2 Randall–Sundrum Brane-Worlds

RS brane-worlds do not rely on compactification to localize gravity at the brane, but on the curvature of the bulk (sometimes called “warped compactification”). What prevents gravity from ‘leaking’ into the extra dimension at low energies is a negative bulk cosmological constant,

6 2 Λ5 = − -2 = − 6μ , (19 ) ℓ
where ℓ is the curvature radius of AdS5 and μ is the corresponding energy scale. The curvature radius determines the magnitude of the Riemann tensor:
1 [ ] (5)RABCD = − -- (5)gAC(5)gBD − (5)gAD (5)gBC . (20 ) ℓ2
The bulk cosmological constant acts to “squeeze” the gravitational field closer to the brane. We can see this clearly in Gaussian normal coordinates XA = (xμ, y) based on the brane at y = 0, for which the AdS 5 metric takes the form
(5) 2 − 2|y|∕ℓ μ ν 2 ds = e ημνdx dx + dy , (21 )
with η μν being the Minkowski metric. The exponential warp factor reflects the confining role of the bulk cosmological constant. The Z2-symmetry about the brane at y = 0 is incorporated via the |y| term. In the bulk, this metric is a solution of the 5D Einstein equations,
(5)GAB = − Λ5(5)gAB, (22 )
i.e., (5)TAB = 0 in Equation (2View Equation). The brane is a flat Minkowski spacetime, gAB (xα,0) = ημνδμA δνB, with self-gravity in the form of brane tension. One can also use Poincare coordinates, which bring the metric into manifestly conformally flat form,
ℓ2 [ ] (5)ds2 = --- ημνdxμdx ν + dz2 , (23 ) z2
where z = ℓey∕ℓ.

The two RS models are distinguished as follows:

RS 2-brane:
There are two branes in this model [359], at y = 0 and y = L, with Z2-symmetry identifications
y ↔ − y, y + L ↔ L − y. (24)
The branes have equal and opposite tensions ± λ, where
3M 2p λ = 4π-ℓ2. (25)
The positive-tension brane has fundamental scale M5 and is “hidden”. Standard model fields are confined on the negative tension (or “visible”) brane. Because of the exponential warping factor, the effective Planck scale on the visible brane at y = L is given by
M 2= M 3ℓ[e2L∕ℓ − 1]. (26) p 5
So the RS 2-brane model gives a new approach to the hierarchy problem: even if M5 ∼ ℓ−1 ∼ TeV, we can recover M ∼ 1016 TeV p by choosing L ∕ℓ large enough. Because of the finite separation between the branes, the KK spectrum is discrete. Furthermore, at low energies gravity on the branes becomes Brans–Dicke-like, with the sign of the Brans–Dicke parameter equal to the sign of the brane tension [155Jump To The Next Citation Point]. In order to recover 4D general relativity at low energies, a mechanism is required to stabilize the inter-brane distance, which corresponds to a scalar field degree of freedom known as the radion [174, 408Jump To The Next Citation Point, 335Jump To The Next Citation Point, 283Jump To The Next Citation Point].
RS 1-brane:
In this model [358Jump To The Next Citation Point], there is only one, positive tension, brane. It may be thought of as arising from sending the negative tension brane off to infinity, L → ∞. Then the energy scales are related via
M 2 M 53= --p. (27) ℓ
The infinite extra dimension makes a finite contribution to the 5D volume because of the warp factor:
∫ ∘ ------ ∫ ∫ ∞ ℓ∫ d5X − (5)g = 2 d4x dye−4y∕ℓ = -- d4x. (28) 0 2
Thus the effective size of the extra dimension probed by the 5D graviton is ℓ.

We will concentrate mainly on RS 1-brane from now on, referring to RS 2-brane occasionally. The RS 1-brane models are in some sense the most simple and geometrically appealing form of a brane-world model, while at the same time providing a framework for the AdS/CFT correspondence [129Jump To The Next Citation Point, 342Jump To The Next Citation Point, 375Jump To The Next Citation Point, 193Jump To The Next Citation Point, 386Jump To The Next Citation Point, 390Jump To The Next Citation Point, 290Jump To The Next Citation Point, 347Jump To The Next Citation Point, 180Jump To The Next Citation Point]. The RS 2-brane introduce the added complication of radion stabilization, as well as possible complications arising from negative tension. However, they remain important and will occasionally be discussed.

 2.1 KK modes in RS 1-brane
 2.2 RS model in string theory

  Go to previous page Go up Go to next page