the effect of higher-order curvature terms on string quantum cosmology

20 July 2000

Ž .Physics Letters B 485 2000 408–421www.elsevier.nlrlocaternpe

The effect of higher-order curvature termson string quantum cosmology

Simon Davis, Hugh LuckockSchool of Mathematics and Statistics, UniÕersity of Sydney, NSW 2006, Australia

Received 11 November 1999; received in revised form 7 June 2000; accepted 9 June 2000Editor: M. Cvetic

Abstract

Several new results regarding the quantum cosmology of higher-order gravity theories derived from superstring effectiveactions are presented. After describing techniques for solving the Wheeler–DeWitt equation with appropriate boundaryconditions, it is shown that this quantum cosmological model may be compared with semiclassical theories of inflationarycosmology. In particular, it should be possible to compute corrections to the standard inflationary model perturbatively abouta stable exponentially expanding classical background. q 2000 Published by Elsevier Science B.V.

MSC: 83C45; 83E30; 83F05

1. Classical cosmological space-times of higher-order gravity theories

Since solutions to the general relativistic fieldequations contain initial curvature singularitieswhenever the dominant energy condition is satisfied,one of the motivations for developing quantum cos-mology has been the theoretical justification of theelimination of the singularities. Curvature singulari-ties predicted by general relativity can be avoided byintroducing boundary conditions in the path integraldefining the quantum theory which restrict the inte-gral over all four-geometries and matter fields toRiemannian metrics g on compact manifoldsmn

bounded by a three-dimensional hypersurface with

Ž .E-mail address: [email protected] S. Davis .

metric h and fields with specified values on thei jw xhypersurface 1 . They might also be eliminated in a

quantum theory of gravity free from divergences atshort distances, since the wavefunctions defined bythe path integral of the theory may represent non-sin-gular geometries at initial times.

Classical cosmological solutions to the equationsof motion for several different types of theoriescontaining higher-order curvature terms have beenanalyzed with regard also to the presence of singular-ities, and non-singular solutions have been obtained.Renormalizability has been obtained with the addi-

w xtion of quadratic terms in the action 2 . Moreover,any theory of superstrings consistent at the quantumlevel will have an effective action containinghigher-order curvature terms. It has been shown, inparticular, that dimensionally continued Lovelock in-variants, giving rise to second-order field equationsfor the metric, arise in superstring effective actions.

0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V.Ž .PII: S0370-2693 00 00610-9

( )S. DaÕis, H. LuckockrPhysics Letters B 485 2000 408–421 409

According to the conventional usage of termsrelated to modifications of the Einstein–Hilbertaction, a Lagrangian which has the formŽ .L g ,E g ,E E g , and is a nonlinear functionmn a mn a b mn

of the curvature tensor, represents a higher-orderw xgravity theory 3 , whereas an action containing third-

and higher-order derivatives of the metric describeshigher-derivative gravity theories.

To determine whether quantum cosmologicalmodels based on theories containing third- andhigher-order derivatives of the metric or powers ofthe second derivative of the metric produce wavefunctions consistent with a non-singular geometry,without imposing a non-singular boundary conditionat initial times, it is useful to start with an actionwhich combines higher-order gravity with scalarfields and possesses singularity-free cosmological

w xsolutions 4,5 . At string tree-level and first-order inthe a

X-expansion of the compactified heterotic stringeffective action in four dimensions, the dynamics ofthe graviton, scalar field S and modulus field T canbe described by the effective Lagrangian

1 DSDS DTDTLL s Rq q3eff . 2 2 22k SqS TqTŽ . Ž .

1 12 ˜q Re S R q Im S RRŽ . Ž .GB8 8

1y V S,S,T ,T 1Ž .Ž .2

If Re T , representing the square of the compactifica-tion radius, is set equal to a constant, and Im T is setequal to zero, the kinetic term for the modulus fieldvanishes. In addition, defining the real part of the

Ž 2 . yFdilaton field to be Re Ss 1rg e restricting4˜attention to isotropic models for which RRs0, and

choosing units such that ks1, one obtains theaction1

1 w xThe sign convention in Ref. 6 is being used for the defini-tion of Re S and it is combined with the expression for LL ineff.

w xRef. 4 . The integral for the action I is deduced after multiplica-tion by an overall factor of 2.

yFe214 'Is d x yg Rq DF qŽ .H 2 24 g4

mnk l mn 2= R R y4R R qR yV FŽ .Ž .mnkl mn

2Ž .The Gauss–Bonnet invariant arises in this action, butit is multiplied now by the factor eyFr4 g 2, where4

g is the four dimensional string coupling constant4

and F is the dilaton field, so that the integral is not atopological invariant. This quadratic curvature com-bination is precisely that required to remove ghostpoles in the perturbative expansion of the propagator,and since the term is dynamical, the theory repre-sents a modification of general relativity which isunitary and has improved renormalizability proper-ties.

