HUnT and Dark HUnT and Dark EnergyEnergy

Cosmophysics GroupCosmophysics Group

IPNS, KEKIPNS, KEK

Hideo KodamaHideo Kodama

Dark Energy ProblemDark Energy Problem

Provided that GR is valid on cosmic Provided that GR is valid on cosmic scales, the total dark energy density scales, the total dark energy density including quantum contributionsincluding quantum contributions is is – positive (positive (Acceleration ProblemAcceleration Problem),),– much smaller than typical characteristic much smaller than typical characteristic

scales of particle physics (scales of particle physics (Hierarchy/Hierarchy/ ProblemProblem), ),

– of the order of the present critical density of the order of the present critical density ((Coincidence ProblemCoincidence Problem). ).

Various ApproachesVarious Approaches

• Quantum GravityQuantum Gravity– Spacetime foams, EPI, baby universeSpacetime foams, EPI, baby universe

• Modification of GravityModification of Gravity– UV: string/M theory (UV: string/M theory (→ → brane(world), landscape)brane(world), landscape)– IR: Lorentz SSB, IR: Lorentz SSB, ff((RR,,rr)-models, TeVeS theory, )-models, TeVeS theory,

DGP modelDGP model

• Scalar Field ModelsScalar Field Models – Quintessence, K-essence, phantom field, Quintessence, K-essence, phantom field,

dilatonic ghost condensate, tachyon field(dilatonic ghost condensate, tachyon field(¾¾ Chaplygin gas), Chaplygin gas),

• Anthropic PrincipleAnthropic Principle

Ref: Copeland, Sami, Tsujikawa: IJMPD15, 1753(2006)

Requirements on the Basic Requirements on the Basic TheoryTheory

1.1. There is no freedom of adding a There is no freedom of adding a cosmological constant to the action.cosmological constant to the action.

2.2. Quantum corrections including zero-Quantum corrections including zero-point energies are under control.point energies are under control.

3.3. It is consistent with all low energy local It is consistent with all low energy local experiments and experiments and astrophysical/cosmological observations.astrophysical/cosmological observations.

4.4. It provides a natural unification of It provides a natural unification of gravity and other fundamental physical gravity and other fundamental physical laws. laws.

SupersymmetrySupersymmetry• Cancellation of UV divergences in the zero Cancellation of UV divergences in the zero

point energy:point energy:

• The vacuum energy of a supersymmetric The vacuum energy of a supersymmetric ground state is non-positive, but SUSY ground state is non-positive, but SUSY breaking adds positive energy:breaking adds positive energy:– Poincare superalgebraPoincare superalgebra

– AdS superalgebra: osp(N|4) AdS superalgebra: osp(N|4) ¾¾ so(3,2) so(3,2)

– no realistic dS superalgebra in four and five no realistic dS superalgebra in four and five dimensionsdimensions

dS/AdS Real Simple dS/AdS Real Simple SuperalgebraSuperalgebra

dSdS AdSAdSDD LL GG LL GG

44 osp(n|1,1;osp(n|1,1;HH)) so*(2n)so*(2n) osp(n-p,p|4;osp(n-p,p|4;RR)) so(n-p,p)so(n-p,p)

osp(4,1|2n;osp(4,1|2n;RR)) sp(n,sp(n,RR)) osp(3,2|2n;osp(3,2|2n;RR)) sp(n,sp(n,RR) )

55 sl(2|n;sl(2|n;HH)) su*(2n)su*(2n) su(2,2|n-p,p)su(2,2|n-p,p) u(n-p,p)u(n-p,p)

sl(2|2;sl(2|2;HH)) so(5,1)so(5,1) su(2,2|4-p,p)su(2,2|4-p,p) su(4-p,p)su(4-p,p)

osp(5,1|2n;osp(5,1|2n;RR)) sp(n,sp(n,RR)) osp(4,2|2n,osp(4,2|2n,RR)) sp(n,sp(n,RR))

