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

Picking up speed Picking up speed in string cosmologyin string cosmology

Diederik Roest December 3, 2009

24th Nordic Network Meeting

Size matters!Size matters!Why is there any relation at all between cosmology

and string theory?

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1.1. Modern cosmologyModern cosmology

2.2. Fundamental physicsFundamental physics

3.3. Flux compactificationsFlux compactifications

4.4. Moduli stabilisationModuli stabilisation

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Cosmological principleCosmological principle

Universe has no structure at large scalesstars -> galaxies -> clusters -> superclusters ->

FRWNo preferred points or directions: homogeneous and

isotropic.

Cosmological principleCosmological 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) / ρcrit(t)

Table of content?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

History of CCHistory of CC

Who ordered Λ? First introduced by Einstein

to counterbalance matter Overtaken by expansion

of universe

Convoluted history through the 20th century.

Age crisesAge 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 cosmologyModern cosmology

Supernovae (SNe)

Cosmic Microwave Background (CMB)

Baryon AcousticOscillations (BAO)

SupernovaeSupernovae

Explosions of fixed brightness

Standard candles Luminosity vs. redshift

plot SNe at high redshift

(z~0.75) appear dimmer Sensitive to ΩM - ΩΛ

[Riess et al (Supernova Search Team Collaboration) ’98][Perlmutter et al (Supernova Cosmology Project Collaboration) ’98]

Cosmic Microwave Cosmic Microwave BackgroundBackground

Primordial radiation from recombination era Blackbody spectrum of T=2.7 K

Anisotropies of 1 in 105

Power spectrum of correlation in δT

Location of first peakis sensitive to ΩM + ΩΛ

[Bennett et al (WMAP collaboration) ’03]

Baryon acoustic oscillationsBaryon 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 ΩM

[Eisenstein et al (SDSS collaboration) ’05] [Cole et al (2dFGRS collaboration) ’05]

Putting it Putting it all togetherall together

Putting it Putting it all togetherall together

Concordance ModelConcordance Model

Nearly flat Universe, 13.7 billion years old.

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

Concordance ModelConcordance 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 timeGoing back in time

InflationInflation

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).

InflationInflation

Modelled by scalar field with non-trivial scalar potential V

Slow-roll parameters:

Measured:

² = 12M 2

P

¡ V0

V

¢2 ¿ 1; ´ = M 2P

V00

V¿ 1:

ns = 1¡ 6² + 2 » 0:951§ 0:016

The future is bright!The future is bright! Beautiful probe of physics

at very high energies (~1016 Gev)

Inflationary properties are now being measured

Planck satellite:– Non-Gaussianities?– Tensor modes?– Constraints on inflation?

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StringsStrings

Quantum gravity No point particles, but small

strings Unique theory Bonus: gauge forces

Unification of four forces of Nature?

……and then some!and then some!

Super-symmetry

Dualities

Many vacua (~10500)?

Extradimensions

Branes& fluxes

String theory has many implications:

How can one extract 4D physics

from this?

CompactificationsCompactifications

Stable compactificationsStable 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.

Fifth-force experimentsFifth-force experiments

V(r) = ¡ Gm1m2r (1+ ®e¡ r = )

[Kapner et al ‘06]

Stable compactificationsStable 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.

Flux compactificationsFlux 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

Geometric fluxes

G-structures

Generalised geometries

Non-geometricfluxes

Today

Lectures by Louis

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“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.


Flux compactificationsFlux 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 reductionToroidal 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:

g¹ º ; B¹ º ; Á

g¹ º ;AM¹ ;ÁM N B¹ º ;AM

¹ ;»M N Á

Gauge fluxesGauge fluxes

Possibility to wrap fluxes around internal cycles:

Corresponds to some internal dependence of gauge potential:

Monodromy: gauge transformation

B = Hmnpxp dxm ^dxn +:::

H = Hmnpdxm ^dxn ^dxp +:::

xp ! xp + 1: B ! B ¡ d(Hmnpxmdxn)

Geometric fluxesGeometric 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.

ds2

= (dxm +f npm xndxp)2 +:::

xn ! xn +1: xm ! xm ¡ f npmxp

T-dualityT-duality

Symmetry of common sector when compactified on a circle.

Requirement: isometry direction x. Explicit Buscher rules relate different

backgrounds:G0xx =

1Gxx

; G0x¹ = ¡

Bx¹

Gxx; B0

x¹ = ¡Gx¹

Gxx

G0¹ º = G¹ º ¡

Gx¹ Gxº ¡ Bx¹ Bxº

Gxx

B0¹ º = B¹ º ¡

Gx¹ Bxº ¡ Bx¹ Gxº

Gxx

eÁ0

=eÁ

pGxx

T-dualityT-duality

T-duality acts in NS-NS sector by raising / lowering indices of fluxes

Gauge and geometric fluxes related via T-duality transformation!

