Supergravity Inflation on the Brane

a r X i v :g r -q c /0204046v 2 25 N o v 2002

Supergravity In?ation on the Brane

M.C.Bento 1,2,O.Bertolami 1and A.A.Sen 3

1

Departamento de F′?sica,Instituto Superior T′e cnico Av.Rovisco Pais 1,1049-001Lisboa,Portugal

2

Centro de F′?sica das Interac?c ?o es Fundamentais,Instituto Superior T′e cnico 3

Centro Multidisciplinar de Astrof′?sica,Instituto Superior T′e cnico and

E-mail addresses:bento@sirius.ist.utl.pt;orfeu@cosmos.ist.utl.pt;anjan@x9.ist.utl.pt

(Dated:February 4,2008)

We study N =1Supergravity in?ation in the context of the braneworld scenario.Particular attention is paid to the problem of the onset of in?ation at sub-Planckian ?eld values and the ensued in?ationary observables.We ?nd that the so-called η-problem encountered in supergravity inspired in?ationary models can be solved in the context of the braneworld scenario,for some range of the parameters involved.Furthermore,we obtain an upper bound on the scale of the

?fth dimension,M 5～<10?3

M P ,in case the in?ationary potential is quadratic in the in?aton ?eld,φ.If the in?ationary potential is cubic in φ,consistency with observational data requires that M 5?9.2×10?4M P .

PACS numbers:98.80.Cq,98.65.Es

Preprint DF/IST-4.2002,FISIST/09-2002/CFIF

I.INTRODUCTION

In supersymmetric theories with a single supersymme-try generator (N =1),complex scalar ?elds are the low-est components,φa ,of chiral super?elds Φa .Masses for ?elds will be generated by spontaneous symmetry break-ing so that the only fundamental mass scale is the re-duced Planck mass,M =M P /

√M 2

+D ?terms ,

(1)

where F A ≡

?W

?ΦA

W

?ΦA ?Φ?B

.(2)

The K¨a hler function,K ,sets the form of the kinetic en-ergy terms of the theory while the superpotential,W ,determines the non-gauge interactions.For canonical ki-netic energy terms,K = A φ?A φA

,the potential takes the relatively simple form V =exp

A

φ?A φA

?φB

2?3|W |2M

+c 2

φ

M

n

.

(5)

The case where the second term is dominant and n =2,a situation typical of chaotic in?ationary models,has al-ready been analysed in Refs.[1,2],where it is shown that speci?c features of the braneworld scenario allow for current observational constraints to be successfully ac-counted for with a single scale at the superpotential level;hence,di?culties with higher order non-renormalizable terms can be quite naturally avoided since it is possible to achieve successful in?ation with sub-Planckian ?eld values.

In this work,we consider the case where the ?rst term is dominant and n =2or n =3;regarding the for-mer case,it is well known that a generic supergravity theory gives contributions of order ±H 2[3]to the in?a-ton mass squared whereas in?ation requires |m 2|?H 2,this is the so-called ηproblem.We will show that,in the braneworld scenario,this problem can be avoided,provided the D =5Planck mass satis?es the condition

M 5～<1016

GeV.This kind of analysis has recently been done for a particular supergravity F-term hybrid in?ation model [4],namely the one of Ref.[5].It is relevant to point out that hybrid in?ationary models [6,7]allow,as a result of the dynamics of two or more scalar ?elds,for successful realizations of the old in?ationary type models,but have,in some instances,di?culties in what concerns initial conditions.However,these problems are shown to

2 be naturally solved in braneworld scenarios in the case

where the potential is dominated by the mass term[8].

We shall also consider the case where the quadratic

term gets cancelled and,therefore,the in?ationary po-

tential is cubic inφ,as is the case of the supergrav-

ity natural in?ation model of Ref.[9].We show that

this model works in the braneworld scenario provided

M5?1.1×1016GeV.

