We summarize our recent results of studying five-dimensional Kasner
cosmologies in a time-dependent Calabi-Yau compactification of M-theory
undergoing a topological flop transition. The dynamics of the additional
states, which become massless at the transition point and give rise to a scalar
potential, helps to stabilize the moduli and triggers short periods of
accelerated cosmological expansion.Comment: 4 pages; contribution to the proceedings of the EURESCO conference
  "What comes beyond the Standard Model?", July 2003, Portoroz, Sloveni

Jarv, Laur

Mohaupt, Thomas

Saueressig, Frank

English

arXiv

We summarize our recent results of studying five-dimensional Kasner
cosmologies in a time-dependent Calabi-Yau compactification of M-theory
undergoing a topological flop transition. The dynamics of the additional
states, which become massless at the transition point and give rise to a scalar
potential, helps to stabilize the moduli and triggers short periods of
accelerated cosmological expansion

University of Liverpool Repository

arXiv:hep-th/0311016v1  4 Nov 2003Singular compactifications and cosmology∗†Laur Ja¨rv, Thomas Mohaupt and Frank SaueressigInstitute of Theoretical Physics, Friedrich-Schiller-University Jena,Max-Wien-Platz 1, D-07743 Jena, GermanyL.Jaerv, T.Mohaupt, F.Saueressig@tpi.uni-jena.deAbstractWe summarize our recent results of studying five-dimensional Kasnercosmologies in a time-dependent Calabi-Yau compactification of M-theory undergoing a topological flop transition. The dynamics of theadditional states, which become massless at the transition point andgive rise to a scalar potential, helps to stabilize the moduli and triggersshort periods of accelerated cosmological expansion.During the last year a lot of effort has been made to explain the as-tronomical evidence for an inflationary epoch of the early universe and thecurrent modest accelerated expansion by invoking a scalar potential derivedfrom string or M-theory compactifications. So far two mechanisms leading topotentials viable to describe accelerated cosmological expansion have beenexplored [1]: (i) compactifications on hyperbolic spaces [2] and (ii) compact-ifications with fluxes [3]. Our recent work [4, 5] gives the first example of(iii) compactification on a singular internal manifold.In the case of smooth compactifications one usually has a moduli spaceof vacua corresponding to the deformations of the internal manifold X andthe background fields. For theories with eight or less supercharges thismoduli space includes special points where X degenerates, rendering thecorresponding low energy effective action (LEEA) discontinuous or singular.However, within the full string or M-theory these singularities are believed tobe artifacts, which result from ignoring some relevant modes of the theory,namely the winding states of strings or branes around the cycles of X.Singularities of X arise when such cycles are contracted to zero volume,∗Work supported by the ‘Schwerpunktprogramm Stringtheorie’ of the DFG.†Talk given by T. Mohaupt at the EURESCO conference “What comes beyond theStandard Model?”, 12. – 17. July 2003, Portorozˇ, Slovenia.1thereby introducing additional massless states. Incorporating these statesleads to a smooth gauged supergravity action which entails a scalar potential.In Calabi-Yau (CY) compactifications of M-theory undergoing a topolog-ical flop transition these additional states (‘transition states’) are given byN charged hypermultiplets which become massless at the transition locus.There are two ways to include the effect of these extra states in the LEEA.The usual LEEA is obtained by dimensional reduction on the smooth CYand contains only states which are generically massless. The flop manifestsitself in a discontinuous change of the vector multiplet couplings at the tran-sition locus. We call this description the ‘Out-picture’ since the extra statesare left out. On the contrary, the ‘In-picture’ is obtained by including thetransition states as dynamical fields in the Lagrangian.In [4] we constructed an In-picture LEEA for a generic M-theory flop bycombining knowledge about the general N = 2,D = 5 gauged supergravityaction with information about the extra massless states.1 While the vectormultiplet sector could be treated exactly we used a toy model based onthe quaternion-Ka¨hler manifolds U(1+N,2)U(1+N)×U(2) to describe the hypermultipletsector. In order to find the gauging describing the flop we worked out themetrics, the Killing vectors, and the moment maps of these spaces. Thisdata enabled us to construct a unique LEEA which has all the propertiesto model a flop: the extra hypermultiplets acquire a mass away from thetransition locus while the universal hypermultiplet remains massless.In [5] we considered an explicit model for a CY compactification under-going a flop with N = 1 and investigated the effect of the transition stateson five-dimensional Kasner cosmologies,2ds2 = −dτ2 + e2α(τ)d~x2 + e2β(τ)dy2 . (1)Comparing the cosmological solutions of the Out- and the In-picture, wefound that the inclusion of the dynamical transition states has drastic con-sequences for moduli stabilization and accelerated expansion.As soon as we allow all light states to be excited the scalar fields nolonger show the usual run-away behavior but are attracted to the flop regionwhere they oscillate around the transition locus. Thus the “almost singular”manifolds