Curvaton reheating: An application to braneworld inflation

PHYSICAL REVIEW D 68, 043517 ~2003!

Curvaton reheating: An application to braneworld inflation

Andrew R. LiddleAstronomy Centre, University of Sussex, Brighton BN1 9QJ, United Kingdom

L. Arturo Urena-LopezAstronomy Centre, University of Sussex, Brighton BN1 9QJ, United Kingdom

and Instituto de Fı´sica de la Universidad de Guanajuato, A. P. E-143, C. P. 37150, Leo´n, Guanajuato, Mexico~Received 5 February 2003; published 29 August 2003!

The curvaton was introduced recently as a distinct inflationary mechanism for generating adiabatic densityperturbations. Implicit in that scenario is that the curvaton offers a new mechanism for reheating after inflation,as it is a form of energy density not diluted by the inflationary expansion. We consider curvaton reheating inthe context of a braneworld inflation model,steep inflation, which features a novel use of the braneworld togive a new mechanism for ending inflation. The original steep inflation model featured reheating by gravita-tional particle production, but the inefficiency of that process causes observational difficulties. We demonstratehere that the phenomenology of steep inflation is much improved by curvaton reheating.

DOI: 10.1103/PhysRevD.68.043517 PACS number~s!: 98.80.Cq, 98.70.Vc

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I. INTRODUCTION

The curvaton@1,2# is an interesting new proposal for explaining the observed large-scale adiabatic density pertutions in the context of inflation. Perturbations are acquireda field other than the inflaton during inflation, whose enedensity is subdominant during inflation but which comesdominate sometime afterward, at which point its initisocurvature perturbations convert to adiabatic@1,3# ~see also@4#!. Decay of the curvaton to conventional matter locksperturbations in as adiabatic, although there can be comphenomenology if some species have already decoufrom the thermal bath by the time this happens@2,5#.

An aspect of this process which has been implicit in mpapers is that this process also offers a new mechanismreheating. The first detailed discussion of this aspectpeared recently in Ref.@6# by Feng and Li, who named icurvaton reheating. The curvaton is a type of matter whosenergy density is not diluted during inflation, and some orof the present material in the Universe may have surviinflation via the curvaton. This reheating mechanism compments the other two known mechanisms, the standardbeing decay of the inflaton energy density into conventiomatter and the rarely used alternative being gravitationalticle production at the end of inflation@7#.

In this paper we will explore an application of curvatoreheating to a particular model of braneworld inflatiknown assteep inflation@8#. This model brings the inflationary epoch to an end in a novel way. The potential chosen~forexample a steep exponential! is so steep that it would nodrive inflation in the standard cosmology, but it may do sothe presence of high-energy braneworld corrections toFriedmann equation, for example as found in the RandSundrum type II scenario. As the energy density decreathose corrections become unimportant, and the scalarmakes a transition into a kinetic energy dominated regbringing inflation to an end. As the inflaton survives thprocess without decay, an alternative reheating mechanisrequired.

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In Ref. @8#, it was proposed that reheating took place vgravitational particle production@7,9#. However this bringsproblems to the scenario. Because of the inefficiency ofprocess, there is a long kinetic energy dominated regiwhich allows short-wavelength gravitational waves to reaexcessive amplitudes@10#. There are likely also to be problems with the low reheat temperature achieved, whichbarely sufficient to allow standard nucleosynthesis to pceed. The scenario also suffers from a separate probwhich is that it tends to predict perturbation spectra whare too far from scale invariance, and too high an amplituof long-wavelength gravitational waves; for the exponenpotential the predicted spectra are in conflict with the mrecent observations.

