Massimo Giovannini- Tracking curvaton(s)?

8/3/2019 Massimo Giovannini- Tracking curvaton(s) ?

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Tracking curvatons?

Massimo Giovannini*Theoretical Physics Division, CERN, CH-1211 Geneva 23, Switzerland

Received 1 October 2003; published 9 April 2004

The ratio of the curvaton energy density to that of the dominant component of the background sources may

be constant during a significant period in the evolution of the Universe. The possibility of having tracking

curvatons, whose decay occurs prior to the nucleosynthesis epoch, is studied. It is argued that the trackingcurvaton dynamics is disfavored since the value of the curvature perturbations prior to curvaton decay is

smaller than the value required by observations. It is also argued, in a related context, that the minimal

inflationary curvature scale compatible with the curvaton paradigm may be lowered in the case of low-scale

quintessential inflation.

DOI: 10.1103/PhysRevD.69.083509 PACS numbers: 98.80.Cq, 98.70.Vc

I. INTRODUCTION

The isocurvature fluctuations generated by a field that islight during the inflationary phase and later on decays, canefficiently produce adiabatic curvature perturbations. This

idea has recently been discussed in different frameworks,ranging from the context of conventional inflationary models1,2 to the case of pre-big-bang models 3,5,4. It is appro-priate to recall that an earlier version of this proposal wasdiscussed in 6, not with the specific purpose of convertingisocurvature into adiabatic modes, but rather with the hopeof providing physical initial conditions for the baryon isocur-vature perturbations.

A common feature of various scenarios 716 is that theenergy density of the homogeneous component of this lightcurvaton field decreases more slowly than the energy densityof the background geometry. In the simplest realization, theUniverse suddenly becomes dominated by radiation, as soon

as inflation ends. During the radiation epoch, the homoge-neous component of the curvaton field is roughly constantdown to the moment when the Hubble parameter is compa-rable with the curvaton mass, i.e. Hm . The energy densityof the oscillating curvaton field then decreases as a3, wherea is the scale factor of a conformally flat Friedmann-Robertson-Walker FRW Universe. Since the energy densityof the background radiation scales as a4, the ratio

r a a

r a 1.1

will increase as a. Both a different post-inflationary historyand a different potential will produce deviations from thisscaling law. In particular, if the potential is not exactly qua-dratic or if the coupling of the curvaton to the geometry isnot exactly minimal 13 r(a) may decrease. If the post-inflationary phase is not immediately dominated by radia-tion, r(a) may increase even faster than a. If, after the end ofinflation the inflaton field turns into a quintessence-like field17 see also 15,18, the background energy density will

be dominated by the kinetic term of the quintessence field. Inthis case r() increases as a3 9.

It is not impossible that the ratio r(a) stays constant dur-ing a significant period in the post-inflationary evolution.This subject will be explored in the present investigation. It

will be argued that this type of tracking phase is disfa-vored because the initial isocurvature mode will rather ge-nerically turn into an adiabatic mode of much smaller am-plitude, making the whole scenario ineffective in correctlyreproducing the amplitude of the large-scale fluctuations.

One of the reasons to speculate on the possibility of track-ing curvatons is that one would like to allow for modelswhere the curvature scale, at the end of inflation, is muchsmaller than the value required, for instance, in single fieldinflationary models, where the curvature fluctuations are di-rectly amplified during the inflationary phase. It will beshown that if the inflationary phase is not followed suddenlyby radiation but rather by a kinetic phase as in the case of

quintessential inflation the minimal allowed curvature scaleis a bit smaller than in the standard case of sudden radiationdomination.

The present paper is organized as follows. In Sec. II thebasic problem will be formulated. In Sec. III it will be shownthat if the curvaton field tracks the evolution of the dominantcomponent of the background the resulting adiabatic mode issmaller then the initial isocurvature mode. In Sec. IV thelower bound on the inflationary curvature scale will be dis-cussed. Finally, Sec. V contains some concluding remarks.

