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

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

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

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:

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

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!

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

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:

RS Brane Cosmology: where = 0 Fine-Tuning

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

The Case of Co-dimension one Brane hep-th/0001197, hep-th/0001206 Consider the Following Action:

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

The equations of motion:

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

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

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

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

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

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

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

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.

* 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.

A fine tuning is still needed! hep-th/0002164 Here

Consider the Case:

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

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.

At two singular points: Two more boundary conditions: Here

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

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

FINE TUNING

The Case of Co-dimension two Brane hep-th/0302067, hep-th/0302129 hep-th/0309042, hep-th/0309050 Consider and action

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

Here A Static and Stable Solution is provided

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

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

This equation has the solution where

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

The geometry of extra two-dimensions

Finally with The brane is always flat for any tension.

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