Duality Invariant Cosmology to All Orders in Α′

PHYSICAL REVIEW D 100, 126011 (2019)

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Duality invariant cosmology to all orders in α0

Olaf Hohm1 and Barton Zwiebach2 1Institute for Physics, Humboldt University Berlin, Zum Großen Windkanal 6, D-12489 Berlin, Germany 2Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

(Received 14 June 2019; published 10 December 2019)

While the classification of α0 corrections of string inspired effective theories remains an unsolved problem, we show how to classify duality invariant α0 corrections for purely time-dependent (cosmological) backgrounds. We determine the most general duality invariant theory to all orders in α0 for the metric, b-field, and dilaton. The resulting Friedmann equations are studied when the spatial metric is a time- dependent scale factor times the Euclidean metric and the b-field vanishes. These equations can be integrated perturbatively to any order in α0. We construct nonperturbative solutions and display duality invariant theories featuring string-frame de Sitter vacua.

DOI: 10.1103/PhysRevD.100.126011

I. INTRODUCTION geometry where the diffeomorphism invariance of general relativity is replaced by a suitably generalized notion of String theory is arguably the most promising candidate diffeomorphisms, which in turn partly determines the α0 for a theory of quantum gravity. As a theory of gravity, the “ ” prospect for a confrontation between string theory and corrections. These ideas play a key role in the chiral observation seems to be particularly promising in the realm string theory of [15], which is the only known gravitational of cosmology, where the effects of fundamental physics at field theory that is exactly duality invariant and has α0 very small scales may be amplified to very large scales. infinitely many corrections. This program, however, Since the early days of string theory there have been has not yet been developed to the point that it can deal with α0 intriguing ideas of how its unique characteristics could play the set of all corrections for bosonic or heterotic string a role in cosmological scenarios [1–3], ideas that have been theory. (See, however, the recent proposal for heterotic revisited and extended recently [4–7]. Two such features of string theory [22].) classical string theory will be central for the present paper: This state of affairs is unsatisfactory since the inclusion α0 (i) the existence of dualities that for cosmological back- of a finite number of higher-derivative corrections is grounds send the scale factor aðtÞ of the universe to 1=aðtÞ, generally insufficient. Gravitational theories with a finite in sharp contrast to Einstein gravity, and (ii) the presence of number of higher derivatives typically display various infinitely many higher-derivative α0 corrections. pathologies that are an artifact of the truncation and not It is a natural idea that the higher-order corrections of present in the full string theory [23]. In this paper we will string theory play a role, for instance, in resolving the big- bypass these difficulties by classifying the higher-derivative α0 bang singularity. The first step is the inclusion of the α0 corrections relevant for cosmology to all orders in . α0 corrections of classical string theory. Such corrections have Rather than finding the complete corrections in general been computed to the first few orders in the 1980s [8–10], dimensions and then assuming a purely time-dependent but a complete computation or classification of these cosmological ansatz, we immediately consider the theory corrections is presently out of reach. In recent years reduced to one dimension (cosmic time) and determine duality-covariant formulations of the string spacetime the complete higher-derivative corrections compatible theories (double field theory [11–14]) have been used to with duality. String theories, being duality invariant, must make progress in describing α0 corrections; see [15–21]. correspond to some particular points in this theory space. These developments hint at a completely novel kind of While we do not know which points those are, the full space of duality invariant theories is interesting in its own right, and may exhibit phenomena that are rather general and apply to string theory. Published by the American Physical Society under the terms of Our analysis is based on the result by Veneziano and the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to Meissner [24], extended by Sen [25], concerning the the author(s) and the published article’s title, journal citation, classical field theory of the metric, the b-field, and the and DOI. Funded by SCOAP3. dilaton arising from D ¼ d þ 1 dimensional string theory.

2470-0010=2019=100(12)=126011(21) 126011-1 Published by the American Physical Society OLAF HOHM and BARTON ZWIEBACH PHYS. REV. D 100, 126011 (2019)

This field theory displays an Oðd; d; RÞ symmetry to all The pattern is clear: the general term at order α0k involves 0 orders in α provided the fields do not depend on the d traces with 2k factors of S_ . Each trace must have an even spatial coordinates. This symmetry, henceforth referred to number of S_ factors, where the even number cannot be “ ” → −1 as duality, contains the scale-factor duality a a . two. The c’s are a priori undetermined coefficients, duality The work of Meissner [26] implies that in terms of standard holding for any value they may take. Except for a few of α0 fields the duality transformations receive corrections, but them, their values for the various string theories are α0 it was shown that to first order in there are new field unknown. In establishing the above result, we made variables in terms of which dualities take the standard form. repeated use of field redefinitions iteratively in increasing We will assume that there are field variables so that duality orders of α0. The result is striking in that only first α0 transformations remain unchanged to all orders in (this derivatives of the fields appear in the action. All duality certainly happens in conventional string field theory invariant terms with more than one time derivative on S can variables [27]). With this assumption we are able to classify be redefined away. All terms with one or more derivatives α0 completely the duality invariant corrections. This work of the dilaton can also be redefined away. The resulting amounts to an extension and elaboration of the results higher derivative actions are in fact actions with high obtained by us in [28]. We use the freedom to perform numbers of fields acted on by one derivative each. This is a duality-covariant field redefinitions to show that only first- major, somehow unexpected, simplification. order time derivatives need to be included and that the In the second part of this paper we investigate this dilaton does not appear nontrivially, thereby arriving at a general α0-complete theory for the simplest cosmological minimal set of duality invariant higher-derivative terms ansatz, a Friedmann-Lemaître-Robertson-Walker (FLRW) α0k to all orders. We prove that at order the number of background whose spatial metric is given by a time- independent invariants, and thus the number of free dependent scale factor times the Euclidean metric and a R parameters not determined by Oðd; d; Þ,isgivenby vanishing b-field. The resulting Friedmann equations are 1 − pðk þ Þ pðkÞ, with pðkÞ the number of partitions of the determined to all orders in α0. While there is of course still integer k. an infinite number of undetermined ck;l parameters in (1.3), Let us briefly summarize the core technical results of the we can write the equations efficiently in terms of a single first part of the paper. The two-derivative spacetime theory function FðHÞ of the Hubble parameter whose Taylor ϕ for the metric g, the b-field b, and the dilaton , restricted to expansion is determined by these coefficients: depend only on time, is described by the one-dimensional action [24] X∞ ≔ 4 −α0 k−122k−1 2k Z FðHÞ d ð Þ ckH ; ð1:4Þ 1 −Φ −Φ_ 2 − S_ 2 k¼1 I0 ¼ dte 8 trð Þ ; ð1:1Þ where, in terms of (1.3), ck ≔ ck;0 þ 2dck;1 þ. The where Φ is the O d; d; R invariant dilaton, defined by Friedmann equations then take the concise form pffiffiffiffiffiffiffiffiffi ð Þ e−Φ ¼ det ge−2ϕ, and we introduced the Oðd; d; RÞ d valued matrix ðe−ΦfðHÞÞ ¼ 0; dt −1 −1 bg g − bg b ̈ 1 S ≡ ; ð1:2Þ Φ þ HfðHÞ¼0; g−1 −g−1b 2 2 Φ_ þ gðHÞ¼0; ð1:5Þ in terms of the spatial components g and b of the metric and b-field. Our classification implies that the most general where the functions fðHÞ and gðHÞ are determined in terms 0 duality invariant α corrections take the form of FðHÞ as Z −Φ 0 _ 4 02 _ 6 f H ≔ F0 H ;gH ≔ HF0 H − F H ; 1:6 I ≡ I0 þ dte α c2;0trðS Þþα c3;0trðS Þ ð Þ ð Þ ð Þ ð Þ ð Þ ð Þ

03 _ 8 _ 4 _ 4 where 0 denotes differentiation with respect to H. þ α ½c4;0trðS Þþc4;1trðS ÞtrðS Þ We show that solutions of the lowest-order equations α04 S_ 10 S_ 6 S_ 4 þ ½c5;0trð Þþc5;1trð Þtrð Þ given by Mueller in [29] can be extended, perturbatively, to 0 05 _ 12 _ 8 _ 4 _ 6 2 arbitrary order in α . We then turn to the arguably most þ α ½c6 0trðS Þþc6 1trðS ÞtrðS Þþc6 2ðtrðS ÞÞ ; ; ; intriguing implication of our results: the potential existence _ 4 3 þ c6;3ðtrðS ÞÞ þ; ð1:3Þ of interesting cosmological solutions that are nonperturbative in α0. We discuss a general nonperturbative initial-value where only first-order time derivatives of S need to be formulation, and we state conditions on the function FðHÞ _ 2 included. Moreover, there are no terms involving trðS Þ. so that the theory permits de Sitter vacua (in string frame).

126011-2 DUALITY INVARIANT COSMOLOGY TO ALL ORDERS IN … PHYS. REV. D 100, 126011 (2019) There are functions satisfying these criteria, and so there are −n2ðtÞ 0 00 gμν ¼ ;bμν ¼ ; duality invariant theories with nonperturbative de Sitter 0 g ðtÞ 0 b ðtÞ vacua, suggesting that string theory may realize de Sitter ij ij in this novel fashion [30]. Note that the Lagrangian for ϕ ¼ ϕðtÞ: ð2:2Þ the theory does not include a cosmological constant term. We also note that in pure gravity the possibility of The resulting one-dimensional two-derivative action then generating inflation through restricted higher-derivative takes the Oðd; dÞ invariant form [24]1 interactions leading to second-order Friedmann equations Z Z has been investigated [31]. Here we do not constrain the 1 1 ≡ −Φ −Φ −Φ_ 2 − S_ 2 number of derivatives in the original theory. Rather, duality I0 dtne L0 ¼ dtne 2 8 trð Þ ; invariance and field redefinitions lead to two-derivative n equations in the cosmological setting. ð2:3Þ The paper is organized as follows. In Sec. II we review the two-derivative theory and its equations of motion and where then tackle the classification problem. We examine carefully the freedom to perform field redefinitions, including bg−1 g − bg−1b 01 those of the lapse function nðtÞ, which is usually set to one S ≡ ηH η ¼ −1 −1 ; ¼ : by a gauge choice. In Sec. III we derive the equations of g −g b 10 motion of the higher-derivative action restricted to the ð2:4Þ single trace terms, and then compute the Noether charges for the global duality symmetry. Section IV specializes to S S2 1 the FLRW metric with zero curvature and derives the The field is constrained: it satisfies ¼ . We have also Φ Friedmann equations to all orders. As it turns out, this result defined the Oðd; dÞ invariant dilaton via is valid even when the action contains the most general qffiffiffiffiffiffiffiffiffiffiffiffi multitrace terms. The solutions in perturbation theory of α0 −Φ −2ϕ e ¼ det gije : ð2:5Þ are determined. In Sec. V we consider nonperturbative solutions. We solve the initial-value problem and show how de Sitter solutions are possible in the space of duality- The action is time-reparametrization invariant, with nðtÞ invariant theories. We conclude this paper with a discussion transforming as a density under time reparametrizations of possible further generalizations. t → t − λðtÞ:

II. CLASSIFICATION OF COSMOLOGICAL δλn ¼ ∂tðλnÞ: ð2:6Þ Oðd;dÞ INVARIANTS Our goal is to classify Oðd; dÞ invariant actions to all A scalar A under this reparametrization transforms as 0 orders in α , up to field redefinitions. This was partially δλA ¼ λ∂tA. The metric, the b-field, and the dilaton Φ done in [28], whose results will here be completed by are scalars under time reparametrizations. Defining the allowing for Oðd; dÞ covariant field redefinitions of the covariant time derivative D, lapse function. 1 ∂ D ≡ ; ð2:7Þ A. Review of two-derivative theory nðtÞ ∂t We start with the two-derivative theory for metric gμν, antisymmetric b-field bμν, and the scalar dilaton ϕ, one can quickly show that if A is a scalar, so is DA. Z The covariant derivative satisfies the usual integration by ffiffiffiffiffiffi 1 2 − 26 D p −2ϕ 2 2 ðD Þ parts rule I0 d x −ge R 4 ∂ϕ − H − ; ¼ þ ð Þ 12 3α0 Z Z Z ð2:1Þ dtnBDA ¼ dt∂tðABÞ − dtnðDBÞA: ð2:8Þ 3∂ where Hμνρ ¼ ½μbνρ. Here we displayed the action for a generally noncritical bosonic string theory in D dimen- The action I0 above can now be written in a manifestly sions. In the following we assume D ¼ 26 but comment reparametrization invariant form on the general case in Sec. VC below. We now drop the ∂ 0 dependence on all spatial coordinates; i.e., we set i ¼ , 1This result generalizes to the compactification of the μ i where x ¼ðt; x Þ, i ¼ 1; …;d, and we subject the fields to D-dimensional theory on a d-torus to D − d dimensions, which the ansatz also exhibits a global Oðd; dÞ symmetry [25,33,34].

126011-3 OLAF HOHM and BARTON ZWIEBACH PHYS. REV. D 100, 126011 (2019) Z −Φ 2 1 2 above right-hand side is indeed equal to δSI. However, (2.16) I0 ¼ dtne −ðDΦÞ − trððDSÞ Þ : ð2:9Þ 8 is not the operational way to derive ES by variation. Applying the above discussion for the case of the action The variation of the dilaton yields the dilaton equation of I0 we now find by direct variation motion EΦ ¼ 0, Z −Φ Z δSI0 ¼ dtne trðδSFSÞ; −Φ δΦI0 ¼ dtne δΦEΦ; 1 ≡ D2S − DΦDS 1 with FS 4 ð Þ: ð2:17Þ ≡ 2D2Φ − DΦ 2 D 2 EΦ ð Þ þ 8 trðð SÞ Þ: ð2:10Þ The equation of motion is now ES ¼ 0 with ES ¼ 1 Let us explain first in general terms how we vary with 2 ðFS − SFSSÞ. It is straightforward to simplify the result- S S respect to to find field equations. Consider an arbitrary ing expression for ES. For this we note from SS ¼ 1 that S dependent action I of the form and DS anticommute: Z I ¼ dtne−ΦLðSÞ; ð2:11Þ SDS ¼ −ðDSÞS: ð2:18Þ Taking a further derivative of this equation we also find that where L could also depend on other fields. To vary I one first varies S as if it would be an unconstrained field with an 2 2 2 SðD SÞS ¼ −D S − 2SðDSÞ : ð2:19Þ unconstrained variation, finding an expression of the form Z It is now a simple matter to show that −Φ δSI ¼ dtne trðδSFSÞ: ð2:12Þ 1 ≡ D2S S DS 2 − DΦ DS ES 4 ð þ ð Þ ð Þð ÞÞ: ð2:20Þ The equation of motion is not the vanishing of FS because δS S2 1 δS is a constrained variation: since ¼ the variation The lapse equation of motion follows by varying with δS −SδSS δ satisfies ¼ . We can write S in terms of an respect to n. The variation is written in the form δ unconstrained variation K as follows: Z δ 1 δ −Φ n δS δ − Sδ S nI0 ¼ dtne En: ð2:21Þ ¼ 2 ð K K Þ: ð2:13Þ n

Note that δK cannot be written in terms of δS; it has extra This is sensible, because one can readily show that a δ information that drops out in the combination shown in the variation n transforms as a density and, as a result, the δ above right-hand side. Substituting δS into the above ratio n=n transforms as a scalar. The quantity En is then a scalar as well. To compute this variation it is simplest to variation δSI we now get Z write the action I0 in terms of ordinary time derivatives 1 using (2.3): δ −Φ δ − S S SI ¼ dtne trð KESÞ;ES ¼ 2 ðFS FS Þ: Z 1 1 −Φ _ 2 _ 2 2:14 I0 ¼ dte −Φ − trðS Þ : ð2:22Þ ð Þ n 8

The equation of motion is indeed ES 0, since δK is ¼ It follows that an unconstrained variation. Note that any ES defining an equation of motion for S satisfies Z −Φ δn 2 1 2 δ I0 ¼ dtne ðDΦÞ þ trððDSÞ Þ : ð2:23Þ n n 8 ES ¼ −SESS: ð2:15Þ Comparing with (2.21) we have therefore found that There is finally an alternative rewriting of δSI that can now be seen to be valid too: 1 Z ≡ DΦ 2 DS 2 En ð Þ þ 8 trðð Þ Þ: ð2:24Þ −Φ δSI ¼ dtne trðδSESÞ: ð2:16Þ

B. Oðd;dÞ invariant α0 corrections Using the expression for ES in terms of FS and the constraint δS ¼ −SδSS one quickly verifies that trðδSESÞ¼ We will discuss a recursive procedure to write the duality trðδSFSÞ, and comparing with (2.12) we conclude that the invariant action in a canonical form. In this we use the

126011-4 DUALITY INVARIANT COSMOLOGY TO ALL ORDERS IN … PHYS. REV. D 100, 126011 (2019) two-derivative part of the action to do field redefinitions Our goal is now to prove that IðkÞ can be brought to the that allow us to simplify the higher-derivative parts of the form action. The four-derivative part of the action is of order α0. Z α0k 2 2 The terms of order will have k þ derivatives. We call I α0k dtne−ΦX DS : 2:30 0k k ¼ ð Þ ð Þ Ik the part of the action of order α , and it is a sum of terms,

XNðkÞ This means that all we can have is a product of traces of DS Ik ¼ Ik;p; ð2:25Þ powers of . The proof will proceed inductively, assum- p¼1 ing that all terms up to Iðk−1Þ are of this form and then using field redefinitions to show that the action IðkÞ can be where each term Ik;p is a product of dilaton derivatives and brought to this form (at the cost of changing the actions S traces of strings of and its derivatives, in order to be Ikþ1, Ikþ2, etc., which will be taken care of in the next duality invariant. There are only a finite number of terms induction step). NðkÞ that can be constructed with the requisite number of To this end we use various simple theorems established derivatives. We have in [28]: Z Y 2 0k −Φ j n trðSÞ¼trðDSÞ¼trðD SÞ¼¼0; ð2:31Þ Ik;p ¼ α ck;p dtne ðD i ΦÞ i Y i q q 2kþ1 k1 m1 k l m l trððDSÞ Þ¼0; for k ¼ 0; 1; …; ð2:32Þ × trððD l SÞ l ðD l SÞ l Þ; ð2:26Þ l trðSðDSÞkÞ¼0; for k ¼ 0; 1; …: ð2:33Þ where ck;p are unit-free constants. For each i in the first α0 product, the values of the indices ji and ni denote the The strategy will be to work at each order of . Assume number of time derivatives and the power of the term. that we have succeeded in casting the action in the Because of the dilaton theorem [35], all nonderivative expected form (2.30) for I2; …;Ik−1. Then we are faced dependence on the dilaton is in the exponential prefactor, with bringing Ik to the desired form. The action can be and we thus take ji ≥ 1. For each l in the second product written as we get a trace factor. Inside the trace there are ql factors, Z each one a fixed power of a multiple time derivative of S. Xk−1 0k −Φ 0kþ1 I ¼ I0 þ Ip þ α dtne XðfDϕg;fSgÞ þ Oðα Þ: This includes the possibility of some kl ¼ 0 which means p¼1 no derivatives of S. This must be accompanied by ml ¼ 1 since S2 ¼ 1. Implicit above is a sum rule that requires that ð2:34Þ the total number of derivatives is 2k þ 2. We will use a notation that makes clear what are the We will then consider field redefinitions of the form arguments of a function X of fields. When we write Φ → Φ þ α0kδΦ; S → S þ α0kδS: ð2:35Þ XðA; B; …Þ; ð2:27Þ To see how I changes, we need only vary I0, because this we mean that X is strictly a function constructed from will generate Oðα0kÞ terms that we are interested in. The A; B; …, without including time derivatives D of these variations of I1; …;Ik−1 generate terms of higher order than functions. When we write α0k that are not relevant, but are of the prescribed form (2.26). The variations of I0 are determined by the field XðfAg; fBg; …Þ; ð2:28Þ equations and take the form Z with braces around the arguments, we mean X is a function 0k −Φ δI0 ¼ α dtne ðδΦEΦ þ trðδKESÞÞ with of A; DA; D2A; …, as well as B; DB; D2B; …. Mixed types of arguments are also possible: XðA; fBgÞ would depend 1 δS δ − Sδ S only on A, but possibly on B; DB; D2B; …. In this notation ¼ 2 ð K K Þ: ð2:36Þ equation (2.26) would be described as saying that Z At each step we use these field redefinitions to remove 0k −Φ undesirable terms from the Oðα Þ part of the action. I ¼ α0k dtne XðfDΦg; fSgÞ; ð2:29Þ k;p We elaborate and systematize some of the derivations of [28], carefully proving several statements that lead to the for some choice of X. desired claim.