II. FORMULATION OF THE PROBLEM

For illustrative purposes it is useful to consider the casewhere the inflationary phase is suddenly replaced by aradiation-dominated phase. During inflation, the curvatonfield should be nearly massless and displaced from theminimum of its potential W():

2W

2Hi

2, i0Hi , 2.1

where the subscript denotes the moment when the cosmo-logically interesting scales left the horizon during inflation.*Electronic address: [email protected]

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Under the assumption that the Universe is dominated by ra-diation, the evolution equations for the background fields canbe written1

MP2H

2

a2

3r, 2.2

MP2H 1

3a 2r

2Wa 2,

2.3

2HW

a 20, 2.4

r4Hr0, 2.5

where it has been assumed that the curvaton is minimallycoupled to the background geometry. This assumption maybe relaxed by requiring, for instance, that the coupling benon-minimal. This would imply the addition, in Eq. 2.4, ofa term going like cR, where RH2 is the Ricci scalar of aradiation-dominated FRW background. The addition of sucha term is expected to lead to a decrease of r(a) 13 evenprior to the true oscillatory phase taking place when Hm . For the purposes of the present section, it is useful toparametrize the evolution of in terms of r/r . In thiscase

2MP2Ha 4 r

a 4 ,

W

a 2

1

a2 a2, 2.6

where the first equation has been obtained by taking the de-rivative of the definition of r and by subsequent use of Eq.2.4.

Provided r(i)1, three possible physical situations mayarise. Ifr0, the curvaton field will become, at some point,dominant over the background density. If r0 , the curva-ton field will never dominate over the background geometry.The third possibility is to have r0, which is the case to bediscussed in the following. Even if examples will be pro-vided in the framework of specific potentials, it is better, atthis stage, to think of the potential as a function of the ratior, as suggested by Eqs. 2.6. The perturbation equations rel-evant in order to study the dynamical evolution of the field are obtained by linearizing the evolution equations to first

order in the amplitude of the metric fluctuations. In generalterms, the scalar fluctuations of a conformally flat metric ofthe type

ga2 , 2.7

can be parametrized as

(1 )g 002a2,

(1 )g i j2a2i jijE,

(1 )g0ia2iB , 2.8

where , , B and E are the four scalar degrees of freedomof the perturbed metric; (1 ) denotes the first order fluctua-tion of the corresponding quantity. Since the infinitesimalcoordinate transformations preserving the scalar nature of thefluctuation have two parameters, out of the four scalar fluc-tuations defined in Eq. 2.8, two gauge-invariant Bardeenpotentials can be defined, i.e.

HBEBE,

HBE. 2.9

The first order perturbation of the Einstein equations, ofthe curvaton equations and of the covariant conservation ofthe energy-momentum tensor of the fluid can be written, re-

spectively, as

(1 )R

1

2(1 )R

1

MP2(1 )T

1

MP2(1 )T

,

2.10

g(1 )(1 )

(1 )

(1 )g

2W

2(1 )0,

2.11

(1 )T(1 )

T

(1 )T(1 )

T

(1 )T

0, 2.12

where T is the energy-momentum tensor of the curvaton

and T is the energy-momentum tensor of the fluid sources.

In terms of the two gauge-invariant Bardeen potentials de-fined in Eq. 2.9 the gauge-invariant fluctuations of theenergy-momentum tensors can be written as

1Units MP(8G)1/2 will be adopted. The variable denotes

the conformal time coordinate while t denotes the cosmic time.

Derivatives with respect to either conformal or cosmic time will be

denoted, respectively, by a prime and by an overdot. Finally, the

expansion rate, in the conformal and cosmic time coordinates will

be denoted by Ha/a and by Ha/a.

