高次元統一理論とダークエネルギー

HUnT and Dark Energy
Cosmophysics Group
IPNS, KEK
Hideo Kodama
Dark Energy Problem
Provided that GR is valid on cosmic scales, the
total dark energy density including quantum
contributions is
– positive (Acceleration Problem),
– much smaller than typical characteristic scales of
particle physics (Hierarchy/ Problem),
– of the order of the present critical density
(Coincidence Problem).
Various Approaches
• Quantum Gravity
– Spacetime foams, EPI, baby universe
• Modification of Gravity
– UV: string/M theory (→ brane(world), landscape)
– IR: Lorentz SSB, f(R,,r)-models, TeVeS theory, DGP
model
• Scalar Field Models
– Quintessence, K-essence, phantom field, dilatonic ghost
condensate, tachyon field(¾ Chaplygin gas),
• Anthropic Principle
Ref: Copeland, Sami, Tsujikawa: IJMPD15, 1753(2006)
Requirements on the Basic Theory
1. There is no freedom of adding a cosmological
constant to the action.
2. Quantum corrections including zero-point
energies are under control.
3. It is consistent with all low energy local
experiments and astrophysical/cosmological
observations.
4. It provides a natural unification of gravity and
other fundamental physical laws.
Supersymmetry
• Cancellation of UV divergences in the zero point
energy:
• The vacuum energy of a supersymmetric ground state
is non-positive, but SUSY breaking adds positive
energy:
– Poincare superalgebra
– AdS superalgebra: osp(N|4) ¾ so(3,2)
– no realistic dS superalgebra in four and five dimensions
dS/AdS Real Simple Superalgebra
dS
AdS
D
L
4
osp(n|1,1;H)
so*(2n)
osp(n-p,p|4;R)
so(n-p,p)
osp(4,1|2n;R)
sp(n,R)
osp(3,2|2n;R)
sp(n,R)
sl(2|n;H)
su*(2n)
su(2,2|n-p,p)
u(n-p,p)
sl(2|2;H)
so(5,1)
su(2,2|4-p,p)
su(4-p,p)
osp(5,1|2n;R)
sp(n,R)
osp(4,2|2n,R)
sp(n,R)
QII(3)
1
QI(3)
1
osp(6,1|2n;R)
sp(n,R)
osp(5,2|2n;R)
sp(n,R)
FIV(4)
su(2)
FIII(4)
su(2)
osp(7,1|2n;R)
sp(n;R)
osp(6,2|2n;R)
sp(n,R)
osp(4|n-p,p;H)
sp(n-p,p)
5
6
7
G
L
G
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 Dimensions
• For D=11, the sugra theory is unique up to 2nd order derivatives (M theory).
In particular, there is no freedom of . [Cremmer, Julia, Scherk 1978]
• For 7· D· 10 or D=4»6 with N¸3, sugra theories are almost unique up to
2nd order derivatives for given D and N, apart from the gauge sector, if we
require that the theory admits Minkowski spacetime as a solution. [Van
Proeyen A: hep-th/0609048]
• Exceptions are
– For D=10, there exist two theories with N=4 : IIA and IIB.
– Further, IIA theory has a massless version and a massive version with  freedom.
[Romans LJ 1986]
– For D=8, there exist two theories with N=4.
• Only for D=4»6 with N=1 or 2, there is the freedom in choices of gauge
kinetic terms, scalar kinetic terms, and superpotential.
Compactification and Landscape
• 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 quasidegenerate 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 Theorem
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 fourdimensional spacetime is allowed.
Proof
For the geometry
from the relation
for any time-like unit vector V on X, we obtain
Hence, if Y is a compact manifold without boundary, h -1
is a smooth function on Y, and the strong energy condition
RV V ¸ 0 is satisfied in the (n+4)-dimensional theory, then
the strong energy condition RV V (X) ¸ 0 is satisfied on X.
Hence, from the celebrated Raychaudhuri equation
the accelerated expansion of the universe cannot occur.
Possible Solutions
1.
Non-compact ‘compactification’
•
•
2.
Dynamical internal space
•
•
3.
Negatively curved internal space [Townsent, Wohlfarth 2003]
S-brane solutions [Chen, Galtsov, Guperle 2002; Ohta 2003]
String/M-theory effective action with higher-order corrections
•
•
4.
Braneworld model: HW(1995)
Cf. RS (1998), DGP(2000)
Singular internal space with branes, flux and instantons: KKLT (2003)
Cf. O’KKLT(2006)
Gauss-Bonnet cosmology, R4 cosmology
Heterotic flux compactification [Becker^2, Fu, Tseng, Yau 2006; Fu,
Yau 2006; Kimura, Yi 2006; Becker, Tseng, Yau 2006]]
Non-perturbative quantum effects
Historical Note
1978
11D sugra model [Cremmer, Julia, Scherk]
1981
1983
Negative analysis for a KK unification by 11D sugra. [Witten]
10D type I sugra model with SYM. [Chapline, Manton]
-
No-Go theorem for the compactification of 10D type I sugra to a
Mink4/dS4/AdS4 . [Freedman, Gibbons, West]
⇒ No non-trivial dimensional reduction.