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(1) A factor D2Φ in an action can be replaced by a factor 1 δK ¼ −XðfDΦg; fSgÞG; ð2:44Þ of QΦ with only first derivatives. 4 0k Consider a generic such term Zk at order α as part of (2.34): and realizing that X is a scalar that can go out of the Z trace, it gives 0k −Φ 2 Z Zk ¼ α dtne XðfDΦg; fSgÞD Φ: ð2:37Þ 0k −Φ δI0 ¼ −α dtne XðfDΦg;fSgÞ

With a dilaton field redefinition equation (2.36) ×tr½GðD2S þ SðDSÞ2 − ðDΦÞðDSÞÞ: ð2:45Þ gives us

Z This variation cancels the Zk term above and 0k −Φ δI0 ¼ α dtne 2δΦ replaces it with Z 1 1 D2Φ − DΦ 2 D 2 Z0 ¼ α0k dtne−ΦXðfDΦg; fSgÞ × 2ð Þ þ 16trðð SÞ Þ : ð2:38Þ k ×trðGð−SðDSÞ2 þðDΦÞðDSÞÞÞ: ð2:46Þ If we choose The net effect is that the following replacement is 2δΦ − DΦ S ¼ Xðf g; f gÞ; ð2:39Þ allowed in the Oðα0kÞ action:

the shift in I0 cancels Z completely and replaces it 2 2 k D S → QS ≡ −SðDSÞ þ DΦDS; ð2:47Þ with

Z with QS having at most first derivatives. 0 α0k −Φ DΦ S Zk ¼ dtne Xðf g; f gÞ (3) Any action can be reduced so that it only has first time derivatives of Φ. 1 1 We already know how to eliminate terms with × ðDΦÞ2 − trððDSÞ2Þ : ð2:40Þ 2 16 two derivatives on the dilaton. Suppose we have an action with more than two derivatives of Φ. This can The net effect is that the following replacement is always be written in the form allowed in the Oðα0kÞ action: Z α0k −Φ DΦ S Dpþ2Φ 1 1 Zk ¼ dtne Xðf g; f gÞ ; D2Φ → ≡ DΦ 2 − D 2 QΦ 2 ð Þ 16 trðð SÞ Þ; ð2:41Þ 0

with QΦ having only first derivatives. (2) A factor of D2S in an action can be replaced by a where the dilaton factor to the right has more than two derivatives. Now, write Dpþ2Φ ¼ DpðD2ΦÞ factor of QS with only first-order derivatives. p 0k and integrate by parts the D , finding Consider a generic term Zk at order α with a second derivative on S: Z 0k p −Φ 2 Z Zk ¼ α dtnð−DÞ ðe XÞD Φ; ð2:49Þ 0k −Φ 2 Zk ¼ α dtne XðfDΦg; fSgÞtrðGD SÞ; suppressing the arguments of X. We can now use 2 ð2:42Þ property 1 to eliminate the D Φ, replacing Zk by Z where G is a matrix of type GðfSgÞ. With a field 0 α0k −D p −Φ redefinition of S, Eq. (2.36) gives Zk ¼ dtnð Þ ðe XÞ Z 1 1 1 DΦ 2 − DS 2 δ α0k −Φ δ D2S S DS 2 × ð Þ trð Þ : ð2:50Þ I0 ¼ dtne tr 4 Kð þ ð Þ 2 16 Now we integrate back the p derivatives of the left − ðDΦÞðDSÞÞ : ð2:43Þ factor, one by one. At each step this will generate a second-order time derivative of the second factor, Choosing the matrix δK [which determines δS ¼ but these can be reduced to first derivatives by 1 2 ðδK − SδKS)] to be properties 1 and 2. We can write this formally

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defining a derivative operator D¯ that acts on func- F. We integrate by parts Dp and then are allowed to 2 tions F of S; DS, and DΦ: replace D S by QS, finding that Zk is replaced by Z D¯ S DS DΦ ð FÞð ; ; Þ 0 α0k −D pF Zk ¼ dtntr½ð Þ QS: ð2:57Þ ≡ D S DS DΦ 2 2 ðFð ; ; ÞÞjD Φ→QΦ;D S→QS : ð2:51Þ

The derivatives are then folded back into QS, one at The replacements here are those in (2.41) and (2.47). D¯ a time, at each step eliminating the second deriva- Note that after taking a derivative, the result is still tives that are generated. As explained in property 3, a function of only S, DS, and DΦ. It follows now each ð−DÞ derivative becomes a D¯ derivative, and that Z in (2.48) is eventually replaced by k we find that Z0 becomes Z k Z 00 0k −Φ ¯ p Z α dtne X DΦ ; S D QΦ: 2:52 k ¼ ðf g f gÞ ð Þ 0 α0k FD¯ p Zk ¼ dtntr½ QS: ð2:58Þ The factor to the right of X contains only first F derivatives of Φ and at most first derivatives of S. Using the definition of we now note that (2.54) Because of (2.33), the factor to the right of X cannot has become contain S; it must be built explicitly with DS only. Z Should X above still contain factors with more 00 α0k −Φ DΦ S G S D¯ p Zk ¼ dtne Xð ; f gÞtr½ ðf gÞ QS; than two derivatives of Φ the procedure is carried through again for each factor. Proceeding in this way ð2:59Þ we can reduce the higher derivative factors in Φ one by one to first order in time derivatives. This step has achieving the desired elimination of the higher shown that the most general form of the action is derivative of S in terms of first derivatives of Φ Z and up to first derivatives of S. This procedure can 0k −Φ S Ik ¼ α dtne XðDΦ; fSgÞ: ð2:53Þ be iterated until all higher derivatives of are eliminated. At his point only S and DS can appear DS This means that, apart from the e−Φ factor, only first and, as explained before, this means that only I derivatives of the dilaton appear. can appear. We have thus shown that terms of k, (4) Any action can be reduced so that it only has first already put in the form (2.53), can be simplified time derivatives of S. further to be set in the form Assume there is a trace factor that has a time Z S 0k −Φ derivative of of a degree higher than 2. Using the Ik ¼ α dtne XðDΦ; DSÞ: ð2:60Þ form of the action (2.53) we already know to be true, we write the action separating out such a term: Z Each term Ik;p in such an action is written in the form Z ¼ α0k dtne−ΦXðDΦ; fSgÞtr½GðfSgÞDpþ2S; k Z 0k −Φ p 0 is equivalent to one without any appearance of DΦ. FðDΦ; fSgÞ ≡ e−ΦXðDΦ; fSgÞGðfSgÞ; ð2:55Þ Suppose we have an action Ik with a term Z −Φ p the action above takes the form Ik;p ¼ dtne ðDΦÞ XlðDSÞ: ð2:62Þ Z 0k pþ2 Zk ¼ α dtntr½FD S: ð2:56Þ Here Xl is an arbitrary (generally multitrace) invariant:

For brevity, we do not indicate the arguments of F, l1 ln XlðDSÞ¼tr½ðDSÞ tr½ðDSÞ ; which are given in (2.55). In this form integration by parts is straightforward and folds the derivatives into l ¼ l1 þþln: ð2:63Þ

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In a moment we will need that the time derivative of Let us note that the denominator on the right-hand side this function has a simple form. As we take a derivative never vanishes for any case of interest. Since the total we generate D2S factors within the traces. Each such number of derivatives in the term is p þ l,foraterm 2 factor can be replaced by QS ¼ −SðDSÞ þ DΦDS, in an action Ik, the denominator takes the value but since traces with a single S multiplying DS factors vanish [Eq. (2.33)], the replacement is effectively ðp þ lÞþl − 3 ¼ 2k þ 2 þ l − 3 ¼ 2k þ l − 1 ≥ 1; D2S → DΦDS. As a result, inside an integral repre- for k ≥ 1: ð2:69Þ senting a term in Ik, the following replacement is allowed: The formula can be used recursively to reduce the power of DΦ in steps of two; in each step p decreases DX ∼ lDΦX ðDSÞ: ð2:64Þ l l by two units and l increasesbytwounits[since 2 tr DS X is an X 2 object], and at no stage will We begin by rewriting (2.62) with one factor of DΦ ð Þ l lþ −Φ the denominator vanish. If p ¼ 2m is even, it follows written as a derivative of e , and then proceed to do from (2.68) that we have integration by parts: Z Z −Φ 2m −Φ p−1 dtne ðDΦÞ XlðDSÞ Ik;p ¼ − dtnDðe ÞðDΦÞ Xl Z Z −Φ 2 m −Φ p−2 2 ∼ C dtne ½trðDSÞ XlðDSÞ; ð2:70Þ ¼ dtne ððp − 1ÞðDΦÞ D ΦXl

DΦ p−1D þð Þ XlÞ: ð2:65Þ where C is a constant that is easily determined. Note also that for p ¼ 1 Eq. (2.68) tells us that D2Φ Replacing by QΦ and using the above evaluation Z of DXl we get dtne−ΦDΦX ðDSÞ ∼ 0: ð2:71Þ Z l I ∼ dtne−Φ ðp − 1ÞðDΦÞp−2 k;p 1 For the case l ¼ , where the prefactor in (2.68) gives 1 1 zero divided by zero, the left-hand side vanishes too DΦ 2 − DS 2 DΦ p × 2 ð Þ 16 trð Þ Xl þ lð Þ Xl because trðDSÞ¼0. Since the recursion relation Z works in steps of two units, the vanishing of terms 1 −Φ 2 − 1 DΦ p with a single DΦ implies that ¼ dtne 2 ðp þ l Þð Þ Xl Z 1 −Φ 2 1 − − 1 DΦ p−2 DS 2 dtne ðDΦÞ mþ X ðDSÞ ∼ 0: ð2:72Þ 16 ðp Þð Þ trð Þ Xl : ð2:66Þ l

The first term is of the same form as the original (2.62), The formula (2.68) also applies when l ¼ 0 and 1 and thus bringing it to the left-hand side we have the Xl ¼ , in which case it gives equivalence Z 1 p − 1 Z −Φ DΦ p ∼ 1 dtne ð Þ 8 − 3 3 − − 2 −Φ DΦ p p 2 ð p lÞ dtne ð Þ Xl Z Z −Φ p−2 2 1 × dtne ðDΦÞ trðDSÞ : − − 1 −Φ DΦ p−2 DS 2 ¼ 16 ðp Þ dtne ð Þ trð Þ Xl; ð2:73Þ ð2:67Þ The denominator in the prefactor only vanishes for and thus in actions we can replace p ¼ 3 which is not relevant for any action Ik.The Z remarks above apply again. When p ¼ 2m is even, dtne−ΦðDΦÞpX ðDSÞ the term can be reduced to one containing no dilaton l DS 2 m Z derivatives and just ½trð Þ .Ifp is odd, the term 1 p − 1 vanishes because it does for p ¼ 1. ∼ dtne−ΦðDΦÞp−2 8 p þ 2l − 3 WehaveshownsofarthatanytermIk;p in the action I takes the form (2.61). Since we have now learned DS 2 DS k ×trð Þ Xlð Þ: ð2:68Þ that powers of derivatives of the dilaton can be