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(gi)Tij

1

a2 2 W

a 2 ,

(gi)T00

1

a 22 W

a2 ,

(gi)Ti0

1

a 2,

(gi)Tij

1

3ri

j

(gi)T00r

(gi)Ti0

4

3riu, 2.13

where

(1 )BE 2.14

is the gauge-invariant fluctuation of the curvaton and

r(1 )rrBE,

u(1 )uBE 2.15

are, respectively, the gauge-invariant fluctuations of the fluidenergy and of the velocity potential. Inserting Eqs. 2.132.15 into the (i,j) component of Eqs. 2.10 the followingrelation is obtained:

ij 0, 2.16

implying . From the (0,0), ( ij) and (0,i) compo-nents of Eq. 2.10 and using the background equations2.22.5, the following equations can be, respectively, ob-tained:

23HH

a 2

2MP2rr

1

2MP2 2 W a 2 ,

2.17

3HH 22H

a2

6MP2rr

1

2MP2 2 W a2 ,

2.18

H

2MP2

2

3MP2

ua 2r , 2.19

where rr/r and where Eq. 2.16 has been used. Fi-nally, using Eqs. 2.132.15 and 2.16 into Eqs. 2.11and 2.12 the explicit form of the remaining equations willbe

2H22W

2a 242

W

a 20,

2.20

r44

32u0, 2.21

u1

4r0, 2.22

where Eq. 2.20 is the perturbed curvaton equation; Eqs.2.21 and 2.22 correspond to the 0 and i components ofEq. 2.12.

Introducing the useful notation xln(/i), Eqs. 2.17,2.18 and 2.20 can be written, in Fourier space,

dk

dxk

r k

2

1

6MP2 k ddx

2

d

dx

dk

dx

e 4xi2W

k , 2.23

d2k

dx 22

dk

dxk

r k2 1

2MP2 k ddx

2

ddx

dk

dxe 4xi

2 W

k ,2.24

d2k

dx 2

dk

dxe 4xi

22W

2k4

d

dx

dk

dx2

W

e4xi

2k0.

2.25

Combining Eqs. 2.23 and 2.24,

d2k

dx23

dk

dx

1

3MP2 ddx

2

d

dx

dk

dx2e 4xi

2W

k

0. 2.26

Imposing isocurvature initial conditions right at the onset ofthe radiation dominated epoch implies, for long-wavelengthmodes:

ki0, ki0, kik(i) , ki0.

2.27

In terms ofRk , the gauge-invariant spatial curvature pertur-bations,

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Rk kHHkkH

2H

, 2.28the initial conditions given in Eq. 2.27 imply Rk(i)0.From the Hamiltonian constraint, recalling Eq. 2.27, therelation between the initial density contrast and the initialcurvaton fluctuation can be obtained

r(i)

kW

i

k(i)

r(i)

, 2.29

where r(i) (k)r(k,i). Using the observation that, from

Eq. 2.21, the quantity r(k)4 k is conserved in the long-wavelength limit the initial value of the isocurvature modecan be related to the final values both of the Bardeen poten-tial and of the curvaton fluctuation. On this basis, usefulanalytical relations will be derived and later compared withthe numerical solutions. From Eq. 2.29 and taking into ac-count Eq. 2.27, the Hamiltonian constraint of Eq. 2.23leads to the relation

r(f)

k4k(f)W

i

k(i)

r(i)

, 2.30

where the superscript f denotes the final constant valueof the corresponding quantity. In the case r0, from Eq.2.6

2HMPr. 2.31

Then Eqs. 2.232.25 admit a solution with constantmode. Equation 2.25 and the combination of Eqs. 2.23

with 2.30 lead, respectively, to the following two relations:

2W

2k

(f)2

W

k

(f)0, 2.32

k(f)

1

6 W

i

k(i)

r(i)

, 2.33

which allow, in turn, k(f) to be determined in terms of k

(i) ,namely,

2W

2 fk(f)

1

3 W

iW

fk

(i)

r(i) . 2.34

Defining now the constant value of r as rc , the previousresults imply

k(f)

rc9

k(i)O rc, 2.35

where kk /MP . From Eq. 2.35 it also follows, usingEq. 2.32 together with Eq. 2.6

k(f)

2rc

9k

(i) . 2.36

Equations 2.35 and 2.36 imply that k(f) is always smaller

by a factor rc1) than the value of the curvaton fluctua-tions at the end of inflation. Hence, even assuming that thecurvaton fluctuations are amplified with flat spectrum, i.e.

k(i)

Hi/(2), the final value of the produced adiabaticmode k

(f) will always be phenomenologically negligible.

