It is demonstrated by explicit solutions of the (4+n)-dimensional vacuum
Einstein equations that accelerating cosmologies in the Einstein conformal
frame can be obtained by a time-dependent compactification of string/M-theory,
even in the case that internal dimensions are Ricci-flat, provided one includes
one or more geometric twists. Such acceleration is transient. When both compact
hyperbolic internal spaces and geometric twists are included, however, the
period of accelerated expansion may be made arbitrarily large.Comment: 5 pages, 1 figure, RevTeX

Bennett

Bergshoeff

Chen

David L. Wiltshire

Emparan

Gibbons

Giddings

Ishwaree P. Neupane

Jarv

Maldacena

Neupane

Ohta

Townsend

Wohlfarth

Physics Letters B

English

arXiv

Crossref

Accelerating cosmologies from compactification with a twist

DOI: 10.1016/j.physletb.2005.06.008It is demonstrated by explicit solutions of the 4+n–dimensional vacuum Einstein equations that
accelerating cosmologies in the Einstein conformal frame can be obtained by a time–dependent compactification
of string/M–theory, even in the case that internal dimensions are Ricci–flat, provided
one includes one or more geometric twists. Such acceleration is transient. When both compact
hyperbolic internal spaces and geometric twists are included, however, the period of accelerated
expansion may be made arbitrarily large

Neupane, I.P.

Wiltshire, D.L.

UC Research Repository (University of Canterbury)

