In the ADD models, more than one extra dimension is required for agreement with experiments, and there is “democracy” amongst the equivalent extra dimensions, which are typically flat. By contrast, the RS models have a “preferred” extra dimension, with other extra dimensions treated as ignorable (i.e., stabilized except at energies near the fundamental scale). Furthermore, this extra dimension is curved or “warped” rather than flat: The bulk is a portion of antide Sitter (
$Ad{S}_{5}$
) spacetime. As in the Horava–Witten solutions, the RS branes are
${Z}_{2}$
symmetric (mirror symmetry), and have a tension, which serves to counter the influence of the negative bulk cosmological constant on the brane. This also means that the selfgravity of the branes is incorporated in the RS models.
The novel feature of the RS models compared to previous higherdimensional models is that the observable 3 dimensions are protected from the large extra dimension (at low energies) by curvature rather than straightforward compactification.
The RS braneworlds and their generalizations (to include matter on the brane, scalar fields in the bulk, etc.) provide phenomenological models that reflect at least some of the features of M theory, and that bring exciting new geometric and particle physics ideas into play. The RS2 models also provide a framework for exploring holographic ideas that have emerged in M theory.
Roughly speaking, holography suggests that higherdimensional gravitational dynamics may be determined from knowledge of the fields on a lowerdimensional boundary. The AdS/CFT correspondence is an example, in which the classical dynamics of the higherdimensional gravitational field are equivalent to the quantum dynamics of a conformal field theory (CFT) on the boundary. The RS2 model with its
$Ad{S}_{5}$
metric satisfies this correspondence to lowest perturbative order [
87]
(see also [
254,
282,
136,
289,
293,
210,
259,
125]
for the AdS/CFT correspondence in a cosmological context).
In this review, I focus on RS braneworlds (mainly RS 1brane) and their generalizations, with the emphasis on geometry and gravitational dynamics (see [
219,
228,
189,
316,
260,
188,
267,
79,
36,
187,
190]
for previous reviews with a broadly similar approach). Other recent reviews focus on stringtheory aspects, e.g., [
101,
230,
68,
264]
, or on particle physics aspects, e.g., [
262,
273,
182,
104,
51]
. Before turning to a more detailed analysis of RS braneworlds, I discuss the notion of Kaluza–Klein (KK) modes of the graviton.
1.3 Heuristics of KK modes
The dilution of gravity via extra dimensions not only weakens gravity on the brane, it also extends the range of graviton modes felt on the brane beyond the massless mode of 4dimensional gravity. For simplicity, consider a flat brane with one flat extra dimension, compactified through the identification
$y\leftrightarrow y+2\pi nL$
, where
$n=0,1,2,...$
. The perturbative 5D graviton amplitude can be Fourier expanded as
$$\begin{array}{c}f({x}^{a},y)={\sum}_{n}{e}^{iny/L}{f}_{n}\left({x}^{a}\right),\end{array}$$ 
(8)

where
${f}_{n}$
are the amplitudes of the KK modes, i.e., the effective 4D modes of the 5D graviton.
To see that these KK modes are massive from the brane viewpoint, we start from the 5D wave equation that the massless 5D field
$f$
satisfies (in a suitable gauge):
$$\begin{array}{c}{}^{\left(5\right)}\square f=0\Rightarrow \square f+{\partial}_{y}^{2}f=0.\end{array}$$ 
(9)

It follows that the KK modes satisfy a 4D Klein–Gordon equation with an effective 4D mass
${m}_{n}$
,
$$\begin{array}{c}\square {f}_{n}={m}_{n}^{2}{f}_{n},{m}_{n}=\frac{n}{L}.\end{array}$$ 
(10)

The massless mode
${f}_{0}$
is the usual 4D graviton mode. But there is a tower of massive modes,
${L}^{1},2{L}^{1},...$
, which imprint the effect of the 5D gravitational field on the 4D brane. Compactness of the extra dimension leads to discreteness of the spectrum. For an infinite extra dimension,
$L\to \infty $
, the separation between the modes disappears and the tower forms a continuous spectrum.
In this case, the coupling of the KK modes to matter must be very weak in order to avoid exciting the lightest massive modes with
$m\gtrsim 0$
.
From a geometric viewpoint, the KK modes can also be understood via the fact that the projection of the null graviton 5momentum
${}^{\left(5\right)}{p}_{A}$
onto the brane is timelike. If the unit normal to the brane is
${n}_{A}$
, then the induced metric on the brane is
$$\begin{array}{c}{g}_{AB}{=}^{\left(5\right)}{g}_{AB}{n}_{A}{n}_{B}{,}^{\left(5\right)}{g}_{AB}{n}^{A}{n}^{B}=1,{g}_{AB}{n}^{B}=0,\end{array}$$ 
(11)

and the 5momentum may be decomposed as
$$\begin{array}{c}{}^{\left(5\right)}{p}_{A}=m{n}_{A}+{p}_{A},{p}_{A}{n}^{A}=0,m{=}^{\left(5\right)}{p}_{A}{n}^{A},\end{array}$$ 
(12)

where
${p}_{A}={{g}_{AB}}^{\left(5\right)}{p}^{B}$
is the projection along the brane, depending on the orientation of the 5momentum relative to the brane. The effective 4momentum of the 5D graviton is thus
${p}_{A}$
.
Expanding
${}^{\left(5\right)}{{g}_{AB}}^{\left(5\right)}{{p}^{A}}^{\left(5\right)}{p}^{B}=0$
, we find that
$$\begin{array}{c}{g}_{AB}{p}^{A}{p}^{B}={m}^{2}.\end{array}$$ 
(13)

It follows that the 5D graviton has an effective mass
$m$
on the brane. The usual 4D graviton corresponds to the zero mode,
$m=0$
, when
${}^{\left(5\right)}{p}_{A}$
is tangent to the brane.
The extra dimensions lead to new scalar and vector degrees of freedom on the brane. In 5D, the spin2 graviton is represented by a metric perturbation
${}^{\left(5\right)}{h}_{AB}$
that is transverse traceless:
$$\begin{array}{c}{}^{\left(5\right)}{h}_{A}^{A}=0={{\partial}_{B}}^{\left(5\right)}{{h}_{A}}^{B}.\end{array}$$ 
(14)

In a suitable gauge,
${}^{\left(5\right)}{h}_{AB}$
contains a 3D transverse traceless perturbation
${h}_{ij}$
, a 3D transverse vector perturbation
${\Sigma}_{i}$
, and a scalar perturbation
$\beta $
, which each satisfy the 5D wave equation [
88]
:
$$\begin{array}{ccc}& {}^{\left(5\right)}{h}_{AB}\u27f6{h}_{ij},{\Sigma}_{i},\beta ,& \end{array}$$ 
(15)

$$\begin{array}{ccc}& {h}_{i}^{i}=0={\partial}_{j}{h}^{ij},& \end{array}$$ 
(16)

$$\begin{array}{ccc}& {\partial}_{i}{\Sigma}^{i}=0,& \end{array}$$ 
(17)

$$\begin{array}{ccc}& (\square +{\partial}_{y}^{2})\left(\begin{array}{c}\beta \\ {\Sigma}_{i}\\ {h}_{ij}\end{array}\right)=0.& \end{array}$$ 
(18)

The other components of
${}^{\left(5\right)}{h}_{AB}$
are determined via constraints once these wave equations are solved. The 5 degrees of freedom (polarizations) in the 5D graviton are thus split into 2 (
${h}_{ij}$
) + 2 (
${\Sigma}_{i}$
) +1 (
$\beta $
) degrees of freedom in 4D. On the brane, the 5D graviton field is felt as

∙
a 4D spin2 graviton
${h}_{ij}$
(2 polarizations),

∙
a 4D spin1 gravivector (graviphoton)
${\Sigma}_{i}$
(2 polarizations), and

∙
a 4D spin0 graviscalar
$\beta $
.
The massive modes of the 5D graviton are represented via massive modes in all 3 of these fields on the brane. The standard 4D graviton corresponds to the massless zeromode of
${h}_{ij}$
.
In the general case of
$d$
extra dimensions, the number of degrees of freedom in the graviton follows from the irreducible tensor representations of the isometry group as
$\frac{1}{2}(d+1)(d+4)$
.
2 Randall–Sundrum BraneWorlds
RS braneworlds 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,
$$\begin{array}{c}{\Lambda}_{5}=\frac{6}{{\ell}^{2}}=6{\mu}^{2},\end{array}$$ 
(19)

where
$\ell $
is the curvature radius of
$Ad{S}_{5}$
and
$\mu $
is the corresponding energy scale. The curvature radius determines the magnitude of the Riemann tensor:
$$\begin{array}{c}{}^{\left(5\right)}{R}_{ABCD}=\frac{1}{{\ell}^{2}}\left[{}^{\left(5\right)}{{g}_{AC}}^{\left(5\right)}{g}_{BD}{}^{\left(5\right)}{{g}_{AD}}^{\left(5\right)}{g}_{BC}\right].\end{array}$$ 
(20)

