PhysicsLettersA 177 (1993) 8—12North-Holland PHYSICSLETTERS A

Stringdilaton cosmologywith exponentialpotential

SalvatoreCapozziello,Ruggierode Ritis andClaudioRubanoDipartimentodi ScienzeFisiche,Universitàdi Napoli, and IstitutoNazionaledi FisicaNucleare,Sezionedi Napoli,Mostra d’Oltremarepad. 19, 80125Naples,Italy

Received15 December1992; revisedmanuscriptreceived8 April 1993; acceptedfor publication8 April 1993Communicatedby J.P.Vigier

Theeffectiveactionof a stringliketheoryis consideredin thecosmologicalcontext.We discussa nonminimallycoupledcos-mologicalmodel andwe find theexistenceofaNoethersymmetryin thepointlikeFriedman—Robertson—Walker(FRW) Lagrang-iandescribingthemodel: thisallowsoneto get interestingcosmologicalsolutions.

1. Introduction enceof matteror scalarfields exceptthe dilaton 0.This fact allowsus to discussa pointlikeLagrangian

The presenceof the dilaton, a scalarfield non- on the samegrounds of a nonminimally coupledminimally coupledwith the Ricci scalarR andthe model and to searchfor Noethersymmetrieswithmatterfields, is a peculiarfeaturethatallowsone to the method already describedin previous paperstreatthestring effectiveactionon the samegrounds [4,5]. We like to stressthat in what follows we doof a scalar—tensortheoryof gravity [11. For exam- not fix a priori any form for the potential V(0) butpie, it is possibleto reducethe low energystringac- the existenceof the Noethersymmetry imposesittion to a form very similar to the Jordan—Brans— mustbeexponentialor constant.Thefirst casecouldDicke (JBD) onebut, in the formercase,thedilaton be of some interest in connectionwith the superis coupledwith all the termsin the action (gravity symmetrybreakingdue,for example,tothe gaugino+matter)while in the JBD caseit is coupledonly condensation[2]; in thesecondwe recoverthescalewith gravity [21. factorduality, that is the invarianceof the Lagrang-

Recently [3], a wide classof scalar—tensortheo- ian (and of the dynamics)underthe transforma-rieshasreceivedthe attentionof physicistsandcos- tions a—~a= ±1/a and O—~0=0— ~In a.mologistsdue to the fact that, in this scheme,it ispossible(i) to implementnaturallyan inflationaryphasein the evolutionof the universeavoiding the 2. Theaction and the equationsof motionshortcomingof the old inflation; (ii) to inducethevaluesof the Newton constantGN andof the now- A stringlike four-dimensionaleffectiveaction,ne-adaysobservedcosmologicalconstantA that, very glectingthe matterandtheotherscalarfields exceptlikely, haveundergonean evolutionfrom the early 0, canbeepochto the presentone. We muststressthat theabovephenomenologyis strictly related to the pres-enceof a scalarfield 0 (the inflaton or the inducing d= $ d4x ~ e 2ø{~[R+4g~Ø.~Ø;p—2V(Ø)]}.field) whosedynamicscould be governedby a po- (1)tential V(Ø).

In this paper,weconsidera stringlikeeffectiveac- This action is nothingbut a particularcase of thetion in four dimensionsand in a standardcosmo- mostgeneralactioninvolving gravity nonminimallylogical scenario(flat FRWmetric)without the pres- coupledwith a scalarfield,

8 0375-9601/93/s06.00© 1993 ElsevierSciencePublishersB.V. All rightsreserved.