There are conformal transformations which mapŽ .f R actions to the Einstein–Hilbert action plus a

w xscalar field with a potential term 7–13 . Other fieldŽ mn .redefinitions can be used to transform f R R tomn

the Ricci scalar plus terms containing scalar fieldsw xand extra tensor modes 14–16 . The transformed

theories should have the same renormalizabilityproperties as the higher-order gravity theories, be-cause field redefinitions do not alter the S-matrix.Such transformations are unnecessary in the present

Ž .case since the field equations obtained from Eq. 2contain at most two derivatives – in effect, althoughthe action contains higher-order curvature terms, theaddition of a boundary term is sufficient to eliminatesecond-order derivatives of the metric from the inte-gral, so that only a conventional quantization of thetheory is required, rather than the Ostrogradski

w xmethod 3 .It is known that most classical string equations of

motion without a dilaton field do not lead to inflationw x17 . Thus, dilaton fields also have been included asthey permit inflation, and a set of higher-order grav-ity theories with a dilaton field has been shown toproduce the required inflationary growth of the

w xFriedmann–Robertson–Walker scale factor 18 .2 w x 2An earlier analysis of R theories 19 and C

w x 2theories 20 has shown that an R term leads toparticle production and inflation with minimal de-pendence on the initial conditions, while the C 2 term

w xgenerates large anisotropy 21 and causes destabi-

( )S. DaÕis, H. LuckockrPhysics Letters B 485 2000 408–421410

lization of positive L metrics. Inflation also has beenderived from higher-derivative terms directly ob-

w xtained as renormalization counterterms 22,23 with-out the inclusion of scalar or inflaton fields.

2. The quantum cosmology of higher-order grav-ity theories

Much of the initial work on higher-derivative andhigher-order quantum cosmology has been devel-oped with only curvature terms and no scalar field inthe action. Nevertheless, scalar fields are probablynecessary in a theory describing gravity and matterinteractions, because they are required for renormal-izability of gauge theories with massive spin-1 vec-

w xtor fields 24 and they result from the conformalwrescaling of the metric in higher-order theories 7–

x13 .The quantum cosmology of standard gravity cou-

pled to a scalar field has been investigated by manyw xauthors 25–28 . The non-zero vacuum expectation

value of the scalar field in grand unified theoriesdrives inflation in semi-classical cosmology, andagain it is found to be useful in obtaining wavefunctions representing inflationary solutions in quan-tum cosmology.

The quantum cosmology of superstring and het-erotic string effective actions in ten dimensions withcurvature terms up to fourth order has been investi-

w xgated previously 29–32 . With the dilaton and mod-ulus fields included, the Wheeler–DeWitt equationsfor both theories, in the minisuperspace of metricswith different scale factors for the physical andinternal spaces, have not been solved in closed form,because the coupling of the curvature and scalarfields leads to quartic terms in the derivatives of thescale factor in the Hamiltonian and fractional powers

w xof the momenta 30 . For the superstring, the differ-ential equation is only reducible to the form of adiffusion equation when the curvatures of the physi-cal and internal spaces are set equal to zero, and the

w xscalar field is set equal to a constant 32 .Even when there are quadratic curvature terms in

the heterotic string effective action, the Hamiltoniancannot be expressed as a simple function of thecanonical momenta, preventing a derivation of adirectly solvable Wheeler–DeWitt equation. For the

model in this paper, a pseudo-differential equation inclosed form, obtained by using the Ostrogradski

Ž .formalism for the heterotic string effective action 2 ,can be converted to a partial differential equation ofhigher order, which may be solved with appropriate

w xboundary conditions 33 .The Hamiltonian is more directly obtained if one

begins by adding a total derivative term to theLagrangian to eliminate second-order derivatives anda different set of momenta conjugate to the scalefactor and dilaton field only is used. While the firstchoice of momenta gives rise to a sixth-orderWheeler–DeWitt equation, the second set of mo-menta results in a Wheeler–DeWitt equation whichreproduces the standard equation for gravity coupledto a scalar field in the limit that the quadratic termvanishes. The wave function can then be determinedperturbatively in the dilaton coupling eyFrg 2.4

Similarly, there is a preference for quantizing atheory that does not include higher-order curvatureterms, because the absence of higher derivatives ofthe metric allows one to avoid superfluous degreesof freedom. However, if the required field redefini-tions depend on the Riemann curvature tensor and itscontractions, it is the set of kinetic terms for theextra scalar fields which now introduce the higherderivatives of the metric, implying that additionaldifficulties arising in quantization would not neces-sarily be circumvented for these types of higher-ordertheories.