QQIIII(3)(3) 11 QQII(3)(3) 11

66 osp(6,1|2n;osp(6,1|2n;RR)) sp(n,sp(n,RR)) osp(5,2|2n;osp(5,2|2n;RR)) sp(n,sp(n,RR))

FFIVIV(4)(4) su(2)su(2) FFIIIIII(4)(4) su(2)su(2)

77 osp(7,1|2n;osp(7,1|2n;RR)) sp(n;sp(n;RR)) osp(6,2|2n;osp(6,2|2n;RR)) sp(n,sp(n,RR))

osp(4|n-p,p;osp(4|n-p,p;HH)) sp(n-p,p)sp(n-p,p)

Parker M: JMP21, 689(1980); Fre P, Trigiante M, Van Proeyen A: CQG19, 4167 (2002); Lukierski J, Nowicki A: PLB151, 382(1985); Pilch K, van Nieuwenhuizen P, Sohnius F: CMP98,, 105(1985)

Supergravities in Various Supergravities in Various DimensionsDimensions

• For D=11, the sugra theory is unique up to 2For D=11, the sugra theory is unique up to 2ndnd order derivatives order derivatives (M theory). In particular, there is no freedom of (M theory). In particular, there is no freedom of .. [Cremmer, [Cremmer, Julia, Scherk 1978]Julia, Scherk 1978]

• For 7For 7·· D D·· 10 or D=4 10 or D=4»»6 with N6 with N¸̧3, sugra theories are almost 3, sugra theories are almost unique up to 2unique up to 2ndnd order derivatives for given D and N, apart from order derivatives for given D and N, apart from the gauge sector, if we require that the theory admits Minkowski the gauge sector, if we require that the theory admits Minkowski spacetime as a solution. spacetime as a solution. [Van Proeyen A: hep-th/0609048][Van Proeyen A: hep-th/0609048]

• Exceptions areExceptions are– For D=10, there exist two theories with N=4For D=10, there exist two theories with N=4 : IIA and IIB. : IIA and IIB. – Further, IIA theory has a massless version and a massive version with Further, IIA theory has a massless version and a massive version with

freedom. freedom. [Romans LJ 1986][Romans LJ 1986]– For D=8, there exist two theories with N=4.For D=8, there exist two theories with N=4.

• Only for D=4Only for D=4»»6 with N=1 or 26 with N=1 or 2, there is the freedom in choices of gauge kinetic terms, scalar kinetic terms, and superpotential.

Compactification and Compactification and LandscapeLandscape

• Vareity of 4D sugras may correspond to large degrees of freedom in compactification of the unique HUnT, including configurations of branes and flux of form fields.– Each compactification corresponds to some supergravity theory in

lower dimensions in general. – In particular, for flux compactification of IIB theory, an infinite

number of quasi-degenerate vacua appear (Landscape problem). – Cf. It is not known whether all lower-dimensional sugras can be

obtained from D=11 sugra by dimensional reductions. – Cf. After compactification, vacuum energies in higher dimensions

turn to potentials for moduli fields describing compactification in 4D.

• Any dS vacuum in the landscape cannot be supersymmetric and absolutely stable in 4D. – It can decay into the higher-dimensional flat solution or a stable

AdS vacuum. Sugra HUnT provides a quite thrilling landscape. [Linde hep-th/0611043; Clifton, Linde, Sivanandam hep-th/0701083]

No-Go TheoremNo-Go Theorem

ProofProofFor the geometryFor the geometry

from the relationfrom the relation

for any time-like unit vector for any time-like unit vector VV on on XX, we obtain, we obtain

Hence, if Hence, if YY is a compact manifold without boundary, is a compact manifold without boundary, h h -1-1 is a smooth function on is a smooth function on YY, and the strong energy , and the strong energy condition condition RRV V V V ¸̧ 0 is satisfied 0 is satisfied 　　 in the (n+4)-dimensional theory, in the (n+4)-dimensional theory, thenthenthe strong energy condition Rthe strong energy condition RV V V V (X) (X) ¸̧ 0 is satisfied on 0 is satisfied on X.X.