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eliminated completely, we have shown that any term in where β is a constant to be determined. The associated Ik takes the claimed form (2.30): variation of I0 is therefore Z Z 1 α0k −Φ DS δ βα0k −Φ DS DΦ 2 DS 2 Ik ¼ dtne Xð Þ: ð2:74Þ nI0 ¼ dtne X2kð Þ ð Þ þ 8 trð Þ : : While this is a great simplification, further simplifi- ð2 80Þ cation can be achieved by using redefinitions of the This counts as an additional term in the action. Note the lapse function nðtÞ. We explore this next. second term in parentheses gives a contribution that could cancel the term in Ik;p above. The first term also does, if we 1. Lapse redefinitions use identity (2.68) which amounts to a further dilaton 2 2 We have confirmed that up to Oðd; dÞ covariant rede- redefinition. This identity with p ¼ and l ¼ k allows us finitions of S and Φ one only needs to consider (arbitrary to write Z powers of) first time derivatives of S and no time 0k −Φ derivatives of Φ. Thus, the most general action takes the δnI0 ¼ βα dtne X2kðDSÞ following form: 1 1 Z DS 2 DS 2 × 8 4 − 1 trð Þ þ 8 trð Þ −Φ 0 0 2 0 3 ð k Þ I ¼ dtne ðL0 þ α L1 þðα Þ L2 þðα Þ L3 þÞ; Z β k 0k −Φ 2 ¼ α dtne X2 ðDSÞtrðDSÞ : ð2:81Þ ð2:75Þ 2ð4k − 1Þ k β where the general forms of L1, L2, and L3 are One can cancel Ik;p in (2.78) by choosing such that βk 2 4 −1 ¼ −1. Since this can always be done, this completes DS 4 DS 2 2 ð k Þ L1 ¼ a1trð Þ þ a2½trð Þ ; our proof that any term with a trðDSÞ2 factor can be set 6 4 2 2 3 L2 ¼ b1trðDSÞ þ b2trðDSÞ trðDSÞ þ b3½trðDSÞ ; to zero. 0 8 4 2 6 2 Given this result, the most general α corrections up to L3 ¼ c1trðDSÞ þ c2½trðDSÞ þ c3trðDSÞ trðDSÞ 3 order ðα0Þ given in (2.76) can be simplified to take the 4 2 2 2 4 þ c4trðDSÞ ½trðDSÞ þ c5½trðDSÞ : ð2:76Þ form

4 We will now show that using lapse redefinitions, together L1 ¼ a1trðDSÞ ; with further dilaton redefinitions, we can remove any term 6 L2 ¼ b1trðDSÞ ; in the action I that has a trðDSÞ2 factor. This leaves, as one DS 8 DS 4 2 can see, the first term in L1, the first term in L2, and the first L3 ¼ c1trð Þ þ c2½trð Þ : ð2:82Þ two terms in L3. As before, we work at fixed orders of α0 recursively: at Thus, refining the classification of [28], we find that the 2 number of independent coefficients in the O α0 k action each fixed order α0k we eliminate terms with a trðDSÞ ðð Þ Þ is given by the number of partitions of k þ 1 that do not factor by doing a redefinition of n of order α0k that use 1, which in the literature is sometimes denoted by contributes through the variation of I0. This variation pðk þ 1; 2Þ. Using Euler’s generating function for the was given in (2.23): unrestricted number pðnÞ of partitions of the integer n, Z δ 1 it is easy to prove that this equals δ −Φ n DΦ 2 DS 2 nI0 ¼ dtne ð Þ þ 8 trð Þ : ð2:77Þ n pðk þ 1; 2Þ¼pðk þ 1Þ − pðkÞ; ð2:83Þ DS 2 Consider now a general term in Ik;p with a trð Þ factor: see, e.g., [36]. This proves our statement in the Introduction. Z α0k −Φ DS DS 2 Ik;p ¼ dtne X2kð Þtrð Þ : ð2:78Þ III. EQUATIONS OF MOTION AND NOETHER CHARGES Here X2kðDSÞ denotes a term with 2k derivatives built from In this section we determine the explicit equations of products of traces of powers of DS. It does not matter if X2k has additional factors of trðDSÞ2; it may. We will consider a motion for the special case that only single trace invariants redefinition of the form are included, which is, however, sufficient for the cosmological applications in the next section. Moreover, we δ n 0k discuss Bianchi identities and the conserved Noether ¼ βα X2 ðDSÞ; ð2:79Þ n k charges corresponding to Oðd; dÞ duality invariance.

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A. Equations of motion where we included the lowest-order term in the sum, with − 1 We now compute the equations of motion obtained c1 ¼ 8. The higher ck coefficients are, of course, only 1 from the action (2.3) upon including single trace partially known (e.g., c2 ¼ 64 for bosonic string theory, 1 1 higher derivative terms. Upon setting n ¼ this action c2 ¼ 128 for heterotic string theory, and c2 ¼ 0 for type II reads string theories). Z X∞ In order to derive the equation of motion for S we follow −Φ _ 2 0k−1 _ 2k I ≡ dte −Φ þ α cktrðS Þ ; ð3:1Þ the steps reviewed in Sec. II A. The variation with respect to k¼1 S of (3.1) reads

Z X∞ δS 0k−1 −Φ d _ 2k−1 δSI ¼ α ck dte ð2kÞtr S 1 dt k¼ X∞ Z 0k−1 −Φ d _ 2k−1 d _ 2k−1 ¼ α ck dte ð2kÞtr ½δSS − δS S 1 dt dt k¼ X∞ Z 0k−1 −Φ _ _ 2k−1 d _ 2k−1 ¼ α ck dte tr δSð2kÞ ΦS − S 1 dt k¼ Z X∞ −Φ −1 _ 2k−1 d _ 2k−1 ¼ dte tr δS α0k c ð2kÞ Φ_ S − S ; ð3:2Þ k dt k¼1

Z X∞ −Φ 2 0k−1 2k where we discarded total time derivatives. The above I ≡ dtne −ðDΦÞ þ α cktrðDSÞ defines FS, and therefore ES is given by 1 Z k¼ 1 X∞ α0k−1 X∞ −Φ 2 _ 2k 1 2 −1 d 2 −1 − Φ_ S α0k−1 2 Φ_ S_ k − S_ k ¼ dte þ 2k−1 cktrð Þ : ð3:7Þ ES ¼ ckð kÞ n n 2 dt k¼1 k¼1 One quickly finds by variation of n and then back to _ 2k−1 d _ 2k−1 − S Φ_ S − S S : ð3:3Þ dt covariant notation that Z X∞ This form can be simplified by using the anticommutativity δn I≡ dtne−Φ ðDΦÞ2 − α0k−1ð2k−1Þc trððDSÞ2kÞ : S S_ n k of and : k¼1 X∞ ð3:8Þ 0k−1 _ _ 2k−1 d _ 2k−1 d _ 2k−1 ES ¼ α c k 2ΦS − S þ S S S : k dt dt k¼1 The object in between big parentheses is En. It is straight- ð3:4Þ forward to compute the Φ variation of (3.1). To recapitulate let us record the general variation. Recall that ES can be This equation can be further simplified as follows: using used directly in the variation [see (2.16)], and we can set ̈ ̈ _ 2 SS ¼ −SS − 2S , which follows by taking the second n ¼ 1 in the En equation, too. We have derivative of S2 ¼ 1, it is straightforward to verify Z −Φ δn δI ¼ dtne δΦEΦ þ trðδSESÞþ En ; ð3:9Þ d _ 2k−1 d _ 2k−1 _ 2k n S S S ¼ − S − 2SS : ð3:5Þ dt dt with the (off-shell) functions Using this in (3.4), ES finally reduces to X∞ ≡ 2Φ̈− Φ_ 2 − α0k−1 S_ 2k EΦ cktrð Þ; X∞ d k¼1 −2 α0k−1 S_ 2k−1 − Φ_ S_ 2k−1 SS_ 2k ES ¼ ckk þ : ð3:6Þ X∞ 1 dt 0k−1 d _ 2k−1 _ _ 2k−1 _ 2k k¼ ES ≡ −2 α kc S − ΦS þ SS ; k dt k¼1 The equation of motion for S is ES ¼ 0. X∞ _ 2 0k−1 _ 2k In order to find the equation of motion for n we restore En ≡ Φ − α ð2k − 1ÞcktrðS Þ: ð3:10Þ this dependence in the action (3.1): k¼1

