1. Picking up speed in string cosmology Diederik Roest December 3, 2009 24th Nordic Network Meeting

2. Size matters! Why is there any relation at all between cosmology and string theory?

4. Outline 1. Modern cosmology 2. Fundamental physics 3. Flux compactifications 4. Moduli stabilisation

5. Outline 1. Modern cosmology 2. Fundamental physics 3. Flux compactifications 4. Moduli stabilisation

6. Cosmological principle Universe has no structure at large scales stars -> galaxies -> clusters -> superclusters -> FRW No preferred points or directions: homogeneous and isotropic.

7. Cosmological principle General Relativity simplifies to:   Space-time described by FRW: – –scale factor a(t) – –curvature k   Matter described by ‘perfect fluids’ with – –energy density ρ(t) – –equation of state parameter w Fractions of critical energy density: Ω(t) = ρ(t) / ρ crit (t)

8. Table of content? What are the ingredients of the universe? Dominant components: w=1/3- radiation / relativistic matter R w=0 - non-relativistic matter M w=-1/3 - curvature C w=-1 - cosmological constant Λ

9. History of CC Who ordered Λ ? First introduced by Einstein to counterbalance matter Overtaken by expansion of universe Convoluted history through the 20th century.

10. Age crises Mid-life crisis? Λ to the rescue!! 1930-40’s: first estimate of Hubble parameter implies a very young universe. Conflict with known ages of stars etc. resolution: better value for Hubble parameter! 1990’s: again tension between estimate of age of universe from Hubble parameter and from ages of stars, galaxies etc. resolution: cosmological constant!

11. Modern cosmology Supernovae (SNe) Cosmic Microwave Background (CMB) Baryon Acoustic Oscillations (BAO)

12. Supernovae Explosions of fixed brightness Standard candles Luminosity vs. redshift plot SNe at high redshift ( z~0.75 ) appear dimmer Sensitive to Ω M - Ω Λ [Riess et al (Supernova Search Team Collaboration) ’98] [Perlmutter et al (Supernova Cosmology Project Collaboration) ’98]

13. Cosmic Microwave Background Primordial radiation from recombination era Blackbody spectrum of T=2.7 K Anisotropies of 1 in 10 5 Power spectrum of correlation in δT Location of first peak is sensitive to Ω M + Ω Λ [Bennett et al (WMAP collaboration) ’03]

14. Baryon acoustic oscillations Anisotropies in CMB are the seeds for structure formation. Acoustic peak also seen in large scale surveys around z=0.35 Sensitive to Ω M [Eisenstein et al (SDSS collaboration) ’05] [Cole et al (2dFGRS collaboration) ’05]

15. Putting it all together

17. Concordance Model Nearly flat Universe, 13.7 billion years old. Present ingredients: 73% dark energy 23% dark matter 4% SM baryons

18. Concordance Model Open questions: What are dark components made of? CC unnaturally small: 30 orders below Planck mass!   Fine-tuning mechanism?   Anthropic reasoning? Cosmic coincidence problem

19. Going back in time

20. Inflation Period of accelerated expansion in very early universe to explain: Cosmological principle Why universe is flat Absence of magnetic monopoles Bonus: quantum fluctuations during inflation act as source for structure formation ( CMB).

21. Inflation Modelled by scalar field with non-trivial scalar potential V Slow-roll parameters: Measured: ² = 1 2 M 2 P ¡ V 0 V ¢ 2 ¿ 1 ; ´ = M 2 P V 00 V ¿ 1 : n s = 1 ¡ 6 ² + 2 ´ » 0 : 951 § 0 : 016

22. The future is bright! Beautiful probe of physics at very high energies (~10 16 Gev) Inflationary properties are now being measured Planck satellite: – –Non-Gaussianities? – –Tensor modes? – –Constraints on inflation?

23. Outline 1. Modern cosmology 2. Fundamental physics 3. Flux compactifications 4. Moduli stabilisation

24. Strings Quantum gravity No point particles, but small strings Unique theory Bonus: gauge forces Unification of four forces of Nature?

25. …and then some! Extra dimensions Many vacua ( ~10 500 )? Dualities Branes & fluxes Super- symmetry String theory has many implications: How can one extract 4D physics from this?

26. Compactifications

27. Stable compactifications Simple compactifications yield massless scalar fields, so-called moduli, in 4D. Would give rise to a new type of force, in addition to gravity and gauge forces. Has not been observed! energy Scalar field simple comp.