The bulk cosmological constant acts to “squeeze” the gravitational field closer to the brane. We can see this clearly in Gaussian normal coordinates
${X}^{A}=({x}^{\mu},y)$
based on the brane at
$y=0$
, for which the
$Ad{S}_{5}$
metric takes the form
$$\begin{array}{c}{}^{\left(5\right)}d{s}^{2}={e}^{2\lefty\right/\ell}{\eta}_{\mu \nu}d{x}^{\mu}d{x}^{\nu}+d{y}^{2},\end{array}$$ 
(21)

with
${\eta}_{\mu \nu}$
being the Minkowski metric. The exponential warp factor reflects the confining role of the bulk cosmological constant. The
${Z}_{2}$
symmetry about the brane at
$y=0$
is incorporated via the
$\lefty\right$
term. In the bulk, this metric is a solution of the 5D Einstein equations,
$$\begin{array}{c}{}^{\left(5\right)}{G}_{AB}={{\Lambda}_{5}}^{\left(5\right)}{g}_{AB},\end{array}$$ 
(22)

i.e.,
${}^{\left(5\right)}{T}_{AB}=0$
in Equation ( 2 ). The brane is a flat Minkowski spacetime,
${g}_{AB}({x}^{\alpha},0)={\eta}_{\mu \nu}{\delta}_{A}^{\mu}{\delta}_{B}^{\nu}$
, with selfgravity in the form of brane tension. One can also use Poincare coordinates, which bring the metric into manifestly conformally flat form,
$$\begin{array}{c}{}^{\left(5\right)}d{s}^{2}=\frac{{\ell}^{2}}{{z}^{2}}\left[{\eta}_{\mu \nu}d{x}^{\mu}d{x}^{\nu}+d{z}^{2}\right],\end{array}$$ 
(23)

where
$z=\ell {e}^{y/\ell}$
.
The two RS models are distinguished as follows:

RS 2brane:
There are two branes in this model [266] , at
$y=0$
and
$y=L$
, with
${Z}_{2}$
symmetry identifications
$$\begin{array}{c}y\leftrightarrow y,y+L\leftrightarrow Ly.\end{array}$$ 
(24)

The branes have equal and opposite tensions
$\pm \lambda $
, where
$$\begin{array}{c}\lambda =\frac{3{M}_{p}^{2}}{4\pi {\ell}^{2}}.\end{array}$$ 
(25)

The positivetension brane has fundamental scale
${M}_{5}$
and is “hidden”. Standard model fields are confined on the negative tension (or “visible”) brane. Because of the exponential warping factor, the effective scale on the visible brane at
$y=L$
is
${M}_{p}$
, where
$$\begin{array}{c}{M}_{p}^{2}={M}_{5}^{3}\ell \left[1{e}^{2L/\ell}\right].\end{array}$$ 
(26)

So the RS 2brane model gives a new approach to the hierarchy problem. Because of the finite separation between the branes, the KK spectrum is discrete. Furthermore, at low energies gravity on the branes becomes Brans–Dickelike, with the sign of the Brans–Dicke parameter equal to the sign of the brane tension [106] . In order to recover 4D general relativity at low energies, a mechanism is required to stabilize the interbrane distance, which corresponds to a scalar field degree of freedom known as the radion [120, 305, 248, 202] .

RS 1brane:
In this model [265] , there is only one, positive tension, brane. It may be thought of as arising from sending the negative tension brane off to infinity,
$L\to \infty $
. Then the energy scales are related via
$$\begin{array}{c}{M}_{5}^{3}=\frac{{M}_{p}^{2}}{\ell}.\end{array}$$ 
(27)

The infinite extra dimension makes a finite contribution to the 5D volume because of the warp factor:
$$\begin{array}{c}\int {d}^{5}X\sqrt{{}^{\left(5\right)}g}=2\int {d}^{4}x{\int}_{0}^{\infty}dy{e}^{4y/\ell}=\frac{\ell}{2}\int {d}^{4}x.\end{array}$$ 
(28)

Thus the effective size of the extra dimension probed by the 5D graviton is
$\ell $
.
I will concentrate mainly on RS 1brane from now on, referring to RS 2brane occasionally. The RS 1brane models are in some sense the most simple and geometrically appealing form of a braneworld model, while at the same time providing a framework for the AdS/CFT correspondence [
87,
254,
282,
136,
289,
293,
210,
259,
125]
.
The RS 2brane 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.
In RS 1brane, the negative
${\Lambda}_{5}$
is offset by the positive brane tension
$\lambda $
. The finetuning in Equation ( 25 ) ensures that there is a zero effective cosmological constant on the brane, so that the brane has the induced geometry of Minkowski spacetime. To see how gravity is localized at low energies, we consider the 5D graviton perturbations of the metric [
265,
106,
117,
81]
,
$$\begin{array}{c}{}^{\left(5\right)}{g}_{AB}{\to}^{\left(5\right)}{g}_{AB}+{{e}^{2\lefty\right/\ell}}^{\left(5\right)}{h}_{AB}{,}^{\left(5\right)}{h}_{Ay}=0{=}^{\left(5\right)}{h}_{\mu}^{\mu}{=}^{\left(5\right)}{h}_{,\nu}^{\mu \nu}\end{array}$$ 
(29)

(see Figure 3 ). This is the RS gauge, which is different from the gauge used in Equation ( 15 ), but which also has no remaining gauge freedom. The 5 polarizations of the 5D graviton are contained in the 5 independent components of
${}^{\left(5\right)}{h}_{\mu \nu}$
in the RS gauge.
Figure 3
: Gravitational field of a small point particle on the brane in RS gauge. (Figure taken from [
106]
.)
We split the amplitude
$f$
of
${}^{\left(5\right)}{h}_{AB}$
into 3D Fourier modes, and the linearized 5D Einstein equations lead to the wave equation (
$y>0$
)
$$\begin{array}{c}\ddot{f}+{k}^{2}f={e}^{2y/\ell}\left[{f}^{\prime \prime}\frac{4}{\ell}{f}^{\prime}\right].\end{array}$$ 
(30)

Separability means that we can write
$$\begin{array}{c}f(t,y)={\sum}_{m}{\phi}_{m}\left(t\right){f}_{m}\left(y\right),\end{array}$$ 
(31)

and the wave equation reduces to
$$\begin{array}{ccc}{\ddot{\phi}}_{m}+({m}^{2}+{k}^{2}){\phi}_{m}& =& 0,\end{array}$$ 
(32)

$$\begin{array}{ccc}{f}_{m}^{\prime \prime}\frac{4}{\ell}{f}_{m}^{\prime}+{e}^{2y/\ell}{m}^{2}{f}_{m}& =& 0.\end{array}$$ 
(33)

The zero mode solution is
$$\begin{array}{ccc}{\phi}_{0}\left(t\right)& =& {A}_{0+}{e}^{+ikt}+{A}_{0}{e}^{ikt},\end{array}$$ 
(34)

$$\begin{array}{ccc}{f}_{0}\left(y\right)& =& {B}_{0}+{C}_{0}{e}^{4y/\ell},\end{array}$$ 
(35)

and the
$m>0$
solutions are
$$\begin{array}{ccc}{\phi}_{m}\left(t\right)& =& {A}_{m+}exp\left(+i\sqrt{{m}^{2}+{k}^{2}}t\right)+{A}_{m}exp\left(i\sqrt{{m}^{2}+{k}^{2}}t\right),\end{array}$$ 
(36)

$$\begin{array}{ccc}{f}_{m}\left(y\right)& =& {e}^{2y/\ell}\left[{B}_{m}{J}_{2}\left(m\ell {e}^{y/\ell}\right)+{C}_{m}{Y}_{2}\left(m\ell {e}^{y/\ell}\right)\right].\end{array}$$ 
(37)

The boundary condition for the perturbations arises from the junction conditions, Equation ( 62 ), discussed below, and leads to
${f}^{\prime}(t,0)=0$
, since the transverse traceless part of the perturbed energymomentum tensor on the brane vanishes. This implies
$$\begin{array}{c}{C}_{0}=0,{C}_{m}=\frac{{J}_{1}(m\ell )}{{Y}_{1}(m\ell )}{B}_{m}.\end{array}$$ 
(38)

The zero mode is normalizable, since
$$\begin{array}{c}\left{\int}_{0}^{\infty}{B}_{0}{e}^{2y/\ell}dy\right<\infty .\end{array}$$ 
(39)