arXiv:hep-th/0502003v3  1 Jul 2005Accelerating cosmologies from compactification with a twistIshwaree P. Neupane1,2, ∗ and David L. Wiltshire1, †1Department of Physics and Astronomy, University of Canterbury,Private Bag 4800, Christchurch, New Zealand and2Central Department of Physics, Tribhuvan University, Kirtipur, Kathmandu, NepalIt is demonstrated by explicit solutions of the 4+ n–dimensional vacuum Einstein equations thataccelerating cosmologies in the Einstein conformal frame can be obtained by a time–dependent com-pactification of string/M–theory, even in the case that internal dimensions are Ricci–flat, providedone includes one or more geometric twists. Such acceleration is transient. When both compacthyperbolic internal spaces and geometric twists are included, however, the period of acceleratedexpansion may be made arbitrarily large.PACS numbers: 98.80.Cq, 11.25.Mj, 11.25.Yb. [ arXiv: hep-th/0502003 ]The observation that the present expansion of the uni-verse is accelerating [1] has proved a challenge to fun-damental theories such as string/M–theory, and its low–energy supergravity limits. Fundamental scalar fields areabundant in such higher–dimensional gravity theories,and potentially provide a natural source for the grav-itationally repulsive “dark energy” that could explainthe present cosmic acceleration, or alternatively at muchhigher energy scales the very early period of cosmic in-flation. However, the constraints imposed by string/M–theory on particular scalar fields, such as the moduli cor-responding to 6 or 7 extra compactified dimensions, aresuch that realistic scenarios giving accelerating cosmolo-gies are very difficult to arrive at.Indeed for many years, it was assumed that cosmicacceleration was ruled out for supergravity compactifica-tions on the basis of a “no go” theorem [2], which forbidsaccelerating cosmologies in the presence of static extradimensions in supergravity compactifications, assumingthat one wishes to stay within the realm of classical su-pergravity rather than resorting to the addition of “quan-tum correction” terms to the action. Recently Townsendand Wohlfarth [3] demonstrated, however, that it waspossible to circumvent the no–go theorem in a time–dependent supergravity compactification using compactextra dimensions with negative curvature. Many addi-tional examples [4]–[10] were subsequently found.The advent of time–dependent compactifications is animportant one, as it has the potential to offer a resolutionto the dilemma posed by the observed cosmic accelera-tion within a natural theoretical framework. Nonetheless,a number of substantial problems remain in the modelsstudied to date. In this Letter we will present new so-lutions arising from compactifications with a geometri-cal twist, which we believe overcome these problems togive time–dependent supergravity compactifications with∗E-mail: ishwaree.neupane@canterbury.ac.nz†E-mail: david.wiltshire@canterbury.ac.nzcompelling physical features.Let us turn to the open issues in time–dependent cos-mological compactifications. Firstly, in the Townsend–Wohlfarth (henceforth TW) model [3] the no–go theo-rem was only circumvented by the choice of negativelycurved internal spaces, which are made compact by pe-riodic identification. While such spaces have particularfeatures, such as the absence of massless Kaluza–Kleinvectors which may have some phenomenological appeal,the restriction on the curvature is a severe one, especiallygiven the view that Ricci–flat internal space compactifi-cations are among the most natural in string theory.The second important issue is the question of thelength of the period of accelerated expansion. The TWmodel and its successors incorporate scalar fields whosepotential energy does not admit a local minimum withpositive vacuum energy. A consequence of this is thatthe period of acceleration is transient, corresponding tojust a few e–foldings. Again if one is to apply the modelto describe the present cosmic evolution, then this is nota major difficulty but does place some constraints on theparameters involved. However, if rather than a model ofthe present accelerated cosmological evolution a modelof very early universe inflation via time–dependent com-pactifications was desired, then this would be a signifi-cant problem.In this Letter we will present model cosmologies fromtime–dependent compactifications which overcome bothof the problems above, while also satisfying a third prop-erty of theoretical naturalness. The limitations with themodels studied to date arise, we believe, from an oversim-plification of the effective potentials that arise from com-pactifications. In particular, when a number of scalarsassociated with different dimensions are present, theirinteractions can give