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Dark Energy as a Manifestation of the Hierarchy

Dark Energy as a Manifestation of the Hierarchy. Joint Davis-NTU Workshop, December 15-18, 2008 . Pisin Chen Department of Physics & Graduate Institute of Astrophysics & Leung Center for Cosmology and Particle Astrophysics National Taiwan University.

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Dark Energy as a Manifestation of the Hierarchy

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  1. Dark Energy as a Manifestation of the Hierarchy Joint Davis-NTU Workshop, December 15-18, 2008 Pisin Chen Department of Physics & Graduate Institute of Astrophysics & Leung Center for Cosmology and Particle Astrophysics National Taiwan University PC, Nucl. Phys. Proc. Suppl.173, 137 (2007). PC and J-A. Gu, Mod. Phys. Lett. A22, 1995 (2007); arXiv:0712.2441

  2. Where we Stand now SN Ia (SNLS, Higher-Z, Essence, low-Z stuff) + WMAP5 +BAO(SDSS)+ HST H0 Brian Schmidt’s talk If it looks like an apple and tastes like an apple, then it must be an apple! - PC Kowalski et al 08 w+ =1

  3. Smallness of Dark Energy • Combination of recent data from WMAP3 + SDSS • determines w = − 0.94 ± 0.09 • for dark energy (DE) equation of state p = wρ. • SNLS gives w = −1.023 ± 0.090 (stat) ± 0.54(sys). DE likely a bone fide CC: w = − 1. If DE never changes in space and time, then it must be associated with fundamental properties of spacetime. • Observations Why much smaller than standard model scale? ρDE1/4 ~ 10-15! MSM

  4. Another Hierarchy in Physics • Gravityis much weaker, or Planck scale (1019 GeV), • much larger, than that of SM gauge interactions: • MPl • MSM ~ 1016. Two well-known solutions: ADD : large (but flat) extra dimensions RS : warped geometry in x-d

  5. A Numerical Coincidence A remarkable numerical coincidence, Perhaps not accidental but implies a deeper connection: Caution: Unlike the 1st hierarchy that links 4 fundamental interaction strengths, DE must be a secondary, derived quantity.

  6. Bohr atom Fundamental energy scale in Schrödinger equation: me Ground state energy suppressed by 2 powers of fine structure constant Dark energy Fundamental energy scale in quantum gravity: MPl Dark energy suppressed by 2 powers of “gravity fine structure constant” Analogy in Atomic Physics

  7. Randall-Sundrum Warped Geometry Gravity lives in the bulk while gauge interactions live on the brane. dxν dxμ Visible brane Hidden brane Y=π Y=0

  8. Randall-Sundrum Warped Geometry Weakness of gravity = smallness of graviton wave function at visible brane dxμ dxν Hidden brane Visible brane MPl ~ 1016TeV MSM~TeV MSM = e-πka MPl ~ 10-16 k~MPl , e-kay Y=π Y=0

  9. Casimir Effect QED vacuum fluctuations

  10. Casimir Energy casimir = vac (||) vac(a  ) ∞ a-4 a (a  )

  11. Casimir Energy vs. Vacuum Energy • Casimir energy:px = ‒ρ, py > 0. • py cannot be tuned away. • Conventional vacuum energy (brane tension): • px = ‒ρ, py = 0. • pycan in principle be tuned away. px py

  12. Strategy for the Smallness of Dark Energy = 0 + vac: fermion boson only on the brane BUT SUSY Brane World Supersymmetry vac(4)~ (mn - mn-1)2mn2

  13. Casimir Energy in RS Geometry • Consider the following action in 5-d action • (Gherghatta & Pomarol, 2001, 2002): • where • Here is the anti- symmetric product of gamma matrices,

  14. Casimir Energy in RS Geometry • The gravitino SUSY transformation is given by • From the action, the bulk gravitino satisfies the 5D • Schwinger-Rarita equation in the AdS background, • Assume separation of variables, we KK-decompose • the 5D gravitino field as • where are defined as even (odd) under • Z2 parity.

  15. SUSY Solution • Solving the equation of motion, it can be shown that • (Gherghatta-Pomarol, 2001) the y-dependent gravitino • wavefunctions are • Jμ, Yμ= Bessel functions and • satisfies the boundary condition • Solving this equation, one finds the 4D KK gravitino mass as

  16. SUSY-Breaking by Higgs on the Brane • Now we invoke Higgs coupling to the gravitino on the TeV brane: • where is the Higgs field and f(n)= f(n)L+ f(n)R.

  17. It can be shown thatthis will induce a KK gravitino mass-shift for the nth mode • Graviton, on the other hand, remains massless (at tree-level).

  18. By definition, the Casimir energy under SUSY breaking is which is The KK mass shift can be shown to scale as • The SUSY-KK graviton/gravitino energy spectrum, on the other hand, goes like

  19. Casimir Energy under SUSY-Breaking Putting all these together, we find

  20. Summary • Dark energy may very well be a cosmological • constant. • The numerical coincidence between the SM-Planck • and the SM-DE hierarchies suggests a deeper • connection between the two. • This approach does not attempt to solve the “old” • CC problem, that is, the problem of 10120 (~Mpl4). • Assuming that this old CC problem will be resolved • someday, our model seems able to solve the “new” • CC problem, i.e., the smallness of CC as inferred • by observations.

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