28. Fifth-force experiments V ( r ) = ¡ G m 1 m 2 r ( 1 + ®e ¡ r = ¸ ) [Kapner et al ‘06]

29. Stable compactifications Simple compactifications yield massless scalar fields, so-called moduli, in 4D. Would give rise to a new type of force, in addition to gravity and gauge forces. Has not been observed! Need to give mass terms to these scalar fields (moduli stabilisation). Extra ingredients of string theory, such as branes and fluxes, are crucial! energy Scalar field with fluxes and branes simple comp.

30. Flux compactifications Lots of progress in understanding moduli stabilisation in string theory (2002-…) Using gauge fluxes one can stabilise the Calabi-Yau moduli Classic results: – –IIB complex structure moduli stabilised by gauge fluxes [1] – –IIB Kahler moduli stabilised by non-perturbative effects [2] – –All IIA moduli stabilised by gauge fluxes [3] But: – –Vacua are supersymmetric AdS (i.e. have a negative cosmological constant) [1: Giddings, Kachru, Polchinski ’02] [2: Kachru, Kallosh, Linde, Trivedi ’03] [3: DeWolfe, Giryavets, Kachru, Taylor ’05]

31. Going beyond flux compactifications Geometric fluxes G- structures Generalised geometries Non- geometric fluxes Today Lectures by Louis

32. Outline 1. Modern cosmology 2. Fundamental physics 3. Flux compactifications 4. Moduli stabilisation

33. Compactifications “Vanilla” compactifications lead to ungauged supergravities: e.g. on torus (N=8) with orientifold(N=4) on Calabi-Yau(N=2) on CY with orientifold(N=1) Problem of massless moduli in 4D, no scalar potential! Need to include additional “bells and whistles” on internal manifold M.

34. Compactifications

35. Flux compactifications Additional “ingredients”: Gauge fluxes (electro-magnetic field lines in M) Geometric fluxes (non-trivial Ricci-curvature on M) Non-geometric fluxes (generalisation due to T-duality) Difference with lectures by Jan Louis: not consider manifolds with non-trivial SU(3) holonomy / structure.

36. Toroidal reduction Example: torus reduction of 10D common sector: Split into 4D space-time and 6D internal space Drop all internal dependence Expansion over non-trivial cycles leads to 4D field content: ^ g ¹º ; ^ B ¹º ; ^ Á g ¹º ; A M ¹ ; Á MN B ¹º ; A M ¹ ; » MN Á

37. Gauge fluxes Possibility to wrap fluxes around internal cycles: Corresponds to some internal dependence of gauge potential: Monodromy: gauge transformation ^ B = H mnp x p d x m ^ d x n + ::: ^ H = H mnp d x m ^ d x n ^ d x p + ::: x p ! x p + 1 : ^ B ! ^ B ¡ d ( H mnp x m d x n )

38. Geometric fluxes Going from torus to twisted torus or group manifold Internal dependence for metric: Monodromy: coordinate transformation Geometric fluxes form structure constants of a group. ^ d s 2 = ( d x m + f np m x n d x p ) 2 + ::: x n ! x n + 1 :x m ! x m ¡ f np m x p

39. T-duality Symmetry of common sector when compactified on a circle. Requirement: isometry direction x. Explicit Buscher rules relate different backgrounds:

40. T-duality T-duality acts in NS-NS sector by raising / lowering indices of fluxes Gauge and geometric fluxes related via T-duality transformation! T p : ( H mnp ! f mn p ; f mn p ! H mnp :

41. Further T-duality Start from single H flux Single T-duality: geometric flux Further T-duality in other isometry direction possible! “Non-geometric flux” Monodromy mixes metric and gauge flux ^ d s 2 = ( d x 1 + f 23 1 x 3 d x 2 ) 2 + ::: H 123 ! f 23 1 ! Q 3 12 ^ B = H 123 x 3 d x 1 ^ d x 2 + :::

42. Yet further T-duality Non-geometric flux Q still locally geometric Formally one could think about performing another T-duality However this is not an isometry direction! Leads to non-geometric flux that does not have any local description H 123 ! f 23 1 ! Q 3 12 ! R 123

43. Effective description What is the resulting 4D description of flux compactifications? gauged supergravities where the fluxes play the role of structure constants specifying the gauging.