The value of k(f) will be even smaller. In the second place

the above equations are derived assuming that, in theasymptotic regime, the following relation holds:

1

2 W

2

f

1

4 2W

2

f

. 2.37

The relation 2.37 holds exactly in the case of exponentialpotentials of the type W()W0e

/. In this case the so-lution for is obtained by solving Eq. 2.4 in a radiation-

dominated background, with the result that

i1ln/i, 2.38

where 14 and rc4( /MP)2. This remark already

rules out a curvaton potential, which is purely exponentialdown to the moment of curvaton decay. In this case Eqs.2.35 and 2.36 imply the smallness of the obtained curva-ture perturbations. However, it is also suggestive to think ofthe case where the curvaton potential is not purely exponen-tial 19,20. The question would be, in this case, if the newfeatures of the potential allow a radically different dynamicsof the fluctuations.

III. EXPLICIT EXAMPLES

In order to discuss a physically realistic situation, con-sider as an example the following potential

WW0 cosh/ 1 , 3.1

which has been studied, for instance, in the context of quin-tessence models 21. In spite of this formal analogy, in thepresent case, the field will decay prior to big-bang nucleo-synthesis and it will not act as a quintessence field. With thiscaveat, also other quintessential potentials see, for instance,22 for a comprehensive review may be used, in the present

context, to construct physical models. The class of potentials3.1 has the property that when /1 the potential ad-mits solutions where the ratio between the curvaton energydensity and the radiation energy density is, momentarily,constant. On the other hand, when /1 the curvatonoscillates. Consider, for concreteness, the simplest case,namely,

WM4 cosh/1 , 3.2

where the oscillations are quadratic since, for /1,


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WM4

222. 3.3

Using the notation

MP,

MP,

M

MP,

Hi

MP, 3.4

the evolution of the curvaton is determined by the followingequation:

d2

dx2

d

dxe4x

4

2sinh/ 0. 3.5

The constraints of Eq. 2.1 imply, in the present case,

M4

2ei /Hi

2, i, iHi . 3.6

The ratio between the curvaton energy density and theradiation energy density can be expressed as

rx 1

6 d

dx

2

4

32e 4x cosh/1 . 3.7

The initial data obeying the conditions 3.6 for the evolu-tion of the homogeneous component of the curvaton field are

set during the inflationary epoch in such a way that i . Since the potential is essentially exponential, in thisregime the field will be swiftly attracted towards the trackingsolution, where the critical fraction of curvatons energy den-sity will be constant. In this regime the solution can be ap-proximated by

04x , 0 ln 8224

. 3.8When H2M4/2M22/2, i.e.

xm1

4 1ln 822

4 , 3.9

the curvaton will start oscillating in the minimum of its po-tential. During the tracking phase r(x)rtr4

21. Dur-

ing the oscillatory regime the energy density of the curvatonwill decrease as a3 so that

rx 42 aa m

, xx m . 3.10Let us now write, as a first step, the evolution equations

for the fluctuations in the case when the background after theend of inflation is suddenly dominated by radiation:

dk

dxk

r k

2

1

6 k ddx 2

d

dx

dk

dx

e 4x4

2sinh

/

k

, 3.11d2k

dx 22

dk

dxk

r k

2

1

2 k d

dx

2

d

dx

dk

dx

e 4x4

2sinh/ k , 3.12

d2k

dx2

dk

dxe4x

4

22cosh/k4

d

dx

dk

dx

24

2e 4xsinh/ k0, 3.13

d

dxr k4 k0. 3.14

Combining Eqs. 2.23 and 2.24

d2k

dx 23

dk

dx

1

3 k ddx 2

d

dx

dk

dx2e 4x

4

2sinh/ k0.