Its contribution to the gravitational potential
$V={\frac{1}{2}}^{\left(5\right)}{h}_{00}$
gives the 4D result,
$V\propto {r}^{1}$
. The contribution of the massive KK modes sums to a correction of the 4D potential. For
$r\ll \ell $
, one obtains
$$\begin{array}{c}V\left(r\right)\approx \frac{GM\ell}{{r}^{2}},\end{array}$$ 
(40)

which simply reflects the fact that the potential becomes truly 5D on small scales. For
$r\gg \ell $
,
$$\begin{array}{c}V\left(r\right)\approx \frac{GM}{r}\left(1+\frac{2{\ell}^{2}}{3{r}^{2}}\right),\end{array}$$ 
(41)

which gives the small correction to 4D gravity at low energies from extradimensional effects.
These effects serve to slightly strengthen the gravitational field, as expected.
Tabletop tests of Newton's laws currently find no deviations down to
$\mathcal{O}({10}^{1}mm)$
, so that
$\ell \lesssim 0.1mm$
in Equation ( 41 ). Then by Equations ( 25 ) and ( 27 ), this leads to lower limits on the brane tension and the fundamental scale of the RS 1brane model:
$$\begin{array}{c}\lambda >(1TeV{)}^{4},{M}_{5}>{10}^{5}TeV.\end{array}$$ 
(42)

These limits do not apply to the 2brane case.
For the 1brane model, the boundary condition, Equation (
38 ), admits a continuous spectrum
$m>0$
of KK modes. In the 2brane model,
${f}^{\prime}(t,L)=0$
must hold in addition to Equation ( 38 ).
This leads to conditions on
$m$
, so that the KK spectrum is discrete:
$$\begin{array}{c}{m}_{n}=\frac{{x}_{n}}{\ell}{e}^{L/\ell}\text{where}{J}_{1}\left({x}_{n}\right)=\frac{{J}_{1}(m\ell )}{{Y}_{1}(m\ell )}{Y}_{1}\left({x}_{n}\right).\end{array}$$ 
(43)

The limit Equation ( 42 ) indicates that there are no observable collider, i.e.,
$\mathcal{O}(TeV)$
, signatures for the RS 1brane model. The 2brane model by contrast, for suitable choice of
$L$
and
$\ell $
so that
${m}_{1}=\mathcal{O}(TeV)$
, does predict collider signatures that are distinct from those of the ADD models [
132,
137]
.
3 Covariant Approach to BraneWorld Geometry and Dynamics
The RS models and the subsequent generalization from a Minkowski brane to a Friedmann–Robertson–Walker (FRW) brane [
27,
181,
155,
162,
128,
243,
149,
99,
105]
were derived as solutions in particular coordinates of the 5D Einstein equations, together with the junction conditions at the
${Z}_{2}$
symmetric brane. A broader perspective, with useful insights into the interplay between 4D and 5D effects, can be obtained via the covariant Shiromizu–Maeda–Sasaki approach [
291]
, in which the brane and bulk metrics remain general. The basic idea is to use the Gauss–Codazzi equations to project the 5D curvature along the brane. (The general formalism for relating the geometries of a spacetime and of hypersurfaces within that spacetime is given in [
315]
.) The 5D field equations determine the 5D curvature tensor; in the bulk, they are
$$\begin{array}{c}{}^{\left(5\right)}{G}_{AB}={{\Lambda}_{5}}^{\left(5\right)}{g}_{AB}+{{\kappa}_{5}^{2}}^{\left(5\right)}{T}_{AB},\end{array}$$ 
(44)

where
${}^{\left(5\right)}{T}_{AB}$
represents any 5D energymomentum of the gravitational sector (e.g., dilaton and moduli scalar fields, form fields).
Let
$y$
be a Gaussian normal coordinate orthogonal to the brane (which is at
$y=0$
without loss of generality), so that
${n}_{A}d{X}^{A}=dy$
, with
${n}^{A}$
being the unit normal. The 5D metric in terms of the induced metric on
$\{y=const.\}$
surfaces is locally given by
$$\begin{array}{c}{}^{\left(5\right)}{g}_{AB}={g}_{AB}+{n}_{A}{n}_{B}{,}^{\left(5\right)}d{s}^{2}={g}_{\mu \nu}({x}^{\alpha},y)d{x}^{\mu}d{x}^{\nu}+d{y}^{2}.\end{array}$$ 
(45)

The extrinsic curvature of
$\{y=const.\}$
surfaces describes the embedding of these surfaces. It can be defined via the Lie derivative or via the covariant derivative:
$$\begin{array}{c}{K}_{AB}=\frac{1}{2}{\text{\$}}_{\mathbf{n}}{g}_{AB}={{{g}_{A}}^{C}}^{\left(5\right)}{\nabla}_{C}{n}_{B},\end{array}$$ 
(46)

so that
$$\begin{array}{c}{K}_{\left[AB\right]}=0={K}_{AB}{n}^{B},\end{array}$$ 
(47)

where square brackets denote antisymmetrization. The Gauss equation gives the 4D curvature tensor in terms of the projection of the 5D curvature, with extrinsic curvature corrections:
$$\begin{array}{c}{R}_{ABCD}{=}^{\left(5\right)}{R}_{EFGH}{{g}_{A}}^{E}{{g}_{B}}^{F}{{g}_{C}}^{G}{{g}_{D}}^{H}+2{K}_{A[C}{K}_{D]B},\end{array}$$ 
(48)

and the Codazzi equation determines the change of
${K}_{AB}$
along
$\{y=const.\}$
via
$$\begin{array}{c}{\nabla}_{B}{K}_{A}^{B}{\nabla}_{A}K{=}^{\left(5\right)}{R}_{BC}{{g}_{A}}^{B}{n}^{C},\end{array}$$ 
(49)

where
$K={K}_{A}^{A}$
.
Some other useful projections of the 5D curvature are:
$$\begin{array}{ccc}{}^{\left(5\right)}{R}_{EFGH}{{g}_{A}}^{E}{{g}_{B}}^{F}{{g}_{C}}^{G}{n}^{H}& =& 2{\nabla}_{[A}{K}_{B]C},\end{array}$$ 
(50)

$$\begin{array}{ccc}{}^{\left(5\right)}{R}_{EFGH}{{g}_{A}}^{E}{n}^{F}{{g}_{B}}^{G}{n}^{H}& =& {\text{\$}}_{\mathbf{n}}{K}_{AB}+{K}_{AC}{K}_{B}^{C},\end{array}$$ 
(51)

$$\begin{array}{ccc}{}^{\left(5\right)}{R}_{CD}{{g}_{A}}^{C}{{g}_{B}}^{D}& =& {R}_{AB}{\text{\$}}_{\mathbf{n}}{K}_{AB}K{K}_{AB}+2{K}_{AC}{K}_{B}^{C}.\end{array}$$ 
(52)

The 5D curvature tensor has Weyl (tracefree) and Ricci parts:
$$\begin{array}{c}{}^{\left(5\right)}{R}_{ABCD}{=}^{\left(5\right)}{C}_{ACBD}+\frac{2}{3}\left({}^{\left(5\right)}{{g}_{A[C}}^{\left(5\right)}{R}_{D]B}{}^{\left(5\right)}{{g}_{B[C}}^{\left(5\right)}{R}_{D]A}\right){\frac{1}{6}}^{\left(5\right)}{{g}_{A[C}}^{\left(5\right)}{{g}_{D]B}}^{\left(5\right)}R.\end{array}$$ 
(53)