rise to effects which are absent ifthe extra dimensions form a single space of constant cur-vature, as in the TW model. Product space compactifi-cations have already been considered, sometimes also inthe presence of higher dimensional form–fields with non–zero fluxes [8]. However, while higher–dimensional fluxesgo a little way towards altering the form of the compact-2ification potential, for example, by allowing the presenceof a local de Sitter minimum [8], in the examples studiedto date there is not enough parameter freedom to allowsignificant periods of accelerated expansion.Here we will explicitly consider cosmologies that arisein models of gravity which correspond to the dimensionalreduction to 4 dimensions of the Einstein equations in10, 11 or generally 4 + n dimensions, where some of theextra dimensions form product spaces with a geometricaltwist. In particular, consider a (4+n)–dimensional metricansatz with n = p+ 3 internal dimensions,ds24+n = e−2Φds2(M4) + r 21 e2φ1ds2(Mpǫ1)+ r 22e2φ2 ds2(M2ǫ2) + r 23e2φ3(dz +̟ǫ2)2, (1)where φi = φi(u), the parameters ri define appropriatecurvature radii, ds2(M4) is the metric of the physicallarge dimensions in the formds2(M4) = −a2δ(u)du2 + a2(u)dΩ2k,3, (2)and δ is a constant, the choice of which fixes the natureof the time coordinate, u. The internal space Mp is ap–dimensional space of constant curvature of sign ǫ1 =0,±1, and the remaining 3 internal dimensions form atwisted product space,M2 ⋉M1, as followsds2(M2+1)= dx2 + sin2 xdy2, ̟+1 = f cosxdy ,ds2(M20) = dx2 + dy2, ̟0=f2(xdy − ydx),ds2(M2−1)= dx2 + sinh2 xdy2, ̟−1 = f coshxdy,(3)when ǫ2 = +1, 0,−1 respectively, f being the twist pa-rameter. One chooses the Einstein conformal frame infour dimensions by setting Φ = pφ1/2 + φ2 + φ3/2, andso Newton’s constant is time–independent.To demonstrate the novel features introduced by a ge-ometric twist, we will begin by presenting a special exactsolution in the case that the physical universe is spa-tially flat, k = 0, and both Mp and M2 are Ricci–flat,i.e., ǫ1 = 0 and ǫ2 = 0, but with non–zero twist, f . Thissolution to the (p+7)–dimensional vacuum Einstein equa-tions is most readily written down in the gauge, δ = 3,used in Ref. [3]. In particular, we finda(u) = a0 e[(3q+4)c+pc1]qu/8[cosh(χ(u− u0))]q/4,φ2 =−φ3q= 12 ln[cosh(χ(u− u0))]+ 14 (p c1 + 3qc)u,φ1 =2pln(fr3a03χr22)− c1u, (4)whereχ ≡ 12√6pq c c1 + 3q(3q + 4)c2 − p[(4− q)p+ 8]c21/q.(5)and u0, c and c1 are integration constants. We haveignored constants which can be absorbed into the ri. Theparameter q denotes the number of twists, and for themetric (1)–(3) we take q = 1 in (4), (5).For reality of χ we require α1−c < c1 < α1+c, whereα1± ≡ [3 ± sgn(c)2√6(3 + 7/p)]/(3p + 8) when q = 1.It is readily seen that the scale factor of the physicaluniverse is monotonic for all values of c1 in this inter-val. We therefore restrict our attention to solutions withc > 0 which correspond to universes which expand as uincreases. Thevalue of u0 is merely a gauge choice andso we set u0 = 0.In terms of the physical cosmic time, (i.e., proper timeof co–moving observers), t = ± ∫ u du¯ a3(u¯), both theearly and late time behaviour of the cosmic scale fac-tor is a∼ t1/3. This is similar to the TW solution withtwo minor differences. Firstly, here the t→∞ limit cor-responds to u → ∞, whereas in the TW solution thislimit corresponds to u → 0−. Secondly, the TW solu-tion decelerates slightly less quickly at late times, witha∼ tn/(n+2). Otherwise, the solution is physically verysimilar to the TW solution. In particular, since the ac-celeration parameter (for q = 1) is given bya5a¨ = 14χ2(1− 32 tanh2 χu)− 132(7c+ p c1) (4χ tanhχu+ 7c+ p c1)(6)where an overdot denotes differentiation w.r.t. cosmictime, t, it follows that solutions will exhibit a period oftransient acceleration provided that α2−c < c1 < α2+c,where α2± ≡ [1± 2√3(3 + 7/p)]/(5p+12). Accelerationoccurs on the interval u− < u < u+, wheretanh(χu±)≡ −(7c+ p c1)± 2√7c2 + 2p c c1 − p(5p+ 12)c126χ.00.050.10.150.20.250.3–0.2 –0.1 0.1 0.2 0.3NαeFIG. 1: The number of e–folds during acceleration epochNe =q8ln[(1+tanhχu+1+tanhχu−)κ−1 (1−tanhχu−1−tanhχu+)κ+1]where κ ≡ [(3q+4)c+pc1]/(2χ), as a function of the parameterα ≡ c1/c, for q = 1, p = 4, d = 11.3The number of e–folds during the period of accelera-tion, Ne ≡ ln a(u+) − ln a(u−) is plotted in Fig. 1. Itreaches a maximum Ne = 0.3041 at c1 = 0 independentlyof p, which corresponds to the special case in which theinternal spaceMp is static with constant φ1. (The p = 0solution is formally equivalent to (4)–(5) with c1 = 0and no φ1.) In the general case φ2 slowly increases —giving a slow “decompactification” as in the TW model[3] — while φ3 decreases, and φ1 decreases (increases) ifc1u > 0 (c1u < 0).The number of e–folds in the