In this paper, we propose that implementation of curvareheating in place of gravitational particle production credress those difficulties. Curvaton reheating can be mmore efficient, allowing a high reheat temperature and pventing short-wavelength gravitational wave dominanFurther, adiabatic perturbations produced via the curvamay be closer to scale invariance, and in particular the ctribution of gravitational waves to the microwave anisotrpies may be considerably suppressed. Two related paperpeared as we were completing this work; Ref.@6# discussescurvaton reheating in a quintessential inflation model@11# inthe standard cosmology, while Ref.@12# considers adding thecurvaton to the steep inflation model, but focusing on pticle physics model building aspects rather than obsertional phenomenology and constraints.

A summary of this paper is as follows. In Sec. II, wreview the main features of steep inflation in the branewoscenario. We present only the main results of these kindmodels, including the production of primordial gravitationwaves. In Sec. III, we study the evolution of a curvaton fieand its perturbations in steep inflation, and we focusattention on all the constraints to be applied upon the fparameters of the model. We consider separately the cthat arise depending on whether the curvaton field dominafter or before it decays into radiation. In Sec. IV we putconstraints together and analyze the parameter space o

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A. R. LIDDLE AND L. A. URENA-LOPEZ PHYSICAL REVIEW D68, 043517 ~2003!

model of curvaton field1 steep inflation which satisfies athe requirements. In the final section we draw some consions and indicate topics to be investigated in future pucations.

II. THE STEEP INFLATION MODEL

In this section we give a brief description of an inflatioary model in the braneworld scenario, assuming that theflaton field is a minimally coupled real scalar fieldf en-dowed with an exponential potential of the form

V~f!5V0e2aAk0f, ~1!

whereV0 , a are free parameters andk058p G58pmPl22

with mPl the Planck mass. More details of the model canfound in Refs.@8,10#. A somewhat related idea has bediscussed in Ref.@13#.

A. Inflationary dynamics

In the (411)-dimensional brane scenario inspired by tRandall-Sundrum model, the standard Friedman equatiomodified to

H25k0

3rS 11

r

2lbD , ~2!

wherelb is the brane tension and, taking the inflaton fieldbe the dominant component,r5f2/21V(f). The scalarfield is confined to the brane so that its equation of motionthe usual Klein-Gordon one

f13Hf1dV

df50, ~3!

where overdots denote temporal derivatives.For large energiesr@lb , the inflaton field experiences a

increased friction as it rolls down its potential, and then ocan expect inflation to occur even for steep potentials~i.e.even fora@1). This can be seen from the slow-roll parameters for the exponential potential@14#, which for large en-ergies read

e5h52a2lb

V, ~4!

and thus inflation can occur wheneverV.2a2lb .The equations of motion can be solved exactly in

slow-roll approximation, from which we find the followingexpressions for the scale factora, the Hubble factorH andthe inflaton fieldf, respectively:

lna

ai5~N11!2e2H f(t2t f), ~5!

H5H fe2H f(t2t f), ~6!

aAk0f~ t !5H f~ t2t f!2 lnVf

V0. ~7!

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Here ‘‘i’’ ~‘‘f’’ ! indicates the initial~final! value of thedifferent quantities during inflation, andN is the number ofe-foldings as can be seen from Eq.~5!. Observe that, in thebraneworld scenario, the slow-roll approximation for the eponential potential is a much better approximation thanstandard cosmology.

Inflation ends by violation of the slow-roll conditionswhen e.h;1, and so the value of the potential energythe end of inflation isVf.2a2lb . Using Eqs.~5!–~7!, wefind that the initial and final values of the Hubble factor athe potential energy, respectively, are related through

H i

H f5

Vi

Vf.N11, ~8!

where

H f5a2A2lbk0

3. ~9!

So far the model has three free parameters, but only twothem are relevant for inflation, the brane tensionlb and theself-interaction of the inflaton fielda.

The amplitude of scalar density perturbations can beculated for the exponential potential, and it is given by@8,10#

As2.

8

75

Vi4

a2lbmPl4

5128

75a6~N11!4

lb

mPl4

. ~10!

In the usual scenario of steep inflation, the values ofinflation parameters can be adjusted so that we get theserved valueAs

2.4310210. In particular, we can determinthe value of the brane tension in terms ofa as

lb.2.3310210

a6

mPl4

~N11!4. ~11!