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These expressions satisfy a Bianchi identity as a conse- Sη − ηSt ¼ 0; ð3:18Þ quence of time reparametrization invariance of I under the field variations: and is also preserved under the variation δτS. We can compute the Noether charges Q by the usual _ _ δξS ¼ ξS; δξΦ ¼ ξΦ; δξn ¼ ∂tðξnÞ: ð3:11Þ trick of promoting the symmetry parameter to be local, τ → τðtÞ, and to compute the variation Specializing (3.9) to these variations and integrating by Z Z parts we then infer _ Z δτI ¼ dttrðτ_QÞ¼− dttrðτQÞ: ð3:19Þ −Φ _ _ Φ −Φ 0 ¼ δξI ¼ dte ξðΦEΦ þ trðSESÞ − e ∂tðe EnÞÞ; An explicit computation of the variation of the action I in ð3:12Þ (3.1) under δτS follows quickly from the first line in (3.2) and gives where we set n ¼ 1, which is legal after variation. Since the integral vanishes for arbitrary ξ, we infer the X∞ −Φ 0k−1 _ 2k−1 Q ¼ e α 4kckSS : ð3:20Þ _ _ d k¼1 Bianchiidentity∶ ΦðEΦ þ E ÞþtrðSESÞ¼ E : ð3:13Þ n dt n Since on-shell the variation (3.19) must vanish for arbitrary _ This identity may also be verified by a direct computation functions τ, we conclude that on-shell Q ¼ 0. Thus Q is using (3.10). It will be instrumental because it implies that conserved, which may also be verified by a quick compu- it is sufficient to solve the equations _ −Φ tation with (3.6), which gives Q ¼ −2e SES, and shows that the conservation of Q is equivalent to the equation of EΦ þ E ¼ 0;ES ¼ 0; ð3:14Þ n motion of S. Moreover, Q takes values in the Lie algebra so Qη ηQt 0 and to impose the lapse equation (or Hamiltonian ðd; dÞ as one quickly sees that þ ¼ , using (3.18) and its time derivative. constraint) En ¼ 0 only on initial data. The lapse equation will hold for all times as a consequence of (3.13). Note that IV. FRIEDMANN EQUATIONS TO ALL ORDERS IN α0 X∞ ̈ 0k−1 _ 2k In this section we evaluate the equations of motion (3.10) EΦ þ En ¼ 2Φ − α ð2kÞcktrðS Þ: ð3:15Þ k¼1 for the spatial metric set equal to a scale factor times the Euclidean metric and with vanishing b-field. This yields the α0 corrected Friedmann equations. In the first subsection we B. Noether charges write the Oðd; dÞ covariant fields in terms of conventional We will now compute the Noether charges correspond- cosmological variables and show that the standard (string) α0 ing to Oðd; dÞ invariance. To this end we consider the Friedmann equations are recovered to zeroth order in . α0 infinitesimal Oðd; dÞ transformations that leave the action In the second subsection we give the corrected equations. invariant and that are given by In the final subsection we show how to integrate these equations to arbitrary order in α0. δτS ¼ τS − Sτ; δτΦ ¼ 0; ð3:16Þ A. Review of two-derivative equations where τ τM ∈ so d; d , thus satisfying ¼ð NÞ ð Þ We now specialize to the FLRW metric with curvature 0 2 δ τη þ ητt ¼ 0: ð3:17Þ k ¼ and vanishing b-field: we set gij ¼ a ðtÞ ij and bij ¼ 0 in (2.4). Together with the definition of the Oðd; dÞ M In using matrix notation we also have S ¼ðS NÞ¼ηH invariant dilaton in (2.5) we then have MN M with η ¼ðη Þ and H ¼ HMN. With h ¼ðh NÞ an t 0 2 Oðd; dÞ group element, we have hηh ¼ η and the Lie- a ðtÞ −Φ −2ϕ SðtÞ¼ ;e¼ðaðtÞÞde ; ð4:1Þ algebra valued infinitesimal parameter τ arises from h ≃ a−2ðtÞ 0 1 þ τ and thus satisfies (3.17). The transformation δτS above is the infinitesimal version of the finite transforma- in terms of the scale factor aðtÞ and the scalar dilaton ϕðtÞ. tion S → S0 ¼ hSh−1, again with h ≃ 1 þ τ. Note that We next introduce the Hubble parameter 2 δτS ¼ −SδτSS and thus the variation preserves S ¼ 1. S H Ht a_ The symmetry constraint on (inherited from ¼ ) H ≡ : ð4:2Þ reads a

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The time derivatives of the dilaton Φ then become Replacing the second equation by the third minus the second, we find Φ_ ¼ −dH þ 2ϕ_ ; Φ̈¼ −dH_ þ 2ϕ̈; ð4:3Þ a_ 2 ä a_ ðd − 1Þ þ − 2 ϕ_ ¼ 0; ðSÞ where a a a ̈ −2ϕ̈ a 0 − Φ ̈ þ d ¼ ; ðnÞ ð Þ _ − 2 a a H ¼ H þ : ð4:4Þ 2 a dðd − 1ÞH2 − 4dHϕ_ þ 4ϕ_ ¼ 0: ðnÞð4:10Þ For the following applications it will be convenient to Using linear combinations of the above equations, it is easy establish some relations for the field SðtÞ and its deriva- to see that they are equivalent to the three standard (string) tives. First, SðtÞ is constrained and satisfies S2 ¼ 1, and is Friedmann equations, as, for instance, given in [37]. hence a “pseudocomplex” structure. Amusingly, its derivative is proportional to a complex structure: B. α0 corrected equations Let us now turn to the computation of the α0 corrected 0 2 _ a ðtÞ 2 S ¼ 2HJ ; J ≡ ; J ¼ −1: Friedmann equations. We will start from the equations in −a−2ðtÞ 0 (3.10) and specialize to the FLRW ansatz with k ¼ 0 shown in (4.1). Although these equations were derived only from ð4:5Þ the general single-trace action (3.1), we will now argue that for the FLRW ansatz, and leaving the coefficients c In turn, the derivative of J is proportional to S: k generic, this is, in fact, the most general action. Specifically, we will show that the inclusion of any J_ 2 S ¼ H : ð4:6Þ multitrace invariants in the action merely renormalizes the _ coefficients c . To this end note that with S ¼ 2HJ and S k Thus, the second time derivative of can be expressed in J 2 ¼ −1 we can immediately evaluate a generic term of S J terms of and : order α0k−1 in the Lagrangian:

S̈ 2 _ J 4 2S _ 2k k 2kþ1 2k ¼ H þ H : ð4:7Þ L ∝ cktrðS Þ¼ð−1Þ 2 ckdH ðtÞ: ð4:11Þ

Let us now evaluate the two-derivative equations of Now, let us compare this with, say, a double trace term with α0 motion following from the action I0 for the above ansatz, the same order of : with the aim to compare with the standard Friedmann _ 2ðk−lÞ _ 2l 2 1 2 2 L ∝ c trðS ÞtrðS Þ¼c ð−1Þk2 kþ 2d H kðtÞ; equations that follow directly from the higher dimensional k;l k;l action for metric and dilaton. After setting nðtÞ¼1, the ð4:12Þ equations of motion of the two-derivative theory are those 1 where ck;l are new coefficients. Comparing with (4.11) we in (3.10), keeping only c1 ¼ − 8 and setting all ck>1 equal to zero: see that they coincide, except for the overall normalization. Thus, the only effect of the inclusion of (4.12) is the ̈ _ 2 _ renormalization S þ SS − Φ_ S ¼ 0; ðSÞ 1 c → c þ 2dc : ð4:13Þ −2Φ̈ Φ_ 2 − S_ 2 0 Φ k k k;l þ 8 tr ¼ ; ð Þ More generally, the inclusion of any multitrace invariant 1 Φ_ 2 S_ 2 0 merely leads to a (d-dependent) renormalization of the þ 8 tr ¼ : ðnÞð4:8Þ coefficients ck. (We have also verified this at the level of the equations of motion.) Thus, it is sufficient to start with S Φ Inserting the above expressions for , , and their Eqs. (3.10) obtained from the single-trace action. derivatives in terms of the scale factor, these equations Let us begin with the S field equation. In order to become evaluate it efficiently we first collect some relations for S _ 2 and J that follow from S ¼ 2HJ and J ¼ −1: a_ 2 ä a_ ðd− 1Þ þ −2 ϕ_ ¼ 0; ðSÞ _ 2k 2 a a a SS ¼ð−1Þkð2HÞ kS; ̈ 2 _ _ 2 ̈ a 2 −1 2 −1 dðd − 1ÞH − 4dHϕ þ 4ϕ − 4ϕ þ 2d ¼ 0; ðΦÞ S_ k S_ ðk ÞS_ −1 k−1 2 2k−1J a ¼ ¼ð Þ ð HÞ : ð4:14Þ − 1 2 − 4 ϕ_ 4ϕ_ 2 0 dðd ÞH dH þ ¼ : ðnÞð4:9Þ We can then evaluate the derivative

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X∞ d _ 2k−1 k−1 2ðk−1Þ _ _ 2 0 k−1 2k 2k S ¼ð2k − 1Þð−1Þ ð2HÞ 2HJ Φ ¼ −2d ð−α Þ 2 ð2k − 1ÞckH ¼ −gðHÞ; dt k¼1 −1 k−1 2 2kS þð Þ ð HÞ ; ð4:15Þ ð4:21Þ and thus compute where we introduced the function X∞ −1 2 2 d _ 2k−1 _ 2k−1 _ 2k ≡ 2 −α0 k 2 k 2 − 1 k S − Φ_ S þ SS gðHÞ d ð Þ ð k ÞckH dt k¼1 k−1 2k−2 _ _ 2 0 5 4 ¼ð−1Þ ð2HÞ ðð2k − 1Þ2H − Φð2HÞÞJ ¼ 8dc1H − α 3 · 2 dc2H þ: ð4:22Þ ¼ð−1Þk−122k−1½ð2k − 1ÞHH_ 2k−2 − Φ_ H2k−1J The dilaton equation is most conveniently written in the d combination (3.14) using the lapse equation. One finds ¼ð−1Þk−122k−1 − Φ_ H2k−1 J : ð4:16Þ dt X∞ ̈ 0 k−1 2k 2k 0 ¼ Φ þ 2d ð−α Þ 2 kckH : ð4:23Þ Note that here all terms proportional to S canceled. This k¼1 means that one obtains only a single equation, not a matrix Hf H equation, as it should be, because we are looking for an On the right-hand side we recognize a multiple of ð Þ. Thus, the above equation can also be written as equation for the single function aðtÞ. Writing ES ¼ ESJ we find 1 Φ̈þ HfðHÞ¼0: ð4:24Þ 2 X∞ 0 k−1 2k d _ 2k−1 ES ¼ − ð−α Þ kc 2 − Φ H Summarizing, the three α0-completed Friedmann k dt k¼1 equations take the form ∞ d X ¼ − − Φ_ ð−α0Þk−1kc 22kH2k−1: ð4:17Þ d dt k ðe−ΦfðHÞÞ ¼ 0; k¼1 dt 1 Φ̈ 0 The last expression suggests the definition of a function þ 2 HfðHÞ¼ ; fðHÞ of the Hubble parameter H. Including some overall 2 Φ_ g H 0; 4:25 constants for future convenience we write þ ð Þ¼ ð Þ in terms of the functions fðHÞ and gðHÞ defined in (4.18) X∞ 0 k−1 2k 2k−1 and (4.22): fðHÞ ≡ 4d ð−α Þ 2 kckH k¼1 X∞ −1 2 2 −1 0 3 4 −α0 k 2 k k −2 O α0 ¼ 16dc1H − 128dα c2H þ; ð4:18Þ fðHÞ¼ d ð Þ kckH ¼ dH þ ð Þ; k¼1 X∞ for then Eqs. (4.17) can simply be written as 0 k−1 2k 2k gðHÞ¼2d ð−α Þ 2 ð2k − 1ÞckH k¼1 d − 2 O α0 − Φ_ ðtÞ fðHÞ¼0: ð4:19Þ ¼ dH þ ð Þ: ð4:26Þ dt These definitions show that f and g are, in fact, closely This drastic simplification of the equations can be under- related. One readily verifies that stood as a consequence of the existence of the Noether g0 H Hf0 H ; 4:27 charges (3.20): for the FLRW ansatz Q can be seen to be ð Þ¼ ð Þ ð Þ where 0 denotes differentiation with respect to H. 1 −1 0 Let us discuss briefly some general properties of its Q ¼ e−ΦðtÞfðHðtÞÞ : ð4:20Þ 2d 0 1 solutions. First, as expected, they are invariant under → 1 the duality transformation a a. Indeed, under this Q_ 0 d −Φ 0 transformation The conservation law ¼ gives dt ðe fðHÞÞ ¼ , which is equivalent to (4.19), as anticipated before. H → −H; Φ → Φ;fðHÞ → −fðHÞ; We now determine the lapse and dilaton equations from → (3.10). For the lapse equation one obtains gðHÞ gðHÞ; ð4:28Þ