3.15

The evolution equations of and of the fluctuations can benumerically integrated. Analytical approximations can alsobe obtained. In Fig. 1 the numerical results are illustrated, fora typical set of parameters, in terms of r(x) as defined in Eq.3.7. It can be appreciated that after a phase where r0,the energy density of the will decrease, implying thatr(x)ex. In Fig. 2 the evolution of (x) is reported. The

FIG. 1. The numerical results for the evolution of r see Eq.

3.7 are reported.


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numerical results full line are compared with the analyticalapproximation dashed line. In fact, a useful analytical ap-proximation to the whole evolution can be obtained bymatching the solution 3.8, valid during the tracking regime,with the exact solution of the approximate potential valid inthe oscillating regime. In order to do so it is useful to write

the evolution equation for as

d2

dy 2

3

2y

d

d y0, ye 2x

2

2, 3.16

whose solution is

x ex/2AJ1/4 e 2x 2

2BY1/4 e2x

2

2 , xxm ,

3.17

where the numerical constants

A

exm/24Y1/4 e2xm22 e 2xm2Y5/4 e2xm2

2 4x mlog 822

4

4,

B

exm/2 4J1/4 e2xm22 e 2xm2J5/4 e2xm2

2 4xmlog 822

4

4, 3.18

have been obtained by continuous matching of the solutionsin x m . In more explicit terms

Ae1/825/8 2Y1/42e 2eY5/42e 1/25/41/4

1.701461/25/41/4,

Be1/825/8 2J1/42e 2eJ5/42e 1/25/41/4

3.984721/25/41/4. 3.19

According to Fig. 2, the analytical approximation, based onthe continuity of the solutions of Eqs. 3.8 and 3.17 com-pares very well with the numerical calculation.

Using the results for the evolution of (x) the amount ofthe fluctuations produced during the oscillating phase can beestimated. In Figs. 3 and 4, the numerical evolution for k

and Rk are reported. After a flat plateau corresponding to thephase where r(x) is constant, the adiabatic fluctuations grow.However, the final value of the adiabatic fluctuations com-puted, for instance, at the moment when decays will al-ways be very small. In the integrations reported in Figs. 3and 4 initial conditions are set according to Eqs. 2.27. Inorder to understand the smallness of the final value of theadiabatic fluctuations, recall that the asymptotic solution ofthe perturbation equations during the tracking can be writtenas

k(m)

2

9k

(i) , k(m)

4

92k

(i) , 3.20

FIG. 2. The full line illustrates the numerical result while the

dashed line shows the analytical approximation based on Eq. 3.17.

FIG. 3. The numerical evolution of k illustrated in the case

where M104MP , 108MP and Hi10

3M. Notice that the

constant value of k is correctly reproduced by the analytical esti-

mate given in Eq. 3.20.


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implying that

R k(m)

3

2k

1

2

dk

dx

1

3k

( i) . 3.21

The values obtained in Eqs. 3.20 and 3.21 are in excellentagreement with the amplitude of the flat plateau occurringprior to x m which is about 6.5 for the parameters chosen inFigs. 3 and 4. Different choice of parameters lead to thesame qualitative features in the evolution of the fluctuations.

During the oscillating regime

dRk

dx

W

k

r, rx 42e (xxm), xxm .

3.22

These solutions hold down to the moment of decay occurringat a typical curvature scale

Hd M2

3

MP2, 3.23

where it has been taken into account that the effective massis, during the oscillating phase M2/. Integrating the evolu-tion equation for Rk it can indeed be obtained using Eqs.3.203.22

R k(d)R k

(m)2

rd

k

(m)

3 1 8

3rdk(i) , 3.24

where rdr(xd) and x d is determined through Eq. 3.23.Since 1 to ensure that does not dominate already dur-ing the tracking phase and since rd1, Eq. 3.24 implies

that the final value R k(d) will be much smaller than k

(i)

/(2).