Hence, from the celebrated Raychaudhuri equationHence, from the celebrated Raychaudhuri equation

the accelerated expansion of the universe cannot the accelerated expansion of the universe cannot occur. occur.

For any (warped) compactification with a compact closed internal space, if the strong energy condition holds in the full theory and all moduli are stabilized, no stationary accelerating expansion of the four-dimensional spacetime is allowed.

Possible SolutionsPossible Solutions

1.1. Non-compact ‘compactification’Non-compact ‘compactification’• Braneworld model: HW(1995) Braneworld model: HW(1995)

Cf. RS (1998), DGP(2000)Cf. RS (1998), DGP(2000)• Singular internal space with branes, flux and instantons: Singular internal space with branes, flux and instantons:

KKLT (2003) KKLT (2003) Cf. O’KKLT(2006)Cf. O’KKLT(2006)

2.2. Dynamical internal spaceDynamical internal space• Negatively curved internal space Negatively curved internal space [Townsent, Wohlfarth 2003][Townsent, Wohlfarth 2003]

• S-brane solutions S-brane solutions [Chen, Galtsov, Guperle 2002; Ohta 2003][Chen, Galtsov, Guperle 2002; Ohta 2003]

3.3. String/M-theory effective action with higher-order String/M-theory effective action with higher-order correctionscorrections

• Gauss-Bonnet cosmology, RGauss-Bonnet cosmology, R44 cosmology cosmology• Heterotic flux compactification [Becker^2, Fu, Tseng, Yau Heterotic flux compactification [Becker^2, Fu, Tseng, Yau

2006; Fu, Yau 2006; Kimura, Yi 2006; Becker, Tseng, Yau 2006; Fu, Yau 2006; Kimura, Yi 2006; Becker, Tseng, Yau 2006]]2006]]

4.4. Non-perturbative quantum effectsNon-perturbative quantum effects

Historical NoteHistorical Note19781978 11D sugra model [Cremmer, Julia, Scherk]11D sugra model [Cremmer, Julia, Scherk]

1981 1981 Negative analysis for a KK unification by 11D Negative analysis for a KK unification by 11D sugra. [Witten]sugra. [Witten]

19831983 10D type I sugra model with SYM. [Chapline, Manton]10D type I sugra model with SYM. [Chapline, Manton]

-- No-Go theoremNo-Go theorem for the compactification of 10D type I for the compactification of 10D type I sugra to a Minksugra to a Mink44/dS/dS44/AdS/AdS44 . [Freedman, Gibbons, West] . [Freedman, Gibbons, West]

⇒　⇒　No non-trivial dimensional reductionNo non-trivial dimensional reduction..

Historical Note (Cont’)Historical Note (Cont’)

1984 1984 No-Go theoremNo-Go theorem for accelerated expansion. [Gibbons] for accelerated expansion. [Gibbons]

M-theoryM-theory

Type IIType II

Type IType I

Historical Note (Cont’)Historical Note (Cont’)19841984 Green-Schwarz mechanismGreen-Schwarz mechanism for the anomaly cancellation for the anomaly cancellation

in the 10D type I sugra.in the 10D type I sugra.

where where 33 is the Chern-Simons form: is the Chern-Simons form:

- - Realistic models by Calabi-Yau compactificationRealistic models by Calabi-Yau compactification of the 10D of the 10D heterotic SST to Minkheterotic SST to Mink4 4 with H with H33=0 and F=0 and F220 . 0 . [Candelas, [Candelas, Horowitz, Strominger, Witten 1985]Horowitz, Strominger, Witten 1985]

Embedding of the SU(3) holonomy of CY to the gauge field.Embedding of the SU(3) holonomy of CY to the gauge field.