126011-13 OLAF HOHM and BARTON ZWIEBACH PHYS. REV. D 100, 126011 (2019) which leaves Eqs. (4.25) invariant. Thus, for any given fðHðtÞÞ ¼ qΩ−1ðtÞ; ð4:34Þ solution aðtÞ there is a “dual” solution a˜ðtÞ ≡ 1=aðtÞ. Equations (4.25) are also invariant under time reversal with q a number, proportional to the Noether charge Q [see t → −t. For a given solution aðtÞ, ΦðtÞ, there is the time- (4.20)]. The question arises whether integration constants reversed solution a˜ðtÞ≡að−tÞ, Φ˜ ðtÞ ≡ Φð−tÞ, with H˜ ðtÞ¼ such as q have to be treated also as expansions in α0 by 0 −Hð−tÞ. writing q ¼ q0 þ α q1 þ. This depends on how we set out to solve the differential equations. If we pose initial Ω 0 Ω_ 0 C. Perturbative solutions conditions, say by specifying ð Þ and ð Þ, then we have to determine all integration constants in terms of these α0 We now discuss how to solve the -corrected Friedmann initial data, in which case the value of q, for example, does α0 equations (4.25) perturbatively in . To this end it is get α0 corrected. Here, however, we will follow a different Ω convenient to introduce the variable as an exponential of prescription that simplifies the analysis. Instead of for- Φ the dilaton : mulating an explicit initial value problem we introduce −Φ integration constants whenever necessary, but allow our- Ω ≡ e : ð4:29Þ selves the freedom to set integration constants to zero 2 “ ” Using the derivatives Ω_ ¼ −ΦΩ_ and Ω̈¼ð−Φ̈þ Φ_ ÞΩ, whose only effect is to renormalize constants that are and taking the difference of the second and third equations already present. As long as these previously introduced in (4.25), the set of equations becomes integration constants are completely general, this procedure does not entail a loss of generality. Then there is no need for d α0 expansions of parameters such as q. X ≡ ðΩfðHÞÞ ¼ 0; dt We now show how to iteratively solve the above α0 Y ≡ Ω̈− hðHÞΩ ¼ 0; equations to any desired order in . Since hðHÞ starts at α0 0 2 order , to lowest order Y ¼ implies Z ≡ Ω_ þ gðHÞΩ2 ¼ 0; ð4:30Þ ̈ Ω0ðtÞ¼0 ⇒ Ω0ðtÞ¼γðt − t0Þ; ð4:35Þ where, for later use, they have been labeled X, Y, Z, and we defined the function hðHÞ as which we solved in terms of two integration constants γ and 1 t0. Then, from the expansion of fðHÞ and (4.34), hðHÞ ≡ HfðHÞ − gðHÞ 2 1 X∞ fðHÞ ≡ 16dc1H0 ¼ q ⇒ 0 k−1 2k 2k γðt − t0Þ ¼ −2d ð−α Þ 2 ðk − 1ÞckH k¼2 q 1 H0ðtÞ¼− ; ð4:36Þ 0 5 4 2 γ − ¼ α 2 dc2H þ: ð4:31Þ d t t0 − 1 Note that h, in contrast to f and g, vanishes to zeroth order where we used the universal c1 ¼ 8. The last equation of in α0, which will be crucial for our subsequent perturbative (4.30), Z ¼ 0, can now be used to to zeroth order in α0 to construction. It will also be crucial to recall that there is establish a relation between q and the other parameters: one Bianchi identity among the three equations; cf. (3.13). 2 For the functions defined in (4.30) the Bianchi identity _ 2 2 2 2 2 2 2 q 0 ¼ Ω0 þ gðHÞΩ0 ¼ γ − dH0ðtÞγ ðt − t0Þ ¼ γ − ; takes the form 4d : dZ ð4 37Þ ¼ HΩX þ 2Ω_ Y: ð4:32Þ dt and thus This is easily verified directly by differentiation recalling pffiffiffi 2 γ that g0ðHÞ¼Hf0ðHÞ. q ¼ dj j: ð4:38Þ We will now show how to solve these equations, perturbatively in α0, by expanding The solutions are then summarized by

0 0 2 ΩðtÞ¼Ω0ðtÞþα Ω1ðtÞþðα Þ Ω2ðtÞþ; sgnðγÞ 1 H0ðtÞ¼∓ pffiffiffi ; Ω0ðtÞ¼γðt − t0Þ: ð4:39Þ 0 0 2 t − t0 HðtÞ¼H0ðtÞþα H1ðtÞþðα Þ H2ðtÞþ: ð4:33Þ d The subscripts in these quantities denote the power of α0 This is the complete solution to zeroth order in α0 that is that accompanies them in the expansion. The first equation, well defined for t>t0. Before turning to next order, let us X ¼ 0, is the Noether conservation, solved once and for express this solution in terms of standard variables. From d all by the definition HðtÞ¼dt lnðaðtÞÞ we obtain by integration

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0 3 sgnðγÞ fðHÞ¼16dc1H − 128dα c2H lnðaðtÞÞ ¼∓ pffiffiffi lnðt − t0Þþconst; ð4:40Þ d 0 3 ¼ 16dc1H0 þ 16dα ðc1H1 − 8c2H0Þ: ð4:48Þ and thus Thus, to first order in α0 we have to solve 1 ∓sgnðγÞpffiffi − d aðtÞ¼Aðt t0Þ ; ð4:41Þ Ω − q 1 − 8 3 16 Ω2 ¼ c1H1 c2H0: ð4:49Þ where A is a new integration constant. For the scalar dilaton d 0 this yields with (4.1) Using (4.39) and (4.46) this can be solved for H1, −2ϕ −d −Φ −d e ¼ðaðtÞÞ e ¼ðaðtÞÞ Ω0 ffiffi p 80sgnðγÞc2 1 ∝ − 1sgnðγÞ d ðt t0Þ : ð4:42Þ H1ðtÞ¼ 3=2 3 : ð4:50Þ d ðt − t0Þ This solution is well known in the literature [see Eqs. (4.1)– (4.4) in [29]], which, for d 25 and the special case that all Given the way we have set up the computation, the ¼ 0 radii are equal, reduces to (4.41) and (4.42) for γ < 0. See Z ¼ equation now needs to be satisfied identically, for also [32] for first-order α0 corrections. (The relation to our we have no free parameters left to fix by this equation. α0 0 results is more difficult to establish due to the ambiguity in An explicit computation to first order in shows that Z ¼ field variables and their truncation to four derivatives.) is indeed satisfied. Let us now turn to the solutions of first order in α0. To this More generally, it is clear that the iterative procedure of order, the equation Y ¼ 0 implies solving the equations can be continued to arbitrary orders as follows. Consider first the equation Y ¼ 0, setting equal 0 ̈ 0 5 4 α0k α Ω1 ¼ hðHÞΩ0 ¼ dðα 2 c2H0ðtÞÞΩ0 the terms that are of order : 1 1 α0 25 γ − α0kΩ̈ Ω ¼ d c2 2 4 ðt t0Þ; ð4:43Þ k ¼ðhðHÞ Þk: ð4:51Þ d ðt − t0Þ Ω … Ω and so we need to solve Suppose we have solved for 0; ; k−1, as well as for 0 H0; …;Hk−1. Since hðHÞ starts at order α , the right-hand γ 1 side only involves Ω ’s and H ’s that are already deter- Ω̈ 25 i i 1 ¼ c2 3 : ð4:44Þ Ω̈ d ðt − t0Þ mined. Thus k is determined, and we can directly integrate twice to determine Ωk. This gives two integration This can be integrated immediately: constants as in (4.45), but as before these just renormalize γ, t0 and so we set them to zero. With Ω0; …; Ωk now 4 γ 1 determined we next use the equation X ¼ 0 in the form Ω1ðtÞ¼c22 þ b1t þ b2; ð4:45Þ d t − t0 (4.34) in order to determine Hk, which completes the next iteration step. Upon following the structural dependence where b1 and b2 are integration constants. Looking back at on the time-dependent factor ðt − t0Þ in this computation the zeroth-order solution (4.39) we infer that these effec- one may, in fact, verify that the perturbative solutions take tively “renormalize” the previous integration constants γ the form and t0. Thus, without loss of generality we can set b1 ¼ b2 ¼ 0: 1 1 1 0 02 H t h0 α h1 α h2 ; ð Þ¼ − þ − 3 þ − 5 þ 4 t t0 ðt t0Þ ðt t0Þ 2 c2γ 1 Ω1 t : 4:46 1 1 ð Þ¼ − ð Þ Ω ω − α0ω α02ω d t t0 ðtÞ¼ 0ðt t0Þþ 1 þ 2 3 þ; t − t0 ðt − t0Þ Next, we have to determine H1ðtÞ. To this end we use ð4:52Þ (4.34) and expand to first order in α0: where h and ω are time-independent coefficients, which −1 −1 0 −1 −1 n n fðHÞ¼qΩ ¼ qΩ0 ð1 þ α Ω0 Ω1Þ can in principle be expressed in terms of the cn, as done Ω above for the first two. Finally, it remains to prove that the Ω−1 − α0 1 ¼ q 0 2 ; ð4:47Þ Z 0 Ω0 ¼ equation is automatically satisfied. To this end we first note that the above expansions of H and Ω in (4.52) 2 which then has to be matched to the definition of f determine the time dependence of Z ¼ Ω_ þ gðHÞΩ2 to be expanded to the same order, of the form