IV. MINIMAL INFLATIONARY SCALE

In the standard curvaton scenario the energy density of thecurvaton increases with time with respect to the energy den-sity of the radiation background. From this aspect of thetheoretical construction, a number of constraints can be de-rived; these include an important aspect of the inflationary

dynamics occurring prior to the curvaton oscillations,namely the minimal curvature scale at the end of inflationcompatible with the curvaton idea. It has been shown, in theprevious sections, that to have a phase of tracking curvaton,unfortunately, does not help. In the present section thebounds on the inflationary curvature scale will be reviewed,with particular attention to the case where the post-inflationary phase is not suddenly dominated by radiation

like in the case of quintessential inflation 17,9.

A. The standard argument

First the standard argument will be reviewed see 14 fora particularly lucid approach to this problem. Suppose, forsimplicity, that the curvaton field has a massive potentialand that its evolution, after the end of inflation, occurs duringa radiation dominated stage of expansion. The field startsoscillating at a typical scale Hmm and the ratio betweenthe curvaton energy density and the energy density of theradiation background is, roughly,

r ti

MP2

a

am , HHm . 4.1

When decays the ratio r gets frozen to its value at decay,i.e. r( t)r( td)rd for t td . Equation 4.1 then implies

mi

2

rdMP. 4.2

The energy density of the background fluid just before decayhas to be larger than the energy density of the decay prod-

ucts, i.e. r(td)Td4 . Since

r tdm2MP2 am

a d 4

, 4.3

the mentioned condition implies

m

Td m

MP1, 4.4

which can also be written, using Eq. 4.2, as

iMP

3

rd3/2 Td

MP . 4.5

Equation 4.5 has to be compared with the restrictions com-ing from the amplitude of the adiabatic perturbations, whichshould be consistent with observations. If decays beforebecoming dominant the curvature perturbations at the time ofdecay are

Rk td1

r

W

tdrd

ki

i. 4.6

Recalling that ki Hi/(2) the power spectrum of curva-

ture perturbations

FIG. 4. The numerical evolution ofRk is reported for the same

set of parameters discussed in Fig. 4.


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P1/2

rdHi

4i5105 4.7

implies, using Eq. 4.5 together with Eq. 4.2,

HiMP

104rd1/2 TdMP1/3

. 4.8

Recalling now that rd1 and mHi , the above inequalityimplies that, at most, Hi10

12MP if Td1 MeV is se-lected. This estimate is, in a sense, general since the specificrelation between Td and Hd is not fixed. The bound 4.8 canbe even more constraining, for certain regions of parameter

space, if the condition TdHd MP is imposed with Hdm3/MP

2 . In this case, Eq. 4.8 implies Hi2108mMP ,

which is more constraining than the previous bound for suf-ficiently large values of the mass, i.e. m104Hi . Thus, inthe present context, the inflationary curvature scale is boundto be in the interval 1012MPHi10

6MP .

B. Different post-inflationary histories

Consider now the case where the inflationary epoch is notimmediately followed by radiation. Different models of thiskind may be constructed. For instance, if the inflaton field isidentified with the quintessence field, a long kinetic phaseoccurred prior to the usual radiation-dominated stage of ex-pansion 17 see also 23 for some other references notdirectly related to the present calculation. The evolution of amassive curvaton field in quintessential inflationary modelshas been recently studied 9 in the simplest scenario wherethe curvaton field is decoupled from the quintessence fieldand it is minimally coupled to the metric. In order to bespecific, suppose that, as in 17, the inflaton potential, V()

is chosen to be a typical power law during inflation and aninverse power during the quintessential regime:

V4M4 , 0,

VM8

4M4, 0, 4.9

where is the inflaton self-coupling and M is the typicalscale of quintessential evolution. The potential of the curva-ton may be taken to be, for simplicity, quadratic. In thismodel the curvaton will evolve, right after the end of infla-tion, in an environment dominated by the kinetic energy of

. The curvaton starts oscillating at Hmm and becomesdominant at a typical curvature scale Hm(i/MP)

2. Dueto the different evolution of the background geometry, theratio r( t) will take the form

r tm2 iMP

2

aam

3

, HHm 4.10

to be compared with Eq. 4.1 valid in the standard case. Forttd , r(t) gets frozen to the value rd whose relation to i isdifferent from the one obtained previously see Eq. 4.2

and valid in the case when relaxes in a radiation dominatedenvironment. In fact, from Eq. 4.10,

mi

rd. 4.11

From the requirement

k tdm2MP

2 ama d 6

Td4, 4.12

it can be inferred, using Eq. 4.11, that

iMP

3/2

rd3/4 Td

MP . 4.13

Following the analysis reported in 9, the amount of pro-duced fluctuations can be computed. The spatial curvatureperturbation can be written, in this model, as

Rk

H

2 2 v

v

, 4.14

where

v

H, 4.15

v

H, 4.16

are, respectively, the canonically normalized fluctuations of and . As discussed in 9 the relevant evolution equations

can be written as

v3Hv1

a22v 2V

2

1

MP2

a3

t a

3

H 2 v

1

MP2a3

t a

3

H v0, 4.17

v3Hv1

a22v 2W

2

1

MP2

a 3

t a

3

H 2 v

1

MP2a 3

ta3

H

v

0. 4.18

Solving Eqs. 4.17 and 4.18 with the appropriate initial

conditions, and using that a3 2/H is constant during the ki-netic phase, Eq. 4.14 can be written as 9

Rk tH

kv

k

W

rd k

(i)

i . 4.19

Recalling that k(i) Hi/(2), the observed value of the

power spectrum, i.e. PR1/25105, implies


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HiMP

104 TdMP

2/3

. 4.20

The same approximations discussed in the standard case willnow be applied to the case of quintessential inflation. Sup-pose that Td1 MeV i.e. the minimal value compatiblewith nucleosynthesis considerations. Thus, from Eq. 4.20Hi10

19MP . This estimate has to be compared with Hi1012 MP which was obtained see Eq. 4.8 in the casewhen relaxes to its minimum in a radiation dominatedenvironment. Suppose now that the decay of is purelygravitational, i.e. m3/MP . If this is the case, as previ-

ously argued, one could also require that TdHdMP. Thisrequirement implies that for masses m10 TeV, Hi104m .

V. CONCLUDING REMARKS

In the present investigation the possibility that the ratiobetween the curvaton energy density and that of the domi-nant component of the background sources is constant during

a significant part in the evolution of the Universe. The vari-

ous estimates of this preliminary analysis seem to suggestthat if r0 for a sufficiently long time the obtained curva-ture perturbations present prior to curvaton decay are muchsmaller than the value required by observations. The possi-bility of having r0 down to sufficiently low curvaturescales would, on the other hand, be interesting to relax thebounds on the minimal inflationary curvature scale, which, inthe standard scenario where r0), is roughly Hi1012MP for the most optimistic set of parameters. If thisbound could be evaded inflation could take place, within theframework discussed in the present paper, also at muchsmaller curvature scales. In this sense, the presence of aphase r0 does not alleviate the problem. Furthermore, ifr0 for a sufficiently long time, the final amount of curva-ture perturbations gets drastically reduced.

In a related perspective the case of low-scale quintessen-tial inflation has been examined. In this case, r0 and thebackground geometry is kinetically dominated down to themoment of curvaton oscillations. The same argument, lead-ing to the standard bound on the minimal inflationary curva-ture scale shows, in this case, that the bounds are a bit re-

laxed and curvature scales Hi1019

MP become plausible.

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Massimo Giovannini- Tracking curvaton(s)?