Notice that any deviation oflb from the value obtained inEq. ~11! would imply either a larger or smaller amplitude fothe scalar spectrum.

As was pointed out in Ref.@10#, the number ofe-foldingscan be unambiguously determined for the exponential potial, for which we obtainN.70. This result is based on Eq~10! and is independent of the value ofAs

2 .

B. After inflation

Soon after inflation ends, the brane term becomes unportant and we recover the standard Friedman equatHence, the friction on the inflaton field diminishes and tinflaton energy is dominated by the kinetic term and the slar energy density falls as stiff matter,rf5rf

(kin)(akin /a)6.The epoch that follows is called the ‘‘kinetic epoch’’ or ‘‘kination’’ @15#, and we shall use ‘‘kin’’ to label the value of thdifferent quantities at the beginning of the kinetic epoch.

Since we are now working within standard cosmology~atenergiesr,lb), the Hubble factor evolves as

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CURVATON REHEATING: AN APPLICATION TO . . . PHYSICAL REVIEW D68, 043517 ~2003!

H5HkinS akin

a D 3

, Hkin2 .

k0

3rf

(kin) . ~12!

The kinetic regime does not commence immediately afterend of inflation, but the values of the Hubble parameterthe end of inflation and at the beginning of the kinetic regiare related by the formula@10#

Hkin

H f50.0852

0.688

a2. ~13!

A small amount of radiation has been produced quantumechanically during inflation with an energy densr r;0.01gpH f

4 , wheregp510–100 is the number of differenspecies created from the vacuum. If the inflaton potendoes not decay just after inflation, then this is the omechanism to reheat the Universe.

With the density of scalar matter falling more rapidly ththat of radiation, the Universe continues evolving accordto Eq. ~12! until the small amount of radiation comesdominate. Since the ratio of scalar matter to radiation atend of inflation is large, (rf /r r) f@1, radiation comes todominate the expansion of the Universe only after a vlong kinetic epoch. This is troublesome since the energy dsity of gravitational waves produced quantum-mechaniccan exceed the nucleosynthesis constraints, as shown in@10#.

In braneworld inflation, the amplitude of gravitationwaveshGW

2 is enhanced with respect to the standard scenaand its value at the end of inflation is given by

hGW2 .3a2~N11!

H i2

mPl2

. ~14!

Gravitational waves behave as massless scalar fields andtheir amplitude remains constant during inflation. During tkinetic epoch, the energy density of gravitational wavevolves as@10#

rg532

3phGW

2 rfS a

akinD 2

. ~15!

From this, the energy density of gravitational waves attime of scalar stiff matter–radiation equality (rf5r r) is

rg

r rU

a5aeq

564

3phGW

2 S aeq

akinD 2

.64

3phGW

2 rf

r rU

f

@1. ~16!

Thus, the energy density of gravitational waves onshortwavelength scales@10# exceeds the contribution of radiationIt happens then that gravitational waves come to dominthe stiff scalar matter well before radiation, which in fanever dominates, and then nucleosynthesis constraintsnot satisfied at all. Therefore, the original steep inflatmodel seems to be ruled out@10#.

However, the production of gravitational waves cankept under control if the kinetic epoch is shortened. Tlatter can be done either by a sufficiently early decay of

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inflaton field or by having a post-inflationary equationstate softer than that of stiff matter. The latter picture is quappealing since the surviving inflaton field could becompart of the dark matter at late times@10,16,17#.