While the coupling eyFrg 2 explicitly determines4

the strength of higher-order terms relative to theRicci scalar, a factor of " can be restored at eachorder in the perturbative expansion of the S-matrix

Žof the heterotic string sigma model. Since the nq.1 -loop computation for the sigma model produces

the nth-order contribution to the heterotic string ef-fective action I Žn., the limit "™0 can be taken,eff.

1after multiplication by a factor of , to obtain the"

classical action for gravity coupled to a scalar field.It is necessary to take the "™0 limit because theeyFrg 2

™0 limit eliminates the dilaton field kinetic4

term. Nevertheless, since additional factors of " andeyFrg 2 occur simultaneously at higher loops in the4

sigma model perturbation expansion, the Wheeler–DeWitt equation for the higher-order gravity theorycan be regarded as a perturbation of the standardsecond-order Wheeler–DeWitt equation, with the so-


lution being expanded in powers of eyFrg 2. This4

provides an approximate wave function for the het-erotic string effective action which includes correc-tions to the wave function used to predict inflation-ary cosmology.

Given a fundamental theory at Planck scale withhigher-order curvature terms, it is appropriate toconsider a boundary located between the Planck eraand the inflationary epoch where the predictions ofquantum cosmology of the higher-derivative theorycould be matched, in principle, to the predictions ofthe quantum theory of standard gravity coupled tomatter fields. The inclusion of this boundary willhave an effect on both the quantum cosmology of themore fundamental theory and the computation ofradiative corrections to the semi-classical inflation-ary model.

The significance of the wave function depends onthe stability of the classical background geometrieswhich represent most probable configurations in thequantum cosmology of the model. Stability at thenonlinear level can be proven if the positive energytheorem holds, which requires that the space-timeadmits a Killing spinor that also must be a supersym-metry parameter. For the four-dimensional compacti-fication of heterotic string theory, the Majorana con-dition on the Killing spinor leads to a differentialequation for the scale factor which is solved by anon-singular cosmological bounce solution. Whenthis scale factor is substituted into the equation ofmotion for the dilaton, it will be shown that there is asolution for F which increases at an approximatelylinear rate at large t. Thus, there is a stable back-ground configuration derived from the heterotic stringeffective action, which describes the exponential ex-pansion of the inflationary epoch and allows for thecomputation of perturbative corrections in the quan-tum cosmological model.

3. Quantum cosmology for a four-dimensionalheterotic string effective action

Ž .The model 2 containing quadratic curvatureterms and the dilaton field can be quantized with theHamiltonian constraint being represented by theWheeler–DeWitt equation. Since the solution to this

equation generally requires a reduction in the num-ber of degrees of freedom in the metric field, it isconvenient to consider only a minisuperspace ofFriedmann–Robertson–Walker metrics

a2 tŽ .2 2g sdiag 1,y ,ya t r ,Ž .mn 2ž 1yKr

ya2 t r 2 sin uŽ . /Ž . Žwith Ks1 closed model , Ks0 spatially flat

. Ž .model or Ksy1 open model .Homogeneity of the minisuperspace model im-

plies that the fields are position-independent on folia-tions of the four-dimensional space-times and theaction per unit volume is a one-dimensional integral

I12 2 3 2˙s dt 6a aq6aa q6aK q a FŽ .¨ ˙H 2

VV

yFe2 3q6 a a qK ya V F 3Ž . Ž . Ž .¨ ˙2g4

where VV is a time-independent volume factor givenŽ . 3Ž . Ž .by VV t ra t , with VV t being the three-di-f f f

mensional volume of the spatial hypersurface at timet .f

Since the field equations are not higher-deriva-tive, the action can be modified by adding a bound-ary term to eliminate factors of a.¨

X yFI I d e2 2s y dt 2 a a q3K q6a aŽ .˙ ˙ ˙H 2dt gVV VV 4

12 3 2˙s dt 6a ya qK q a FŽ .˙H 2

yFe2 3˙q2 F a a q3K ya V F 4Ž . Ž . Ž .˙ ˙2g4

The conjugate momenta are then

eyF

2˙P sy12 aaq6 F a qKŽ .˙ ˙a 2g4

eyF

3 2˙P sa Fq2 a a q3K 5Ž . Ž .˙ ˙F 2g4


Expanding in powers of eyFrg 2, expressions for4˙a and F to first order may be obtained˙

1 eyF 1 1 1a,y P q P P P qK˙ a F a a2 4 ž /12 a 144a ag 2 a4

1 eyF 1 1 1F, P q P P P q3KF a a a3 2 4 ž /144a aa g 6a4

6Ž .