3.1 Field equations on the brane
Using Equations ( 44 ) and ( 48 ), it follows that
$$\begin{array}{ccc}{G}_{\mu \nu}& =& \frac{1}{2}{\Lambda}_{5}{g}_{\mu \nu}+\frac{2}{3}{\kappa}_{5}^{2}\left[{}^{\left(5\right)}{T}_{AB}{{g}_{\mu}}^{A}{{g}_{\nu}}^{B}+\left({}^{\left(5\right)}{T}_{AB}{n}^{A}{n}^{B}{\frac{1}{4}}^{\left(5\right)}T\right){g}_{\mu \nu}\right]\end{array}$$  
$$\begin{array}{ccc}& & +K{K}_{\mu \nu}{{K}_{\mu}}^{\alpha}{K}_{\alpha \nu}+\frac{1}{2}\left[{K}^{\alpha \beta}{K}_{\alpha \beta}{K}^{2}\right]{g}_{\mu \nu}{\mathcal{\mathcal{E}}}_{\mu \nu},\end{array}$$ 
(54)

where
${}^{\left(5\right)}T{=}^{\left(5\right)}{T}_{A}^{A}$
, and where
$$\begin{array}{c}{\mathcal{\mathcal{E}}}_{\mu \nu}{=}^{\left(5\right)}{C}_{ACBD}{n}^{C}{n}^{D}{{g}_{\mu}}^{A}{{g}_{\nu}}^{B},\end{array}$$ 
(55)

is the projection of the bulk Weyl tensor orthogonal to
${n}^{A}$
. This tensor satisfies
$$\begin{array}{c}{\mathcal{\mathcal{E}}}_{AB}{n}^{B}=0={\mathcal{\mathcal{E}}}_{\left[AB\right]}={{\mathcal{\mathcal{E}}}_{A}}^{A},\end{array}$$ 
(56)

by virtue of the Weyl tensor symmetries. Evaluating Equation ( 54 ) on the brane (strictly, as
$y\to \pm 0$
, since
${\mathcal{\mathcal{E}}}_{AB}$
is not defined on the brane [
291]
) will give the field equations on the brane.
First, we need to determine
${K}_{\mu \nu}$
at the brane from the junction conditions. The total energymomentum tensor on the brane is
$$\begin{array}{c}{T}_{\mu \nu}^{brane}={T}_{\mu \nu}\lambda {g}_{\mu \nu},\end{array}$$ 
(57)

where
${T}_{\mu \nu}$
is the energymomentum tensor of particles and fields confined to the brane (so that
${T}_{AB}{n}^{B}=0$
). The 5D field equations, including explicitly the contribution of the brane, are then
$$\begin{array}{c}{}^{\left(5\right)}{G}_{AB}={{\Lambda}_{5}}^{\left(5\right)}{g}_{AB}+{\kappa}_{5}^{2}\left[{}^{\left(5\right)}{T}_{AB}+{T}_{AB}^{brane}\delta \left(y\right)\right].\end{array}$$ 
(58)

Here the delta function enforces in the classical theory the string theory idea that Standard Model fields are confined to the brane. This is not a gravitational confinement, since there is in general a nonzero acceleration of particles normal to the brane [
218]
.
Integrating Equation (
58 ) along the extra dimension from
$y=\epsilon $
to
$y=+\epsilon $
, and taking the limit
$\epsilon \to 0$
, leads to the Israel–Darmois junction conditions at the brane,
$$\begin{array}{ccc}{g}_{\mu \nu}^{+}{g}_{\mu \nu}^{}& =& 0,\end{array}$$ 
(59)

$$\begin{array}{ccc}{K}_{\mu \nu}^{+}{K}_{\mu \nu}^{}& =& {\kappa}_{5}^{2}\left[{T}_{\mu \nu}^{brane}\frac{1}{3}{T}^{brane}{g}_{\mu \nu}\right],\end{array}$$ 
(60)

where
${T}^{brane}={g}^{\mu \nu}{T}_{\mu \nu}^{brane}$
. The
${Z}_{2}$
symmetry means that when you approach the brane from one side and go through it, you emerge into a bulk that looks the same, but with the normal reversed,
${n}^{A}\to {n}^{A}$
. Then Equation ( 46 ) implies that
$$\begin{array}{c}{K}_{\mu \nu}^{}={K}_{\mu \nu}^{+},\end{array}$$ 
(61)

so that we can use the junction condition Equation ( 60 ) to determine the extrinsic curvature on the brane:
$$\begin{array}{c}{K}_{\mu \nu}=\frac{1}{2}{\kappa}_{5}^{2}\left[{T}_{\mu \nu}+\frac{1}{3}\left(\lambda T\right){g}_{\mu \nu}\right],\end{array}$$ 
(62)

where
$T={T}_{\mu}^{\mu}$
, where we have dropped the
$(+)$
, and where we evaluate quantities on the brane by taking the limit
$y\to +0$
.
Finally we arrive at the induced field equations on the brane, by substituting Equation (
62 ) into Equation ( 54 ):
$$\begin{array}{c}{G}_{\mu \nu}=\Lambda {g}_{\mu \nu}+{\kappa}^{2}{T}_{\mu \nu}+6\frac{{\kappa}^{2}}{\lambda}{\mathcal{S}}_{\mu \nu}{\mathcal{\mathcal{E}}}_{\mu \nu}+4\frac{{\kappa}^{2}}{\lambda}{\mathcal{\mathcal{F}}}_{\mu \nu}.\end{array}$$ 
(63)

The 4D gravitational constant is an effective coupling constant inherited from the fundamental coupling constant, and the 4D cosmological constant is nonzero when the RS balance between the bulk cosmological constant and the brane tension is broken:
$$\begin{array}{ccc}{\kappa}^{2}& \equiv & {\kappa}_{4}^{2}=\frac{1}{6}\lambda {\kappa}_{5}^{4},\end{array}$$ 
(64)

$$\begin{array}{ccc}\Lambda & =& \frac{1}{2}\left[{\Lambda}_{5}+{\kappa}^{2}\lambda \right].\end{array}$$ 
(65)

The first correction term relative to Einstein's theory is quadratic in the energymomentum tensor, arising from the extrinsic curvature terms in the projected Einstein tensor:
$$\begin{array}{c}{\mathcal{S}}_{\mu \nu}=\frac{1}{12}T{T}_{\mu \nu}\frac{1}{4}{T}_{\mu \alpha}{T}_{\nu}^{\alpha}+\frac{1}{24}{g}_{\mu \nu}\left[3{T}_{\alpha \beta}{T}^{\alpha \beta}{T}^{2}\right].\end{array}$$ 
(66)

The second correction term is the projected Weyl term. The last correction term on the right of Equation ( 63 ), which generalizes the field equations in [
291]
, is
$$\begin{array}{c}{\mathcal{\mathcal{F}}}_{\mu \nu}{=}^{\left(5\right)}{T}_{AB}{{g}_{\mu}}^{A}{{g}_{\nu}}^{B}+\left[{}^{\left(5\right)}{T}_{AB}{n}^{A}{n}^{B}{\frac{1}{4}}^{\left(5\right)}T\right]{g}_{\mu \nu},\end{array}$$ 
(67)

where
${}^{\left(5\right)}{T}_{AB}$
describes any stresses in the bulk apart from the cosmological constant (see [
225]
for the case of a scalar field).
What about the conservation equations? Using Equations (
44 ), ( 49 ) and ( 62 ), one obtains
$$\begin{array}{c}{\nabla}^{\nu}{T}_{\mu \nu}={2}^{\left(5\right)}{T}_{AB}{n}^{A}{g}_{\mu}^{B}.\end{array}$$ 
(68)

Thus in general there is exchange of energymomentum between the bulk and the brane. From now on, we will assume that
$$\begin{array}{c}{}^{\left(5\right)}{T}_{AB}=0\Rightarrow {\mathcal{\mathcal{F}}}_{\mu \nu}=0,\end{array}$$ 
(69)

so that
$$\begin{array}{ccc}{}^{\left(5\right)}{G}_{AB}& =& {{\Lambda}_{5}}^{\left(5\right)}{g}_{AB}\text{in the bulk},\end{array}$$ 
(70)

$$\begin{array}{ccc}{G}_{\mu \nu}& =& \Lambda {g}_{\mu \nu}+{\kappa}^{2}{T}_{\mu \nu}+6\frac{{\kappa}^{2}}{\lambda}{\mathcal{S}}_{\mu \nu}{\mathcal{\mathcal{E}}}_{\mu \nu}\text{on the brane}.\end{array}$$ 
(71)

One then recovers from Equation ( 68 ) the standard 4D conservation equations,
$$\begin{array}{c}{\nabla}^{\nu}{T}_{\mu \nu}=0.\end{array}$$ 
(72)

This means that there is no exchange of energymomentum between the bulk and the brane; their interaction is purely gravitational. Then the 4D contracted Bianchi identities (
${\nabla}^{\nu}{G}_{\mu \nu}=0$
), applied to Equation ( 63 ), lead to
$$\begin{array}{c}{\nabla}^{\mu}{\mathcal{\mathcal{E}}}_{\mu \nu}=\frac{6{\kappa}^{2}}{\lambda}{\nabla}^{\mu}{\mathcal{S}}_{\mu \nu},\end{array}$$ 
(73)

which shows qualitatively how
$1+3$
spacetime variations in the matterradiation on the brane can source KK modes.
The induced field equations (
71 ) show two key modifications to the standard 4D Einstein field equations arising from extradimensional effects:

∙
${\mathcal{S}}_{\mu \nu}\sim ({T}_{\mu \nu}{)}^{2}$
is the highenergy correction term, which is negligible for
$\rho \ll \lambda $
, but dominant for
$\rho \gg \lambda $
:
$$\begin{array}{c}\frac{{\kappa}^{2}{\mathcal{S}}_{\mu \nu}/\lambda }{\left{\kappa}^{2}{T}_{\mu \nu}\right}\sim \frac{\left{T}_{\mu \nu}\right}{\lambda}\sim \frac{\rho}{\lambda}.\end{array}$$ 
(74)


