HUnT%20and%20Dark%20Energy

1
HUnT and Dark Energy

• Cosmophysics Group
• IPNS, KEK
• Hideo Kodama

2
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).

3
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)
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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.

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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(N4) ¾ so(3,2)
• no realistic dS superalgebra in four and five
  dimensions

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dS/AdS Real Simple Superalgebra
dS dS AdS AdS
D L G L G
4 osp(n1,1H) so(2n) osp(n-p,p4R) so(n-p,p)
osp(4,12nR) sp(n,R) osp(3,22nR) sp(n,R)
5 sl(2nH) su(2n) su(2,2n-p,p) u(n-p,p)
sl(22H) so(5,1) su(2,24-p,p) su(4-p,p)
osp(5,12nR) sp(n,R) osp(4,22n,R) sp(n,R)
QII(3) 1 QI(3) 1
6 osp(6,12nR) sp(n,R) osp(5,22nR) sp(n,R)
FIV(4) su(2) FIII(4) su(2)
7 osp(7,12nR) sp(nR) osp(6,22nR) sp(n,R)
osp(4n-p,pH) 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)
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Supergravities in Various Dimensions

• For D11, 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 D46 with N3, 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 D10, there exist two theories with N4
  IIA and IIB.
• Further, IIA theory has a massless version and a
  massive version with ? freedom. Romans LJ 1986
• For D8, there exist two theories with N4.
• Only for D46 with N1 or 2, there is the
  freedom in choices of gauge kinetic terms, scalar
  kinetic terms, and superpotential.

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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 quasi-degenerate
  vacua appear (Landscape problem).
• Cf. It is not known whether all
  lower-dimensional sugras can be obtained from
  D11 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

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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 four-dimensional 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 (n4)-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.

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Possible Solutions

• Non-compact compactification
• Braneworld model HW(1995)
• Cf. RS (1998), DGP(2000)
• Singular internal space with branes, flux and
  instantons KKLT (2003)
• Cf. OKKLT(2006)
• Dynamical internal space
• 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
• Gauss-Bonnet cosmology, R4 cosmology
• Heterotic flux compactification Becker2, Fu,
  Tseng, Yau 2006 Fu, Yau 2006 Kimura, Yi 2006
  Becker, Tseng, Yau 2006
• Non-perturbative quantum effects

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Historical Note

• 11D sugra model Cremmer, Julia, Scherk
• 1981 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.

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Historical Note (Cont)

• 1984 No-Go theorem for accelerated expansion.
  Gibbons
• M-theory
• Type II
• Type I

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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 H30 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

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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.
• 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 H30
  and anomaly cancelation.
• 1996-- 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.

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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

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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., F3H3F50, 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.

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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.
• ?gt0 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

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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 non-perturbative effects
  as well as geometry of extra-dimensions.