The Cosmological Constant Problem & Self-tuning Mechanism

Rong-Gen Cai

Institute of Theoretical Physics Chinese Academy of Sciences

The Cosmological Constant:

18

2R g R g GT

(A. Einstein, 1917)

The Static Universe; “Greatest Blunder”

The Old Cosmological Constant Problem:

Quantum Field Theory Vacuum Energy Density The Cosmological Constant

4 3 4~ ( ) ~ (10 )SUSYE Gev

QUESTION: Why ?0

4 19 4~ ( ) ~ (10 )QGE Gev

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:

0 /p 1 1/ 3

The New Cosmological Constant Problem:

3 4 29 3~ ~ (10 ) ~ 10 /ev g cm

QUESTION: 1) why ? 2) why ?

0

~

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!

To Solve Those Problems Including

the Cosmological Constant Problem

One Needs CRAZY Ideas

(M. S. Turner)

Brane World Scenario:

y

X 1) N. Arkani-Hamed et al, 1998 factorizable product

2) L. Randall and R. Sundrum, 1999 warped product in AdS_5

4 x nM T

14 2

4

x S /

x R

M Z

M

RS1:

RS2:

RS Brane Cosmology:

2 242 3 44 5

8 4( ) ( )

3 3 3H

M M a

where

24 53 3

5 5

25

4 5

4 4( )

3

3( )

4

M M

MM M

= 0

Fine-Tuning

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.

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

sysy

2 ( )a 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:

0 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.

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: 0

One Solution with Assumption:

' '/ 3a 5 50, 0x x

,i ic dHere are integration constants.

1) Continuity at x_5=0 determines one of them, say, d_2.

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:

2 13 / 4, 3 / 4x c x c

Two more boundary conditions:

Here 1/ 3

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/0302129hep-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

This solution describes two-sphere, but a wedge is removed and opposite sides are identified.

By a coordinate transformation, the solution becomes

where

and

The geometry of extra two-dimensions

Finally

with

The brane is always flat for any tension.

THANK YOU