and the Hamiltonian is

1H ,y6a y Pa12 a

2yFe 1 1 1q P P P qKF a a2 4 ž /12 a 12 ag 2 a4

yFe 1qK q6 PF2 3g a4

yFe 1 1 1q P P P q3Ka a a2 4 ž /12 a 12 ag 6a4

1= y Pa½ 12 a

2yFe 1 1 1q P P P qKF a a2 4 ž /12 a 12 ag 2 a4

1qK y Pa5 12 a

yFe 1 1 1q P P P qKF a a2 4 ž /12 a 12 ag 2 a4

3 yFa 1 e 1q P q PF a3 2 42 a g 6a4

21 1

= P P q3Ka až /12 a 12 a

qa3V F 7Ž . Ž .In a Lorentzian space-time, the differential opera-

tor representing the Hamiltonian is obtained by mak-

E Eing the substitutions P ™yi and P ™yi .a FE a EF

Choosing the operator-ordering parameter to be equal1 2 1 E 1 Eto y1, so that P ™y , the "™0 limitŽ .aa a E a a E a

produces the equation

1 E 1 E 1 E 2

H Cs y0 3 2ž 24 E a a E a 2 a EF

y6aKqa3V F CŽ . /s0 8Ž .

This is equivalent to the standard Wheeler–DeWittequation for standard gravity plus a scalar field

21 E E 1 Ep 2a y ya U a,f Cs0,Ž .p 2 2a E a E a a Ef

U a,f s1ya2 V f ,Ž . Ž .V f sscalar potential, 9Ž . Ž .where, as above, the operator-ordering parameter pis set equal to y1, using the rescalings a2

1 2 1 Ž .™ a , F ™ F and V F ™ 72 =' '12 K 2 3

3r2 Ž .K V F .While this choice of operator ordering is conve-

nient for obtaining closed solutions to the Wheeler–Ž .DeWitt Eq. 8 , other values of this parameter also

can be used. The parameter p impacts on the regu-Ž .larity of the wave function C F ,a as a™0; when

pG1, regularity in the a™0 limit requires thew xno-boundary wavefunction 1 , whereas the diver-

gence in the tunneling wavefunction in the limitw xa™0 is regulated by a prefactor when pF0 34 .

Non-singular solutions with Lorentzian signature willoccur if regularity is maintained in this limit.

There are additional operator-ordering ambiguitiesat order O eyFrg 2 , because terms containing theŽ .4

1 yF 2cube of P and the product of e rg and P area 4 Fa1 2

included in the Hamiltonian. Suppose P isŽ .aa

replaced by

1 E E 1 1 E 1 Epy a sypq2 pq1 1yŽ pq1.E a E a a E a E aa a a

Ž .to obtain the second-order Wheeler–DeWitt Eq. 9 ,1 E 1where represents the ordering ofpq1 1yŽ pq1.E aa a

1 Ethe product of and . The second expressiona E a


1provides a way of obtaining higher powers of Paa

through iteration, without the introduction of non-de-rivative terms. For example,

31 1 1 E3 pP ™ yi aŽ .a pq1ž / ž /a a E aa

=1 E E

papq1ž /E a E aa

1 E 1 E Eps i apq2 E a a E a E aa

1 3represents a substitution for P which does notŽ .aa

involve non-derivative terms and therefore has aŽ .form similar to the differential operator in Eq. 9 .

1More generally, powers of can be placed to thea

right of an operator product containing P , givinga

rise to non-derivative terms. Indeed, one can con-sider an arbitrary linear combination of the twentyoperators OO , not all linearly independent, obtainedi

by permutation of the factors in the product1 1 1 20P P P . This combination Ý a OO can bea a a is1 i ia a a

1 E 1 E 1 E 1shown to be equal to i , where p, q, rp q r sa E a a E a a E a a

and s may be expressed in terms of the coefficientsŽ yF 2 .a , and pqqqrqss3. Similarly, e rg Pi 4 F

EyF 2 yF 2may be replaced by yi e rg q t e rg ,Ž .4 4EF

implying the existence of another operator-orderingparameter t in the Wheeler–DeWitt equation HCs0.

The powers p, q, r and s are constrained byconsistency with the algebra of supersymmetry con-straints and hermiticity. While the potential in theheterotic string effective action is also determined bysupersymmetry, it would be modified by the additionof any term not involving the derivative with respect