∙
${\mathcal{\mathcal{E}}}_{\mu \nu}$
is the projection of the bulk Weyl tensor on the brane, and encodes corrections from 5D graviton effects (the KK modes in the linearized case). From the braneobserver viewpoint, the energymomentum corrections in
${\mathcal{S}}_{\mu \nu}$
are local, whereas the KK corrections in
${\mathcal{\mathcal{E}}}_{\mu \nu}$
are nonlocal, since they incorporate 5D gravity wave modes. These nonlocal corrections cannot be determined purely from data on the brane. In the perturbative analysis of RS 1brane which leads to the corrections in the gravitational potential, Equation ( 41 ), the KK modes that generate this correction are responsible for a nonzero
${\mathcal{\mathcal{E}}}_{\mu \nu}$
; this term is what carries the modification to the weakfield field equations. The 9 independent components in the tracefree
${\mathcal{\mathcal{E}}}_{\mu \nu}$
are reduced to 5 degrees of freedom by Equation ( 73 ); these arise from the 5 polarizations of the 5D graviton. Note that the covariant formalism applies also to the twobrane case. In that case, the gravitational influence of the second brane is felt via its contribution to
${\mathcal{\mathcal{E}}}_{\mu \nu}$
.
3.2 5dimensional equations and the initialvalue problem
The effective field equations are not a closed system. One needs to supplement them by 5D equations governing
${\mathcal{\mathcal{E}}}_{\mu \nu}$
, which are obtained from the 5D Einstein and Bianchi equations. This leads to the coupled system [
281]
$$\begin{array}{ccc}{\text{\$}}_{\mathbf{n}}{K}_{\mu \nu}& =& {K}_{\mu \alpha}{K}_{\nu}^{\alpha}{\mathcal{\mathcal{E}}}_{\mu \nu}\frac{1}{6}{\Lambda}_{5}{g}_{\mu \nu},\end{array}$$ 
(75)

$$\begin{array}{ccc}{\text{\$}}_{\mathbf{n}}{\mathcal{\mathcal{E}}}_{\mu \nu}& =& {\nabla}^{\alpha}{\mathcal{\mathcal{B}}}_{\alpha \left(\mu \nu \right)}+\frac{1}{6}{\Lambda}_{5}\left({K}_{\mu \nu}{g}_{\mu \nu}K\right)+{K}^{\alpha \beta}{R}_{\mu \alpha \nu \beta}+3{K}_{(\mu}^{\alpha}{\mathcal{\mathcal{E}}}_{\nu )\alpha}K{\mathcal{\mathcal{E}}}_{\mu \nu}\end{array}$$  
$$\begin{array}{ccc}& & +\left({K}_{\mu \alpha}{K}_{\nu \beta}{K}_{\alpha \beta}{K}_{\mu \nu}\right){K}^{\alpha \beta},\end{array}$$ 
(76)

$$\begin{array}{ccc}{\text{\$}}_{\mathbf{n}}{\mathcal{\mathcal{B}}}_{\mu \nu \alpha}& =& 2{\nabla}_{[\mu}{\mathcal{\mathcal{E}}}_{\nu ]\alpha}+{{K}_{\alpha}}^{\beta}{\mathcal{\mathcal{B}}}_{\mu \nu \beta}2{\mathcal{\mathcal{B}}}_{\alpha \beta [\mu}{{K}_{\nu ]}}^{\beta},\end{array}$$ 
(77)

$$\begin{array}{ccc}{\text{\$}}_{\mathbf{n}}{R}_{\mu \nu \alpha \beta}& =& 2{R}_{\mu \nu \gamma [\alpha}{{K}_{\beta ]}}^{\gamma}{\nabla}_{\mu}{\mathcal{\mathcal{B}}}_{\alpha \beta \nu}+{\nabla}_{\mu}{\mathcal{\mathcal{B}}}_{\beta \alpha \nu},\end{array}$$ 
(78)

where the “magnetic” part of the bulk Weyl tensor, counterpart to the “electric” part
${\mathcal{\mathcal{E}}}_{\mu \nu}$
, is
$$\begin{array}{c}{\mathcal{\mathcal{B}}}_{\mu \nu \alpha}={{g}_{\mu}}^{A}{{g}_{\nu}}^{B}{{{g}_{\alpha}}^{C}}^{\left(5\right)}{C}_{ABCD}{n}^{D}.\end{array}$$ 
(79)

These equations are to be solved subject to the boundary conditions at the brane,
$$\begin{array}{ccc}{\nabla}^{\mu}{\mathcal{\mathcal{E}}}_{\mu \nu}& =.& {\kappa}_{5}^{4}{\nabla}^{\mu}{\mathcal{S}}_{\mu \nu},\end{array}$$ 
(80)

$$\begin{array}{ccc}{\mathcal{\mathcal{B}}}_{\mu \nu \alpha}& =.& 2{\nabla}_{[\mu}{K}_{\nu ]\alpha}=.{\kappa}_{5}^{2}{\nabla}_{[\mu}\left({T}_{\nu ]\alpha}\frac{1}{3}{g}_{\nu ]\alpha}T\right),\end{array}$$ 
(81)

where
$A=.B$
denotes
$A{}_{brane}=B{}_{brane}$
.
The above equations have been used to develop a covariant analysis of the weak field [
281]
. They can also be used to develop a Taylor expansion of the metric about the brane. In Gaussian normal coordinates, Equation ( 45 ), we have
${\text{\$}}_{\mathbf{n}}=\partial /\partial y$
. Then we find
$$\begin{array}{ccc}{g}_{\mu \nu}(x,y)& =& {g}_{\mu \nu}(x,0){\kappa}_{5}^{2}{\left[{T}_{\mu \nu}+\frac{1}{3}(\lambda T){g}_{\mu \nu}\right]}_{y=0+}\lefty\right\end{array}$$  
$$\begin{array}{ccc}& & +{\left[{\mathcal{\mathcal{E}}}_{\mu \nu}+\frac{1}{4}{\kappa}_{5}^{4}\left({T}_{\mu \alpha}{T}_{\nu}^{\alpha}+\frac{2}{3}(\lambda T){T}_{\mu \nu}\right)+\frac{1}{6}\left(\frac{1}{6}{\kappa}_{5}^{4}(\lambda T{)}^{2}{\Lambda}_{5}\right){g}_{\mu \nu}\right]}_{y=0+}{y}^{2}+...\end{array}$$  
$$\begin{array}{ccc}& & \end{array}$$ 
(82)

In a noncovariant approach based on a specific form of the bulk metric in particular coordinates, the 5D Bianchi identities would be avoided and the equivalent problem would be one of solving the 5D field equations, subject to suitable 5D initial conditions and to the boundary conditions Equation ( 62 ) on the metric. The advantage of the covariant splitting of the field equations and Bianchi identities along and normal to the brane is the clear insight that it gives into the interplay between the 4D and 5D gravitational fields. The disadvantage is that the splitting is not well suited to dynamical evolution of the equations. Evolution off the timelike brane in the spacelike normal direction does not in general constitute a welldefined initial value problem [
8]
.
One needs to specify initial data on a 4D spacelike (or null) surface, with boundary conditions at the brane(s) ensuring a consistent evolution [
242,
147]
. Clearly the evolution of the observed universe is dependent upon initial conditions which are inaccessible to branebound observers; this is simply another aspect of the fact that the brane dynamics is not determined by 4D but by 5D equations. The initial conditions on a 4D surface could arise from models for creation of the 5D universe [
105,
178,
6,
31,
284]
, from dynamical attractor behaviour [
247]
or from suitable conditions (such as no incoming gravitational radiation) at the past Cauchy horizon if the bulk is asymptotically AdS.
3.3 The brane viewpoint: A 1 + 3covariant analysis
Following [
218]
, a systematic analysis can be developed from the viewpoint of a branebound observer. The effects of bulk gravity are conveyed, from a brane observer viewpoint, via the local (
${\mathcal{S}}_{\mu \nu}$
) and nonlocal (
${\mathcal{\mathcal{E}}}_{\mu \nu}$
) corrections to Einstein's equations. (In the more general case, bulk effects on the brane are also carried by
${\mathcal{\mathcal{F}}}_{\mu \nu}$
, which describes any 5D fields.) The
${\mathcal{\mathcal{E}}}_{\mu \nu}$
term cannot in general be determined from data on the brane, and the 5D equations above (or their equivalent) need to be solved in order to find
${\mathcal{\mathcal{E}}}_{\mu \nu}$
.
The general form of the brane energymomentum tensor for any matter fields (scalar fields, perfect fluids, kinetic gases, dissipative fluids, etc.), including a combination of different fields, can be covariantly given in terms of a chosen 4velocity
${u}^{\mu}$
as
$$\begin{array}{c}{T}_{\mu \nu}=\rho {u}_{\mu}{u}_{\nu}+p{h}_{\mu \nu}+{\pi}_{\mu \nu}+{q}_{\mu}{u}_{\nu}+{q}_{\nu}{u}_{\mu}.\end{array}$$ 
(83)