Volume 177,number! PHYSICSLETTERSA 31 May 1993

d= J~ ~ W(~)], that is

(2) 2

3(s) _6~+2Ø2+V(O)=0. (13)when we takethe positions a a

~=2e0, F(W)=AW2=~e20,

W(w)=e20V(O). (3) 3. The Noethersymmetries

As usual,the EinsteinequationsareIn orderto solveeqs. (9) and (10) satisfyingthe

F(~i)G~~= — ~T~—g,~,EF(W) +F(W);pv, (4) constraint(13),we look for a Noethersymmetryfor

with Lagrangian(8) like in refs. [4,5]. The transforma-tions on the tangentspaceof the dynamicalsystem

~ =R,4~— ~ (5) can be obtainedby constructingthevector

and a dcxa+d/JaX=a(a,Ø)~-+fl(a,Ø)~+-~-~- dt8~

~ . (6)(14)

TheKlein—GordonequationisA Noethersymmetryexistsif the condition

D~—RF’(~)+W’(yi)=0, (7)(15)

where0 is thebox operatorandtheprime indicatesthe derivativewith respectto w. This equation is holds. L~is the Lie derivativeof the Lagrangian2’equivalentto the contractedBianchi identity [45] along X. Equation (15) gives the systemof differ-

In a FRW flat metric, action (1) gives rise to a ential equationsLagrangiandensity äa 2~ 0 (16)

a—2a/J+2a——2a2’=e~

20[3â2a—6àa2q+2a3ç2~a3V(Ø)] , (8) äa

wherea(t) is thescalefactorof theuniverseandthe 3a—2afl—3 +2a =0, (17)dotis thetimederivative.TheEuler—Lagrangeequa-tions are äa ôa 8

2 . 6a—6a/J+3a —3 2a2~~1~+3a~=0,2~+(~_4_2~+2~2+V(O)=0, (9) 0 (18)

a ~aj a

and 3ctV(çb)+afl[V’(Ø)—2V(çb)]=0, (19)2 that allows one to determinethe vectorX. The po-

4~4ø2+ 12 ~~—6~ _6(?) tential V(Ø) is the solutionof the equationa a ~aj

V’3—2V”V’V+2v2v’—2V”V2=O, (20)+V’(Ø)—2V(Ø)=0. (10)

They canbe recoveredfrom (4) and (7) imposing obtained insertingeqs. (16), (17), and (19) intohomogeneityandisotropy. Thefirst order Einstein eq. (18). We have two particular interestingequationis obtainedfrom the energyfunctionasso- situations:ciatedwith the Lagrangian2’, (i) the system(l6)—( 19) is satisfiedwhen

a=/30~a~e

0,fl=J30a~e

0, (21)82’ 32’

E~=—-â+—~-Ø—2’, (11) and8a 8

by imposingthe condition V(Ø) =~. e~, (22)

E.~=0, (12) where

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Volume 177, number1 PHYSICSLETTERSA 31 May 1993

3 ~ x2+A(ii+v)e~=0, (33)ç=~(2—~), ~=—l±~/~ (23)

(34)(ii) otherwiseit is satisfiedwhen x

a=~f30, fl=/3~, V(Ø)=A, (24) E0 is a constant of the motion.

The systemof differential equations(33), (34)Pb andA are constants.

admits thegeneralsolutionsz(t) =c1 {c2 — [c3(t—t0)+c]c4}+zo , (35)

4. The solutionsand

The existenceof the vector X tells us that, inLagrangian(8), there mustexist a new couple of w(t)= ~ln[c3(t—t0)+c] (36)variables, one of them being cyclic [4,5]. That is we

wheresearch for new variables 2xz=z(a, 0) , w=w(a,0) , (25) c=e~~’°,c

1 = c2=e±2/~~0,

zbeing cyclic iff the contractions/Llo

=—+2, (37)i~(dW)0, i~(dz)=l (26) C

3———, c~

hold. Equations (26) admit the particular solutions and w0, z0 arethe initial data. Equation (32) gives

w=Ø—~’ln a, z=ij0a~e°, (27) the condition

where io=—~—e~)~. (38)

1 2(~+l)