This requires higher-order corrections of the formThis requires higher-order corrections of the form

Historical Note (Cont’)Historical Note (Cont’)19851985 The Gauss-Bonnet conjectureThe Gauss-Bonnet conjecture for the first leading correction for the first leading correction

O(RO(R22). [Zwiebach]). [Zwiebach]

Dynamics of the Gauss-Bonnet cosmologyDynamics of the Gauss-Bonnet cosmology– Flat and AdS solutions. The latter is unstable. [Boulware, Deser Flat and AdS solutions. The latter is unstable. [Boulware, Deser

1985]1985]– Transiently inflationary solutions with contracting internal space. Transiently inflationary solutions with contracting internal space.

The solutions are asymptotically Kasner. [Ishihara H 1986] The solutions are asymptotically Kasner. [Ishihara H 1986] – Vast work recently.Vast work recently.

1985- 1990 Calculations of higher-order corrections up to O(R1985- 1990 Calculations of higher-order corrections up to O(R44) for ) for bosonic string and superstring theories.bosonic string and superstring theories.

– No RNo R22 or R or R33 correction appears in Type II SST and M theory. correction appears in Type II SST and M theory.– RR22 correction is cancelled by the gauge contribution for CY correction is cancelled by the gauge contribution for CY

compactifications with Hcompactifications with H33=0 and anomaly cancelation.=0 and anomaly cancelation.

1996-- 1996-- Cosmology taking account of O(RCosmology taking account of O(R44) corrections. ) corrections. [Bento, Bertolami 1996; Maeda, Ohta 2004, 2005; Akune, Maeda, [Bento, Bertolami 1996; Maeda, Ohta 2004, 2005; Akune, Maeda, Ohta 2006; Elizalde et al 2007]Ohta 2006; Elizalde et al 2007]

– Some inflationary/DE solutions were found, but most models are Some inflationary/DE solutions were found, but most models are not realistic. not realistic.

Higher-Order Higher-Order CorrectionsCorrections

• How to calculate?How to calculate?– String S-matrix calculations: String S-matrix calculations: NNEE expansion expansion – -functions for 2D CFTs: -functions for 2D CFTs: ’ (’ (NNLL ) expansion. ) expansion.– Classification of all higher-order SUSY invariantsClassification of all higher-order SUSY invariants– Superspace approachSuperspace approach

• ProblemsProblems– Field redefinition ambiguities: X Field redefinition ambiguities: X !! X+ f(X) X+ f(X)– Terms proportional to on-shell equations cannot be Terms proportional to on-shell equations cannot be

determined. E.g. determined. E.g. RR, , RR. . This can be a serious problem for This can be a serious problem for cosmology.cosmology.

– Terms containing R-R fields are difficult to determine. Full Terms containing R-R fields are difficult to determine. Full susy completion, including the corrections for local susy susy completion, including the corrections for local susy transformations, is also very difficult. [de Roo, Suelmann, transformations, is also very difficult. [de Roo, Suelmann, Wiedemann 1993; Tseytlin 1996; Peeters, Vanhove, Wiedemann 1993; Tseytlin 1996; Peeters, Vanhove, Westerberg 2001, 2004]Westerberg 2001, 2004]

Moduli StabilisationModuli Stabilisation• Compactification satisfiying the field equations has large moduli Compactification satisfiying the field equations has large moduli

degrees of freedom describing the shape and size of the internal degrees of freedom describing the shape and size of the internal space in general.space in general.

• These moduli parameters determine the coupling constants among These moduli parameters determine the coupling constants among zero modes, i.e., particles at low energyies, including the zero modes, i.e., particles at low energyies, including the gravitational constant and gauge and higgs coupling constants.gravitational constant and gauge and higgs coupling constants.

