We study the effect of higher-curvature terms in the string low-energy
effective actions on the cosmological solutions of the theory, up to
corrections quartic in the curvatures, for the bosonic and heterotic strings as
well as the type II superstring. We find that cosmological solutions exist for
all string types but they always disappear when the dilaton field is included,
a conclusion that can be avoided if string-loop effects are taken into account.Comment: 7 pages, plain Tex with panda.tex macro (included), no figure

Bento

Boulware

Damour

Grisaru

Gross

Jack

Kikuchi

M.C. Bento

Madore

Metsaev

Müller-Hoissen

O. Bertolami

Park

Wheeler

Wiltshire

Zwiebach

Physics Letters B

English

arXiv

We study the effect of higher-curvature terms in the string low-energy effective actions on the cosmological solutions of the theory, up to corrections quartic in the curvatures, for the bosonic and heterotic strings as well as the type II superstring. We find that cosmological solutions exist for all string types but they always disappear when the dilaton field is included, a conclusion that can be avoided if string-loop effects are taken into account

Bento, M.C.

Bertolami, O.

CERN Document Server

arXiv:gr-qc/9503057v1  29 Mar 1995CERN-TH/95-63DFTT 19/95Cosmological Solutions of Higher-Curvature String Effective Theorieswith DilatonsM. C. Bento∗CERN, Theory DivisionCH-1211 Geneva 23SwitzerlandandO. Bertolami∗†INFN-Sezione TorinoVia Pietro Giuria 1I-10125 Turin, ItalyAbstractWe study the effect of higher-curvature terms in the string low-energy effective actionson the cosmological solutions of the theory, up to corrections quartic in the curvatures,for the bosonic and heterotic strings as well as for the type II superstring. We find thatcosmological solutions exist for all string types but they always disappear when the dilatonfield is included, a conclusion that can be avoided if string-loop effects are taken intoaccount.∗ On leave of absence from Departamento de F´ısica, Instituto Superior Te´cnico, Av. RoviscoPais, 1096 Lisboa Codex, Portugal† Also at Theory Division, CERN.CERN-TH/95-63DFTT 19/95March 1995The fact that string theories, in their low-energy limit, give rise to effective theories ofgravity containing higher-curvature corrections to the usual scalar curvature term, has ledto renewed interest in such extensions of Einstein gravity [1 – 2]. The leading quadraticcorrection, proportional to the Gauss-Bonnet combination, in particular, has ben studiedin some detail in [3], where it is shown that there are cosmological (de Sitter) solutions inaddition to Minkowski space and that they are unstable. For the bosonic string, the next-to-leading corrections are cubic in the curvatures and it has ben shown that cosmologicalsolutions remain unstable at this order [2]. It is, however, possible that higher curvatureterms could give rise to stable de Sitter solutions.In this article, we study the effect of higher order corrections to string low-energyeffective actions, up to the quartic curvature level, on the existence and stability of thecosmological solutions to these theories. For the type II superstring there are neitherquadratic nor cubic-curvature corrections whereas for the heterotic string only the cubicterms are absent so that the quartic curvatures are actually the leading and next-to-leading corrections to these theories, respectively. We shall study also the consequences ofthe inclusion of the dilaton in the effective action; for the quadratic correction, it is knownthat the inclusion of the dilaton has dramatic consequences for the cosmological branches,which become excluded at this level [4]. Finally, we shall analyse the effect of introducingstring-loop corrections to the string effective action.We start with the action in the so-called s-parametrisation:S =∫dDx√−g[Rk+ c1α′e−2φL2 + c2α′2e−4φL3 + c3α′3e−6φL4 + ...](1)where k is Newton’s constant in D dimensions, α′ is the string expansion parameter andthe dots denote terms involving derivatives of the dilaton which will not be relevant forour discussion. Here and throughout the article we use the signature (−,+, ...,+) and theconventions: Rµναβ = Γµνβ,α− ..., Rµν = Rλµλν . The Li terms involve appropriate powersof the curvature, which have been found by comparison of the amplitudes generated bythe action (1) with the string amplitudes (equivalent results have been obtained