1. 1 Cosmological Constant as a Manifestation of the Hierarchy Pisin Chen Leung Center for Cosmology and Particle Astrophysics National Taiwan University & Kavli Institute for Particle Astrophysics and Cosmology Stanford University 3rd Biennial Leopoldina Conference on Dark Energy, Munich, Oct. 7-11, 2008 PC, Nucl. Phys. Proc. Suppl.173, 137 (2007). PC and J-A. Gu, Mod. Phys. Lett. A22, 1995 (2007); arXiv:0712.2441

2. 2 SN Ia (SNLS, Higher-Z, Essence, low-Z stuff) + WMAP5 +BAO(SDSS)+ HST H 0 Kowalski et al 08  w  +    =1 Where we Stand now Brian Schmidt’s talk If it looks like an apple and tastes like an apple, then it must be an apple! - PC

3. 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). Why much smaller than standard model scale? DE likely a bone fide CC: w = − 1. Observations ~ 10 -15 ! ρ DE 1/4 M SM If DE never changes in space and time, then it must be associated with fundamental properties of spacetime.

4. 4 Another Hierarchy in Physics Gravity is much weaker, or Planck scale (10 19 GeV), much larger, than that of SM gauge interactions: M Pl M SM Two well-known solutions: ADD : large (but flat) extra dimensions RS : warped geometry in x-d ~ 10 16.

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

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

7. 7 Y=0Y=π Visible braneHidden brane dx μ dx ν Gravity lives in the bulk while gauge interactions live on the brane.

8. 8 Casimir Effect QED vacuum fluctuations

9. 9 (a   ) a Casimir Energy  casimir =  vac ( | | )   vac (a   ) ∞ a -4

10. 10 Casimir Energy vs. Vacuum Energy pypy pxpx Casimir energy: p x = ‒ ρ, p y > 0. p y cannot be tuned away. Conventional vacuum energy (brane tension): p x = ‒ ρ, p y = 0. p y can in principle be tuned away.

11. 11 Strategy for the Smallness of Dark Energy Supersymmetry Brane World BUT SUSY only on the brane = 0  vac : boson fermion +  vac (4) ~ (m n - m n-1 ) 2  m n 2

12. 12 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

13. 13 Casimir Energy under SUSY-Breaking Putting all these together, we find

14. 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, i.e., the problem of 10 120. Assuming that this old CC problem will be resovled someday, our model seems able to solve the “new” CC problem, i.e., the smallness of CC as inferred by observations.