44. Gauged supergravity Gauged supergravity Supergravity Supergravity has many scalar fields that could be used for e.g. cosmology. A priori massless scalar fields. Only possibility of introducing masses is via specific scalar potential energies. Fully specified by gaugings: part of the global symmetries are made local. Depends on global symmetry and number of vectors gauged supergravity.

45. Gauged supergravity Example: maximal N=8 supergravity has global symmetry group and 28 gauge vectors. Ungauged theory: gauge algebra is U(1) 28. Vanishing scalar potential, Minkowski vacuum. Gauged theory: gauge algebra is e.g. SO(8). Complicated scalar potential, Anti-De Sitter vacuum. Other possibility: gauge algebra is e.g. SO(4,4). Complicated scalar potential, De Sitter vacuum. SL ( 8 ) ½ E 7 ( 7 )

46. Gauged supergravity Generically gives rise to negative potential energy. Corresponding vacuum is Anti-De Sitter space (AdS). Scalar potentials of gauged supergravity play important role in AdS/CFT correspondence. By careful finetuning one can also build scalar potentials that are interesting for cosmology, e.g. with a positive potential energy. Corresponding vacuum is De Sitter space (dS). What is the gauge algebra from flux compactifications? Does this allow for e.g. stable De Sitter vacua?

47. Gauge algebra Without fluxes, a compactification of the common sector leads to 12 gauge vectors with gauge group U(1) 12 : Gauge and geometric flux leads to non-Abelian algebra [1]: How does this change when including other fluxes? Find out by doing T-dualities! X m :x m ! x m + ¸ m ; Z m : ^ B ! ^ B + d¸ m : [ X m ; X n ] = f mn p X p + H mnp Z p ; [ X m ; Z n ] = f mp n Z p ; [ Z m ; Z n ] = 0 : [1: Kaloper, Myers ’99]

48. Gauge algebra X and Z are interchanged under T-duality: Indices raised and lowered as for fluxes. Proposal to include all NS-NS fluxes [1]: T p : X p $ Z p : [ X m ; X n ] = f mn p X p + H mnp Z p ; [ X m ; Z n ] = f mp n Z p + Q m np Z p ; [ Z m ; Z n ] = Q p mn Z p + R mnp X p : [ X m ; X n ] = f mn p X p + H mnp Z p ; [ X m ; Z n ] = f mp n Z p ; [ Z m ; Z n ] = 0 : [1: Shelton, Taylor, Wecht ’05]

49. IIB with O3-planes Convenient duality frame: can always be reached by T-duality transformations. Only allowed NS-NS fluxes are gauge and non- geometric fluxes: All fluxes locally geometric! Proposed algebra reduces to H mnp ; Q m np : [ X m ; X n ] = H mnp Z p ; [ X m ; Z n ] = Q m np Z p ; [ Z m ; Z n ] = Q p mn Z p :

50. Puzzle: compare duality frames ? Type I has R-R instead of NS-NS three-form!!! Type I = IIB / = IIB with O9 IIB with O3 [ X m ; X n ] = f mn p X p + H mnp Z p ; [ X m ; Z n ] = f mp n Z p ; [ Z m ; Z n ] = 0 : [ X m ; X n ] = H mnp Z p ; [ X m ; Z n ] = Q m np Z p ; [ Z m ; Z n ] = Q p mn Z p : T 4 ¢¢¢ 9 ­

51. Correct gauge algebra Starting point should be F is Ramond-Ramond and behaves differently under T-duality: Six-tuple T-duality takes us to Also derived by [1] on different grounds. T p : ( F m 1 ¢¢¢ m n ! F m 1 ¢¢¢ m n p ; F m 1 ¢¢¢ m n p ! F m 1 ¢¢¢ m n ; [ X m ; X n ] = f mn p X p + F mnp Z p ; [ X m ; Z n ] = f mp n Z p ; [ Z m ; Z n ] = 0 : [ X m ; X n ] = 0 ; [ X m ; Z n ] = Q m nn X p ; [ Z m ; Z n ] = Q p mn Z p + ² mnpqrs F qrs X p : [1: Aldazabal, Camara, Rosabal ’08]

52. Correct gauge algebra So far we have been concerned with the algebra spanned by the electric part of the gauge vectors. Relevant fluxes: non-geometric Q and gauge F. Also possibility to gauge with magnetic parts. S-duals fluxes: non-geometric P and gauge H. Constraints to ensure orthogonality of charges. Elec:Magn: NS-NS:QH R-R:FP