III. THE CURVATON FIELD

In this section we explore the possibility that the inflatfield is not responsible for providing the primordial fluctutions required for the formation of structure in the late Unverse, as in the case where the value of the brane tensiolower than that in Eq.~11!. Instead, the primordial fluctuations are generated by a second scalar fields, usually calledthe curvaton field, through initially isocurvature perturbations @1#. For simplicity, we assume here that the curvatfield has a quadratic scalar potentialU(s)5m2s2/2, andhence obeys the Klein-Gordon equation

s13Hs1m2s50, ~17!

wherem is the curvaton mass.We will now follow the evolution of the curvaton field

through different stages and find the constraints upon theparameters of the model in order to have a viable curvascenario together with steep inflation. In doing this, we shfollow a similar procedure to that in Ref.@18#. We begin witha brief overview of the dynamics.

First of all, it is assumed that the curvaton field coexiwith the inflaton field during inflation, during which the inflaton energy density is the dominant component. The curton field should survive the rapid expansion of the Univerand for that it has to be effectively massless. As we shallbelow, this condition is translated into a constraint on tcurvaton massm. If this constraint is satisfied, the curvatofield will remain constant at its initial value.

The following stage is that when the curvaton field bcomes effectively massive, and this will generally happduring the kinetic epoch. In order to prevent a periodcurvaton-driven inflation, the Universe must remain inflatdominated until this time. The latter condition imposesconstraint on the initial values the curvaton field can tas i . Once effectively massive, the curvaton field oscillaaround the minimum of its potential and its energy densevolves as non-relativistic matter.

The final stage is that of curvaton decay into radiatioand the standard big bang cosmology is recovered afterwIn the general case, curvaton decay should occur beforecleosynthesis, but other constraints arise depending on eof the decay, governed by the decay parameterGs . There aretwo scenarios to be considered, depending whether thevaton field decays before or after it becomes the domincomponent of the Universe. Whatever the case, the enfrom gravitational waves should be maintained under ctrol, which will actually become an extra constraint involing different parameters of the model. Also, we calculatecurvaton perturbations for each case which will help usreduce the number of free parameters. At this point,Gaussianity conditions on the perturbations will be takinto account too.

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The introduction of the curvaton field adds three freerameters to the steep inflation model: the curvaton masm,the initial value of the curvaton fields i and the curvatondecay parameterGs . However, a successful model needssatisfy the conditions outlined above which are stroenough to restrict the allowed curvaton parameters.

A. Inflation

We first study the evolution of the curvaton field durinbraneworld inflation with an exponential inflaton potentialdescribed in Sec. II. We change the independent variablz[3H/H f , for which the equation of motion of the curvatofield Eq. ~17! reads

z2s91z~12z!s81m2

H f2

s50, ~18!

where primes denote derivatives with respect toz.Assuming that the curvaton field is effectively massle

during inflation m!H f , the last term in Eq.~18! is smallcompared to the others, so that the curvaton equation oftion is

z2s91z~12z!s8.0, ~19!

whose exact solutions are

s~z!5s i1s i8zie2zi@Ei~z!2Ei~zi!#, ~20!

s85s i8zi

zexp~z2zi!. ~21!

Here, s i and s i8 are the initial values of the field and itderivative, and Ei(z) is the exponential integral. In the casthat N570, so thatz runs in the rangezi5213→zf53 @seeEq. ~8!#, the ratio of the final value of the curvaton field to iinitial value is

s f

s i5121.005

s i8

s i. ~22!

We see that the final values f depends on the initial conditions. The interesting case is that in which the curvaenergy is equally distributed into kinetic and potential format the beginning of inflation, so that we can tas i

2;m2s i2 . Thus, froms i5H fzis i8 we obtain

s i8~z!

s i~z!;

m

H fzi5

1

3~N11!

m

H f. ~23!

Therefore, the conditionm!H f makes the curvaton field remain constant under general initial conditions, and wewrite s f.s i . Notice also thats f50.

B. Kinetic epoch

As we anticipated at the beginning of this section, tdynamics of the curvaton field after inflation are differefrom the scenarios considered before in the literature. T

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difference is mainly because the curvaton field coexists wthe surviving inflaton field, whose energy density is by fthe dominant one at the end of inflation. The consequenof this will be explored in this section and our immediaconcern will be to work out the corresponding gravitationwave constraint.

