Picking up speed in string cosmology

Diederik Roest December 3, 2009 24th Nordic Network Meeting

Size matters!

cosmology and string theory?

Outline

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Modern cosmology Fundamental physics Flux compactifications Moduli stabilisation

Outline

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Modern cosmology

Fundamental physics Flux compactifications Moduli stabilisation

Cosmological principle

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

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)

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 w=-1

- curvature C - cosmological constant

Λ

History of CC

Who ordered

Λ

?

 First introduced by Einstein  to counterbalance matter Overtaken by expansion of universe Convoluted history through the 20th century.

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!

Modern cosmology

Supernovae (SNe) Baryon Acoustic Oscillations (BAO) Cosmic Microwave Background (CMB)

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]

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]

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]

Putting it all together

Putting it all together

Concordance Model

Nearly flat Universe, 13.7 billion years old.

Present ingredients:  73% dark energy   23% dark matter 4% SM baryons

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

Going back in time

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).

Inflation

Modelled by scalar field with non-trivial scalar potential

V

Slow-roll parameters: Measured: ² = 1 2 M 2 P ¡ V 0 ¢ 2 V ¿ 1 ; ´ = M 2 P V 00 V ¿ 1 : n s = 1 ¡ 6² + 2´ » 0:951 § 0:016

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?

Outline

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Modern cosmology

Fundamental physics

Flux compactifications Moduli stabilisation

Strings

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

String theory has many implications:

…and then some!

Super symmetry Extra dimensions Branes & fluxes Dualities Many vacua (

~10 500

)?

How can one extract 4D physics from this?

Compactifications

Stable compactifications

energy

 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!

simple comp.

Scalar field

Fifth-force experiments

V (r ) = ¡ G m 1 m 2 r (1 + ®e ¡ r =¸ ) [Kapner et al ‘06]

Stable compactifications

energy

 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!

Scalar field simple comp.

with fluxes and branes

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]

Going beyond flux compactifications

Today Geometric fluxes Non-geometric fluxes Generalised geometries G-structures Lectures by Louis

Outline

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Modern cosmology Fundamental physics

Flux compactifications

Moduli stabilisation

Compactifications

“Vanilla” compactifications lead to ungauged supergravities: e.g. on torus with orientifold on Calabi-Yau on CY with orientifold (N=8) (N=4) (N=2) (N=1) Problem of massless moduli in 4D, no scalar potential!

Need to include additional “bells and whistles” on internal manifold M.

Compactifications

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.

Toroidal reduction

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

Gauge fluxes

 Possibility to wrap fluxes around internal cycles: ^ m n p dx m ^ dx n ^ dx p + : : :  Corresponds to some internal dependence of gauge potential: ^ m n p x p dx m ^ dx n + : : :  Monodromy: gauge transformation x p !

x p + 1 : ¡ d(H m n p x m dx n )

Geometric fluxes

  Going from torus to twisted torus or group manifold Internal dependence for metric: ^ 2 = (dx m + f n p m x n dx p ) 2 + : : :  Monodromy: coordinate transformation x n !

x n + 1 : x m !

x m ¡ f n p m x p  Geometric fluxes form structure constants of a group.

T-duality

   Symmetry of common sector when compactified on a circle.

Requirement: isometry direction x.

Explicit Buscher rules relate different backgrounds: 1 G 0 x x G 0 ¹ º B 0 ¹ º = = = G x x G ¹ º B ¹ º ¡ ¡ ; G x ¹ G x ¹ G 0 x ¹ G x º = ¡ ¡ B B G x x x B x º G x x ¡ B x ¹ ¹ x ¹ B G ; x x º º G x x B 0 x ¹ = ¡ G x ¹ G x x e Á e Á 0 = p G x x

T-duality

 T-duality acts in NS-NS sector by raising / lowering indices of fluxes T p : ( f H m n p m n p !

!

f m n p H m n p ; :  Gauge and geometric fluxes related via T-duality transformation!

Further T-duality

 ^ 123 x 3 1 ^ dx 2 + : : :  Single T-duality: geometric flux = (dx 1 + f 23 1 x 3 dx 2 ) 2 + : : :  Further T-duality in other isometry direction possible!

H 123 !

f 23 1 !

Q 3 12   “Non-geometric flux” Monodromy mixes metric and gauge flux

Yet further T-duality

  Non-geometric flux Q still locally geometric Formally one could think about performing another T-duality H 123 !

f 23 1 !

Q 3 12 !

R 123   However this is not an isometry direction!

Leads to non-geometric flux that does not have any local description

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.

Gauged supergravity

Supergravity Gauged 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.

Gauged supergravity

Example: maximal N=8 supergravity has global symmetry group SL (8) ½ E 7( 7) 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.