examples is possibly toolow to be consistent with observational bounds, especiallywhen one notes that any additional matter obeying thestrong energy condition would have a decelerating effectthat may decrease the period of transient acceleration.However, the number of e–folds can be increased by in-creasing the dimension of the twisted space. In particu-lar, consider the case n = p + 2q + 1 when the internalspace is now the product of a p–dimensional torus Mp,(i.e., ǫ1 = 0), and the (2q+1)–dimensional twisted spaceds2T 1,··· ,1 = r22e2φ2q∑i=1(dx2i + dy2i)+ r23e2φ3[dz +f2q∑i=1(xidyi − yidxi)]2, (7)which has topology (T2 × · · · × T2) ⋉ S1. Eqns (4)–(5) with q arbitrary in fact represent the solution tothe (4 + p+ 2q + 1)–dimensional vacuum Einstein equa-tions with this more general metric ansatz, where nowΦ = pφ1/2 + qφ2 + φ3/2. For small values q > 1 thequalitative features are the same as the q = 1 case, withtransient acceleration and a maximum number of e–foldsat c1 = 0, independently of p. For example, the maxi-mum number of e–folds for q = 2, 3, 4 are respectivelyNe = 0.3689, 0.4019, 0.4221, a marginal increase.Next we turn our attention to solutions with a periodof acceleration which is not merely transient. In additionto a twist, we will take the curvature of the internal spaceto be non–zero. To this end, it is convenient to set δ = 0,so that the metric (2) takes the standard Friedmann–Robertson–Walker form, and u becomes the cosmic time,t. For the metric (1), with arbitrary ǫi, the field equationsmay be written in the formΦ¨ + 3HΦ˙− 2K − 3 a¨a= 0,Φ¨ + 3HΦ˙− 2(a˙a)2− a¨a− 2ka2= 0, (8)φ¨i+ 3Hφ˙i− δ1Vi− δ2VF= 0, i = 1, 2, 3where δ1 = 2/p, δ2 = 0 (for i = 1), δ1 = 1, δ2 = 2 (fori = 2), δ1 = 0, δ2 = −2 (for i = 3), H ≡ a˙/a is theHubble parameter, k is the spatial curvature, and thekinetic and potential terms are respectivelyK = p(p+2)4 φ˙21 + 2φ˙22 +34 φ˙23 + pφ˙1φ˙2 + φ˙2φ˙3 +p2 φ˙3φ˙1,V = V1 + V2 + VF = Λ1e−(p+2)φ1−2φ2−φ3+ Λ2e−pφ1−4φ2−φ3 + F 2e−pφ1−6φ2+φ3 , (9)where Λ1 = −p(p − 1)ǫ1/(2r21), Λ2 = −ǫ2/r22 and F ≡(f/2)(r3/r22). Even with ǫi = 0, the volume modulusfield has a non-zero (positive) potential, and so the cos-mological solutions with f > 0 circumvent the no-go the-orem [2], while retaining Ricci-flat internal spaces. Manyadditional examples are given in [11].In terms of alternative canonically normalizedscalars, which may be defined by ϕ1 ≡ −√p+5p+4 φ3,ϕ2 ≡√p(p+4)2(φ1 +1p+4φ3), ϕ3 ≡ p2φ1 + 2φ2 + 12φ3,the field equations areϕ¨i + 3Hϕ˙i +dVdϕi= 0, (10)H˙ +K − ka−2 = 0, (11)along with the Friedmann (constraint) equationH2 = 13 (K + V )− ka−2, (12)where K = 12∑3i ϕ˙i2. The resulting scalar potential isV = Λ1 e−ϕ2/β−ϕ3 + Λ2 e−2ϕ3 + F 2 e−√5−β2ϕ1+βϕ2−3ϕ3(13)where β ≡ √p/√p+ 4 < 1. As above we will confineour attention to k = 0 cosmologies. Solutions with notwist, F 2 = 0, and compact hyperbolic product spaces,i.e., Λi > 0, have been previously discussed [8], and givetransient acceleration without a local minimum in thepotential. However, if there is a non–trivial twist andat least one of Λi is positive, then it may be possible tofind models with a large number of e-foldings. Indeed,if we choose Λi such that the potential (13) is strictlynon-negative then it always has a minimum with respectto a subset of the ϕi directions.Even with k = 0 the general solution to (10)–(13) ishighly non–trivial. To demonstrate the general physicaleffects, we will take Λ2 = 0, so thatM2 is a 2-torus, andspecialize to the case in which d = 11 (or p = 4) and ϕ3 =b1 = const. An explicit exact solution can then be foundin terms of a new logarithmic time variable τ , defined byτ =∫ tdt¯ exp[−ϕ2(t¯)/√2], or ϕ2 = √2 ln(dt/dτ ). Theexplicit solution is then(ln a)′= 1√12√V0(ξ + ξ−1), ϕ′2 =12√V0(ξ − ξ−1) ,ϕ1 = ϕ2 +√2b03, ξ ≡√6−1√5tanh√5V08(τ − τ0), (14)where ′ ≡ d/dτ , τ0 is a constant,V0 ≡ Λ1e−b1 + F 2e−(b0+3b1) = 3F 2e−(b0+3b1). (15)4and b0 is fixed once b1 is chosen. At late times we finda ∝ t2, with the corresponding value of ω < −2/3, whereω ≡ (K − V )/(K + V ). In the limit ϕ1 → ϕ2 we findφ1 → −φ2/2 and φ3 → −2φ2. Thus two of the extradimensions associated with the spaceM2 may grow withtime while other dimensions shrink (or vice versa).The potential V has a minimum with respect to ϕ2 atϕ(0)2 = ϕ1 + (√2/3) ln(2Λ1/F2) + 2√2b0/3, withV (ϕ0) =(274 Λ1F4e−7b0)1/3e−√2ϕ1 = V,ϕ2ϕ2(ϕ0). (16)Since the minimum has the curious feature, special to po-tentials with Λ2 = 0, that V,ϕ2ϕ2(ϕ0) = V (ϕ0), particle–like states will have a mass m2 = V (ϕ0). If this valuewere to be assigned to the vacuum energy at the presentepoch, i.e., V (ϕ0)∼ 10−120 in Planck units, then both thevacuum energy and scalar excitations about the vacuumare ultra–light. The degree to which the model wouldneed to be fine–tuned to achieve such an outcome re-mains