1. The Cosmological Constant Problem & Self-tuning Mechanism
Rong-Gen Cai
Institute of Theoretical Physics
Chinese Academy of Sciences

2. The Cosmological Constant:
(A. Einstein, 1917)
The Static Universe; “Greatest Blunder”

3. The Old Cosmological Constant Problem:
Quantum Field Theory
Vacuum Energy Density
The Cosmological Constant
QUESTION: Why ?

4. 近年的天文观测支持: 暴涨模型⊕暗物质 ⊕ 暗能量 22% ⊕ 73% 挑战:暴涨模型 ? 暗物质 ? 暗能量 ?
Inflation Model: A. Guth, 1981
Dark Matter:
Dark Energy:

5. The New Cosmological Constant Problem:
QUESTION:
1) why ?
2) why ?
Dark Energy: Quintenssence ?

6. IF THE COSMOLOGICAL CONSTANT EXISTS:
Cosmological Event Horizon:
Entropy:
Finite Degrees of Freedom:
Consistent With String Theory?
T. Banks, 2000:
The Cosmological Constant is an Input
of the Fundamental Theory!

7. One Needs CRAZY Ideas To Solve Those Problems Including
the Cosmological Constant Problem
One Needs CRAZY Ideas
(M. S. Turner)

8. Brane World Scenario: RS1: RS2: N. Arkani-Hamed et al, 1998
factorizable product
2) L. Randall and R. Sundrum, 1999
warped product in AdS_5 y
RS1:
RS2:

9. RS Brane Cosmology:
where
= 0
Fine-Tuning

10. The Self-tuning Mechanism
The New Approach to
the Cosmological Constant Problem
in the Brane World Scenario
The Self-tuning Mechanism

11. The Case of Co-dimension one Brane
hep-th/ , hep-th/
Consider the Following Action:

12. To incorporate the effects of SM quantum loops,
one may consider the effective action:

13. The equations of motion:

14. Consider the following 5D metric
with Poincare symmetry:
And the SM matters:

15. The equations of motion in the bulk:
where
Consider the delta function source on the brane and
Z_2 symmetry, y ---> -y:

16. Key point:
With the variable , the equations of motion are completely independent of the effective potential V_extremal.

17. the de Sitter Symmetry and Anti-de Sitter Symmetry on the Brane
Recalling the conformal coupling
It Pohibits both
the de Sitter Symmetry and
Anti-de Sitter Symmetry
on the Brane
The Flat Domain Wall Solution is the Unique One,
for any Value of the Brane Tension

18. Some Remarks:
1) There is a naked curvature singularity at y

19. 2) Finite 4D Planck Scale
The zero mode tensor fluctuations correspond to
a massless 4D graviton with finite Planck scale

20. 3) Why it works The bulk action has a shift symmetry:
results in an associated conserved current:

21. However, the coupling to the brane tension breaks this symmetry.
The SM vacuum energy is converted into
a current emerging on the brane
and ending in the singularity region.

22. * when a=2b=3/4, the action agrees with tree level
More general coupling to the brane tension:
with
* when a=2b=3/4, the action agrees with tree level
string theory with phi as the dilaton.

23. A fine tuning is still needed!
hep-th/
Here

24. Consider the Case:

25. One Solution with Assumption:
are integration constants.
Here
1) Continuity at x_5=0 determines one of them, say, d_2.

26. The Solution Does Exist for any Value of V and b.
2) The condition on the first derivatives at x_5=0
determines c_1 and c_2:
The Solution Does Exist for
any Value of V and b.

27. At two singular points:
Two more boundary conditions:
Here

28. IF cutting off the fifth dimension by defining
The boundary conditions then reduce to:

29. The Brane Contributions to
the 4D Cosmological Constant:
As a result:

30. FINE TUNING

31. The Case of Co-dimension two Brane
hep-th/ , hep-th/
hep-th/ , hep-th/
Consider
and action

32. The Maxwell Field Has the Solution
The Einstein’s Equations:
The Stress-Energy Tensor:

33. Here
A Static and Stable Solution is
provided

34. Now Add Brane to the System
with
The Stress-Energy Tensor of Branes:

35. Rewrite the metric of two-dim. sphere
Two branes at r=0 and r= infinity.
obeys the following equation:

36. This equation has the solution
where

37. By a coordinate transformation, the solution becomes
where
and
This solution describes two-sphere, but a wedge
is removed and opposite sides are identified.

38. The geometry of extra two-dimensions

39. Finally
with
The brane is always flat for any tension.

40. THANK YOU