1 3to a in the expansion of P .Ž .aa

Based on the substitutions

1 yi Ep1P ™ aa p q11a E aa

21 1 E 1 E 1P ™ya p q rž / 2 2 2a a E a a E a a

p qq qr s22 2 2

31 i E 1 E 1 E 1P ™a p q r sž / 3 3 3 3a a E a a E a a E a a

p qq qr qs s3 10Ž .3 3 3 3

the non-derivative term is

1 r q qr q1 eyFŽ .2 2 2q t3 2½ 24 a g4

1= s s qr q1 q qr qs q2Ž . Ž .3 3 3 3 3 39864 a

q6 q qr qs r qs r qs q2Ž . Ž .Ž .3 3 3 3 3 3 3

3q r qs r qs q2 q qr qs q1Ž . Ž .Ž .3 3 3 3 3 3 32

qs r qs q1 y6r r qq q1 q30Ž . Ž .3 3 3 2 2 2

yFK 7y4 p eŽ .1q y5 24 a g4

s r qs q1 q qr qs q2Ž . Ž .3 3 3 3 3 3= C 11Ž .9 51728 a

A specific operator-ordering, such as normal or-dering, should be used uniformly for all of the termsin a Lagrangian field theory. In this case, setting p1

equal to p, it follows that p spq1,q s1,r sˆ ˆ2 2 2

yp, p spq1,q s1,r s1,s syp so that theˆ ˆ ˆ3 3 3 3

non-derivative term becomes

1 p py2Ž .ˆ ˆ3½ 24 a

yFe 1 173qt y p2 9 ž 2g 864 a4

81 93 K 7y4 pŽ .ˆ2q p y56 pq qˆ ˆ 5/2 2 4 a

eyF p py2 py4Ž . Ž .ˆ ˆ ˆq C 12Ž .2 9 5g 1728 a4

When the factor of " is included in P , the first termaŽ . Ž .in the expression 12 has the same order as U a,F ,

Ž .whereas the second term represents an O " contri-bution. While this does represent a modification ofŽ .U a,F , invariance of the entire Lagrangian under

supersymmetry transformations implies that consis-tency with the supersymmetry constraints would notbe affected by rearrangement of terms in the Hamil-

Ž .tonian. The O 1 part of the non-derivative term


should be set equal to zero to minimize the correc-Ž .tion to U a,F , and this requires ps0,2, when theˆ

Ž .operator-ordering 10 is applied uniformly. In termsof the original operator-ordering parameter p, thesevalues correspond to p s y1,2. Non-derivative

Ž .terms can be removed at O 1 for more general1 2

values of p only if the substitution P ™Ž .aa1 2p1 E Ey a is used instead of P ™Ž .aa

pq2 E a E aap1 E 1 Ey a .

pq1ˆ aE a E aaThe heterotic string potentia is

byKK a y1Vse KK KK KK q3Ž . a b

2< <KKs ln SqS q3 ln TqT y ln WW SŽ . Ž . Ž .E KK

KK sa aEf

eyF

sRe Ss Re Tse 13Ž .2g4

� a4where s is set equal to a constant, s , and f0

represents the fields S and T , with the other matterw x Ž .fields set equal to zero 35–38 , and WW S is the

superpotential.The equation HCs0 can be solved approxi-

mately by using e'eyFrg 2 as an expansion pa-4

rameter and noting that CsC qeC is a solution0 1Ž . Ž .to H qeH Cs0 to order O e if H C s0 1 0 1

Ž .yH C . Given the Hamiltonian 7 , the first-order1 0

correction is

1 E 1 E 1 E 1 Ey y qK4 ž /4 E a EF 144a E a a E aa

1 E 1 E 1 E 1 Ey y qK3 ž /EF 144a E a a E a a E a4a

1 E 1 E 1 E 1 Eq y qK3 ž /EF 144a E a a E a a E a2 a

1 E 1 E 1 E 1 Ey y1 y4ž / ž12 EF E a 144a E a a E aa

1 E 1 E 1 Eq3K y y q3K/ ž /12 a E a 144a E a a E a

=1 E

14Ž .3 EFa

and

1 K 1 EC0H C s y1 0 4 4ž /4 E aa 576a

1 E 2C 1 E 3C0 0q y7 2 6 3576a E a 1728a E a

1 7K 35 EC 1 E 2C0 0q y q5 4 8ž /4 EF E aEFa 576a 24a

1 E 3C 1 E 4C0 0y q 15Ž .7 2 6 364a E a EF 864a E a EF

In the classically allowed range, the ‘no bound-w xary’ wavefunction 1 , for example, is

3 1y2 2 424K a V

C sexp y10 NB ž /V 6K� 0=

33

2 224K a V p2cos y1 y 16Ž .ž /V 6K 4

and

H C1 0 NB

3

21 24Ks exp3 Va � 0

7y2 2 45 V a V

= y13 ž /144 6K2K

13 2 2a V a V 42yK q2 y1ž /6K 6K

33

2 224K a V p2Psin y1 yž /V 6K 4


3

21 24Kq exp3 Va � 0

13y3 2 45 V a V

= y1ž /24 24K 6K

5y2 2 4V a V a V

q y2 y1ž /48 6K 6K

33

2 224K a V p2Pcos y1 yž /V 6K 4

V X

qO 17Ž .ž /V

where terms involving the derivative of the potentialwith respect to F may be evaluated using the opera-