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1 1 _ α0 α02 assuming Ω ≠ 0 as we will do from now on. The case ZðtÞ¼z0 þ z1 2 þ z2 4 þ; ð4:53Þ _ ðt − t0Þ ðt − t0Þ Ω ¼ 0 will be treated separately and is a bit singular. Let us first see that the initial conditions fix the value where the zi are time-independent parameters. Now, since Hð0Þ of the Hubble parameter at t ¼ 0. For this consider 0 we have satisfied X ¼ Y ¼ in finding (4.52), the Bianchi the second equation in (5.2) which at t ¼ 0 gives identity (4.32) implies that Z does not depend on time, so 0 actually z1 ¼ z2 ¼¼ . Therefore, ZðtÞ¼z0, but since γ2 we fixed parameters in (4.37) so that z0 0,wehave α0g H 0 − : : ¼ ð ð ÞÞ ¼ ω2 ð5 4Þ established Z ¼ 0 to all orders in α0. We close this section with a few remarks. The above Note that we could pass to a new function g˜ defined so that solutions (4.52) seem to suggest that the perturbative expansion breaks down in the high curvature region, since pffiffiffiffi pffiffiffiffi pffiffiffiffi α0 ˜ α0 − α0 2 O α0 4 the terms become more and more singular as t → t0. gðHÞ¼gð HÞ¼ dð HÞ þ ðð HÞ Þ; ð5:5Þ However, it is conceivable that the entire series converges − 1 − which is a unit-free function of the unit-free argument to a regular function (as, for instance, e t t0 is regular at pffiffiffiffi α0 t → t0 although every term in its power series diverges). H. Equation (5.4) then becomes Moreover, for late times t the solutions seem to imply that ffiffiffiffi 2 H is driven to zero and thus to Minkowski space, but p γ g˜ð α0Hð0ÞÞ ¼ − : ð5:6Þ this, of course, may change once matter contributions are ω2 included. In any case, it is important to investigate the possible nonperturbative solutions, to which we turn in The possible values of Hð0Þ are the roots of this equation. 2 the next section. In this way Ωð0Þ and Ω_ ð0Þ have determined Hð0Þ up to a discrete ambiguity. Equation (5.5) makes it clear that we V. NONPERTURBATIVE SOLUTIONS can always solve perturbatively for Hð0Þ but, of course, it could be that nonperturbatively there are no solutions. In this section we discuss a few aspects of possible Now that we have the initial value of H we use the X ¼ 0 solutions that are nonperturbative in α0. In the first sub- Ω_ Ω section we solve the initial-value formulation for the equation to solve for = as a function of H: general equations derived above, for generic functions _ fðHÞ and gðHÞ. In the second subsection we give con- Ω f0ðHÞ ¼ − H:_ ð5:7Þ ditions on the functions fðHÞ and gðHÞ so that the resulting Ω fðHÞ theory permits de Sitter vacua. The Z ¼ 0 equation can be rewritten as A. Initial-value formulation _ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Let us now show how to obtain the general solution Ω ¼ −gðHÞ: ð5:8Þ given initial conditions at time t ¼ 0 of the form Ω 1 Ωð0Þ¼ω; Ω_ ð0Þ¼pffiffiffiffi γ: ð5:1Þ The last two equations imply α0 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi −Φ f ðHÞ Note that since Ω ¼ e is unit free, ω and γ are unit-free H_ ¼∓ −gðHÞ; ð5:9Þ constants. Our strategy will be to solve the first and last fðHÞ equations in (4.30), which can be rewritten as ≡ d Ω 0 X ð fðHÞÞ ¼ ; 0 dt f ðpHÞffiffiffiffiffiffiffiffiffiffiffiffiffiffidH ∓ _ 2 2 ¼ dt: ð5:10Þ Z ≡ Ω þ gðHÞΩ ¼ 0: ð5:2Þ fðHÞ −gðHÞ The equation Y ¼ 0 will then hold due to the Bianchi Integrating from t ¼ 0 to t then gives identity (4.32), Z dZ HðtÞ f0ðHÞdH ¼ HΩX þ 2Ω_ Y; ð5:3Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼∓ t: ð5:11Þ dt Hð0Þ fðHÞ −gðHÞ

2We thank Robert Brandenberger and Jean-Luc Lehners for This is a complete solution. If the integral is done by discussions on these points. finding a WðHÞ such that

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0 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2 f ðpHÞffiffiffiffiffiffiffiffiffiffiffiffiffiffidH fðH0Þ¼ dH0 [see (4.26)] and therefore the only sol- dWðHÞ¼ ¼ − d −gðHÞ; ð5:12Þ 0 fðHÞ −gðHÞ HfðHÞ ution is H0 ¼ , which is not of interest. There are, however, nonperturbative dS solutions con- where the second form follows by g0 ¼ Hf0 [cf. (4.27)], the sistent with duality. That is, as we demonstrate below, there solution is of the form exist functions fðHÞ such that dS solutions exist. To find a solution to (5.18) we integrate the relation g0ðHÞ¼Hf0ðHÞ WðHðtÞÞ − WðHð0ÞÞ ¼∓ t: ð5:13Þ to find Z H This implicitly determines H as a function of t. The value of gðHÞ¼HfðHÞ − fðH0ÞdH0; ð5:19Þ ΩðtÞ now follows from the X ¼ 0 equation in that 0 g 0 0 g q fðHð0ÞÞ where we used ð Þ¼ , as required by the definition of . ΩðtÞ¼ ¼ ω ; ð5:14Þ To get a vanishing gðH0Þ when fðH0Þ¼0 we need that fðHðtÞÞ fðHðtÞÞ the integral of fðHÞ from zero to H0 vanishes. This, for example, happens at the nonvanishing zeros of the sine where in the second step the constant of the motion q was function. More explicitly, we take evaluated using the initial condition Ωð0Þ¼ω and the 0 value of Hð Þ. 2d pffiffiffiffi Consider now the somewhat singular case of vanishing fðHÞ ≡ − pffiffiffiffi sinð α0HÞ α0 Ω_ .IfΩ_ ≡ 0 for all t, the Z ¼ 0 equation implies gðHÞ¼0, X∞ 1 which fixes HðtÞ to be a constant equal to a zero of g. ¼ −2d ð−α0Þk−1 H2k−1: ð5:20Þ 0 2k − 1 ! We will consider this case in the next subsection. If t ¼ is k¼1 ð Þ the only time for which Ω_ ¼ 0, then this fixes the initial condition for H. This fðHÞ¼−2dH þ Oðα0Þ is consistent with the known two-derivative theory. From the definition (4.18) it is clear that there are coefficients c so that this holds (in fact, B. Nonperturbative de Sitter solutions k − 1 ck ¼ 2k). With (5.19) we then get We now turn to a discussion of possible de Sitter (dS) ð2k!Þ2 solutions. To this end consider the dS metric 2d pffiffiffiffi pffiffiffiffi pffiffiffiffi g H − α0H sin α0H cos α0H − 1 : ð Þ¼ α0 ð ð Þþ ð Þ Þ ds2 ¼ −dt2 þ e2H0tdx2; ð5:15Þ ð5:21Þ

H0t leading to the scale factor aðtÞ¼e , with H ¼ H0 ≠ 0 0 const. Consider the Friedmann equations (4.25) copied here We now see that fðH0Þ¼gðH0Þ¼ is satisfied for for convenience: pffiffiffiffi 0 α H0 ¼ 2πn; n ∈ Z;n≠ 0: ð5:22Þ d ðe−ΦfðHÞÞ ¼ 0; dt This is a discrete infinity of dS solutions. 1 Although (5.20) is surely not the function arising in any Φ̈ 0 þ 2 HfðHÞ¼ ; string theory, the actual functions could have the properties Φ_ 2 0 required in order to lead to dS vacua in the string frame. þ gðHÞ¼ : ð5:16Þ The conditions for this to happen can be stated more concisely as follows. Instead of fðHÞ we consider its integral With H constant, the first equation then implies that Φ is Z ∞ also constant: H X 0 0 0 k−1 2k−1 2k FðHÞ ≡ fðH ÞdH ¼ 4d ð−α Þ 2 ckH ; 0 Φ_ ¼ 0 → Φ ¼ const: ð5:17Þ k¼1 ð5:23Þ Since H0 ≠ 0, the second equation implies the vanishing of fðH0Þ and the third the vanishing of gðH0Þ: where the evaluation used the expansion (4.26) of fðHÞ. Then, from (5.19), we have that gðHÞ is given by fðH0Þ¼0;gðH0Þ¼0: ð5:18Þ gðHÞ¼HF0ðHÞ − FðHÞ: ð5:24Þ Note that while the two functions fðHÞ and gðHÞ are 0 different, fðHÞ determines gðHÞ via the relations g ðHÞ¼ We now infer that if there is an H0 ≠ 0 so that F and its first Hf0ðHÞ and gð0Þ¼0. In the two-derivative approximation derivative vanish at this point,