‘in = — ~ ~ (28) The scalefactor iswhen the vector Xis given by (21) and the potential a(t)=’i~’( [c3(t—to)+c]”~ “~“

is (22). Lagrangian(8) assumesthe form ci{c2_Ec3(t_to)+c1c4}+zo) (39)

(29) where

where 2 (2)(~+l)Y1=~ ~ (40)

2(2~+1)(2—~) ~(2—~) (30) The scalarfield is

and 0(t) =Y2 ln(’io/zo) —Y2 ln(c1 {c2 — [c3(t—t0)+c]”})

4(~+l)(2_~)(5~2_8~+2) — ~ln[c3(t—t0)+c] , (41)3(~2+2)2

(31) where

Constants(30) and(31) assumevaluesdepending 2(~+1)on~=—l±\,

5.OfcourseLagrangian(29)doesnotY2 ~2+l (42)dependon z.Theenergyfunction associatedwith 2’ ~(2—

(43)Y3~ ~2+2

~ (32)Let us assume, as special case, that .~o=0. From (34),

The Euler—Lagrangeequationsare we get that w=w0=const,and from (32) that2=0,

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Volume177, number1 PHYSICSLETFERSA 31 May 1993

thatisthe potentialis zero.In this caseusingeq. (33) from (SOb), choosingp (a) = —2/3a (we are ex-we immediatelyfind eludingthe line a= 0 in the space(a, 0)~which is

quite obvious), we find that Z has to solve thez(t)=zo+~

0(t—to). (44)equation

Going backto the variables(a, 0)~we get2 8z 8~ 2

a(t) = zo+~o(t—to)\ 8a — _ (51)( ‘ioe~ ) ‘ (45) This equationhasthe particularsolution

wherea=—(s+~4)’,and 1

(52)Ø(t)=wo+ (z~—t~\ a

‘ioe~ (46) (of course, choosingp(a)=2/3a we would get

Finally,weremarkthat from (44), if weassumethat ~= — 1/a).

= 0 then the solutionsof system(33), (34) are It is clear that the i-variablecanbe chosenin aw= const,z= const,which give,goingbackto theold moregeneralway. The conclusionis the following:variables,the solutionsa= const,0= const [2]. thereexistsa Noethersymmetrywhich gives riseto

In the secondcase,that is eqs.(24), system(26) a new set of variables(z, w) which transformsthegives, asparticularsolutions,the transformations startingLagrangian(8) into a new Lagrangian(in

thiscase(48)) that is cyclic in z. Furthermore,wew=Ø—~ln a, z=ln a (47) canreconstructthe transformationa—Z= ±1/a us-

andLagrangian(8) becomes ing the NoethervectorX and requiring that ~ is avariableadaptedto the foliation given by X. Using

2’=e2’~’(—3~2+4~2—2A), (48) ~ we do not have the cyclicity in the transformed

which is cyclic in the variablez. If we use only the Lagrangianbut the duality invariance.first of transformations(47) we have

2’=e2~~’[—3(â/a)2+4~2~2A]‘ (49) 5. Discussion

which is invariantunderthe changea—p±1/a (scalefactor duality). The equationsof motion deduced The asymptoticbehaviorof eq. (39),when t—+ ~,

from (49) (or (48)) are solvedin ref. [2]. representsapolelikeinflation [7], a(t)~t~’°1,orNow the issueis how to connectthe scalefactor a power law expansion[8], a(t)~t’31,depending

duality (a discrete symmetry) with the Noether on the value c~= — 1 +~J~or ~2 = —1 —~J~respec-symmetry (a continuoussymmetry).Actually the tively. The 0 field is asymptoticallydecreasingor in-waywehavefollowedto find thechangeof variables creasingsothe nonminimalcouplingcanbe increas-(47), usingthe Noethersymmetry connectedwith ing or decreasingdependingon the behaviorof theX, can be generalized (see ref. [6]). That is, in potential. In theparticularcase.�~=0,wehavethatsearchingfor a new set of variables,which we call the scalefactor evolveslike a (t) t 095 in the ~ -

now (VP, ~),it is possible,insteadof therequirement case,and like a(t) ~t°” in the ~2-case.

