We obtain a new class of exact cosmological solutions for multi-scalar fields
with exponential potentials. We generalize the assisted inflation solutions
previously obtained, and demonstrate how they are modified when there exist
cross-couplings between the fields, such as occur in supergravity inspired
cosmological models.Comment: 5 page

A.B. Burd

A.R. Liddle

Anupam Mazumdar

C. Wetterich

D. Wands

E. J. Copeland

E.J. Copeland

F. Lucchin

H. Reall

J.J. Halliwell

K. Malik

M. Bremer

N. J. Nunes

P. Kanti

P.G. Ferreira

S.W. Hawking

T. Barreiro

English

arXiv

We obtain a new class of exact cosmological solutions for multi-scalar fields with exponential potentials. We generalize the assisted inflation solutions previously obtained, and demonstrate how they are modified when there exist cross-couplings between the fields, such as occur in supergravity inspired cosmological models

J. Copeland, E.

Mazumdar, Anupam

J. Nunes, N.

Lancaster E-Prints

PHYSICAL REVIEW D, VOLUME 60, 083506Generalized assisted inflationE. J. CopelandCentre for Theoretical Physics, University of Sussex, Falmer, Brighton BN1 9QJ, United KingdomAnupam MazumdarAstrophysics Group, Blackett Laboratory, Imperial College, London SW7 2BZ, United KingdomN. J. NunesCentre for Theoretical Physics, University of Sussex, Falmer, Brighton BN1 9QJ, United Kingdom~Received 22 April 1999; published 15 September 1999!We obtain a new class of exact cosmological solutions for multiscalar fields with exponential potentials. Wegeneralize the assisted inflation solutions previously obtained, and demonstrate how they are modified whenthere exist cross couplings between the fields, such as occur in supergravity inspired cosmological models.@S0556-2821~99!04216-2#PACS number~s!: 98.80.CqI. INTRODUCTIONScalar field theory has become the generic playground forbuilding cosmological models related to particle physics, inparticular for obtaining inflationary cosmologies. One suchclass of models involves exponential potentials@exp(A16p/pmPl2 f)# , which lead to power law inflation, a}tp, with p.1, for sufficiently flat potentials @1–3#. A num-ber of related features have also been discovered for suchpotentials: in a universe containing a perfect fluid and such ascalar field, then for a wide range of parameters the scalarfield ‘‘mimics’’ the perfect fluid, adopting its equation ofstate @4–6# and leading to attractor scaling solutions at latetime @7#. These solutions offer a plausible mechanism forstabilizing the dilaton field in models of gaugino condensa-tion arising in supersymmetry breaking @8#.It is generally assumed that even if there are many scalarfields present, only one of them will dominate the dynamicsand roll down the potential slowly. However, recentlyLiddle, Mazumdar, and Schunck ~LMS! @9# have demon-strated in a particular example that multiple scalar fields,each with an exponential potential, can lead to inflationarysolutions, even if the individual field potentials are too steepfor inflation. There exists a cumulative effect of all the fieldsthat can give rise to inflationary behavior—a result theytermed ‘‘assisted inflation.’’ Malik and Wands have demon-strated that the associated attractor solution could be identi-fied through a rotation in field space, with a hybrid modelwhere the vacuum energy had an exponential dependenceupon the dilaton field @10#. Multiple exponential potentialsdo arise in modern Kaluza-Klein theories. Indeed, they are anatural outcome of the compactification of higher dimen-sional theories down to 311 dimensions. With this in mind itis worth investigating such potentials in a bit more detail.Indeed, Kanti and Olive have recently proposed a possiblerealization of assisted inflation based on the compactificationof a five-dimensional Kaluza-Klein model @11#. It also raisesthe question, could inflation arise out of the 11-dimensionalsupergravity models compactified on squashed seven spheresfor example? Such models have been investigated by a num-0556-2821/99/60~8!