close to the flop are dynamically preferred. This is somewhat sur-prising, because the potential has still many unlifted flat directions meaningthere is no energy barrier which prevents the system from running away.1This strategy was first applied in [6] in the case of SU(2) enhancement.2This setup was previously considered in [7], but there the hypermultiplet manifoldwas taken to be hyper-Ka¨hler which is not consistent with local supersymmetry.2Hence this effect cannot be predicted by just analyzing the critical points ofthe superpotential. The following thermodynamic analogy helps to explainthe situation. Generically the available energy of the system is distributedequally among all the light modes (“thermalization”). Thus near the flopline the additional degrees of freedom get their natural share of it. Oncethis has happened, it becomes very unlikely that the system “finds” theflat directions and “escapes” from the flop region. Our numerical solutionsconfirm this picture: irrespective of the initial conditions the system finallysettles down in a state where all the fields either approach finite values or os-cillate around the transition region with comparable and small amplitudes.From time to time one sees “fluctuations from equilibrium”, i.e., some modepicks up a bigger share of the energy for a while, but the system eventuallythermalizes again. In an ideal scenario of moduli stabilization, however, onewould like to have a damped system so that the moduli converge to fixedpoint values.The second important aspect is that the scalar potential of the In-pictureinduces short periods of accelerated expansion in the three-space. Yet thenet effect of the accelerating periods on cosmic expansion is not very signif-icant. Again, this feature can be understood qualitatively in terms of theproperties of the scalar potential. The point is that the potential is onlyflat along the unlifted directions while along the non-flat ones it is too steepto support sustained accelerated expansion. Transient periods of accelera-tion occur when the scalar fields pass through their collective turning point,where running “uphill” the potential turns into running “downhill” and thepotential energy momentarily dominates over the kinetic energy.3 To get aconsiderable amount of inflation via a slow-roll mechanism, one would needto lift some of the flat directions gently without making them too steep.In summary we see that the dynamics of the transition states is in-teresting and relevant, and can be part of the solution of the problems ofmoduli stabilization and inflation. One direction for further investigationsis to consider more general gaugings of our five-dimensional model. Oncegaugings leading to interesting cosmological solutions are found, one shouldclarify whether these can be derived from string or M-theory where theycorrespond to adding fluxes or branes. Another direction is to extend ourconstruction to other topological transitions. In particular it would be inter-esting to study the effect of transition states on four-dimensional cosmologiesarising, e.g., from type II compactifications on singular CY manifolds. It is3This behavior is also common to the models of hyperbolic and flux compactifications,where likewise the acceleration is not pronounced enough for primordial inflation.3conceivable that a realistic cosmology derived from sting or M-theory willhave to include both the effects of fluxes and branes, and the possibility ofinternal manifolds becoming singular.References[1] R. Emparan, J. Garriga: A note on accelerating cosmologies from com-pactifications and S branes, JHEP 0305 (2003) 026; P.K. Townsend:Cosmic acceleration and M-theory, hep-th/0308149.[2] P. Townsend, M.R.N. Wohlfarth: Accelerating cosmologies from com-pactification, Phys. Rev. Lett. 91 (2003) 061302; N. Ohta: Acceler-ating cosmologies from S-branes, Phys. Rev. Lett. 91 (2003) 061303;S. Roy: Accelerating cosmologies from M/string theory compactifica-tions, Phys. Lett. B567 (2003) 322; M.R.N. Wohlfarth: Acceleratingcosmologies and a phase transition in M theory, Phys. Lett. B563(2003) 1; N. Ohta: A study of accelerating cosmologies from super-string / M theories, Prog. Theor. Phys. 110 (2003) 269; C.-M. Chen,P.-M. Ho, I.P. Neupane, J.E. Wang: A note on acceleration from prod-uct space compactification, JHEP 0307 (2003) 017; C.-M. Chen, P.-M. Ho, I.P. Neupane, N. Ohta, J.E. Wang: Hyperbolic space cosmolo-gies, JHEP 0310 (2003) 058; M.R.N. Wohlfarth, Inflationary cosmolo-gies from compactification?, hep-th/0307179; I.P. Neupane: Inflationfrom string/M-theory compactification?, hep-th/0309139.[3] L. Cornalba, M.S. Costa: A new cosmological scenario in string theory,Phys. Rev. D66 (2002) 066001; A.R. Frey, A. Mazumdar: 3-form in-duced potentials, dilaton stabilization, and running moduli, Phys. Rev.D67 (2003) 046006.[4] L. Ja¨rv, T. Mohaupt, F. Saueressig: Effective supergravity actions forflop transitions, hep-th/0310173.[5] L. Ja¨rv, T. Mohaupt, F. Saueressig: M-theory cosmologies from singu-lar Calabi-Yau compactifications, hep-th/0310174.[6] T. Mohaupt, M. Zagermann: Gauged supergravity and singular Calabi-Yau manifolds, JHEP 0112 (2001) 026; J. Louis, T. Mohaupt, M. Za-germann: Effective actions near singularities, JHEP 0302 (2003) 053.[7] M. Bra¨ndle, A. Lukas: Flop Transitions in M-theory Cosmology,Phys. Rev. D68 (2003) 024030.4

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