Ž .tor 15 .w xIn the classically forbidden region 1,2 ,

1y2 41 a V

C , 1y0 NB 2 6K

=

33

2 224K a V 2exp 1y 1y 18Ž .ž /V 6Kž /� 0

and

13° y3 2 41 5 aV a V~H C s y 1y1 0 3NB ž /24 24K 6K2 a ¢7

y2 2 45 V a Vy 1y3 ž /144 6K

2K

5y2 2 4V a V a V

q 2y 1yž /48 6K 6K

1¶3 2 2a V a V 42 •qK 2q 1yž /6K 6K ß

=

33

2 224K a V 2exp 1y 1yž /V 6Kž /� 0

V X

qO 19Ž .ž /V

Similarly, the tunneling wave function in the clas-w xsically acceptable region 39–41 is

1yip 2 4a V

4C s2e y10T 6K

33

2 224K a V 2=exp 1y i y1 20Ž .ž /V 6Kž /

and

ip2

4H C s e1 0 3T a

13° y3 2 45 V a V~= y1ž /24 24K 6K¢7

y2 2 45i V a Vq y13 ž /144 6K

2K

5y2 2 4V a V a V

q 2y y1ž /48 6K 6K


1¶3 2 2a V a V 42 •yiK q2 y1ž /6K 6K ß3

32 224K a V 2

Pexp 1y i y1ž /V 6Kž /� 0V X

qO 21Ž .ž /V

whereas the tunneling wave function in the classi-w xcally forbidden region 41 is

1y2 4a V

C , 1y0T 6K

33

2 224K a V 2= exp 1y 1yž /V 6Kž /� 0

33

2 224K a V 2qiexp 1q 1y 22Ž .ž /V 6Kž /� 0

and

13° y3 2 41 5 aV a V~H C s y 1y1 0 3T ž /24 24K 6Ka ¢7

y2 2 45 V a Vy 1y3 ž /144 6K

2K

5y2 2 4V a V a V

q 2y P 1yž /48 6K 6K

1¶3 2 2a V a V 42 •qK q2 1yž /6K 6K ß

33

2 224K a V 2Pexp 1y 1yž /V 6Kž /� 0

13° y3 2 42 i 5 aV a V~q y 1y3 ž /24 24K 6Ka ¢7

y2 2 45 V a Vq 1y3 ž /144 6K

2K5

y2 2 4V a V a Vq 2y 1yž /48 6K 6K

1¶3 2 2a V a V 42 •yK q2 1yž /6K 6K ß3

32 224K a V 2

Pexp 1q 1yž /V 6Kž /� 0V X

qO 23Ž .ž /VXŽ .Terms containing V f are negligible if

< y1 XŽ . <V V f <1 in Planck units, which holds forscalar potentials in standard inflationary models. Instring cosmology, slow-roll inflation occurs becauseof the flatness of the potential in the chiral field

w xdirections 42 , but it can be verified that the termsinvolving the derivative with respect to the dilaton,

XŽ .V F , are comparable in magnitude to the otherŽ .terms in H C when V F does not contain an1 0

effective cosmological constant in addition to ex-pressions multiplied by powers of eyFrg 2. While4

V X

Ž .the O terms can be written down explicitly, theyV

are omitted here so that compact expressions can begiven for H C and H C .2

1 0 1 0T NB

Nevertheless, in the parameter range where thederivative with respect to F can be ignored, the

2 XŽ .The complete calculation of the terms containing V F canbe found in gr-qcr9911043.


general solution to the equation H C syH C ,0 1 1 0

given two linearly independent solutions of the ho-mogeneous equation,

1 d 1 d3y6aKqa V F C ,0,Ž . 0ž /24 da a da

would be

H C1 0C fC C qC C y24C C adaH1 1 0 2 0 0 01 2 2 1 W

H C1 0q24C C ada 24Ž .H0 01 2 W

where

d dWsC C yC C .0 0 0 01 2 2 1da da

Substituting H C or H C into this formula and1 0 1 0NB T

imposing the appropriate boundary conditions on C1Ž yF 2 .provides the correction of order e rg C to the4 1

standard wave function C .0

Amongst the solutions to the equations of motionof a one-loop effective Lagrangian, having the same

Ž .form as LL in Eq. 1 and containing only theeff.

dilaton field, are the homogeneous and isotropicsolutions that begin with a Gauss–Bonnet phase

v 11y t 3Ž . Ž .a t ;e and continue to the FRW phase a t ; tw x43 . The stability of these solutions with respect to

w xlinearized tensor perturbations 44 is determined by2 2 w xthe effective adiabatic index Gs 1yaara 45 .Ž .¨ ˙3

5The Gauss–Bonnet phase is unstable since G- ,3

and the background geometry is eventually describedby a stable FRW metric with Gs2.