/083506~5!/$15.00 60 0835ber of authors as low-energy cosmologies from string or Mtheory @12–15#. Most ‘‘realistic’’ models of dimensional re-duction lead to steep potentials, which generally do not leadto inflation. In this paper we consider a more general class ofexponential potentials, which can include those generallyfound in supergravity compactifications, and obtain exactcosmological solutions for them. In particular we demon-strate how difficult it is to obtain assisted inflation whenthere exist cross couplings between the scalar fields in thepotential, a result also discussed in @14# and @11#. We firstrecall the model discussed by LMS, before generalizing theirpotential to exponentials involving cross-coupling terms anddemonstrating that the attractor behavior of the scalar fieldsstill exists, leading to scaling solutions for the generalizedpotential. We then turn our attention to the case of potentialsinvolving multiple exponential terms containing the samescalar fields, and relate the solutions to those arising in su-pergravity models.II. THE DYNAMICS OF ASSISTED INFLATIONLiddle et al. @9#, considered n scalar fields, f i,i51, . . . n , each with exponential potentials decoupled fromeach other:Vi~f i!5V0exp~a if i!, ~1!where a i is the slope of the individual field with dimensionsof the inverse Planck mass. Although the fields are not di-rectly coupled through the potential, they are coupledthrough the Friedmann equation, which implies that the com-bined role of the fields affect the expansion rate of the uni-verse:H258p3mPl2 (i51n FVi~f i!112f˙ i2G , ~2!f¨ i523Hf˙ i2dVi~f i!df i, ~3!©1999 The American Physical Society06-1E. J. COPELAND, ANUPAM MAZUMDAR, AND N. J. NUNES PHYSICAL REVIEW D 60 083506where H5a˙ /a is Hubble’s constant and a is the scale factorof the flat Friedmann-Robertson-Walker ~FRW! universe.The solution to this is the modified power law @9#a~ t !}tp, ~4!where p is given byp516pmPl2 (i51n 1a i2 . ~5!Inflationary solutions exist provided p.1, hence even ifeach of the a i’s are too steep to individually satisfy the con-dition for inflation, as long as n is large enough, the inequal-ity p.1 can be satisfied. These solutions, and the inflation-ary ones we shall present below are eternal, they do notpossess an exit from the inflationary epoch. Realistic modelswould of course have to possess such an exit in order to enterthe radiation- and matter-dominated epochs of our universe.The particular example of Eq. ~5! suggests that it is worthinvestigating whether or not such assisted inflation existswith more general exponential potentials. An alternative ap-proach with interesting results has been adopted in @11#,where they have applied the assistance method to the case ofpolynomial scalar potentials.Exponential potentials with coupled scalar fieldsTo begin with we consider the most natural generalizationof the single field exponential, namely the case of twocoupled fields:V~f ,c!5V0eaf1bc , ~6!where a and b are the slopes for the fields f and c . We seefrom Eq. ~2! that dimensionally, the right-hand side shoulddecrease as t22, because H2}t22}Vi(f i). We further as-sume that, for our potential Eq. ~6!,eaf5katc, ebc5kbt22c, ~7!where ka , kb are dimensional constants and c is a dimen-sionless constant. Substituting the power-law solution Eq. ~4!into Eq. ~2! we obtainp25H2t258p3mPl2 FV0kakb112 S ca D2112 S 22cb D2G , ~8!which when coupled with Eq. ~3! for the f and c fields leadstoV0kakb5~3p21 !ca2, V0kakb5~3p21 !