According to our hypothesis, it is during the kinetic epothat the curvaton field becomes effectively massive. Thappens at a time whenm.H, from which we obtain@seeEq. ~12!#

m

Hkin5

akin3

am3

, ~24!

where ‘‘m’’ labels the quantities at the time when the curvton becomes massive. Up to this point, the curvaton firemained effectively massless and we can considersm.s i .

In order to prevent a period of curvaton-driven inflatiothe Universe must still be dominated by the scalar stiff mter at this point. Hence, using Eqs.~12! and~24!, we arrive atthe restriction

m2s i2

2rf(m)

54p

3

m2s i2

mPl2 m2

!1⇒s i2!

3

4pmPl

2 . ~25!

Equation~25! is sufficient to achieve subdominance at ttime when the curvaton field becomes massive. But thevaton energy should also be subdominant at the end of intion, which implies

U f

Vf5

m2s i2

4a2lb

!3m2mPl

2

16pa2lb

5a2m2

H f2

, ~26!

where we have used Eqs.~9! and ~25!. Thus, the curvatonmass should obey the stronger constraint

m!a21H f . ~27!

After the curvaton field becomes effectively massive, its eergy decays as non-relativistic matter in the form

rs5m2s i

2

2

am3

a3. ~28!

C. Curvaton domination

We calculate here the constraints to be applied oncurvaton model for the case in which the curvaton fiecomes to dominate the cosmic expansion, dealing first wthe gravitational wave constraint. Since the curvaton eneredshifts as in Eq.~28!, we find at the time of stiff scalarmatter–curvaton matter equality (rs5rf) that1

1Note that the labelaeq has different meanings in Eqs.~16!, ~29!and ~35!.

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rs

rfU

a5aeq

54p

3

m2s i2

mPl2 Hkin

2

am3

akin3

aeq3

akin3

51. ~29!

Using Eqs.~24! and ~29! in Eq. ~15!, we can calculate theenergy density of gravitational waves at this time, whishould obey the constraint

rg

rsU

a5aeq

564

3phGW

2 S 3

4p

mPl2

s i2

Hkin

m D 2/3

!1. ~30!

To finish with, we derive the conditions to be appliedthe decay parameterGs . On the one hand, we require thcurvaton field to decay before nucleosynthesis, and tHnucl510240mPl,Gs . On the other hand, we also requithe curvaton to decay after domination, and thenGs,Heq,whereHeq is found by using Eqs.~12!, ~24! and ~29!:

Heq5Hkin

akin3

aeq3

54p

3

s i2

mPl2

m. ~31!

Hence, the constraint upon the decay parameter is

10240mPl,Gs,4p

3

s i2

mPl2

m. ~32!

D. Curvaton decay before domination

We now turn our attention to the case in which the curton field decays before it dominates the cosmological expsion, but after it becomes massive so that we can use~28!. As before, we calculate first the gravitational wave costraint. The curvaton field decays at a time whenGs5H andthen

Gs

Hkin5

akin3

ad3

, ~33!

where ‘‘d’’ labels the different quantities at the time of cuvaton decay. The radiation produced from curvaton dehas an energy density

r r(s)5

m2s i2

2

am3

ad3

ad4

a4. ~34!

We will now assume that this radiation energy densitymuch larger than that produced by inflation, a necessaryquirement if we are to maintain the gravitational waves uder control. From Eq.~34!, radiation equals the stiff scalamatter (r r

(s)5rf) at a time given by

r r(s)

rfU

a5aeq

54p

3

m2s i2

mPl2 Hkin

2

am3

akin3

aeq2

akin2

ad

akin51. ~35!