II.REQUIREMENTS ON THE INFLATIONARY

POTENTIAL

We shall consider the?ve-dimensional brane scenario,

where one assumes that Einstein equations with a nega-

tive cosmological constant hold(an anti-De-Sitter space

is required)in D-dimensions and that matter?elds are

con?ned to the3-brane;then,the4-dimensional Einstein

equation is given by[10]:

Gμν=?Λgμν+8π

M35 2Sμν?Eμν,(6)

where Tμνis the energy-momentum on the brane,Sμνis a tensor that contains contributions that are quadratic in Tμνand Eμνcorresponds to the projection of the5-dimensional Weyl tensor on the3-brane(physically,for a perfect?uid,it is associated to non-local contributions to the pressure and energy?ux).In a cosmological frame-work,where the3-brane resembles our universe and the metric projected onto the brane is an homogeneous and isotropic?at Friedmann-Robertson-Walker(FRW)met-ric,the Friedmann equation becomes[10,11]:

H2=Λ

3M2P ρ+

4πa4,(7)

where?is an integration constant.The four and?ve-dimensional cosmological constants are related by

Λ=4π

3M35

λ2 ,(8)

whereλis the3-brane tension,and the four and?ve-dimensional Planck scales through

M P= 4πM35λ.(9)

Assuming that,as required by observations,the cos-mological constant is negligible in the early universe and since the last term in Eq.(7)rapidly becomes unim-portant after in?ation sets in,the Friedmann equation becomes

H2=

8π

2λ (10)

= 4πρ ,(11) with

ρc=

3

M2

P

.(12)

Hence,the new term inρ2is dominant at high energies, compared toλ1/4,i.e.ρ>ρc,but quickly decays at lower energies,and the usual four-dimensional FRW cosmology is recovered.

Consistently,we shall assume that the scalar?eld is con?ned to the brane,so that its?eld equation has the standard form

¨φ+3H˙φ=?dV

16π V′(2+V/λ)2,(14)η≡

M2P

V 1

M2P φfφi V2λ

dφ,(17)

in the slow-roll approximation.We see that,as a result of the modi?cation in the Friedmann equation,the ex-pansion rate is increased,at high energies,by a factor V/2λ.

The amplitude of scalar perturbations is given by[1] A2s? 512πV′2 1+V

where the right-hand side should be evaluated as the co-moving scale equals the Hubble radius during in?ation, k=aH.Thus the amplitude of scalar perturbations is increased relative to the standard result at a?xed value ofφfor a given potential.

The scale-dependence of the perturbations is described by the spectral tilt[1]

n s?1≡d ln A2s

A s 2?3M2P V

22λ

2

m2φ2,(21)

and assume that the?rst term is dominant.

In supergravity,e?ective mass squared contributions of?elds are given by[7]

1

M2

P

≈3H2(22)

since the horizon of the in?ationary De Sitter phase has a Hawking temperature given by T H=H/2π[3]. Contributions like the ones of Eq.(22)lead toη≡M2P V′′/8πV?2;however,the onset of in?ation requires η?1.Within the braneworld scenario,however,ηis modi?ed,at high energies,by a factorλ/V(see Eq.(15)). Hence,if the quantity,α≡V0/λ,is su?ciently large,this problem is automatically solved by the brane corrections, as in this case,η?4/αfor largeα.

We shall now see that a constraint on M5can be ob-tained from the requirement that the magnitude of the FIG.1:Contours of the in?ationary observables n s and r in theα,βplane.The requirement that the observational bounds on these quantities(cf.Eq.(27))be respected,leads to the shaded regions;after the condition that there is su?cient in?ation is applied,N≥70,only the dark grey region remains allowed.We also show contours corresponding to di?erent values of M5.

energy density perturbations ensuing from our in?ation-ary setup explains the anisotropies in the CMB radiation observed by COBE.

The number of e-foldings,N,in terms ofαis given by N=α 1φF +2πM4P(φ4I?φ4F) .(23)

Using the high energy approximation and V?V0in Eq.(18),we obtain for A2s

A2s?64πλ3m4φ

?