throughthe computation of the relevant σ-model β-functions), as follows [5]1L2 = Ω2, (2)L3 = 2 Ω3 +RµναβRαβλρRλρµν , (3)L4 = ζ(3)[RµνρσRανρβ(RµγδβRδσαγ − 2RµγδαR δσβγ)]− ξH[18(RµναβRµναβ)2+14R γδµν Rρσγδ Rαβρσ Rµναβ− 12R αβµν Rρσαβ RµσγδRνγδρ −R αβµν R ρναβ R γδρσ R µσγδ]− 12ξB[(RµναβRµναβ)2 − 10RµναβRµνασRσγδρRβγδρ−RµναβRµνρσRβσγδR αδγρ], (4)where ξH(B) = 1 for the heterotic (bosonic) string and vanishes for the other string typesand Ω2, Ω3 are the second and third order Euler densities:Ω2 = R2µναβ − 4R2µν +R2, (5)Ω3 = RµναβRαβλρRλρµν − 2RµναβR λβρν Rαρµλ +34RR2µναβ+ 6RµναβRαµRβν + 4RµνRναRαµ − 6RR2αβ +14R3. (6)Coefficients (c1, c2, c3) are different for different string theoriesbosonic :(14,148,18), (7)heterotic :(18, 0,18), (8)superstringII :(0, 0,18). (9)For maximally symmetric spacesRµνλσ = Λ (δµλgνσ − δµσgνλ) , (10)2and the dilaton is constant. Hence, the gravitational and scalar field equations reduce to,respectivelyf(Λ) ≡ Λ(c3ρΛ3 + c2σΛ2 + c1βΛ + α) = 0, (11)g(Λ) ≡ Λ2(3c3ρΛ2 + 2c2σΛ+ c1β) = 0, (12)whereα = (D − 2)(D − 1), (13)β = α′ke−2φ(D − 4)(D − 3)(D − 2)(D − 1), (14)σ = (α′k)2e−4φ2(D − 6)(D − 1)[2 + (D − 5)(D − 4)(D − 3)(D − 2)], (15)ρ = (α′k)3e−6φ(D − 1)(D − 2)[−3ζ(3)(D − 3) + ξH(D − 2)(D − 9)− 2ξB[(D − 2)(D − 11)− 1]], (16)In the absence of the dilaton, only eq. (11) has to be satisfied; clearly, Minkowskispace is always a solution and, with our conventions, de Sitter (anti-de Sitter) solutionscorrespond to the real positive (negative) roots of (11). We find that, for the superstringII in the critical number of dimensions (D = 10), there is only one positive real root,Λs =(−αc3ρ)1/3. For the heterotic string, there is only one positive real root in D = 4and three real roots in D = 10 (two negative and one positive). For the bosonic string(D = 26) there is only one positive real root.Regarding the stability of these solutions for small graviton excitations about thebackground, notice that it can be deduced immediately from the observation that the signof δ2S is determined by the sign of f ′(Λ) at the roots [3]; in this way, we have checkedthat all cosmological branches are unstable except for one of the anti-de Sitter branches ofthe heterotic string in D = 10.When the dilaton is included, eq. (12) has to be taken into account as well; this is astrong constraint and, indeed, it is easy to check that the cosmological branches we foundabove do not survive and only Minkowsky space remains a solution. We shall see in the3following that this conclusion can be evaded if string-loop effects are taken into account.In the so called σ-parametrization of the effective action, namelyS(σ) =∫dDx√−ge−2φ[Rk+ c1α′L2 + c2α′2L3 + c3α′3L4 + ...], (17)these can be included in a rather simple way, i.e. substituting the exponential e−2φ by theseries in the string coupling gS ≡ e2φ [6]:B(φ) ≡ e−2φ + a0 + a1e2φ + a2e4φ + ..., (18)where the coefficients a0, a1, ... are, at the present level of understanding of string theory,unknown.The s-parametrisation of action (1), is obtained from the σ-parametrization, eq. (17),through the conformal transformation gµν → e−2φgµν , or, if string-loop effects are included:gµν → B(φ)gµν , (19)which, in turn, translates into the substitutionse−2nφ → Bn(φ), n = 1, 2, 3, ... (20)in (1). As a consequence, eqs. (11) and (12) now becomef¯(Λ¯) ≡ Λ¯(ρ¯c3Λ¯3 + σ¯c2Λ¯2 + β¯c1Λ¯ + α) = 0, (21)g¯(Λ¯) ≡ B′(φ)B(φ)Λ¯2(3ρ¯c3Λ¯2 + 2σ¯c2Λ¯ + β¯c1) = 0, (22)where ρ¯ = B3(φ)e6φρ, σ¯ = B2(φ)e4φσ and β¯ = B(φ)e2φβ. Notice now the appearence ofthe factor B′(φ) in front of the dilaton equation of motion; this allows us to require thatthis factor vanishes in order to satisfy eq. (22), which fixes the value of the dilaton field,φ = φ0, so that there is only one algebraic equation for Λ¯. The analysis then proceeds asfor the solutions to eq. (11), except that now results depend on the value of B(φ0). Forinstance, for the superstring II, where Λ¯s = B−1(φ0)Λs and f′(Λ¯s) = −3α, so that the4sign of the solution indeed depends on B(φ0) but the stability analysis does not changeas compared to the case with no dilaton. The same result holds for the heterotic string inD = 4.We stress that our conclusions regarding the existence and stability of cosmologicalsolutions to higher-curvature string effective theories, upon inclusion of next-to-leadingcorrections, support and reinforce the ones that were already pointed out at the leading-order level, namely