Here
$\rho $
and
$p$
are the energy density and isotropic pressure, respectively, and
$$\begin{array}{c}{h}_{\mu \nu}={g}_{\mu \nu}+{u}_{\mu}{u}_{\nu}{=}^{\left(5\right)}{g}_{\mu \nu}{n}_{\mu}{n}_{\nu}+{u}_{\mu}{u}_{\nu}\end{array}$$ 
(84)

projects into the comoving rest space orthogonal to
${u}^{\mu}$
on the brane. The momentum density and anisotropic stress obey
$$\begin{array}{c}{q}_{\mu}={q}_{\langle \mu \rangle},{\pi}_{\mu \nu}={\pi}_{\langle \mu \nu \rangle},\end{array}$$ 
(85)

where angled brackets denote the spatially projected, symmetric, and tracefree part:
$$\begin{array}{c}{V}_{\langle \mu \rangle}={{h}_{\mu}}^{\nu}{V}_{\nu},{W}_{\langle \mu \nu \rangle}=\left[{{h}_{(\mu}}^{\alpha}{{h}_{\nu )}}^{\beta}\frac{1}{3}{h}^{\alpha \beta}{h}_{\mu \nu}\right]{W}_{\alpha \beta}.\end{array}$$ 
(86)

In an inertial frame at any point on the brane, we have
$$\begin{array}{c}{u}^{\mu}=(1,\stackrel{\u20d7}{0}),{h}_{\mu \nu}=diag(0,1,1,1),{V}_{\mu}=(0,{V}_{i}),{W}_{\mu 0}=0=\sum {W}_{ii}={W}_{ij}{W}_{ji},\end{array}$$ 
(87)

where
$i,j=1,2,3$
.
The tensor
${\mathcal{S}}_{\mu \nu}$
, which carries local bulk effects onto the brane, may then be irreducibly decomposed as
$$\begin{array}{ccc}{\mathcal{S}}_{\mu \nu}& =& \frac{1}{24}\left[2{\rho}^{2}3{\pi}_{\alpha \beta}{\pi}^{\alpha \beta}\right]{u}_{\mu}{u}_{\nu}+\frac{1}{24}\left[2{\rho}^{2}+4\rho p+{\pi}_{\alpha \beta}{\pi}^{\alpha \beta}4{q}_{\alpha}{q}^{\alpha}\right]{h}_{\mu \nu}\end{array}$$  
$$\begin{array}{ccc}& & \frac{1}{12}(\rho +2p){\pi}_{\mu \nu}+{\pi}_{\alpha \langle \mu}{{\pi}_{\nu \rangle}}^{\alpha}+{q}_{\langle \mu}{q}_{\nu \rangle}+\frac{1}{3}\rho {q}_{(\mu}{u}_{\nu )}\frac{1}{12}{q}^{\alpha}{\pi}_{\alpha (\mu}{u}_{\nu )}.\end{array}$$ 
(88)

This simplifies for a perfect fluid or minimallycoupled scalar field to
$$\begin{array}{c}{\mathcal{S}}_{\mu \nu}=\frac{1}{12}\rho \left[\rho {u}_{\mu}{u}_{\nu}+\left(\rho +2p\right){h}_{\mu \nu}\right].\end{array}$$ 
(89)

The tracefree
${\mathcal{\mathcal{E}}}_{\mu \nu}$
carries nonlocal bulk effects onto the brane, and contributes an effective “dark” radiative energymomentum on the brane, with energy density
${\rho}_{\mathcal{\mathcal{E}}}$
, pressure
${\rho}_{\mathcal{\mathcal{E}}}/3$
, momentum density
${q}_{\mu}^{\mathcal{\mathcal{E}}}$
, and anisotropic stress
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
:
$$\begin{array}{c}\frac{1}{{\kappa}^{2}}{\mathcal{\mathcal{E}}}_{\mu \nu}={\rho}_{\mathcal{\mathcal{E}}}\left({u}_{\mu}{u}_{\nu}+\frac{1}{3}{h}_{\mu \nu}\right)+{q}_{\mu}^{\mathcal{\mathcal{E}}}{u}_{\nu}+{q}_{\nu}^{\mathcal{\mathcal{E}}}{u}_{\mu}+{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}.\end{array}$$ 
(90)

We can think of this as a KK or Weyl “fluid”. The brane “feels” the bulk gravitational field through this effective fluid. More specifically:

∙
The KK (or Weyl) anisotropic stress
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
incorporates the scalar or spin0 (“Coulomb”), the vector (transverse) or spin1 (gravimagnetic), and the tensor (transverse traceless) or spin2 (gravitational wave) 4D modes of the spin2 5D graviton.

∙
The KK momentum density
${q}_{\mu}^{\mathcal{\mathcal{E}}}$
incorporates spin0 and spin1 modes, and defines a velocity
${v}_{\mu}^{\mathcal{\mathcal{E}}}$
of the Weyl fluid relative to
${u}^{\mu}$
via
${q}_{\mu}^{\mathcal{\mathcal{E}}}={\rho}_{\mathcal{\mathcal{E}}}{v}_{\mu}^{\mathcal{\mathcal{E}}}$
.

∙
The KK energy density
${\rho}_{\mathcal{\mathcal{E}}}$
, often called the “dark radiation”, incorporates the spin0 mode.
In special cases, symmetry will impose simplifications on this tensor. For example, it must vanish for a conformally flat bulk, including
$Ad{S}_{5}$
,
$$\begin{array}{c}{}^{\left(5\right)}{g}_{AB}\text{conformally flat}\Rightarrow {\mathcal{\mathcal{E}}}_{\mu \nu}=0.\end{array}$$ 
(91)

The RS models have a Minkowski brane in an
$Ad{S}_{5}$
bulk. This bulk is also compatible with an FRW brane. However, the most general vacuum bulk with a Friedmann brane is Schwarzschildantide Sitter spacetime [
249,
32]
. Then it follows from the FRW symmetries that
$$\begin{array}{c}{\text{Schwarzschild AdS}}_{5}\text{bulk, FRW brane:}{q}_{\mu}^{\mathcal{\mathcal{E}}}=0={\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}},\end{array}$$ 
(92)

where
${\rho}_{\mathcal{\mathcal{E}}}=0$
only if the mass of the black hole in the bulk is zero. The presence of the bulk black hole generates via Coulomb effects the dark radiation on the brane.
For a static spherically symmetric brane (e.g., the exterior of a static star or black hole) [
72]
,
$$\begin{array}{c}\text{static spherical brane:}{q}_{\mu}^{\mathcal{\mathcal{E}}}=0.\end{array}$$ 
(93)

This condition also holds for a Bianchi I brane [
221]
. In these cases,
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
is not determined by the symmetries, but by the 5D field equations. By contrast, the symmetries of a Gödel brane fix
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
[
20]
.
The braneworld corrections can conveniently be consolidated into an effective total energy density, pressure, momentum density, and anisotropic stress:
$$\begin{array}{ccc}{\rho}_{\text{tot}}& =& \rho +\frac{1}{4\lambda}\left(2{\rho}^{2}3{\pi}_{\mu \nu}{\pi}^{\mu \nu}\right)+{\rho}_{\mathcal{\mathcal{E}}},\end{array}$$ 
(94)

$$\begin{array}{ccc}{p}_{\text{tot}}& =& p+\frac{1}{4\lambda}\left(2{\rho}^{2}+4\rho p+{\pi}_{\mu \nu}{\pi}^{\mu \nu}4{q}_{\mu}{q}^{\mu}\right)+\frac{{\rho}_{\mathcal{\mathcal{E}}}}{3},\end{array}$$ 
(95)

$$\begin{array}{ccc}{q}_{\mu}^{\text{tot}}& =& {q}_{\mu}+\frac{1}{2\lambda}\left(2\rho {q}_{\mu}3{\pi}_{\mu \nu}{q}^{\nu}\right)+{q}_{\mu}^{\mathcal{\mathcal{E}}},\end{array}$$ 
(96)

$$\begin{array}{ccc}{\pi}_{\mu \nu}^{\text{tot}}& =& {\pi}_{\mu \nu}+\frac{1}{2\lambda}\left[(\rho +3p){\pi}_{\mu \nu}+3{\pi}_{\alpha \langle \mu}{{\pi}_{\nu \rangle}}^{\alpha}+3{q}_{\langle \mu}{q}_{\nu \rangle}\right]+{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}.\end{array}$$ 
(97)