Historical Note (Cont’)
1984 No-Go theorem for accelerated expansion. [Gibbons]
M-theory
Type II
Type I
Historical Note (Cont’)
1984
Green-Schwarz mechanism for the anomaly cancellation in the 10D type I
sugra.
where 3 is the Chern-Simons form:
-
Realistic models by Calabi-Yau compactification of the 10D heterotic SST to
Mink4 with H3=0 and F20 . [Candelas, Horowitz, Strominger, Witten 1985]
Embedding of the SU(3) holonomy of CY to the gauge field.
This requires higher-order corrections of the form
Historical Note (Cont’)
1985
The Gauss-Bonnet conjecture for the first leading correction O(R2).
[Zwiebach]
Dynamics of the Gauss-Bonnet cosmology
–
–
–
Flat and AdS solutions. The latter is unstable. [Boulware, Deser 1985]
Transiently inflationary solutions with contracting internal space. The solutions
are asymptotically Kasner. [Ishihara H 1986]
Vast work recently.
1985- 1990 Calculations of higher-order corrections up to O(R4) for bosonic string and
superstring theories.
–
–
1996--
No R2 or R3 correction appears in Type II SST and M theory.
R2 correction is cancelled by the gauge contribution for CY compactifications
with H3=0 and anomaly cancelation.
Cosmology taking account of O(R4) corrections.
[Bento, Bertolami 1996; Maeda, Ohta 2004, 2005; Akune, Maeda, Ohta 2006;
Elizalde et al 2007]
–
Some inflationary/DE solutions were found, but most models are not realistic.
Higher-Order Corrections
• How to calculate?
–
–
–
–
String S-matrix calculations: NE expansion
 -functions for 2D CFTs: ’ (NL ) expansion.
Classification of all higher-order SUSY invariants
Superspace approach
• Problems
– Field redefinition ambiguities: X ! X+ f(X)
– Terms proportional to on-shell equations cannot be determined. E.g. R,
R . This can be a serious problem for cosmology.
– Terms containing R-R fields are difficult to determine. Full susy
completion, including the corrections for local susy transformations, is
also very difficult. [de Roo, Suelmann, Wiedemann 1993; Tseytlin 1996;
Peeters, Vanhove, Westerberg 2001, 2004]
Moduli Stabilisation
• Compactification satisfiying the field equations has large moduli degrees of
freedom describing the shape and size of the internal space in general.
• These moduli parameters determine the coupling constants among zero
modes, i.e., particles at low energyies, including the gravitational constant and
gauge and higgs coupling constants.
• Further, if the action is independent of the moduli parameters, they produce
massless particles, whose existence contradicts observations in general.
• Hence, all moduli must acquire potentials that fix their values at sufficiently
large energy scales.
• Such moduli stabilisation is not easily realised. For example, if there is no
form flux, i.e., F3=H3=F5=0, the moduli for supersymmetric CY
compactification of IIB theory have no potential.
• However, if 3-form flux does not vanish, all complex moduli are fixed in IIB
theory.
Flux Compactification
• IIB model (KKLT)
– There are supersymmetric compactifications with all moduli being stabilised. The
vacuum energy for them is negative.
– Non-perturbative effects are required for the Kahler moduli stabilisation. Further,
such stabilisation is realised only for special CYs.
– >0 is realised by uplifting utilising anti D-branes, which are singular objects in
flux classically. This singular feature is essential to circumvent the No-Go theorem.
– To derive the SM at low energies, we have to live in a (low dimensional) brane,
but a consistent braneworld model in this framework has not been constructed.
• Heterotic model
– There are CY compactifications with no flux giving MSSM with 3 generations and
hidden sector susy breaking utilising CY instantons. [e.g. Bouchard, Donagi 2006]
– Higher-order corrections are essential for construction of a consistent model.
– The internal space has to be non-Kahler, whose moduli structure is not known in
general. [Strominger 1986]
– 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.
– There exists a consistent flux compactification with smooth internal space, but
not all moduli may be fixed. [Becker, Tseng, Yau 2006]
Summary
• Supersymmetric HUnTs are very natural candidates of the
fundamental theory to resolve the dark energy problem.
• However, they are hampered by two serious problems: the
moduli stabilisation problem and the No-Go theorem against
cosmic acceleration.
• At present, it appears that constructing a realistic cosmological
model is much more difficult to derive MSSM in HUnT.
• In order to resolve these problems, we have to learn more about
the effect of higher-order quantum corrections and nonperturbative effects as well as geometry of extra-dimensions.

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