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?
OutlineOutline
1.1. Modern cosmologyModern cosmology
2.2. Fundamental physicsFundamental physics
3.3. Flux compactificationsFlux compactifications
4.4. Moduli stabilisationModuli stabilisation
OutlineOutline
1.1. Modern cosmologyModern cosmology
2.2. Fundamental physicsFundamental physics
3.3. Flux compactificationsFlux compactifications
4.4. Moduli stabilisationModuli stabilisation
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?
OutlineOutline
1.1. Modern cosmologyModern cosmology
2.2. Fundamental physicsFundamental physics
3.3. Flux compactificationsFlux compactifications
4.4. Moduli stabilisationModuli stabilisation
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
OutlineOutline
1.1. Modern cosmologyModern cosmology
2.2. Fundamental physicsFundamental physics
3.3. Flux compactificationsFlux compactifications
4.4. Moduli stabilisationModuli stabilisation
CompactificationsCompactifications
“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.
CompactificationsCompactifications
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.
Gauged supergravityGauged 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)
Gauged supergravityGauged 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?
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;X n]= f mnpX p + HmnpZp ;
[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;X n]= f mnpX p + HmnpZp ;
[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
OutlineOutline
1.1. Modern cosmologyModern cosmology
2.2. Fundamental physicsFundamental physics
3.3. Flux compactificationsFlux compactifications
4.4. Moduli stabilisationModuli stabilisation
Higher-dimensional origin?Higher-dimensional origin?
10D stringtheory
4D gauged supergravity
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
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
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
OutlineOutline
1.1. Modern cosmologyModern cosmology
2.2. Fundamental physicsFundamental physics
3.3. Flux compactificationsFlux compactifications
4.4. Moduli stabilisationModuli stabilisation
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