These general expressions simplify in the case of a perfect fluid (or minimally coupled scalar field, or isotropic oneparticle distribution function), i.e., for
${q}_{\mu}=0={\pi}_{\mu \nu}$
, to
$$\begin{array}{ccc}{\rho}_{\text{tot}}& =& \rho \left(1+\frac{\rho}{2\lambda}+\frac{{\rho}_{\mathcal{\mathcal{E}}}}{\rho}\right),\end{array}$$ 
(98)

$$\begin{array}{ccc}{p}_{\text{tot}}& =& p+\frac{\rho}{2\lambda}(2p+\rho )+\frac{{\rho}_{\mathcal{\mathcal{E}}}}{3},\end{array}$$ 
(99)

$$\begin{array}{ccc}{q}_{\mu}^{\text{tot}}& =& {q}_{\mu}^{\mathcal{\mathcal{E}}},\end{array}$$ 
(100)

$$\begin{array}{ccc}{\pi}_{\mu \nu}^{\text{tot}}& =& {\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}.\end{array}$$ 
(101)

Note that nonlocal bulk effects can contribute to effective imperfect fluid terms even when the matter on the brane has perfect fluid form: There is in general an effective momentum density and anisotropic stress induced on the brane by massive KK modes of the 5D graviton.
The effective total equation of state and sound speed follow from Equations (
98 ) and ( 99 ) as
$$\begin{array}{ccc}{w}_{\text{tot}}& \equiv & \frac{{p}_{\text{tot}}}{{\rho}_{\text{tot}}}=\frac{w+(1+2w)\rho /2\lambda +{\rho}_{\mathcal{\mathcal{E}}}/3\rho}{1+\rho /2\lambda +{\rho}_{\mathcal{\mathcal{E}}}/\rho},\end{array}$$ 
(102)

$$\begin{array}{ccc}{c}_{\text{tot}}^{2}& \equiv & \frac{{\dot{p}}_{\text{tot}}}{{\dot{\rho}}_{\text{tot}}}=\left[{c}_{s}^{2}+\frac{\rho +p}{\rho +\lambda}+\frac{4{\rho}_{\mathcal{\mathcal{E}}}}{9(\rho +p)(1+\rho /\lambda )}\right]{\left[1+\frac{4{\rho}_{\mathcal{\mathcal{E}}}}{3(\rho +p)(1+\rho /\lambda )}\right]}^{1},\end{array}$$ 
(103)

where
$w=p/\rho $
and
${c}_{s}^{2}=\dot{p}/\dot{\rho}$
. At very high energies, i.e.,
$\rho \gg \lambda $
, we can generally neglect
${\rho}_{\mathcal{\mathcal{E}}}$
(e.g., in an inflating cosmology), and the effective equation of state and sound speed are stiffened:
$$\begin{array}{c}{w}_{\text{tot}}\approx 2w+1,{c}_{\text{tot}}^{2}\approx {c}_{s}^{2}+w+1.\end{array}$$ 
(104)

This can have important consequences in the early universe and during gravitational collapse. For example, in a very highenergy radiation era,
$w=1/3$
, the effective cosmological equation of state is ultrastiff:
${w}_{\text{tot}}\approx 5/3$
. In latestage gravitational collapse of pressureless matter,
$w=0$
, the effective equation of state is stiff,
${w}_{\text{tot}}\approx 1$
, and the effective pressure is nonzero and dynamically important.
3.4 Conservation equations
Conservation of
${T}_{\mu \nu}$
gives the standard general relativity energy and momentum conservation equations, in the general, nonlinear case:
$$\begin{array}{ccc}\dot{\rho}+\Theta (\rho +p)+{\stackrel{\u20d7}{\nabla}}^{\mu}{q}_{\mu}+2{A}^{\mu}{q}_{\mu}+{\sigma}^{\mu \nu}{\pi}_{\mu \nu}& =& 0,\end{array}$$ 
(105)

$$\begin{array}{ccc}{\dot{q}}_{\langle \mu \rangle}+\frac{4}{3}\Theta {q}_{\mu}+{\stackrel{\u20d7}{\nabla}}_{\mu}p+(\rho +p){A}_{\mu}+{\stackrel{\u20d7}{\nabla}}^{\nu}{\pi}_{\mu \nu}+{A}^{\nu}{\pi}_{\mu \nu}+{\sigma}_{\mu \nu}{q}^{\nu}{\varepsilon}_{\mu \nu \alpha}{\omega}^{\nu}{q}^{\alpha}& =& 0.\end{array}$$ 
(106)

In these equations, an overdot denotes
${u}^{\nu}{\nabla}_{\nu}$
,
$\Theta ={\nabla}^{\mu}{u}_{\mu}$
is the volume expansion rate of the
${u}^{\mu}$
worldlines,
${A}_{\mu}={\dot{u}}_{\mu}={A}_{\langle \mu \rangle}$
is their 4acceleration,
${\sigma}_{\mu \nu}={\stackrel{\u20d7}{\nabla}}_{\langle \mu}{u}_{\nu \rangle}$
is their shear rate, and
${\omega}_{\mu}=\frac{1}{2}curl{u}_{\mu}={\omega}_{\langle \mu \rangle}$
is their vorticity rate.
On a Friedmann brane, we get
$$\begin{array}{c}{A}_{\mu}={\omega}_{\mu}={\sigma}_{\mu \nu}=0,\Theta =3H,\end{array}$$ 
(107)

where
$H=\dot{a}/a$
is the Hubble rate. The covariant spatial curl is given by
$$\begin{array}{c}curl{V}_{\mu}={\varepsilon}_{\mu \alpha \beta}{\stackrel{\u20d7}{\nabla}}^{\alpha}{V}^{\beta},curl{W}_{\mu \nu}={\varepsilon}_{\alpha \beta (\mu}{\stackrel{\u20d7}{\nabla}}^{\alpha}{W}_{\nu )}^{\beta},\end{array}$$ 
(108)

where
${\varepsilon}_{\mu \alpha \beta}$
is the projection orthogonal to
${u}^{\mu}$
of the 4D brane alternating tensor, and
${\stackrel{\u20d7}{\nabla}}_{\mu}$
is the projected part of the brane covariant derivative, defined by
$$\begin{array}{c}{\stackrel{\u20d7}{\nabla}}_{\mu}{F}_{...\beta}^{\alpha ...}={\left({\nabla}_{\mu}{F}_{...\beta}^{\alpha ...}\right)}_{\perp u}={{h}_{\mu}}^{\nu}{h}_{\gamma}^{\alpha}...{{h}_{\beta}}^{\delta}{\nabla}_{\nu}{F}_{...\delta}^{\gamma ...}.\end{array}$$ 
(109)

In a local inertial frame at a point on the brane, with
${u}^{\mu}={\delta}_{0}^{\mu}$
, we have:
$0={A}_{0}={\omega}_{0}={\sigma}_{0\mu}={\varepsilon}_{0\alpha \beta}=curl{V}_{0}=curl{W}_{0\mu}$
, and
$$\begin{array}{c}{\stackrel{\u20d7}{\nabla}}_{\mu}{F}_{...\beta}^{\alpha ...}={{\delta}_{\mu}}^{i}{\delta}_{j}^{\alpha}...{{\delta}_{\beta}}^{k}{\nabla}_{i}{F}_{...k}^{j...}\text{(local inertial frame)},\end{array}$$ 
(110)

where
$i,j,k=1,2,3$
.
The absence of bulk source terms in the conservation equations is a consequence of having
${\Lambda}_{5}$
as the only 5D source in the bulk. For example, if there is a bulk scalar field, then there is energymomentum exchange between the brane and bulk (in addition to the gravitational interaction) [
225,
16,
236,
97,
194,
98,
35]
.
Equation (
73 ) may be called the “nonlocal conservation equation”. Projecting along
${u}^{\mu}$
gives the nonlocal energy conservation equation, which is a propagation equation for
${\rho}_{\mathcal{\mathcal{E}}}$
. In the general, nonlinear case, this gives
$$\begin{array}{ccc}& & {\dot{\rho}}_{\mathcal{\mathcal{E}}}+\frac{4}{3}\Theta {\rho}_{\mathcal{\mathcal{E}}}+{\stackrel{\u20d7}{\nabla}}^{\mu}{q}_{\mu}^{\mathcal{\mathcal{E}}}+2{A}^{\mu}{q}_{\mu}^{\mathcal{\mathcal{E}}}+{\sigma}^{\mu \nu}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}=\end{array}$$  
$$\begin{array}{ccc}& & \frac{1}{4\lambda}[6{\pi}^{\mu \nu}{\dot{\pi}}_{\mu \nu}+6(\rho +p){\sigma}^{\mu \nu}{\pi}_{\mu \nu}+2\Theta \left(2{q}^{\mu}{q}_{\mu}+{\pi}^{\mu \nu}{\pi}_{\mu \nu}\right)+2{A}^{\mu}{q}^{\nu}{\pi}_{\mu \nu}\end{array}$$  
$$\begin{array}{ccc}& & 4{q}^{\mu}{\stackrel{\u20d7}{\nabla}}_{\mu}\rho +{q}^{\mu}{\stackrel{\u20d7}{\nabla}}^{\nu}{\pi}_{\mu \nu}+{\pi}^{\mu \nu}{\stackrel{\u20d7}{\nabla}}_{\mu}{q}_{\nu}2{\sigma}^{\mu \nu}{\pi}_{\alpha \mu}{{\pi}_{\nu}}^{\alpha}2{\sigma}^{\mu \nu}{q}_{\mu}{q}_{\nu}].\end{array}$$ 
(111)

