Generalized assisted inflation

Generalized assisted inflation

E. J. Copeland

Centre for Theoretical Physics, University of Sussex, Falmer, Brighton BN1 9QJ, United Kingdom

Anupam Mazumdar

Astrophysics Group, Blackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom

N. J. Nunes

Centre for Theoretical Physics, University of Sussex, Falmer, Brighton BN1 9QJ, United Kingdom

共Received 22 April 1999; published 15 September 1999兲

We obtain a new class of exact cosmological solutions for multiscalar 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. 关S0556-2821共99兲04216-2兴

PACS number共s兲: 98.80.Cq

I. INTRODUCTION

Scalar field theory has become the generic playground for building cosmological models related to particle physics, in particular for obtaining inflationary cosmologies. One such class of models involves exponential potentials 关exp(

冑

16␲/ pmPl

2_␾

)兴, which lead to power law inflation, a ⬀tp, with p⬎1, for sufficiently flat potentials关1–3兴. A num-ber of related features have also been discovered for such potentials: in a universe containing a perfect fluid and such a scalar field, then for a wide range of parameters the scalar field ‘‘mimics’’ the perfect fluid, adopting its equation of state 关4–6兴and leading to attractor scaling solutions at late time 关7兴. These solutions offer a plausible mechanism for stabilizing the dilaton field in models of gaugino condensa-tion arising in supersymmetry breaking 关8兴.

It is generally assumed that even if there are many scalar fields present, only one of them will dominate the dynamics and roll down the potential slowly. However, recently Liddle, Mazumdar, and Schunck 共LMS兲 关9兴 have demon-strated in a particular example that multiple scalar fields, each with an exponential potential, can lead to inflationary solutions, even if the individual field potentials are too steep for inflation. There exists a cumulative effect of all the fields that can give rise to inflationary behavior—a result they termed ‘‘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 model where the vacuum energy had an exponential dependence upon the dilaton field 关10兴. Multiple exponential potentials do arise in modern Kaluza-Klein theories. Indeed, they are a natural outcome of the compactification of higher dimen-sional theories down to 3⫹1 dimensions. With this in mind it is worth investigating such potentials in a bit more detail. Indeed, Kanti and Olive have recently proposed a possible realization of assisted inflation based on the compactification of a five-dimensional Kaluza-Klein model关11兴. It also raises the question, could inflation arise out of the 11-dimensional supergravity models compactified on squashed seven spheres for example? Such models have been investigated by a

num-ber of authors as low-energy cosmologies from string or M theory关12–15兴. Most ‘‘realistic’’ models of dimensional re-duction lead to steep potentials, which generally do not lead to inflation. In this paper we consider a more general class of exponential potentials, which can include those generally found in supergravity compactifications, and obtain exact cosmological solutions for them. In particular we demon-strate how difficult it is to obtain assisted inflation when there exist cross couplings between the scalar fields in the potential, a result also discussed in 关14兴 and 关11兴. We first recall the model discussed by LMS, before generalizing their potential to exponentials involving cross-coupling terms and demonstrating that the attractor behavior of the scalar fields still exists, leading to scaling solutions for the generalized potential. We then turn our attention to the case of potentials involving multiple exponential terms containing the same scalar fields, and relate the solutions to those arising in su-pergravity models.

II. THE DYNAMICS OF ASSISTED INFLATION

Liddle et al. 关9兴, considered n scalar fields, ␾_i,i

⫽1, . . . n, each with exponential potentials decoupled from each other:

Vi共␾i兲⫽V0exp共␣i␾i兲, 共1兲

where␣iis the slope of the individual field with dimensions

of the inverse Planck mass. Although the fields are not di-rectly coupled through the potential, they are coupled through the Friedmann equation, which implies that the com-bined role of the fields affect the expansion rate of the uni-verse:

H2⫽ 8␲ 3m_Pl2 i

兺

⫽1

n

冋

Vi共␾i兲⫹

1

2␾˙i 2

册

, 共2兲

␾¨_i⫽⫺3H␾˙_i⫺dVi共␾i兲

where H⫽a˙/a is Hubble’s constant and a is the scale factor of the flat Friedmann-Robertson-Walker 共FRW兲 universe. The solution to this is the modified power law关9兴

a共t兲⬀tp, 共4兲

where p is given by

p⫽16␲ m_Pl2 i

兺

_⫽1

n

1 ␣i

2. 共5兲

Inflationary solutions exist provided p⬎1, hence even if each of the␣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 not possess an exit from the inflationary epoch. Realistic models would of course have to possess such an exit in order to enter the radiation- and matter-dominated epochs of our universe. The particular example of Eq. 共5兲 suggests that it is worth investigating whether or not such assisted inflation exists with more general exponential potentials. An alternative ap-proach with interesting results has been adopted in 关11兴, where they have applied the assistance method to the case of polynomial scalar potentials.

Exponential potentials with coupled scalar fields

To begin with we consider the most natural generalization of the single field exponential, namely the case of two coupled fields:

V共␾,␺兲⫽V₀e␣␾⫹␤␺, 共6兲 where␣ and␤are the slopes for the fields␾and␺. We see from Eq. 共2兲that dimensionally, the right-hand side should decrease as t⫺2, because H2⬀t⫺2⬀Vi(␾i). We further

as-sume that, for our potential Eq.共6兲,

e␣␾⫽k␣ tc, e

␤␺_⫽ k␤

t2⫺c, 共7兲

where k_␣, k_␤ are dimensional constants and c is a dimen-sionless constant. Substituting the power-law solution Eq.共4兲 into Eq.共2兲we obtain

p2_⫽H2_t2_⫽ 8␲

3m_Pl2

冋

V0k␣k␤⫹ 1

2

冉

c ␣

冊

2

⫹1

2

冉

2⫺c

␤

冊

2

册

, 共8兲

which when coupled with Eq.共3兲for the␾and␺ fields leads to

V0k␣k␤⫽

共3 p⫺1兲c

␣2 , V0k␣k␤⫽

共3 p⫺1兲共2⫺c兲 ␤2 , 共9兲

hence

V0k␣k␤⫽

2共3 p⫺1兲

␣2_⫹␤2 . 共10兲

Using Eqs.共9兲and共10兲in Eq.共8兲, we obtain a simple scaling solution between the two fields:

p⫽16␲ m_Pl2

1

␣2_⫹␤2, 共11兲

冉

␾˙ ␺˙

冊

2

⫽

冉

␣_␤

冊

2

. 共12兲

An important problematic feature for inflationary solutions confronts us immediately in Eq. 共11兲, namely the coupling between the two fields reduces the rate of expansion of the universe, a point also made in关14,11兴. We will return to this point again later. An alternative method which would also lead to Eq. 共11兲 is described in关10兴 in terms of field rota-tions, which results in the introduction of two orthogonal fields, one of which is massless and the other posseses an exponential potential.

Having demonstrated that it is possible to obtain a scaling solution without using slow roll approximations, we now generalize this simple case to include an arbitrary number of fields and exponential terms making up the overall potential.

General exponential potentials with coupled scalar fields

We now consider a potential where we have multiple sca-lar fields but their corresponding exponential potentials can contain arbitrary combinations of the fields with different slopes. The potential we will consider is

V⫽

兺

s_⫽1

n

Vs⫽V0

兺

s_⫽1

n

exp

冉

兺

j_⫽1

m_s

␣s j␾s j

冊

, 共13兲

with the corresponding Friedmann equation

H2⫽ 8␲

3m_Pl2

冋

V⫹s

兺

⫽1

n

兺

j⫽1

m_s

1

2␾˙s j

2

册

, 共14兲

where, from now on q,r,s stands for index terms in the potential and i, j ,k,l stands for field indexes, hence, ␾s j

stands for the jth field in the sth potential term. In other words, there are a total of 兺_sn_⫽1m_s fields distributed in groups of m_s through the terms of the potential. We obtain the solution to this problem by generalizing the assumption Eq. 共7兲. There exists an attractor region with a power-law solution, which from Eq. 共14兲, dimensionally satisfies H2 ⬀t⫺2⬀Vi. Hence, we write

e␣s j␾s j⫽ks j

tcs j

, 共15兲

兺

j⫽1

m_s

c_{s j}⫽2, 共16兲

where ks j are dimensional and cs j are dimensionless

␾¨_{s j⫹3H␾}˙_{s j⫹} ⳵V ⳵␾s j

⫽0, 共17兲

result in

共3 p⫺1兲cs j⫽␣s j

2 V0

兿

k⫽1

m_s

ksk, 共18兲

from which we find, using Eqs.共16兲and共17兲:

V₀

兿

k⫽1

m_s

k_sk⫽2共3 p⫺1兲

兺

j⫽1

m_s

␣s j

2 ,

冉

c_{s j} ␣s j

冊

2

⫽ 4␣s j

2

冉

k

兺

⫽1

m_s

␣sk

2

冊

2. 共19兲

When substituted into Eq.共14兲this leads to a key result, the generalization of the original assisted inflation result given by Eq. 共5兲:

p⫽16␲ m_Pl2 s

兺

_⫽1

n

1

兺

j⫽1

m_s

␣s j

2

. 共20兲

We also note that the generalization of the scaling solution found in Eq. 共11兲, quickly follows for the case of any two scalar fields,␾s j and␾ql:

冉

␾˙

s j

␾˙

ql

冊

2

⫽

冉

␣s j

␣ql

冊

2

冉

兺

i⫽1

m_q

␣qi

2

兺

k⫽1

m_s

␣sk

2

冊

2

. 共21兲

It is directly evident that Eqs.共20兲and共21兲reduce to Eq.共5兲 for the example of n exponential terms each containing just one field, and Eq. 共11兲for the case of one exponential term but containing two fields. We again see the inhibiting affect that multiplicative coupling of the fields共i.e., m_s⬎1) has for obtaining inflationary solutions. However, in this case, as with the original version of assisted inflation, this can be compensated for if there are enough exponential terms present in the potential 共i.e., if n is large enough兲 关9兴.

There is another feature of the potentials we have been discussing so far that makes them rather unphysical in gen-eral. We have been demanding that any two fields present cannot be the same共i.e.,␾s j⫽␾ql). In other words they can

only appear once in the full potential. Nearly all realistic models which emerge from compactifications arising in su-pergravity models have the same field appearing in at least two separate exponential terms. In the following section, we turn our attention to this case.

III. EXPONENTIAL POTENTIALS INSPIRED BY SUPERGRAVITY MODELS

To set the scene, we generalize Eq.共6兲to the case where the scalar field potential takes the following form:

V共␾1,␾2兲⫽z1e␣11␾1⫹␣12␾2⫹z2e␣21␾1⫹␣22␾2, 共22兲 where, ␣s j, are dimensional constants which can take any

real value and z_s⬎0. The occurrence of such forms of the potential are quite common in dimensionally reduced super-gravity models 关13,14兴. Remarkably we can solve this sys-tem to obtain scaling solutions in a manner analogous to those already presented in Eqs.共7兲–共11兲obtaining the unique late time scaling solution for the fields␾1 and␾2,

p⫽16␲ m_Pl2

冋

共␣21⫺␣11兲2⫹共␣22⫺␣12兲2 共␣11␣22⫺␣21␣12兲2

册

. 共23兲

This simple result reduces to the particular cases we have already investigated when the appropriate limits are taken. For example, when␣21⫽␣12⫽0, we reproduce the result Eq. 共5兲. The equivalent of the assisted inflation result Eq. 共11兲 follows by setting␣11␣21⫹␣12␣22⫽0, in which case we find

p⫽16␲ m_Pl2

冋

1 ␣11

2_⫹␣

12

2 ⫹

1 ␣21

2 _⫹␣

22

2

册

. 共24兲

We shall now generalize the potential to n such exponen-tial potenexponen-tials and m combinations of linear fields in the ex-ponent关explicitly calculating for the simple case of 2 terms