Tp :

(Hmnp ! fmn

p ;fmn

p ! Hmnp :

Further T-dualityFurther 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

ds2

= (dx1 + f 231x3dx2)2 + :::

H123 ! f 231 ! Q3

12

B = H123x3 dx1 ^dx2 +:::

Yet further T-dualityYet 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

H123 ! f 231 ! Q3

12 ! R123

Effective descriptionEffective 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 supergravityGauged supergravity

Gaugedsupergravity

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.


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) ½E7(7)


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?

Gauge algebraGauge 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 : xm ! xm +¸m ;

Zm : B ! B + d m :

[X m;X n]= f mnpX p + HmnpZp ;

[X m;Zn]= f mpnZp ;

[Zm;Zn]= 0:

[1: Kaloper, Myers ’99]

Gauge algebraGauge algebra

X and Z are interchanged under T-duality:

Indices raised and lowered as for fluxes.

Proposal to include all NS-NS fluxes [1]:

Tp : X p $ Zp :

[X m;X n] = f mnpX p + HmnpZp ;

[X m;Zn] = f mpnZp + Qm

npZp ;

[Zm;Zn] = QpmnZp + RmnpX p :


[X m;Zn]= f mpnZp ;

[Zm;Zn]= 0:

[1: Shelton, Taylor, Wecht ’05]

IIB with O3-planesIIB 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

Hmnp ; Qmnp :

[X m;X n] = HmnpZp ;

[X m;Zn] = QmnpZp ;

[Zm;Zn] = QpmnZp :

Puzzle: compare duality Puzzle: compare duality framesframes

?

Type I has R-R instead of NS-NS three-form!!!

Type I = IIB / = IIB with O9

IIB with O3


[X m;Zn]= f mpnZp ;

[Zm;Zn]= 0:

[X m;X n] = HmnpZp ;

[X m;Zn] = QmnpZp ;

[Zm;Zn] = QpmnZp :

T4¢¢¢9

Correct gauge algebraCorrect 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.

Tp :

(Fm1¢¢¢mn ! Fm1¢¢¢mn p ;Fm1¢¢¢mn p ! Fm1¢¢¢mn

;

[X m;X n]= f mnpX p + FmnpZp ;

[X m;Zn]= f mpnZp ;

[Zm;Zn]= 0:

[X m;X n] = 0;

[X m;Zn] = QmnnX p ;

[Zm;Zn] = QpmnZp +²mnpqrsFqrsX p :

[1: Aldazabal, Camara, Rosabal ’08]

Correct gauge algebraCorrect 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 ensureorthogonality of charges.

Elec: Magn:

NS-NS:

Q H

R-R: F P

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Higher-dimensional origin?Higher-dimensional origin?

10D stringtheory

4D gauged supergravity


Which of these two sets contain (stable) De Sitter vacua?

Compactification with gauge and (non-)geometric fluxes

N=4 gauged supergravityN=4 gauged supergravity

Most common gauging: gauge group is direct product of factors

G = G1 x G2 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.

VV ((ÁÁ;; ~~'' )) == eeÁÁVV00((~~'' ))

[1: De Roo, Wagemans ’85]

De Sitter in N=4De Sitter in N=4 Known De Sitter vacua in N = 4:

split up in two 6D gauge factors G = G1 x G2 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]

Fate of dS in Fate of dS in compactifications?compactifications?

This year it was shown that one can build up gaugings of the form

G = G1 x G2

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 Fate of dS in compactifications?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]

groupcontractio

nsflux compac-tifications [2,3]

Higher-dimensional origin?Higher-dimensional origin?

10D stringtheory



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

Semi-direct product Semi-direct product gaugingsgaugings

“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 Semi-direct product gaugingsgaugings

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

fluxesVacua can be either AdS / Minkowski / dS!

SO(4) SO(3,1)ISO(3) SO(3) x U(1)3

Nilpotent U(1)6

[Zm;Zn]= QpmnZp

G+ n G ¡

The ISO(3) caseThe ISO(3) case

N=4

Gauge and geometric fluxes [1]

[1: Caviezel, Koerber, Kors, Lust, Wrase, Zagermann ‘08]

The SO(3,1) caseThe SO(3,1) case

N=4

Gauge and non-geometric fluxes

stable

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ConclusionsConclusions

Modern cosmology (CMB, SNe and BAO) involves inflation and dark energy

Link with fundamental physics: string cosmology.

ConclusionsConclusions 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!Thanks for your attention!

Diederik Roest December 3, 2009

24th Nordic Network Meeting

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