• Further, if the action is independent of the moduli parameters, they Further, if the action is independent of the moduli parameters, they produce massless particles, whose existence contradicts produce massless particles, whose existence contradicts observations in general.observations in general.

• Hence, all moduli must acquire potentials thHence, all moduli must acquire potentials thatat fix their values at fix their values at sufficiently large energy scales.sufficiently large energy scales.

• Such moduli stabilisation is not easily realised. For example, if Such moduli stabilisation is not easily realised. For example, if there is no form flux, i.e., Fthere is no form flux, i.e., F33=H=H33=F=F55=0, the moduli for =0, the moduli for supersymmetric CY compactification of IIB theory have no supersymmetric CY compactification of IIB theory have no potential. potential.

• However, if 3-form flux does not vanish, all complex moduli are However, if 3-form flux does not vanish, all complex moduli are fixed in IIB theory.fixed in IIB theory.

Flux CompactificationFlux Compactification• IIB model (KKLT)IIB model (KKLT)

– There are supersymmetric compactifications with all moduli being There are supersymmetric compactifications with all moduli being stabilised. The vacuum energy for them is negative.stabilised. The vacuum energy for them is negative.

– Non-perturbative effects are required for the Kahler moduli Non-perturbative effects are required for the Kahler moduli stabilisation. Further, such stabilisation is realised only for special CYs.stabilisation. Further, such stabilisation is realised only for special CYs.

– 0 is realised by uplifting utilising anti D-branes, which are singular 0 is realised by uplifting utilising anti D-branes, which are singular objects in flux classically. This singular feature is essential to objects in flux classically. This singular feature is essential to circumvent the No-Go theorem.circumvent the No-Go theorem.

– To derive the SM at low energies, we have to live in a (low To derive the SM at low energies, we have to live in a (low dimensional) brane, but a consistent braneworld model in this dimensional) brane, but a consistent braneworld model in this framework has not been constructed. framework has not been constructed.

• Heterotic modelHeterotic model – There are CY compactifications with no flux giving MSSM with 3 There are CY compactifications with no flux giving MSSM with 3

generations and hidden sector susy breaking utilising CY instantons. generations and hidden sector susy breaking utilising CY instantons. [e.g. Bouchard, Donagi 2006][e.g. Bouchard, Donagi 2006]

– Higher-order corrections are essential for construction of a consistent Higher-order corrections are essential for construction of a consistent model. model.

– The internal space has to be non-Kahler, whose moduli structure is not The internal space has to be non-Kahler, whose moduli structure is not known in general. known in general. [Strominger 1986][Strominger 1986]

– Branes are not required in deriving the low energy SM, but the simple Branes are not required in deriving the low energy SM, but the simple KK reduction does not work due to the warp of the geometry. KK reduction does not work due to the warp of the geometry.

– There exists a consistent flux compactification with smooth internal There exists a consistent flux compactification with smooth internal space, but not all moduli may be fixed. space, but not all moduli may be fixed. [Becker, Tseng, Yau 2006][Becker, Tseng, Yau 2006]

SummarySummary

• Supersymmetric HUnTs are very natural Supersymmetric HUnTs are very natural candidates of the fundamental theory to resolve candidates of the fundamental theory to resolve the dark energy problem.the dark energy problem.

• However, they are hampered by two serious However, they are hampered by two serious problems: the moduli stabilisation problem and problems: the moduli stabilisation problem and the No-Go theorem against cosmic acceleration. the No-Go theorem against cosmic acceleration.

• At present, it appears that constructing a At present, it appears that constructing a realistic cosmological model is much more realistic cosmological model is much more difficult to derive MSSM in HUnT.difficult to derive MSSM in HUnT.

• In order to resolve these problems, we have to In order to resolve these problems, we have to learn more about the effect of higher-order learn more about the effect of higher-order quantum corrections and non-perturbative quantum corrections and non-perturbative effects as well as geometry of extra-dimensions. effects as well as geometry of extra-dimensions.