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.

De Sitter vacua?

 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).

Gauge algebra

 Without fluxes, a compactification of the common sector leads to 12 gauge vectors with gauge group U(1) 12 : X m : x m !

x m + ¸ m ; Z m : ^ ¸ m :  Gauge and geometric flux leads to non-Abelian algebra [1]: [X m ; X n ] = f [X m ; Z n ] = f m n p X p m p n Z p ; + H m n p Z p [Z m ; Z n ] = 0 : ;  How does this change when including other fluxes? Find out by doing T-dualities!

[1: Kaloper, Myers ’99]

Gauge algebra

 X and Z are interchanged under T-duality: T p : X p $ Z p : Indices raised and lowered as for fluxes.

 Proposal to include all NS-NS fluxes [1]: [X m ; X n ] = f [X m ; Z n ] = f m n p X p m p n Z p ; + H m n p Z p [Z m ; Z n ] = 0 : ; [X m ; X n ] = f [X m ; Z n ] = f m n p X p m p n Z p + H m n p Z p ; + Q m n p Z p ; [Z m ; Z n ] = Q p m n Z p + R m n p X p : [1: Shelton, Taylor, Wecht ’05]

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: m n p ; Q m n p : All fluxes locally geometric!

Proposed algebra reduces to [X m ; X n ] = H m n p Z p [X m ; Z n ] = Q m n p Z p ; ; [Z m ; Z n ] = Q p m n Z p :

Puzzle: compare duality frames

T 4 ¢¢¢9 Type I = IIB / = IIB with O9 IIB with O3 [X m ; X n ] = f [X m ; Z n ] = f m n p X p m p n Z p ; + H m n p Z p [Z m ; Z n ] = 0 : ; ?

[X m ; X n ] = H m n p Z p [X m ; Z n ] = Q m n p Z p [Z m ; Z n ] = Q p m n Z p ; : ; Type I has R-R instead of NS-NS three-form!!!

Correct gauge algebra

 Starting point should be [X m ; X n ] = f [X m ; Z n ] = f [Z m ; Z n ] = 0 : m n p X p m p n Z p ; + F m n p Z p ;  F is Ramond-Ramond and behaves differently under T-duality: T p : F m F m 1 1 ¢¢¢m ¢¢¢m n n p !

!

F m 1 ¢¢¢m n F m 1 ¢¢¢m n p ; ;  Six-tuple T-duality takes us to [X m ; X n ] = 0 ; [X m ; Z n ] = Q m n n X p [Z m ; Z n ] = Q p m n Z p ; + ² m n pqr s F qr s X p :  Also derived by [1] on different grounds.

[1: Aldazabal, Camara, Rosabal ’08]

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.

NS-NS: R-R: Elec: Q F Magn: H P

Outline

1.

2.

3.

4.

Modern cosmology Fundamental physics Flux compactifications

Moduli stabilisation

Higher-dimensional origin?

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

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: Á V 0 ' ) Impossible to stabilise moduli in dS.

[1: De Roo, Wagemans ’85]

De Sitter in N=4

 Known De Sitter vacua in N = 4: split up in two 6D gauge factors G = G 1 SO(4), SO(3,1) or SO(2,2).

x G 2 given by [1]    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]

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]

Fate of dS in compactifications?

Factors of gauge groups given by: group contractions SO(4) / SO(3,1) / SO(2,2) ISO(3) / ISO(2,1) CSO(2,0,2) / CSO(1,1,2) CSO(1,0,3) U(1) 6 - De Sitter vacua [1] flux compac tifications [2,3] 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]

Higher-dimensional origin?

10D string theory Compactification with gauge and (non-)geometric fluxes G + n G ¡ 4D gauged supergravity 4D gauged supergravity G + £ G ¡ Which of these two sets contain De Sitter vacua?

[1: Dibitetto, Linares, D.R. - in progress] [2: De Carlos, Guarino, Moreno ’09]

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]

Semi-direct product gaugings

 Classification based on subalgebra spanned by Q-fluxes [Z m ; Z n ] = Q p m n Z p SO(4) ISO(3) Nilpotent SO(3,1) SO(3) x U(1) U(1) 6 3  product gauging of the form G + n G ¡  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!

The ISO(3) case

N=4 Gauge and geometric fluxes [1] [1: Caviezel, Koerber, Kors, Lust, Wrase, Zagermann ‘08]

N=4

The SO(3,1) case

stable Gauge and non-geometric fluxes

Outline

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Modern cosmology Fundamental physics Flux compactifications Moduli stabilisation

Conclusions

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

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!

Thanks for your attention!

Diederik Roest December 3, 2009 24th Nordic Network Meeting