~22c !b2, ~9!henceV0kakb52~3p21 !a21b2. ~10!08350Using Eqs. ~9! and ~10! in Eq. ~8!, we obtain a simple scalingsolution between the two fields:p516pmPl21a21b2, ~11!S f˙c˙D 25S ab D 2. ~12!An important problematic feature for inflationary solutionsconfronts us immediately in Eq. ~11!, namely the couplingbetween the two fields reduces the rate of expansion of theuniverse, a point also made in @14,11#. We will return to thispoint again later. An alternative method which would alsolead to Eq. ~11! is described in @10# in terms of field rota-tions, which results in the introduction of two orthogonalfields, one of which is massless and the other posseses anexponential potential.Having demonstrated that it is possible to obtain a scalingsolution without using slow roll approximations, we nowgeneralize this simple case to include an arbitrary number offields and exponential terms making up the overall potential.General exponential potentials with coupled scalar fieldsWe now consider a potential where we have multiple sca-lar fields but their corresponding exponential potentials cancontain arbitrary combinations of the fields with differentslopes. The potential we will consider isV5(s51nVs5V0(s51nexpS (j51msas jfs jD , ~13!with the corresponding Friedmann equationH258p3mPl2 FV1(s51n(j51ms 12f˙s j2 G , ~14!where, from now on q ,r ,s stands for index terms in thepotential and i , j ,k ,l stands for field indexes, hence, fs jstands for the j th field in the sth potential term. In otherwords, there are a total of (s51n ms fields distributed ingroups of ms through the terms of the potential. We obtainthe solution to this problem by generalizing the assumptionEq. ~7!. There exists an attractor region with a power-lawsolution, which from Eq. ~14!, dimensionally satisfies H2}t22}Vi . Hence, we writeeas jfs j5ks jtcs j, ~15!(j51mscs j52, ~16!where ks j are dimensional and cs j are dimensionless con-stants, respectively. Equation ~15!, coupled with the equa-tions of motion6-2GENERALIZED ASSISTED INFLATION PHYSICAL REVIEW D 60 083506f¨ s j13Hf˙ s j1]V]fs j50, ~17!result in~3p21 !cs j5as j2 V0)k51msksk , ~18!from which we find, using Eqs. ~16! and ~17!:V0)k51msksk52~3p21 !(j51msas j2,S cs jas j D254as j2S (k51msask2 D 2 . ~19!When substituted into Eq. ~14! this leads to a key result, thegeneralization of the original assisted inflation result givenby Eq. ~5!:p516pmPl2 (s51n 1(j51msas j2. ~20!We also note that the generalization of the scaling solutionfound in Eq. ~11!, quickly follows for the case of any twoscalar fields, fs j and fql :S f˙ s jf˙ qlD 25S as jaqlD 2S (i51mq aqi2(k51msask2 D 2 . ~21!It is directly evident that Eqs. ~20! and ~21! reduce to Eq. ~5!for the example of n exponential terms each containing justone field, and Eq. ~11! for the case of one exponential termbut containing two fields. We again see the inhibiting affectthat multiplicative coupling of the fields ~i.e., ms.1) has forobtaining inflationary solutions. However, in this case, aswith the original version of assisted inflation, this can becompensated for if there are enough exponential termspresent in the potential ~i.e., if n is large enough! @9#.There is another feature of the potentials we have beendiscussing so far that makes them rather unphysical in gen-eral. We have been demanding that any two fields presentcannot be the same ~i.e., fs jÞfql). In other words they canonly appear once in the full potential. Nearly all realisticmodels which emerge from compactifications arising in su-pergravity models have the same field appearing in at leasttwo separate exponential terms. In the following section, weturn our attention to this case.08350III. EXPONENTIAL POTENTIALS INSPIREDBY SUPERGRAVITY MODELSTo set the scene, we generalize Eq. ~6! to the case wherethe scalar field potential takes the following form:V~f1 ,f2!5z1ea11f11a12f21z2ea21f11a22f2, ~22!where, as j , are dimensional constants which can take anyreal value and zs.0. The occurrence of such forms of thepotential are quite common in dimensionally reduced super-gravity models @13,14#. Remarkably we can solve this sys-tem to obtain scaling solutions in a manner analogous tothose already presented in Eqs. ~7!