⫻2 fields of Eq.共22兲兴. The generalized Eq.共22兲is then

V⫽

兺

s⫽1

n

zsexp

冉

兺

j⫽1

m

␣s j␾j

冊

. 共25兲

Of course, we are allowing here for the possibility that ␣s j

⫽0 for some combination of s j . We assume that for late times the fields have an attractor solution, given by

zsexp

冉

兺

j⫽1

m

␣s j␾j

冊

⫽

k_s

t2, 共26兲

and, following Eqs.共15兲and共16兲we write

␾j⫽aj⫺

cs j

␣s j

ln t, 共27兲

where, aj is a constant depending on the initial conditions

and兺m_j⫽1cs j⫽2, s⫽1, . . . ,n. Substituting Eq.共27兲into the

equation of motion Eq.共3兲, we obtain the constraint equation for the j th field, which follows from assuming the existence of a power-law solution

共3 p⫺1兲cs j ␣s j⫽q

兺

⫽1

n

␣q jkq. 共28兲

兺

j_⫽1

m

␣s j

冉

兺

q_⫽1

n

␣q jkq

冊

⫽2共3 p⫺1兲, 共29兲

which is equivalent to writing

兺

q⫽1

n

A_sqk_q⫽2共3 p⫺1兲, 共30兲

where

Asq⫽

兺

j⫽1

m

␣s j␣q j. 共31兲

Since s is a free index, we have a set of n equations that can be written as

Ak⫽2共3 p⫺1兲, 共32兲

where A is the n⫻n matrix with elements A_sq and k

⫽(k1, . . . ,kn) a column vector. For the 2⫻2 case of Eq.

共22兲we obtain

A⫽

冉

␣11

2_⫹␣

12

2 _␣

11␣21⫹␣12␣22 ␣21␣11⫹␣22␣12 ␣21

2_⫹␣

22

2

冊

. 共33兲

The solution to this system is

k⫽A⫺12共3 p⫺1兲, 共34兲 with

A⫺1⫽ACOF

T

det A, 共35兲

where ACOF

T

is the transpose of the cofactor matrix of A. To simplify notation we will write B⬅A_COFT and the sum of the elements in row s of B as Bs⬅兺_qn_⫽1Bsq, hence, each ks is

ks⫽

2共3 p⫺1兲 det A B

s_{. 共36兲}

For the 2⫻2 case this yields

B⫽

冉

␣21

2_⫹␣

22

2 _⫺␣

11␣21⫺␣12␣22

⫺␣21␣11⫺␣22␣12 ␣11

2 _⫹␣

12

2

冊

, 共37兲

k1⫽2共3 p⫺1兲 ␣21

2_⫹␣

22

2 _⫺␣

11␣21⫺␣12␣22 共␣11␣22⫺␣12␣21兲2

, 共38兲

k2⫽2共3 p⫺1兲

⫺␣21␣11⫺␣22␣12⫹␣11

2_⫹␣

12 2

共␣11␣22⫺␣12␣21兲

2 . 共39兲

From Eqs. 共27兲 and 共28兲the late time ratio between the kinetic terms of two different fields becomes

冉

␾˙_j ␾˙

l

冊

2

⫽

冉

q

兺

⫽1

n

␣q jBq

兺

r⫽1

n

␣rlBr

冊

2

. 共40兲

Substitution of Eqs. 共36兲 and 共28兲 into the Friedmann equation yields

p2⫽ 8␲ 3m_Pl2

冋

s

兺

⫽1

n

ks⫹

1

2 j

兺

⫽1

m

冉

cs j

␣s j

冊

2

册

⫽ 8␲

3m_Pl2

冋

2共3 p⫺1兲s

兺

⫽1

n

Bs

det A⫹2

兺

j⫽1

m

冉

兺

q⫽1

n

␣q jBq

det A

冊

2

册

. After some algebra, we obtain the simple result for p as the ratio between the sum of all the elements in the cofactor matrix of A and its determinant

p⫽16␲ m_Pl2

兺

s n

兺

q n

B_sq

det A . 共41兲

The reader should have no problem showing that for the 2

⫻2 case this reduces to Eq.共23兲.

