1. Dark Energy as a Manifestation of the Hierarchy
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:
Joint Davis-NTU Workshop, December 15-18, 2008

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 ± (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
~ !
MSM

4. Another Hierarchy in Physics
Gravity is much weaker, or Planck scale (1019 GeV),
much larger, than that of SM gauge interactions:
MPl
MSM ~
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. Analogy in Atomic Physics
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”

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 = vac (| |)  vac(a  ) ∞ a-4
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.
py can in principle be tuned away. px py

12. Strategy for the Smallness of Dark Energy
Supersymmetry
Brane World
= 0
vac:
boson
fermion +
only on the brane
BUT SUSY
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 Jμ , Yμ = Bessel functions and
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 that this 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 (~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.