expressedby system(26), to imposethe moregen- Thislast caseseemsto us particularlyinterestingeralconditions when it is comparedwith the Friedman-dustevo-

lution of the scalefactor (a(t) ~ t2’3 t°~6).

i~dw=0, (50a) Thetechniquewehaveshownenablesonetosearch

i~dZ=p (a), (SOb) forNoethersymmetriesinapointstringlikeLagrang-ian in which we havenot specifiedthe dilatonp0-

wherep(a) is a genericfunction of a. Technically tential a priori and,as output, an exponentialpo-speaking,we are searchingfor a new coupleof van- tential is meaningful. Actually, the Noethersym-ables(~,Z) adaptedtothefoliationgivenby X. From metryexistsalso in the caseV(0) =A = const,butthe(50a), we seethat a solution is si~=w= 0—~in a; vectorXandthesolutionshavequitedifferentforms.

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Volume177, number1 PHYSICSLETFERSA 31 May 1993

Furthermore,we haveshown that the scalefactor M. Gasperini,J. Maharanaand0. Veneziano,Phys.Lett. B

dualitycanberevisitedfromthe pointof view of the 272 (1991)277;K.A. Meissnerand0. Veneziano,Phys.Lett. B 267 (1991)

Noethersymmetry. This meansthat the vectorX33.

gives rise to variable transformations which makes [3] D. La andP.J.Steinhardt,Phys.Rev.Lett. 62 (1989)376;

the starting Lagrangian either cyclic or duality D. La, P.J. SteinhardtandE.W. Bertschinger,Phys.Let!. Binvariant. 231 (1989) 231;

A.D. Linde,Phys.Let!. B 238 (1990) 160;R.Holman,E.W. Kolb, S. VadasandY. Wang,Phys.Rev. D43(1991)995.

Acknowledgement [4] R. deRitis, G. Marmo,G. Platania,C. Rubano,P.ScudellaroandC.Stornaiolo,Phys.Rev.D 42 (1990) 1091;R. de Ritis, G. Platania,C. RubanoandR. Sabatino,Phys.

We like to thank our friendsG. Marmo, P. Scu-Lett.A 161 (1991)230;deilaroandA. Simonifor the fruitful discussionwe M. Demianski, R. de Ritis, 0. Marmo, G. Platania,C.

have had together. Rubano,P. Scudellaroand C. Stornaiolo,Phys.Rev. D 44(1991)3136.

[5] S.CapozzielloandR. deRitis, submittedto Class.QuantumGray.(1992); Phys.Let!. A 177 (1993) 1.

References [6] R. Abraham and J. Marsden,Foundationof mechanics(Benjamin,NewYork, 1978);G. Marmo,E.J. Saletan,A. SimoniandB. Vitale,Dynamical

[I] D.W. Sciama,Mon. Not. R. Astrom.Soc. 113 (1953)34;C.BransandR.H.Dicke, Phys.Rev. 124 (1961)925; systems.A differential geometricapproachto symmetryand

reduction(Wiley, New York, 1985).A. Zee,Phys.Rev.Let!. 42 (1979)417;

[7] F. LucchinandS. Matarrese,Phys.Rev.D 32 (1985)1316.L. Smolin, NucI.Phys.B 160 (1979)253;

[8] E.W. KoIb, D.LindleyandD. Seckel,Phys.Rev.D 30 (1984)S. Adler, Phys.Rev.Lett. 44 (1980) 1567. 1205;

[2] A.A. TseytlinandC.Vafa, Nucl. Phys.B 372 (1992)443; R.B.Abbott,S.M. BarrandS.D. Ellis,Phys.Rev.D 30 (1984)A.A. Tseytlin,DAMPT-1S-l992;

720.G.Veneziano,Phys.Let!. B 265 (1991)287;

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