–~11! obtaining the uniquelate time scaling solution for the fields f1 and f2,p516pmPl2 F ~a212a11!21~a222a12!2~a11a222a21a12!2 G . ~23!This simple result reduces to the particular cases we havealready investigated when the appropriate limits are taken.For example, when a215a1250, we reproduce the result Eq.~5!. The equivalent of the assisted inflation result Eq. ~11!follows by setting a11a211a12a2250, in which case we findp516pmPl2 F 1a112 1a122 1 1a212 1a222 G . ~24!We shall now generalize the potential to n such exponen-tial potentials and m combinations of linear fields in the ex-ponent @explicitly calculating for the simple case of 2 terms32 fields of Eq. ~22!#. The generalized Eq. ~22! is thenV5(s51nzsexpS (j51mas jf jD . ~25!Of course, we are allowing here for the possibility that as j50 for some combination of s j . We assume that for latetimes the fields have an attractor solution, given byzsexpS (j51mas jf jD 5 kst2, ~26!and, following Eqs. ~15! and ~16! we writef j5a j2cs jas jln t , ~27!where, a j is a constant depending on the initial conditionsand ( j51m cs j52, s51, . . . ,n . Substituting Eq. ~27! into theequation of motion Eq. ~3!, we obtain the constraint equationfor the j th field, which follows from assuming the existenceof a power-law solution~3p21 !cs jas j5 (q51naq jkq . ~28!Again, using ( j51m ci j52 we obtain6-3E. J. COPELAND, ANUPAM MAZUMDAR, AND N. J. NUNES PHYSICAL REVIEW D 60 083506(j51mas jS (q51naq jkqD 52~3p21 !, ~29!which is equivalent to writing(q51nAsqkq52~3p21 !, ~30!whereAsq5(j51mas jaq j . ~31!Since s is a free index, we have a set of n equations that canbe written asAk52~3p21 !, ~32!where A is the n3n matrix with elements Asq and k5(k1 , . . . ,kn) a column vector. For the 232 case of Eq.~22! we obtainA5S a112 1a122 a11a211a12a22a21a111a22a12 a212 1a222 D . ~33!The solution to this system isk5A212~3p21 !, ~34!withA215ACOFTdet A , ~35!where ACOFT is the transpose of the cofactor matrix of A. Tosimplify notation we will write B[ACOFT and the sum of theelements in row s of B as Bs[(q51n Bsq , hence, each ks isks52~3p21 !det A Bs. ~36!For the 232 case this yieldsB5S a212 1a222 2a11a212a12a222a21a112a22a12 a112 1a122 D , ~37!k152~3p21 !a212 1a222 2a11a212a12a22~a11a222a12a21!2 , ~38!k252~3p21 !2a21a112a22a121a112 1a122~a11a222a12a21!2 . ~39!From Eqs. ~27! and ~28! the late time ratio between thekinetic terms of two different fields becomes08350S f˙ jf˙ lD 25S (q51n aq jBq(r51narlBrD 2 . ~40!Substitution of Eqs. ~36! and ~28! into the Friedmannequation yieldsp258p3mPl2 F (s51nks112 (j51m S cs jas j D2G58p3mPl2F 2~3p21 !(s51n Bsdet A 12(j51m S (q51n aq jBqdet AD 2G.After some algebra, we obtain the simple result for p as theratio between the sum of all the elements in the cofactormatrix of A and its determinantp516pmPl2(sn(qnBsqdet A . ~41!The reader should have no problem showing that for the 232 case this reduces to Eq. ~23!.It is instructive to rewrite Eq. ~41! in another form. FromEqs. ~30!, ~36!, and ~41!,p516pmPl2 (s51n 1(q51nAsqkq /ks, ~42!and using Eq. ~31! with q5s and Eq. ~36! we obtainp516pmPl2 (s51n 1(j51mas j2 1(qÞsnAsqBq/Bs. ~43!A number of points need to be made about Eq. ~43!. It issimilar in form to Eq. ~20!, which should not be too surpris-ing, the additional terms in the denominator arising from thefact that we have allowed for fields to appear more than oncein the potential, hence leading effectively to ‘‘self-interaction’’-type contributions. Indeed if these terms wereturned off we would reproduce the result in Eq. ~20!. In the232 case, it is the constraint leading to Eq. ~24!. Due