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0 FðH0Þ¼F ðH0Þ¼0; ð5:25Þ Einstein frame. We will now discuss some aspects of the Einstein-frame equations.3 then H0 isazeroofbothf and g, and hence the theory permits Denoting the Einstein frame metric by Gμν, the required dS solutions. The conditions above require F to attain value Weyl rescaling reads zero at maxima or minima. Such functions are easily − 4ϕ constructed, as we will show below. Since the leading term Gμν ¼ e D−2gμν;D¼ d þ 1; ð5:28Þ in fðHÞ is fixed by the two-derivative theory, we also 2 0 which implies for the Einstein frame scale factor aEðtÞ¼ F H −dH O α 2 have ð Þ¼ þ ð Þ. − ϕðtÞ Let us emphasize that the dS solutions discussed above e d−1 aðtÞ. Thus, the Einstein frame Hubble parameter are nonperturbative in α0: they are not constructed from a reads solution of the two-derivative equations which is then α0 _ 2 1 aEðtÞ _ _ corrected. The complete series of all α0 corrections needs to H ≡ ¼ − ϕ þ H ¼ − ðΦ þ HÞ; ð5:29Þ E a ðtÞ d − 1 d − 1 be used in order to obtain a solution. E We demonstrated that the two-derivative theory does not where we used the relation (4.3) between dilaton have dS solutions. The inclusion of the leading α0 correc- derivatives: tions does not change this. But with suitable order α02 _ _ corrections, dS solutions appear. This corresponds to 2ϕ ¼ Φ þ dH: ð5:30Þ fðHÞ ∼ H þ α0H3 þ α02H5, and therefore FðHÞ ∼ H2 þ 4 2 6 We next have to recall that the Weyl rescaling (5.28) α0H þ α0 H . It is simple to see that this happens for implies 2 2 2 2 FðHÞ¼−dH ð1 − γ α0H Þ ; ð5:26Þ − 4ϕ − 4ϕ G00 ¼ e D−2g00 ¼ −e D−2; ð5:31Þ where γ is an arbitrary constant. The solution here is − 2 −1 pffiffiffiffi since we have set g00 ¼ n ¼ . To identify a de Sitter 0 α H0 ¼1=γ. Such solutions, however curious, cannot solution in the standard form (5.15) we need G00 ¼ −1, and be trusted in string theory, which carries an infinite number hence we need to reparametrize time. Let t0ðtÞ be a time 0 0 0 of α corrections. A reliable solution should either solve the such that G00ðt Þ¼−1 and thus two-derivative equations (and then be corrected perturba- 2 2 2 − 2ϕ 0 0 0 0 0 −1 0 tively) or be nonperturbative in α . G00ðtÞdt ¼ G00ðt Þdt ¼ −dt → e d dt ¼ dt : ð5:32Þ A general class of functions FðHÞ admitting dS solutions is easily obtained. For a polynomial wðxÞ the condition This redefinition has no effect on the gij components of the wðx0Þ¼0 means that wðxÞ¼ðx − x0ÞuðxÞ, where uðxÞ is metric other than reparametrizing the time dependence, so 0 another polynomial. The additional condition w ðx0Þ¼0 we have 2 implies that uðx0Þ¼0, and thus wðxÞ¼ðx − x0Þ vðxÞ for a0 ðt0Þ¼a ðtðt0ÞÞ: ð5:33Þ some polynomial vðxÞ. While FðHÞ is an infinite series in E E 2 H , not a polynomial, the above remarks show how to Therefore the Hubble parameter of the solution in the produce consistent solutions. A general class of FðHÞ with Einstein frame with coordinates ðt0;xiÞ is ð1Þ ðkÞ possible dS Hubble parameters H0 ; …; H0 takes the 1 0 0 1 2ϕ daEðt Þ dt daEðtÞ form 0 0 d−1 0 HEðt Þ¼ 0 0 0 ¼ 0 · ¼ e HEðtðt ÞÞ aEðt Þ dt dt aEðtÞ dt X∞ Yk 2 2 2ϕ 1 2 0p 2p H _ FðHÞ¼−dH 1 þ d α H 1 − : ¼ −ed−1 ðΦ þ HÞ; ð5:34Þ p ðiÞ d− 1 p¼1 i¼1 H0

ð5:27Þ where we used HEðtÞ computed in (5.29). On the right- hand side dot is a t derivative, and all terms are evaluated at The first factor in parentheses is an arbitrary series in H2 tðt0Þ. Since e2ϕ ¼ðaðtÞÞdeΦ, we then have with a leading term equal to one. 1 We close with a few general remarks on the possible dS 0 0 d Φ _ H ðt Þ¼−ðaðtÞÞd−1ed−1 ðΦ þ HÞ: ð5:35Þ solutions. First, they are most likely insufficient as phe- E d − 1 nomenologically viable models, because the natural values _ of the cosmological constant Λ ∼ 1=α0 obtained by this Our solutions with Φ ¼ 0 and H ¼ H0 would lead to a 0 ∼ H0t mechanism are many orders of magnitude too large. This is time-dependent HE through the scale factor aðtÞ e . the cosmological constant problem. Second, they are dS solutions in the string frame, while the match with the 3We thank M. Gasperini and G. Veneziano for explanations on observable dark energy is conventionally done in the how to relate the two frames.

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Thus, these solutions do not lead to dS vacua in the Note that while EΦ and En are modified, their sum is not Einstein frame. and is still given by (3.15). It is then straightforward to The general condition for dS vacua in the Einstein frame is specialize these equations to the FLRW background: d 0 0 0 0 HEðt Þ¼ and thus, by the chain rule, that the right-hand dt d side of (5.34) or (5.35) is independent of t.Wehavenot ðe−ΦfðHÞÞ ¼ 0; investigated the general problem whether there are solutions dt with this property, but as an illustration for the usefulness 1 Φ̈þ HfðHÞ¼0; of this framework we present a simple no-go result: there are 2 no dS solutions in the Einstein frame with constant dilaton ϕ. 2ðD − 26Þ _ Φ_ 2 g H − 0; 5:39 To this end we observe that ϕ ¼ 0 implies with (5.30) that þ ð Þ 3α0 ¼ ð Þ Φ_ ¼ −dH,whichwith(5.34) yields which generalizes (5.16). 2ϕ 0 0 −1 We can now state the conditions for de Sitter vacua for HEðt ðtÞÞ ¼ ed HðtÞ: ð5:36Þ which H ¼ H0 ¼ const. As before, we see that in the two- ϕ 0 derivative theory there are no de Sitter vacua: the first Since we assume to be constant, HE is constant if and only if H is constant. With H constant, however, we equation of (5.39) implies that Φ is constant, Φ_ ¼ 0, which are back to our previous analysis of dS vacua in the string with fðHÞ¼−2dH and the second equation implies that 0 frame, where we concluded that the Friedmann equa- H0 ¼ 0, leading to flat space. Including all α corrections, tions (5.16) imply Φ_ ¼ 0 and hence, recalling Φ_ ¼ −dH, the condition that (5.39) permits de Sitter solutions can be 0 0 0 that H ¼ , leading to HE ¼ and flat space in the Einstein stated in terms of the function FðHÞ defined in (5.23): frame. This proves that there are no dS vacua in the Einstein 2 − 26 frame with constant ϕ. ðD Þ 0 F H0 − ;FH0 0: 5:40 ð Þ¼ 3α0 ð Þ¼ ð Þ C. de Sitter solutions in noncritical string theory Indeed, one then finds We now briefly comment on the general case of 0 noncritical string theory in arbitrary dimension D in which fðH0Þ¼F ðH0Þ¼0; case the last term in (2.1) needs to be included. Note that 2 − 26 0 ðD Þ this term is not a cosmological constant term since it is gðH0Þ¼H0F ðH0Þ − FðH0Þ¼ ; ð5:41Þ written in string frame. Rather, passing to the Einstein 3α0 frame, this term gives rise to a scalar potential for ϕ. The so that (5.39) is satisfied for constant Φ. Thus, duality theory including at most two derivatives still does not allow invariant theories with de Sitter vacua exist provided there for de Sitter vacua, as we confirm momentarily. − 2 O 4 Let us now analyze the dynamics of FLRW backgrounds is a function FðHÞ¼ dH þ ðH Þ with an extremum at − 2ðD−26Þ in noncritical string theory. As argued around (4.11) it is a nonzero H0 where it takes value 3α0 . Clearly, these sufficient to include single-trace invariants, and so we conditions can be met. As a toy example we display a consider the action that simply extends (3.1) by the addi- quartic polynomial with these properties: tional term (in the gauge n ¼ 1): 1 2 Z − 2 1 − H X∞ 2 − 26 FðHÞ¼ dH ; ð5:42Þ −Φ 2 −1 2k ðD Þ 2 H0 I ≡ dte −Φ_ α0k c tr S_ − : þ k ð Þ 3α0 k 1 ¼ for which ð5:37Þ 2 2 dH0 0 H The equations of motion (3.10) are unaffected for S, but are FðH0Þ¼− ;FðHÞ¼−2dH 1 − ; 2 H0 modified for Φ and n: ∞ 2 D − 26 X ð5:43Þ ≡ 2Φ̈− Φ_ 2 ð Þ − α0k−1 S_ 2k EΦ þ cktrð Þ; qffiffiffiffiffiffiffiffi 3α0 pffiffiffiffi k¼1 0 d−25 so that (5.40) holds for α H0 2 . X∞ ¼ 3d 0k−1 d _ 2k−1 _ _ 2k−1 _ 2k ES ≡ −2 α kc S − ΦS þ SS ; k dt k¼1 2 − 26 X∞ VI. CONCLUSIONS AND OUTLOOK 2 ðD Þ −1 _ 2k E ≡ Φ_ − − α0k ð2k − 1Þc trðS Þ: n 3α0 k In this paper we have discussed, to all orders in α0, the k¼1 most general duality-invariant spacetime actions for a ð5:38Þ metric, b-field, and dilaton that depend only on time.

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This allowed us to study some generic features for the initial singularity, leading to a pre-big-bang cosmological backgrounds when all α0 corrections are phase that is smoothly connected to the post-big-bang included. In particular, we discussed solutions such as phase [32]. With the framework developed here it de Sitter vacua that are nonperturbative in α0. We close with should be possible to investigate whether this is a list of natural follow-up projects and open problems: possible in principle. (i) The cosmological backgrounds investigated here (iv) In our classification of higher derivative terms, field involved only a single scale factor aðtÞ in addition redefinitions at each order in α0 allowed us to to the dilaton. In general it should be straightforward eliminate any terms in which the field S had more to include independent “rolling radii” in different than one time derivative or the field Φ had any dimensions, as analyzed to lowest order in α0 long number of derivatives. How general is this phe- ago [29]. Moreover, it would be important to include nomenon in time-dependent field dynamics? a b-field. Qualitatively new phenomena may appear (v) Arguably a central problem is to determine for each in this case. string theory the free coefficients defining FðHÞ, (ii) Eventually it would be crucial to arrive at (semi) which determines the functions fðHÞ and gðHÞ that realistic cosmological solutions. In particular, one has appear in the α0-complete Friedmann equations. to make sure that the string coupling given by the Perhaps the significantly simplified setup developed dilaton, which here is generally time dependent, does here will suggest a viable strategy. not become too large, for otherwise the classical string theory can no longer be trusted. Moreover, matter ACKNOWLEDGMENTS fields should be included in a duality covariant fashion, as in [2]. We are very grateful to Robert Brandenberger, Maurizio (iii) An important challenge for any theory of quantum Gasperini, Jean-Luc Lehners, Diego Marques, Krzysztof gravity is to show how the big-bang singularity Meissner, Mark Mueller, Ashoke Sen, Andrew Tolley, and could be resolved. (Related work includes [38,39].) Gabriele Veneziano for useful discussions and correspon- One possibility is that the higher-derivative α0 dence. The work of O. H. is supported by the ERC corrections of classical string theory could resolve Consolidator Grant “Symmetries & Cosmology.”

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