folding number taking into accountthe modification of the background evolution due to thecurvaton. First we consider the case with small curvatoninitial amplitude (i.e., X 1). In this case, as discussed inthe section II E, the modification of the background evolu-tion is relatively small. Thus, in generating the inflationmodels, we adopt the original range of the e-folding num-ber 40 � Ne � 70 even for the case with the curvaton.

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10-4

10-3

10-2

10-1

100

-0.4 -0.2 0 0.2 0.4

r

ns − 1

FIG. 6 (color online). Inflation models generated by the flowequation in the ns � 1 vs r plane. Models in the standardscenario are shown in red (circles) while the curvaton caseswith ~f � 5 are shown in blue (triangles).

TAKEO MOROI AND TOMO TAKAHASHI PHYSICAL REVIEW D 72, 023505 (2005)

In Fig. 6, we show how the distribution on the ns vs rplane is modified. In the figure, distribution for the stan-dard case (without the curvaton) is shown in red color (orcircles) while that in the curvaton scenario with ~f � 5 isindicated in blue color (or triangles). We can see someclustering structures of generated models near the thesecond class of the fixed point for the standard case.Importantly, most of such clustering region is excludedby the WMAP data because of too small ns and/or toolarge r. With the curvaton, the clustering occurs at theregion with smaller value of r. Since r always decreasesin the curvaton scenario, the cluster is shifted to lower rregion with curvaton. However, the change of the spectralindex due to the curvaton is not significant enough and

-0.04

-0.02

0

0.02

0.04

-0.4 -0.2 0 0.2 0.4

dns

/ dln

k

ns −1

FIG. 7 (color online). Inflation models generated by the flowequation in the ns � 1 vs d lnns=dk plane. Models in the stan-dard and curvaton scenarios are shown in red (circles) and blue(triangles), respectively.

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hence, if we take 40 � Ne � 70, the large class of gener-ated inflation models are excluded falling into the cluster-ing region.

In Fig. 7, we also show the distribution on the ns vsd lnns=dk plane. As one can see, running of the indexd lnns=dk is quite small even with the curvaton.

It is notable that the clustering structure shows up be-cause of relatively large value of Ne. If a second inflationoccurs with the energy density of the curvaton, however,this result may be changed. With the second inflation,significant expansion after the first inflation is possibleand hence Ne may drastically decrease. When the initialamplitude of the curvaton becomes as large as Mpl, this canbe the case. Then, Ne much smaller than 40� 70 may berealized. For some inflation models, reduced value of Ne istoo small to make the point �ns � 1; r� into the clusteringregion and, consequently, the resulting distribution on thens vs r plane becomes more scattered than the previouscase. This fact has some importance for models with�4�H � 2�H > 1 (which means that, for some scale, theprimordial spectrum is blue-tilted) because the fixed pointlocates in the region where ns � 1< 0. For models wherethe initial values of the slow-roll parameters satisfy therelation �4�H � 2�H > 1, the value of �4�H � 2�H canbe reduced to make ns � 1 ’ 1 in the course of approach-ing to the fixed point. As one can see, in the case with thecurvaton, larger number of points fall into the region con-sistent with the WMAP data (i.e., ns ’ 1 with r & 1) inFig. 8 than in Fig. 6.

To be more quantitative, we perform the numericalanalysis with smaller values of Ne. The second inflationdriven by the curvaton may decrease the number ofe-folding $Ne � 20� 30. Thus we take the range of the

10-4

10-3

10-2

10-1

100

-0.4 -0.2 0 0.2 0.4

r

ns −1

FIG. 8 (color online). The same as Fig. 6 except that we take10 � Ne � 40 and ~f � 3 for the curvaton case in this figure.(Notice that the data points for the standard case without thecurvaton are unchanged.)

-8

-0.04

-0.02

0

0.02

0.04

-0.4 -0.2 0 0.2 0.4

dns

/ dln

k

ns −1

FIG. 9 (color online). The same as Fig. 7 except that we take10 � Ne � 40 for the curvaton case in this figure.

IMPLICATIONS OF THE CURVATON ON. . . PHYSICAL REVIEW D 72, 023505 (2005)

initial e-folding number as 10 � Ne � 40 assuming thesecond inflation due to the curvaton. (In fact, the presenthorizon scale has to exit the horizon during the first in-flation, which gives an upper bound on the initial amplitudeof the curvaton. Consequently, in the case with the secondinflation, the ~f parameter cannot be so large; seeEq. (2.20).) The results with the mild value ~f � 3 areshown in Figs. 8 and 9 in the ns vs r and ns vsd lnns=d lnk planes, respectively. As one can see, com-pared to the case with 40 � Ne � 70, the distribution ismore scattered for 10 � Ne � 40. In particular, sizablenumber of the inflation models are moved into the regionwith ns ’ 1, the allowed region from the WMAP data,which is not the case for the case without the secondinflation. In fact, in the case with the second inflation, thechange of the distribution is mostly from the reduction ofthe e-folding number in the first inflation. Thus, the effectsof the curvaton fluctuation (i.e., the values of ~f) on the

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distribution of models in the ns vs r plane is not sosignificant compared with the case with small curvatoninitial amplitude.

V. SUMMARY

In this paper, we discussed the effects of the curvaton oninflation models from some general points of view. For thispurpose, first we classify inflation models in the ns vs rplane into three categories: the ‘‘small-field’’ models, ‘‘-large-field’’ models and ‘‘hybrid-type’’ models.

For the case that the initial amplitude of the curvaton issmall (�init <Mpl) and that the change of Ne being negli-gible, we have shown how the scalar spectral index and thetensor-to-scalar ratio are modified for each models. In the‘‘small-field’’ models, both the spectral index and thetensor-to-scalar ratio are not affected by the curvatonmuch (although slight increase in ns can be seen). In the‘‘large-field’’ models, the spectral index ns always in-creases and the tensor-to-scalar ratio is largely suppressed.In the ‘‘hybrid-type’’ models, the spectral index alwaysdecreases which is the opposite effect compared to thecases with the small-field and large-field models.

We also investigated the effects of the curvaton oninflation models generated by the inflationary flow equa-tion. Since we gave a general argument how the observablequantities are affected by the curvaton, we can easilyunderstand the modification of the structure of distributionof inflation models generated by the inflationary flowequation.

ACKNOWLEDGMENTS

T. T. would like to thank the Japan Society for Promotionof Science for financial support. The work of T. M. issupported by the Grants-in Aid of the Ministry ofEducation, Science, Sports, and Culture of Japan No.15540247.

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