Stability of the geometry, against perturbations ofthe metric that do not change the topology, followsfrom the positive-energy theorem, which is applica-ble when the metric admits Killing spinors. Therequirement of covariance and spatial homogeneityimplies the following form for the Killing spinorequation:

k tŽ .= hq i g hs0 25Ž .m m2

Ž .for some function k t . Considering the spatial com-Ž .ponents of Eq. 25 and applying the commutator of

the modified covariant derivative to the spinor h, thew xintegrability relation 46 is

k 2 tŽ .1 abR g hs g h ,i j a b i j4 2

implying that the four-dimensional space-time shouldbe foliated by three-dimensional spaces satisfyingthe condition

k 2 tŽ .R s g g yg gŽ .i jk l i k jl i l jk2

and

k 2 t say2 Kqa2Ž . Ž .˙.

The time-space component of the commutatorcondition is

1 10 j jkR g q R g hŽ .0 i 0 j 0 i jk2 4

i k 2 tŽ .˙sy k t g hq g h 26Ž . Ž .i 0 i2 2

which implies that

1 a t k 2 t iŽ . Ž .¨ ˙g hs g hy k t g h 27Ž . Ž .0 i 0 i i2 a t 2 2Ž .Using the Dirac representation of the gamma matri-

I 0ces in four dimensions, g s , and setting0 ž /0 yIh 01

h equal to c qc , it follows that1 2 hž /ž / 20a tŽ .¨ h12 ˙c yk t q ik tŽ . Ž .1 ž /ž /a tŽ . 0

a tŽ .¨ 02 ˙qc y qk t q ik t s0 28Ž . Ž . Ž .2 hž /ž / 2a tŽ .There are three types of solutions:

Ž .a t¨ 2˙Ž . Ž . Ž .i c ,c /0, k t s0, yk t s0,Ž .1 2 a tŽ .a t¨ 2 ˙Ž . Ž . Ž .ii c s0, yk t q ik t s0,Ž .2 a t

Ž .a t¨ 2 ˙Ž . Ž . Ž .iii c s0,y qk t q ik t s0.Ž .1 a t

None of these conditions are satisfied by the scaleŽ . yv 1 r t Ž . 1r3factors a t ;e or a t ; t . This implies

that the positive-energy theorem is not applicable tothese background geometries, allowing for the possi-bility that they are unstable at a non-perturbativelevel. Moreover, these classical solutions may repre-sent only local extrema of the action, providing


sub-dominant contributions to the string path inte-gral, since the absence of stability at the non-per-turbative level suggests that there is tunneling fromthese solutions to more stable background geome-tries.

The Majorana condition implies that c ,c /01 2

and the Killing spinor belongs to first category.When Ks1, both constraints are satisfied by the

t 1Ž . Ž . Ž .scale factor a t sa cosh ,k t s , represent-0 a a0 0

ing a cosmological bounce solution. When Ks0,Ž . ltthe Killing spinor conditions require a t sa e ,0

tŽ . Ž .and if Ksy1, they imply that a t sa sinh or0 a0tŽ . Ž .a t sa sin .0 a0

Since the positive-energy theorem is applicable tothese space-times, they will be stable at the non-per-turbative level within the class of metrics represent-ing the same topology. It may be noted that theKs0 and Ks1 solutions do not have an initialsingularity and that they provide adequate classicalcosmological models through the inflationary epoch,

Ž .as the scale factors a t increase exponentially withtime.

It is customary to attribute physical significanceto a wave function only when there is an appropriateclassical limit. While the metric with an initialGauss–Bonnet phase and a perturbatively stableFRW phase represents a cosmological model whichis realistic at later times, the stable, non-singular,exponentially expanding space-times are preferableas background geometries in this quantum cosmolog-ical model, because they describe the inflationaryepoch, where quantum effects are still relevant.

The Euler–Lagrange equation of motion for F is

2 2 2 yFa Kqa a a q3K eŽ . Ž .¨ ˙ ˙ ˙Fq6 y3 4 2a a g4

qV XF s0 29Ž . Ž .

tŽ .With the scale factor a t sa cosh , and theŽ .0 a0

Ž .heterotic string potential 13 , this equation becomes

6 t t eyF

2 2Fq sech 1y2tanh4 2ž / ž /ž /a aa g0 00 4

2Xy3 s yF °0e e d 1 1 WW SŽ .2~ < <y WW S y2Ž .2 ¢16 dS S S WW Sg Ž .4

=

y12X XX ¶1 WW S yWW S WW SŽ . Ž . Ž . •y q2 s02 2 ßS WW SŽ .30Ž .

w xUsing the superpotential 47

3S3SyWW S scqh 1q e ,Ž . 2 b0ž /b0

the equation of motion becomes

6 t t eyF


2°y3 s yF0 3Se e 1 3hSy~y y cqh q eŽ . 2 b02 216 bg S¢ 04

23S 3S1 18hcS 3h 6Sy yq y e q 1q e2 b 2 b0 02 ž /S b bb 0 00

2X1 WW SŽ .= y2

S WW SŽ .y12X XX1 WW S yWW S WW SŽ . Ž . Ž .