that there exist de Sitter branches that are, however, erased in thepresence of the dilaton; given that this is basically due to the fact that the dilaton equationof motion represents an additional strong constraint on possible maximally symmetricsolutions, a feature that remains when higher order corrections are included, it is thereforenatural to conjecture that this may be a rather general result. Our analysis shows, however,that this conclusion is not unescapable if string-loop corrections are taken into account.Thus, our results show that string theory has no intrinsic cosmological problem, atleast in D = 4. Furthermore, it is rather easy to show that the inclusion of an ex-plicit cosmological constant may change considerably our conclusions as stable cosmolog-ical branches can be then found. This implies, for instance, that in order to achieve asufficiently long lasting period of inflation a potential for the dilaton with at least onesingle local minimum is required. This indicates the prominance of the issue of super-symmetry breaking and the ensued cosmological constant problem in any realistic stringycosmological scenario.There is, nonethess, a basic ambiguity in our analysis, associated with the fact thatthe form we have chosen for the higher-curvature corrections, eqs. (2)–(4), is just oneamong various possible parametrizations that are related by metric redefinitions of thetype gµν → gµν + α′(a1Rµν + a2gµνR) + α′2(RµαβγR αβγν + ...) + ..., which do not changethe S-matrix. In fact, it is possible to show that there are both ambiguous and fixed termsaccording to whether they are changed by field redefinitions or not. At order α′, only theR2µνρσ term is fixed by the string amplitudes and the Gauss-Bonnet form is singled out onlyif we introduce the further requirement that the effective theory be unitary at this level [7].However, it has been shown that at cubic and higher orders, it is not possible to write theeffective action only in terms of Euler densities as required by unitarity. The ambiguous5terms are then often chosen so as to keep the effective action as simple as possible at eachorder; however, even this choice is not unique since such terms cannot, in general, be allset to zero simultaneously given that there are relations among their variations. We havestudied other possible forms for the effective action at the cubic curvature level, where acomplete field redefinition analysis has been performed in Refs. [8, 9], and we find that ourconclusions do not basically change, i.e. although there indeed exist different cosmologicalsolutions for different actions, they are still always erased in the presence of the dilatonand saved by the inclusion of string-loop effects.AcknowledgementsWe would like to thank Prof. V. de Alfaro for discussions on the subject of this paper.References[1] J. Madore, Phys. Lett. A110 (1985) 289; A111 (1985) 283; F. Mu¨ller-Hoissen, Phys.Lett. B163 (1985) 106; J.T. Wheeler, Nucl. Phys. B268 (1986) 36; D.L. Wiltshire,Phys. Lett. B169 (1986) 36.[2] M.C. Bento and O. Bertolami, Phys. Lett. B228 (1989) 348.[3] D.G. Boulware and S. Deser, Phys. Rev. Lett. 55 (1985) 2656.[4] D.G. Boulware and S. Deser, Phys. Lett. B175 (1986) 409.[5] D.G. Gross and E. Witten, Nucl. Phys. B277 (1986) 1; Y. Kikuchi, C. Marzban andY.J. Ng, Phys. Lett. B176 (1986) 57; M.T. Grisaru and D. Zanon, Phys.Lett. B277(1986) 199; Q.-H. Park and D. Zanon, Phys. Rev. D35 (1987) 4038; R.R. Metsaev andA.A. Tseytlin, Phys. Lett. B185 (1987) 52; I. Jack, D.R.T. Jones and N. Mohammedi,Phys. Lett. B220 (1989) 171.[6] T. Damour and A.M. Polyakov, Nucl. Phys. B423 (1994) 532.[7] B. Zwiebach, Phys. Lett. B156 (1985) 315.[8] M.C. Bento, O. Bertolami, A.B. Henriques and J.C. Roma˜o, Phys. Lett. B218 (1989)162; B220 (1989) 113.[9] M.C. Bento, O. Bertolami and J.C. Roma˜o, Phys. Lett. B252 (1990) 401; Int. J.Mod. Phys. A6 (1991) 5099.6

Cosmological solutions of higher-curvature string effective theories with dilatons

Bento, M C

Bertolami, O

Maximally symmetric cosmological solutions of higher-curvature string effective theories with dilatons

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Cosmological Solutions of Higher-Curvature String Effective Theories
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Cosmological Solutions of Higher-Curvature String Effective Theories with Dilatons

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