This equation defines the ratioakin /aeq to be used in Eq.~15!. Hence, the constraint from gravitational waves nreads

04351

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-n-q.-

y

se--

rg

r rU

a5aeq

516

p2hGW

2 A3 Gs

Hkin

m

Hkin

mPl2

s i2

!1, ~36!

where we have also used Eqs.~24! and ~33!.Finally, we derive the new constraints onGs . The curva-

ton field should decay after it becomes massive, soGs,m; and before it dominates the expansion of the Uverse,Gs.Heq @see Eq.~31!#. Therefore, the constraints othe decay parameter are

4p

3

s i2

mPl2

m,Gs,m. ~37!

E. The surviving inflaton field

For completeness, we should add a small note here athe evolution of the inflaton field after curvaton dominatioand/or decay. It is well known that a steep exponentialtential has a tracker behavior where it evolves with the saequation of state as the dominant component@19#. However,during the kination regime, the field is fast rolling, and thtypically takes it to very low energy densities, so that it cjoin the tracker only late in the cosmological evolution. Dtailed analysis, following Ref.@16#, would be needed to determine its precise evolution, but it is unlikely to have asignificant cosmological consequences unless the expotial form of the potential becomes modified at low energi

F. Curvaton fluctuations

Equations~30! and ~37! are the constraints on the freparameters of the model coming from gravitational wavbut their actual form is not useful yet because of the tehGW

2 . The latter is indeed related to other parameters ofmodel through the curvaton fluctuations, as we shall showthis section.

The curvaton fluctuationsdsk obey the linearized Klein-Gordon equation

dsk13Hdsk1F k2

a21m2Gdsk50, ~38!

wherek is the comoving wave number.During the time the fluctuations are inside the horizo

they obey the same differential equation as do the inflafluctuations, from which we conclude that they acquire tamplitudeds i.H i/2p. Once the fluctuations are out of thhorizon, they obey the same differential equation as doesunperturbed curvaton field and then we expect that theymain constant during inflation, under quite general initconditions.

Up to this point, the evolution of the curvaton fluctuatioresembles that of previous scenarios already present inliterature. But, in a similar manner to our analysis of thomogeneous curvaton field in Secs. III C and III D, wshould now take into account that the inflaton field survivthe inflationary period, and study how the final spectrumperturbations could be modified by this.

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Following previous work, we can calculate the spectruof the Bardeen parameterPz , whose observed value is abo231029. When decay occursafter curvaton domination, theproduced perturbation is@1,2,18#

P z.1

9p2

H i2

s i2

. ~39!

This case is by far the simplest one. The spectrumfluctuations is automatically Gaussian sinces i

2@H i2/4p2,

and is independent ofGs , a feature that will simplify theanalysis of the parameter space. Moreover, even thoughcurvaton field coexists with the inflaton field up to this eoch, the spectrum of fluctuations is the same as in the sdard scenario.

In the limit in which the curvaton decays when subdomnant, the Bardeen parameter is given by@1,2#

P z.r d

2

36p2

H i2

s i2

, ~40!

where the normalization takes into account that the domincomponent at that time is the scalar stiff matter. Herer d isthe ratio of curvaton energy density to stiff scalar mattercurvaton decay, which is found by using Eqs.~24! and ~33!in

r d5rs

rfU

a5ad

54p

3

m2s i2

mPl2 Hkin

2

am3

akin3

ad3

akin3

54p

3

m

Gs

s i2

mPl2

.

~41!

Contrary to the previous case, Eqs.~40! and ~41! showexplicitly the influence of the surviving inflaton field, anthey are markedly different from the standard scenariowhich the curvaton field coexists only with radiation.

IV. MODEL CONSTRAINTS

In this section, we use the different constraints oncurvaton field to study under which conditions it can be ustogether with steep inflation. The two cases that were worout before will be treated separately.

A. Curvaton decay after domination

In this case, Eqs.~8!, ~9!, and ~39! allow us to fix lb interms ofs i anda as

lb

mPl4

527p

16

Pz

~N11!2a4

s i2

mPl2

, ~42!

and the amplitude of primordial gravitational waves Eq.~14!can be written in terms ofPz , s i anda as

hGW2 527p2a2~N11!P z

s i2

mPl2

. ~43!