= y q22 2S WW SŽ .

23S1 3hS

yq cqh q eŽ . 2 b0S b0

X1 WW SŽ .= 2 y2ž /S WW SŽ .

2X1 WW SŽ .q2 y2ž /S WW SŽ .

y22XX X1 WW S WW S yWW SŽ . Ž . Ž .= q22 2ž /S WW SŽ .


1 WW XXX S WW X S WW XX SŽ . Ž . Ž .= y q y33 2ž WW SS Ž . WW SŽ .

3X ¶WW SŽ . •q2 3 /ßWW SŽ .

s0 31Ž .which reduces to

6 t t eyF


27 h2ey3 s 0 ey2F

y3Fy qO e s0 32Ž . Ž .3 48 b g0 4

when c is set equal to yh. Based on the leading-orderXŽ .term in V F , the dilaton field at large times is

approximately

2 2 y3s 03 3 a h e0F t , ln 1q 1qŽ . )2 2 382 a g bž /0 4 0

'=cosh C t ,Ž .4

=Cs 33Ž .2a0

and

'F t ™ C tŽ .2 2 y3s 03 3 a h e0

q ln 1q 1q)2 2 384a g bž /0 4 0

'' C tqD

at large t, representing the linear dilaton solutionw x48,49 .

Ž .The equation of motion for a t is

eyF

2 2 2¨ ˙30a y6Kq6 FyF a qKŽ .Ž .˙ ˙ 2g4

eyF a eyF˙2˙ ˙y12 ay F a ay18 F a qKŽ .˙ ¨ ˙2 2ž / ag g4 4

3 2 2 2˙y a F q3a V F s0 34Ž . Ž .2

Given the string potential

g 2eFey3 s 04

V F sŽ .16

=

2yF eyF3h ey3cqh q eŽ . 2 b g 20 42b g0 4

2y2F eyF9h ey3= cqhq e 2 b g 20 42 4ž /b g0 4

yF3h hqc eŽ .2= cqh qŽ . 2b g0 4

y1y2F1 h 50hq81c eŽ .y 35Ž .2 44 b g0 4

'yŽ C r2. tŽ .it is convenient to set cqh ske insteadof equating it to zero.

tŽ .When Ks1 and a t sa cosh , to leadingŽ .0 a0

Ž .order, Eq. 34 implies

33 2 2 2 y3s 2 D018y a Cq a g e k e s0.0 0 42 16

While there is a positive solution for C when k isreal, it is not consistent with the value required tosolve the dilaton equation of motion. A similar result

Ž . ltoccurs when Ksy1. If Ks0 and a t sa e ,0

there is again a linear dilaton solution

2 y3s 027 h e 'F t ; ln cosh C ty t .Ž . Ž .Ž .03 44C b g0 4

To leading order, the equation of motion for thescale factor gives rise to the relation

332 2 y3s 2 D018l y Cq g e k e s0,42 16

which yields a positive value of C that can beadjusted with l.

Since a linear growth for F and an exponentialŽ .scale factor a t have been obtained from an approx-

imate solution to the equations of motion for theŽ .action 4 derived from heterotic string theory, these

properties should characterize configurations that are


relatively more probable in the minisuperspace� Ž . Ž .4a t ,F t . This allows one to compare directlypredictions of the quantum cosmological model withstandard inflationary cosmology. Moreover, super-symmetry ensures that the FRW universes with scale

t tltfactors a cosh , a e and a sinh are stable,Ž . Ž .0 0 0a a0 0

so that inflation occurring in such exponentially ex-panding space-times would be terminated only whenthis symmetry is broken. Other scale factors, consis-tent with astrophysical observations, can be intro-duced consistently within this model for later epochs,once supersymmetry is broken.

The same techniques can be applied to othersuperstring effective actions with higher-order curva-ture terms, and again, the conjugate momenta andthe Hamiltonian may be derived, with the solution tothe Wheeler–DeWitt equation satisfying specifiedboundary conditions. Since the calculation of thewave function is equivalent to the evaluation of thepath-integral over 4-metrics between initial and finaltimes, the conclusions are consistent with the space-time foam picture, and theoretical expectation valuesof functions of the minisuperspace coordinates canbe compared to observations of these cosmologicalvariables.

Acknowledgements

Research on this project has been partly supportedby an ARC Small Grant.

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the effect of higher-order curvature terms on string quantum cosmology

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