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,
where
is the curvature radius of AdS5 and
is the corresponding energy scale. The curvature radius
determines the magnitude of the Riemann tensor:
The bulk cosmological constant acts to “squeeze” the gravitational field closer to the brane. We can see this
clearly in Gaussian normal coordinates
based on the brane at
, for which the AdS
5
metric takes the form
with
being the Minkowski metric. The exponential warp factor reflects the confining role of the bulk
cosmological constant. The Z2-symmetry about the brane at
is incorporated via the
term. In
the bulk, this metric is a solution of the 5D Einstein equations,
i.e.,
in Equation (2). The brane is a flat Minkowski spacetime,
, with
self-gravity in the form of brane tension. One can also use Poincare coordinates, which bring the metric into
manifestly conformally flat form,
where
.
The two RS models are distinguished as follows:
-
RS 2-brane:
- There are two branes in this model [359], at
and
, with Z2-symmetry
identifications
The branes have equal and opposite tensions
, where
The positive-tension brane has fundamental scale
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
is given by
So the RS 2-brane model gives a new approach to the hierarchy problem: even if
,
we can recover
by choosing
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 [155
]. 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, 408
, 335
, 283
].
-
RS 1-brane:
- In this model [358
], there is only one, positive tension, brane. It may be thought of as arising
from sending the negative tension brane off to infinity,
. Then the energy scales are related
via
The infinite extra dimension makes a finite contribution to the 5D volume because of the warp factor:
Thus the effective size of the extra dimension probed by the 5D graviton is
.
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 [129
, 342
, 375
, 193
, 386
, 390
, 290
, 347
, 180
]. 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.