,(24)

whereφ?is the value ofφwhen scales corresponding to large-angle CMB anisotropies,as observed by COBE,left the Hubble radius during in?ation.For N?≈55,φI=φ?andφF=βM in Eq.(23),we get

φ??0.2βexp 220

4

critically depends on the value of the in?aton ?eld,φF ,and hence on β,at the time the instabilities arise.No-tice that these instabilities are quite necessary in order

to end in?ation as ?=64πφ2/αM 2

P <<1for α>>1and sub-Planckian ?eld values.Moreover,it is clear that the condition ??1cannot be met subsequently as the φ?eld is decreasing.

We also mention that the problem of initial conditions for

hybrid models,discussed in Ref.[8],can also be solved in this

model due to the brane corrections.

Inserting Eq.(25)into Eq.(24)and using the fact that the observed value from COBE is A s =2×10?5,we obtain for M 5:

M 35?

8π

A s

2

,as given by Eqs.(19)and (20),in terms of

parameters αand β.The shaded regions in Figure 1are obtained by requiring that the bounds resulting from lat-est CMB data from BOOMERANG [13],and MAXIMA [14]and DASI [15]

0.8

are satis?ed.Also shown is the contour corresponding to N =70(we have chosen the initial value of the in?a-ton ?eld to be sub-Planckian,φI =0.2M P )and,after applying the condition that there is su?cient in?ation,N ≥70,only the darker region remains allowed.This analysis clearly leads to a constraint on parameter α,α～>140.Finally,also in Figure 1,we have superposed contours of the scale M 5,from which we can derive an upper bound on this quantity

M 5～<10?3

M P .

(28)

Finally,we mention that ?(or instead V 0)in Eq.(5)can be estimated from the bound on the reheating tem-perature so as to avoid the gravitino problem (see Ref.[16]and references therein)

T RH ～<2×109GeV ,6×109

GeV ,for

m 3/2=1TeV ,10TeV ,(29)

leading to ?～<6.71×10?4

M P ,as the reheating tem-perature is given by [17]

T RH ?5.5×10

?1

?3

M

3

.(31)

As mentioned before,we shall assume that the ?rst term

is dominant.For instance,the model of Ref.[9]corre-sponds to precisely this case,with c 3=?4.We shall consider a generic c 3and adapt our results for this par-ticular example.

We start by computing the slow-roll parameters ?and η:

??18c 23

M 4,(32)

η??

12c 3

M

.(33)

where α≡

?4

18c 23

1/4

M (34)

while,from |η|?1,we obtain

φF ?

α

6|c 3|

1

φF

=

αM

|c 3|

M ,(37)

yielding,for α=26and c 3=?4:

φI <7.5×10?2M P .

(38)

In order to match CMB anisotropies as observed by

COBE,we compute the value ofφ,φ?,that corresponds to N=55:

φ?=

αM

75M6

P

α3 V

5400π2c23φ4?

,(40) which determines M5,using Eqs.(9)and(39):

M5=7.3×10?2 A s

c23 1/4M P.(42) Requiring that the reheating temperature,computed using Eq.(30)and the above result,obeys the bounds necessary to avoid the gravitino problem,Eq.(29),leads, in turn,to the following bounds onα(for|c3|=4)

α～<3.4,14.7,

for m3/2=1TeV,10TeV.(43) Notice that for such low values ofα,the relevant prescrip-tion for the end of in?ation is the one given by Eq.(33). Moreover,we?nd that,within our approximations,n s does not depend onαand we get,for|c3|=4,

n s?0.93(44) and

r=2.5×10?8

α3

(2000)259.

[13]https://www.sodocs.net/doc/ad606675.html,ter?eld Pryke,et al.,“A measurement by

BOOMERANG of multiple peaks in the angular power spectrum of the cosmic microwave background”,as-tro/ph0104460.

[14]A.T.Lee,et al.,“A High Spatial Resolution Analy-

sis of the MAXIMA-1Cosmic Microwave Background

Anisotropy Data”,astro/ph0104459.

[15]C.Pryke,et al.,“Cosmological Parameter Extraction

from the First Season of Observations with DASI”,as-tro/ph0104490.

[16]M.C.Bento,O.Bertolami,Phys.Lett.B384(1996)98.

[17]G.G.Ross,S.Sarkar,Nucl.Phys.B461(1995)597.