53. Outline 1. Modern cosmology 2. Fundamental physics 3. Flux compactifications 4. Moduli stabilisation

54. Higher-dimensional origin? 10D string theory 4D gauged supergravity 4D gauged supergravity Which of these two sets contain (stable) De Sitter vacua? Compactification with gauge and (non-)geometric fluxes

55. N=4 gauged supergravity Most common gauging: gauge group is direct product of factors G = G 1 x G 2 x … Crucial for moduli stabilisation [1]: both electric and magnetic factor in gauge group. If entire gauge group is electric, the scalar potential has runaway directions: Impossible to stabilise moduli in dS. V ( Á ; ~ ' ) = eÁV 0 ( ~ ' ) [1: De Roo, Wagemans ’85]

56. De Sitter in N=4 Known De Sitter vacua in N = 4: split up in two 6D gauge factors G = G 1 x G 2 given by [1] SO(4), SO(3,1) or SO(2,2). Plus some exceptional cases with 3+9 split. All unstable: tachyonic directions with -1 < η 0. ≥ No stable De Sitter vacua are expected for N ≥ 4 - proof? [2] [1: De Roo, Westra, Panda ’06] [2: Gomez-Reino, (Louis), Scrucca ’06, ’07, ’08]

57. Fate of dS in compactifications? This year it was shown that one can build up gaugings of the form G = G 1 x G 2 in this way [1]. But these fluxes are not enough to build up any of the products of simple gauge groups with dS vacua [2]. Crucially depends on correct form of gauge algebra!! [1: D.R. ’09, Dall’Agata, Villadoro, Zwirner ‘09] [2: Dibitetto, Linares, D.R. - in progress]

58. Fate of dS in compactifications? Factors of gauge groups given by: SO(4) / SO(3,1) / SO(2,2)- De Sitter vacua [1] ISO(3) / ISO(2,1) CSO(2,0,2) / CSO(1,1,2) CSO(1,0,3) U(1) 6 where CSO(p,q,r) is a (contraction) r of SO(p,q+r). [1: De Roo, Westra, Panda ’06] [2: Dibitetto, Linares, D.R. - in progress] [3: D.R. ‘09] group contractions flux compac- tifications [2,3]

59. Higher-dimensional origin? 10D string theory 4D gauged supergravity 4D gauged supergravity Which of these two sets contain De Sitter vacua? New ones [2] Known ones [1] G + £ G ¡ G + n G ¡ [1: Dibitetto, Linares, D.R. - in progress] [2: De Carlos, Guarino, Moreno ’09] Compactification with gauge and (non-)geometric fluxes

60. Semi-direct product gaugings “Complete classification of Minkowski vacua in generalised flux models” [1] Use fluxes F, H and Q (and exclude P) Restrict to “isometric” truncation of fluxes Both in N=1 and N=4 [1: De Carlos, Guarino, Moreno ’09]

61. Semi-direct product gaugings Classification based on subalgebra spanned by Q-fluxes Full gauge algebra, including gauge fluxes F and H, is semi-direct product gauging of the form Each case has two remaining flux parameters. Allows for: – –ISO(3): unstable N=1 with purely geometric fluxes – –SO(3,1): unstable N=4 and stable N=1 with non-geom fluxes Vacua can be either AdS / Minkowski / dS! SO(4)SO(3,1) ISO(3) SO(3) x U(1) 3 NilpotentU(1) 6 [ Z m ; Z n ] = Q p mn Z p G + n G ¡

62. The ISO(3) case N=4 Gauge and geometric fluxes [1] [1: Caviezel, Koerber, Kors, Lust, Wrase, Zagermann ‘08]

63. The SO(3,1) case N=4 Gauge and non-geometric fluxes stable

64. Outline 1. Modern cosmology 2. Fundamental physics 3. Flux compactifications 4. Moduli stabilisation

65. Conclusions Modern cosmology (CMB, SNe and BAO) involves inflation and dark energy Link with fundamental physics: string cosmology.

66. Conclusions Flux compactifications & moduli stabilisation – –Gauge and (non-)geometric fluxes Stabilise the moduli of string theory in a De Sitter vacuum: – –None of known gaugings! – –Unstable N=1 from geometric fluxes – –Unstable N=4 and stable N=1 from non-geometric fluxes What about semi-direct product gaugings, requirements for dS in gauged supergravity, corresponding string backgrounds, P-flux, G-structure, inflation, …? Many interesting (future) developments!

67. Thanks for your attention! Diederik Roest December 3, 2009 24th Nordic Network Meeting