to thepresence of these ‘‘self-interaction’’ terms, p could increaseabove the value in Eq. ~20! if there happened to be a com-bination of positive and negative slopes in Eq. ~43!.An issue emerges when considering these more compli-cated potentials. For the two field, two term case of Eq. ~22!,if a11.a12 then, a necessary ~but not sufficient! conditionfor the second term to be comparable to the first term at latetimes is, a21,a22 . By comparable we mean that the poten-tial terms reach a constant ratio. If this were not the case,6-4GENERALIZED ASSISTED INFLATION PHYSICAL REVIEW D 60 083506then one of the two terms would quickly dominate the over-all dynamics. One way to check if a combination of terms ina given potential will be comparable at late times is to useEq. ~36! to obtain the ratios ks /kq for these terms. From Eq.~26! it follows that for consistency we require them to bepositive, with p.1/3. In general, the surviving terms ~i.e.,those which remain comparable! will be the ones with thesmallest slopes, corresponding to the largest values of p.IV. CONCLUSIONSIn this paper we have derived a new class of exact cos-mological solutions involving exponential potentials. In do-ing so we have been able to generalize the assisted inflationsolutions discussed by LMS @9# to the case of multiple ex-ponential terms involving many fields. Such potentials aremore likely to arise in realistic models of particle physicswhere individual fields will occur in a number of separateexponential terms, leading to cross couplings between theterms. In general, it transpires that it is more difficult toobtain assisted inflation in such models, the fields in any one08350exponential term tend to conspire to act against one anotherrather than assist each other, a result also noticed in @14,11#.This is the real reason why such models tend to fail to pro-duce inflationary solutions in supergravity models compacti-fied on squashed seven spheres @14#. We also investigatedthe case where a number of exponential terms contained thesame scalar fields and demonstrated that a number of novelfeatures emerged, including the possibility of increasing therate of expansion when there exists a mixture of positive andnegative slopes in the potential.ACKNOWLEDGMENTSWe would like to thank Andrew Liddle, Karim Malik,David Wands, and Orfeu Bertolami for useful discussions.E.J.C. was supported by PPARC and is particularly gratefulto David Wands for conversations on the nature of the gen-eralized solutions. N.J.N. was supported by FCT ~Portugal!under Contract No. PRAXIS XXI BD/15736/98 and A.M.was supported by the INLAKS and the ORS.@1# F. Lucchin and S. Matarrese, Phys. Rev. D 32, 1316 ~1985!.@2# J.J. Halliwell, Phys. Lett. B 185, 341 ~1987!.@3# A.B. Burd and J.D. Barrow, Nucl. Phys. B308, 929 ~1988!.@4# C. Wetterich, Nucl. Phys. B302, 668 ~1988!.@5# D. Wands, E.J. Copeland, and A.R. Liddle, Ann. ~N.Y.! Acad.Sci. 688, 647 ~1993!.@6# P.G. Ferreira and M. Joyce, Phys. Rev. D 58, 023503 ~1998!.@7# E.J. Copeland, A.R. Liddle, and D. Wands, Phys. Rev. D 57,4686 ~1998!.@8# T. Barreiro, B. de Carlos, and E.J. Copeland, Phys. Rev. D 58,083513 ~1998!.@9# A.R. Liddle, A. Mazumdar, and F.E. Schunck, Phys. Rev. D58, 061301 ~1998!.@10# K. Malik and D. Wands, Phys. Rev. D 59, 123501 ~1999!.@11# P. Kanti and K.A. Olive, Phys. Rev. D 60, 043502 ~1999!.@12# A. Lukas, B. Ovrut, and D. Waldram, hep-th/9806022,hep-th/9902071.@13# M. Bremer, M.J. Duff, H. Lu, C.N. Pope, and K.S. Stelle,Nucl. Phys. B543, 321 ~1999!.@14# S.W. Hawking and H. Reall, Phys. Rev. D 59, 023502 ~1999!.@15# H. Reall, Phys. Rev. D 59, 103506 ~1999!.6-5

Generalized assisted inflation

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Generalized Assisted Inflation

Generalized Assisted Inflation

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