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Thus, Eqs.~30! and ~27! are transformed into the followingconstraints on the curvaton mass:

m

mPl@103Ap3~N11!a~2.6a2221!P z

2s i

2

mPl2

, ~44!

m

mPl!

3pP z1/2

a~N11!

s i

mPl, ~45!

where we have used Eq.~13!.The initial value of the curvaton fields i is restricted by

Eq. ~25! and by another restriction coming from the assumtion that the inflaton fluctuations are not relevant. The latis obtained by using Eqs.~11! and ~42!, which give

s i

mPl,

5

A72p

1

a~N11!

As

APz

. ~46!

In other words, Eq.~46! just means that the brane tension ha lower value than that of Eq.~11!.

Finally, Eq.~32! restricts the value of the decay parameGs , which can be transformed into another constraint upm ands i as

m

mPl

s i2

mPl2

@3

4p310240. ~47!

The different conditions on the model of steep inflation1curvaton field constrain the values oflb , s i andm, but notof a and N which remain free parameters of the modTherefore, we can draw the allowed region for the formera plot ofm versuss i by fixing the value ofa. Such a plot isshown in Fig. 1 fora515 andN570, where we can see thathere are viable models satisfying all the requirements.

FIG. 1. Allowed region of parameter space of the curvatosteep inflation model for the case of curvaton domination befdecay. Shown are the constraints Eqs.~25!, ~44!, ~45!, ~46! and~47!usinga515 andN570. The regions excluded by each constraare indicated by the arrows, and the allowed region is shaded.allowed region is reduced for a larger value ofa.

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ton

achn is


However, the allowed region of the parameter spacereduced for larger values of the self-interaction of the inflton fielda@1, as we can see from Eqs.~44! and~45!. Thus,it is not always possible to find a successful model of curton field 1 steep inflation, and this is mainly because of tgravitational wave constraint Eq.~36!.

B. Curvaton decay before domination

For this case, it is convenient to use Eqs.~40! and~41! towrite s i in terms of the other quantities as

s i2

mPl2

581

4

Pz

~N11!2

Hkin2

H f2

mPl2

Hkin2

Gs2

m2. ~48!

On the other hand, the amplitude of primordial gravitationwaves Eq.~14! is rewritten as

hGW2 53a2~N11!3

H f2

Hkin2

Hkin2

mPl2

, ~49!

and then the gravitational wave constraint Eq.~36! reads

64a2

27p2Pz

~N11!5H f

4

Hkin4

Hkin4

mPl4 S Hkin

GsD 5/3 m3

Hkin3

!1. ~50!

In the same manner, Eq.~37! is now written as

27pPz

~N11!2

Hkin2

H f2

mPl2

Hkin2

Gs2

m,Gs,m. ~51!

Notice that, from Eqs.~25! and ~51! respectively, there arethree similar constraints on the curvaton mass of the for

m

mPl.SA27pPz

N11

Hkin

H f

mPl

HkinD Gs

mPl, ~52!

m

mPl.SA27pPz

N11

Hkin

H f

mPl

HkinD 2 Gs

mPl, ~53!

m

mPl.

Gs

mPl, ~54!

while Eqs.~27! and ~50! give the following constraints, respectively:

m

mPl!

1

a

H f

Hkin

Hkin

mPl, ~55!

m

mPl!F 27p2P z

64a2~N11!5

Hkin4

H f4 G 1/3S mPl

8

Hkin8

Gs5

mPl5 D 1/9

. ~56!

At this point, we should take into account the Gaussiancondition upon the curvaton perturbations, since it is nottomatically satisfied as in Sec. IV A. The Gaussianity contion implies that the perturbations must be small compare

04351

is-

-

l

y--to

the mean value of the curvaton field, and thens i2@H i

2/4p.Using Eqs.~8!, ~13! and ~48!, we obtain

m

mPl!