It is instructive to rewrite Eq.共41兲in another form. From Eqs. 共30兲,共36兲, and 共41兲,

p⫽16␲ m_Pl2 s

兺

⫽1

n

1

兺

q_⫽1

n

Asqkq/ks

, 共42兲

and using Eq. 共31兲with q⫽s and Eq.共36兲we obtain

p⫽16␲ m_Pl2 s

兺

⫽1

n

1

兺

j⫽1

m

␣s j

2_⫹

兺

q⫽s n

AsqBq/Bs

. 共43兲

A number of points need to be made about Eq.共43兲. It is similar in form to Eq.共20兲, which should not be too surpris-ing, the additional terms in the denominator arising from the fact that we have allowed for fields to appear more than once in the potential, hence leading effectively to ‘‘self-interaction’’-type contributions. Indeed if these terms were turned off we would reproduce the result in Eq.共20兲. In the 2⫻2 case, it is the constraint leading to Eq.共24兲. Due to the presence of these ‘‘self-interaction’’ terms, p could increase above the value in Eq. 共20兲if there happened to be a com-bination of positive and negative slopes in Eq.共43兲.

then one of the two terms would quickly dominate the over-all dynamics. One way to check if a combination of terms in a given potential will be comparable at late times is to use Eq.共36兲to obtain the ratios ks/kqfor these terms. From Eq.

共26兲 it follows that for consistency we require them to be positive, with p⬎1/3. In general, the surviving terms 共i.e., those which remain comparable兲 will be the ones with the smallest slopes, corresponding to the largest values of p.

IV. CONCLUSIONS

In 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 inflation solutions discussed by LMS 关9兴to the case of multiple ex-ponential terms involving many fields. Such potentials are more likely to arise in realistic models of particle physics where individual fields will occur in a number of separate exponential terms, leading to cross couplings between the terms. In general, it transpires that it is more difficult to obtain assisted inflation in such models, the fields in any one

exponential term tend to conspire to act against one another rather than assist each other, a result also noticed in关14,11兴. 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 关14兴. We also investigated the case where a number of exponential terms contained the same scalar fields and demonstrated that a number of novel features emerged, including the possibility of increasing the rate of expansion when there exists a mixture of positive and negative slopes in the potential.

ACKNOWLEDGMENTS

We 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 grateful to David Wands for conversations on the nature of the gen-eralized solutions. N.J.N. was supported by FCT 共Portugal兲 under Contract No. PRAXIS XXI BD/15736/98 and A.M. was supported by the INLAKS and the ORS.

关1兴F. Lucchin and S. Matarrese, Phys. Rev. D 32, 1316共1985兲. 关2兴J.J. Halliwell, Phys. Lett. B 185, 341共1987兲.

关3兴A.B. Burd and J.D. Barrow, Nucl. Phys. B308, 929共1988兲. 关4兴C. Wetterich, Nucl. Phys. B302, 668共1988兲.

关5兴D. Wands, E.J. Copeland, and A.R. Liddle, Ann.共N.Y.兲Acad. Sci. 688, 647共1993兲.

关6兴P.G. Ferreira and M. Joyce, Phys. Rev. D 58, 023503共1998兲. 关7兴E.J. Copeland, A.R. Liddle, and D. Wands, Phys. Rev. D 57,

4686共1998兲.

关8兴T. Barreiro, B. de Carlos, and E.J. Copeland, Phys. Rev. D 58, 083513共1998兲.

关9兴A.R. Liddle, A. Mazumdar, and F.E. Schunck, Phys. Rev. D

58, 061301共1998兲.

关10兴K. Malik and D. Wands, Phys. Rev. D 59, 123501共1999兲. 关11兴P. Kanti and K.A. Olive, Phys. Rev. D 60, 043502共1999兲.

关12兴A. Lukas, B. Ovrut, and D. Waldram, hep-th/9806022,

hep-th/9902071.

关13兴M. Bremer, M.J. Duff, H. Lu, C.N. Pope, and K.S. Stelle, Nucl. Phys. B543, 321共1999兲.