to be investigated.Another alternative would be to take V (ϕ0) to have thevastly higher energy scale associated with the epoch ofinflation in the very early universe. Let us assume that ϕ1starts initially at ϕ1 = ϕ(0)1 , and that Λ1 and F both areof order unity (in Planck units), then when ϕ1∼ϕ(0)1 +47the number of e–folds is Ne∼ 65. The actual relationbetween Ne and the shift in ϕ1 can be different, however,depending upon the precise values of the compactificationscale,√Λ1, and the twist parameter, F .A point also worth emphasizing is that, especially in aflat universe, the de Sitter stage can be transient. If thisis the case in our model, we must allow also ϕ3 to rollwith t. Different choices of vacua could lead to differentasymptotic expansion. More specifically, for the poten-tial (13), when k = 0 and p = 4 the late time behaviourof the scale factor is characterized by a(t) ∝ tγ , whereγ =13/19 (Λ2 = 0, Λ1 6= 0, F 2 6= 0)3/5 (Λ1 = 0, Λ2 6= 0, F 2 6= 0)3/4 (F 2 = 0, Λi 6= 0)7/9 (Λi 6= 0, F 2 6= 0).(17)It is expected the contribution of dust or radiation couldmodify the above asymptotic behaviour.Let us summarize why we believe that exponentialpotentials arising from compactifications of supergrav-ity models on a twisted product space of time-varyingvolume are a potentially significant source for dynami-cal dark energy at the present epoch. Firstly, Townsendand Wohlfarth [3] argued that the no–go theorem for ac-celerating cosmologies in supergravity compactificationscould only be circumvented by including compact hyper-bolic internal spaces, and would not work in the Ricci–flatcases. We have shown on the contrary that the no–go the-orem is also circumvented by twisted Ricci–flat spaces byexplicit construction of exact solutions with the internalspace Tp×(T2×· · ·×T2)⋉S1. Since Ricci–flat spaces arenatural in the string/M–theory context, this is an impor-tant development. Secondly, we have demonstrated thatit is possible to construct a metastable de Sitter vacuumin the general framework of [8] by incorporating one ormore geometric twists in the internal space.As mentioned above, some of the extra dimensions mayeventually grow large, though slowly, without bound anddecompactify, a feature that our transient solutions sharewith those of the TW model. One might argue that pro-vided that the relative size of the extra dimensions andtheir rate of change remain sufficiently small as to beconsistent with observation over any epoch of physicalrelevance, then the eschatological consequences of such“decompactifications” are no more severe than those ofa universe whose ordinary three dimensions accelerateforever. Nonetheless, given that the natural endpoint ofthe solutions is a fundamentally higher–dimensional run-away epoch [12], it is clear that we would have additionalcosmic coincidence constraints in explaining why the uni-verse appears to have three large spatial dimensions atthe present epoch. However, before arriving at any suchconclusions it is important to include additional matter,such as pressureless dust. Matter obeying the strong en-ergy condition will certainly affect the length of any tran-sient acceleration, and may also change the evolution ofthe scalar fields.Finally, we remark that geometric twists add richnessto string/M–theory cosmology, and may potentially leadto the realization of new cosmological scenarios.Acknowledgements: This work was supported in partby the Mardsen Fund of the Royal Society of NewZealand. I.P.N. thanks CERN-TH for hospitality whilepart of this work was done.[1] C.L. Bennett et al., Astrophys. J. Suppl. 148, 1 (2003).[2] G. W. Gibbons, in Supersymmetry, Supergravity and Re-lated Topics, eds. F. del Aguila, J. A. de Azcarragaand L. E. Ibanez, (World Scientific, Singapore, 1985),pp. 123–146; J. M. Maldacena and C. Nunez, Int. J. Mod.Phys. A 16, 822 (2001).[3] P. K. Townsend and M.N.R. Wohlfarth, Phys. Rev. Lett.91 (2003) 061302.[4] C.-M. Chen, P.-M. Ho, I. P. Neupane, and J. E. Wang,JHEP 0307, 017 (2003).[5] N. Ohta, Phys. Rev. Lett. 91, 061303 (2003).[6] M. N. R. Wohlfarth, Phys. Lett. B 563, 1 (2003); S. Roy,Phys. Lett. B 567, 322 (2003).[7] R. Emparan and J. Garriga, JHEP 0305 (2003) 028.[8] C.-M. Chen, P.-M. Ho, I. P. Neupane, N. Ohta andJ. E. Wang, JHEP 0310, 058 (2003).[9] I. P. Neupane, Class. Quantum Grav. 21, 4383 (2004);Mod. Phys. Lett. A 19, 1093 (2004); L. Jarv, T. Mohauptand F. Saueressig, JCAP 0408, 016 (2004).5[10] E. Bergshoeff, A. Collinucci, U. Gran, M. Nielsen andD. Roest, Class. Quantum Grav. 21, 1947 (2004).[11] I. P. Neupane and D. L. Wiltshire, hep-th/0504135.[12] S. B. Giddings and R. C. Myers, Phys. Rev. D 70, 046005(2004).

Accelerating cosmologies from compactification with a twist

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