9ApPz

~N11!2

Hkin2

H f2

mPl2

Hkin2

Gs

mPl. ~57!

All these constraints can be satisfied ifm,Gs→0 indepen-dently of the value ofHkin , but we know that this should nobe possible since we still expect to recover the standardbang scenario at nucleosynthesis. Hence, we will alsopose the restrictionGs.10240mPl .

The final constraint has to do with the inflaton fluctutions, which should be negligible compared to the curvaones. In principle, this is a constraint uponlb through Eq.~11!, but Eqs.~9! and ~13! allow us to transform it into aconstraint onHkin in the form

Hkin

mPl!

6.2131025

a3~N11!2~0.085a220.688!. ~58!

To unravel the parameter space allowed by the differconstraints, we proceed as follows. We will take as freerameters the inflationary ones,a, N andHkin , but the latterwill take only values permitted by Eq.~58!. Once we fix thevalues ofa, N andHkin by hand, we plot only the strongesconstraint of Eqs.~52!–~54! ~the other two will be automati-cally satisfied! and the constraints in Eqs.~55!, ~56! and~57!.An example of this is shown in Fig. 2 for the valuesa515andN570, where we also assumed that inflaton fluctuatiocontribute as much as the 10% of the Cosmic BackgroExplorer ~COBE! signal.

The different constraints permit some room for viabmodels, although the allowed region is smaller for smavalues of Hkin , i.e., smaller values oflb , and for larger

FIG. 2. Curvaton constraints for the case in which the curvafield decays before domination, usinga515, N570 and(Hkin /mPl)52.13310211 in Eqs.~50!, ~52!–~57!, ~58! and the nu-cleosynthesis constraint. As in Fig. 1, the regions excluded by econstraint are indicated by the arrows, and the allowed regioshaded. The allowed region would be smaller for a smaller~larger!value ofHkin (a).

7-7

iner

iointieinevesc

caeaongl

ais

li

es,he

atit-m

ina-e-be-ichtheheaseuin-ntry.on.

t.


values ofa. Thus, we conclude that a realistic scenariowhich the curvaton field decays before domination is vrestricted by the different constraints.

V. CONCLUSIONS

The steep inflation model is of interest as it ends inflatin a novel manner, through reduction of inflation-sustainbraneworld corrections, and because it generates inflafrom the steep exponential potentials often found in supgravity phenomenology. Unfortunately, the original steepflation model appears unable to match observations, duan excessive amplitude of short-scale gravitational waand because its perturbations are not close enough toinvariance.

In this paper we have shown that curvaton reheatingalleviate these difficulties, by dramatically raising the rehtemperature~as compared to gravitational particle productireheating!, while simultaneously suppressing the large-angravitational wave contribution. The model is subject tocomplex network of constraints, but viable models do exprovided the brane tensionlb and the exponential slopeaare not too large. We should stress that our analysis app

04351

y

ngonr--tos

ale

nt

e

t,

es

only to the case where the inflaton survives until late timsince an early decay of the inflaton field would lead to tstandard scenario already studied in the literature@1,2,18#.

An interesting feature of the steep inflation model is ththe inflaton field may survive to the present, and with suable modification of the potential from its exponential forat low energies it could act as either dark matter@16,17# orquintessence@8,11,20#. This resembles the scenarios givenRef. @2# concerning residual isocurvature matter perturbtions. For instance, if cold dark matter CDM is created bfore curvaton decay, then there is a maximal correlationtween curvature and CDM-isocurvature perturbations whis at variance with current observations. We expect thatinflaton perturbations will be affected in the same way if tinflaton field is to be the dark matter at late times. The cof quintessence should not be troublesome, since the qtessence perturbations should decay upon horizon eHowever, these situations merit more detailed investigati

ACKNOWLEDGMENTS

L.A.U.-L. was supported by CONACyT, Me´xico undergrant 010385, and A.R.L. in part by the Leverhulme Trus

hys.

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Curvaton reheating: An application to braneworld inflation