Projecting into the comoving rest space gives the nonlocal momentum conservation equation, which is a propagation equation for
${q}_{\mu}^{\mathcal{\mathcal{E}}}$
:
$$\begin{array}{ccc}& & {\dot{q}}_{\langle \mu \rangle}^{\mathcal{\mathcal{E}}}+\frac{4}{3}\Theta {q}_{\mu}^{\mathcal{\mathcal{E}}}+\frac{1}{3}{\stackrel{\u20d7}{\nabla}}_{\mu}{\rho}_{\mathcal{\mathcal{E}}}+\frac{4}{3}{\rho}_{\mathcal{\mathcal{E}}}{A}_{\mu}+{\stackrel{\u20d7}{\nabla}}^{\nu}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}+{A}^{\nu}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}+{{\sigma}_{\mu}}^{\nu}{q}_{\nu}^{\mathcal{\mathcal{E}}}{{\varepsilon}_{\mu}}^{\nu \alpha}{\omega}_{\nu}{q}_{\alpha}^{\mathcal{\mathcal{E}}}=\end{array}$$  
$$\begin{array}{ccc}& & \frac{1}{4\lambda}[4(\rho +p){\stackrel{\u20d7}{\nabla}}_{\mu}\rho +6(\rho +p){\stackrel{\u20d7}{\nabla}}^{\nu}{\pi}_{\mu \nu}+{q}^{\nu}{\dot{\pi}}_{\langle \mu \nu \rangle}+{{\pi}_{\mu}}^{\nu}{\stackrel{\u20d7}{\nabla}}_{\nu}(2\rho +5p)\end{array}$$  
$$\begin{array}{ccc}& & \frac{2}{3}{\pi}^{\alpha \beta}\left({\stackrel{\u20d7}{\nabla}}_{\mu}{\pi}_{\alpha \beta}+3{\stackrel{\u20d7}{\nabla}}_{\alpha}{\pi}_{\beta \mu}\right)3{\pi}_{\mu \alpha}{\stackrel{\u20d7}{\nabla}}_{\beta}{\pi}^{\alpha \beta}+\frac{28}{3}{q}^{\nu}{\stackrel{\u20d7}{\nabla}}_{\mu}{q}_{\nu}\end{array}$$  
$$\begin{array}{ccc}& & +4\rho {A}^{\nu}{\pi}_{\mu \nu}3{\pi}_{\mu \alpha}{A}_{\beta}{\pi}^{\alpha \beta}+\frac{8}{3}{A}_{\mu}{\pi}^{\alpha \beta}{\pi}_{\alpha \beta}{\pi}_{\mu \alpha}{\sigma}^{\alpha \beta}{q}_{\beta}\end{array}$$  
$$\begin{array}{ccc}& & +{\sigma}_{\mu \alpha}{\pi}^{\alpha \beta}{q}_{\beta}+{\pi}_{\mu \nu}{\varepsilon}^{\nu \alpha \beta}{\omega}_{\alpha}{q}_{\beta}{\varepsilon}_{\mu \alpha \beta}{\omega}^{\alpha}{\pi}^{\beta \nu}{q}_{\nu}+4(\rho +p)\Theta {q}_{\mu}\end{array}$$  
$$\begin{array}{ccc}& & +6{q}_{\mu}{A}^{\nu}{q}_{\nu}+\frac{14}{3}{A}_{\mu}{q}^{\nu}{q}_{\nu}+4{q}_{\mu}{\sigma}^{\alpha \beta}{\pi}_{\alpha \beta}].\end{array}$$ 
(112)

The
$1+3$
covariant decomposition shows two key features:

∙
Inhomogeneous and anisotropic effects from the 4D matterradiation distribution on the brane are a source for the 5D Weyl tensor, which nonlocally “backreacts” on the brane via its projection
${\mathcal{\mathcal{E}}}_{\mu \nu}$
.

∙
There are evolution equations for the dark radiative (nonlocal, Weyl) energy (
${\rho}_{\mathcal{\mathcal{E}}}$
) and momentum (
${q}_{\mu}^{\mathcal{\mathcal{E}}}$
) densities (carrying scalar and vector modes from bulk gravitons), but there is no evolution equation for the dark radiative anisotropic stress (
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
) (carrying tensor, as well as scalar and vector, modes), which arises in both evolution equations.
In particular cases, the Weyl anisotropic stress
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
may drop out of the nonlocal conservation equations, i.e., when we can neglect
${\sigma}^{\mu \nu}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
,
${\stackrel{\u20d7}{\nabla}}^{\nu}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
, and
${A}^{\nu}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
. This is the case when we consider linearized perturbations about an FRW background (which remove the first and last of these terms) and further when we can neglect gradient terms on large scales (which removes the second term).
This case is discussed in Section
6 . But in general, and especially in astrophysical contexts, the
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
terms cannot be neglected. Even when we can neglect these terms,
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
arises in the field equations on the brane.
All of the matter source terms on the right of these two equations, except for the first term on the right of Equation (
112 ), are imperfect fluid terms, and most of these terms are quadratic in the imperfect quantities
${q}_{\mu}$
and
${\pi}_{\mu \nu}$
. For a single perfect fluid or scalar field, only the
${\stackrel{\u20d7}{\nabla}}_{\mu}\rho $
term on the right of Equation ( 112 ) survives, but in realistic cosmological and astrophysical models, further terms will survive. For example, terms linear in
${\pi}_{\mu \nu}$
will carry the photon quadrupole in cosmology or the shear viscous stress in stellar models. If there are two fluids (even if both fluids are perfect), then there will be a relative velocity
${v}_{\mu}$
generating a momentum density
${q}_{\mu}=\rho {v}_{\mu}$
, which will serve to source nonlocal effects.
In general, the 4 independent equations in Equations (
111 ) and ( 112 ) constrain 4 of the 9 independent components of
${\mathcal{\mathcal{E}}}_{\mu \nu}$
on the brane. What is missing is an evolution equation for
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
, which has up to 5 independent components. These 5 degrees of freedom correspond to the 5 polarizations of the 5D graviton. Thus in general, the projection of the 5dimensional field equations onto the brane does not lead to a closed system, as expected, since there are bulk degrees of freedom whose impact on the brane cannot be predicted by brane observers. The KK anisotropic stress
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
encodes the nonlocality.
In special cases the missing equation does not matter. For example, if
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}=0$
by symmetry, as in the case of an FRW brane, then the evolution of
${\mathcal{\mathcal{E}}}_{\mu \nu}$
is determined by Equations ( 111 ) and ( 112 ). If the brane is stationary (with Killing vector parallel to
${u}^{\mu}$
), then evolution equations are not needed for
${\mathcal{\mathcal{E}}}_{\mu \nu}$
, although in general
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
will still be undetermined. However, small perturbations of these special cases will immediately restore the problem of missing information.
If the matter on the brane has a perfectfluid or scalarfield energymomentum tensor, the local conservation equations (
105 ) and ( 106 ) reduce to
$$\begin{array}{ccc}\dot{\rho}+\Theta (\rho +p)& =& 0,\end{array}$$ 
(113)

$$\begin{array}{ccc}{\stackrel{\u20d7}{\nabla}}_{\mu}p+(\rho +p){A}_{\mu}& =& 0,\end{array}$$ 
(114)

while the nonlocal conservation equations ( 111 ) and ( 112 ) reduce to
$$\begin{array}{ccc}\dot{{\rho}_{\mathcal{\mathcal{E}}}}+\frac{4}{3}\Theta {\rho}_{\mathcal{\mathcal{E}}}+{\stackrel{\u20d7}{\nabla}}^{\mu}{q}_{\mu}^{\mathcal{\mathcal{E}}}+2{A}^{\mu}{q}_{\mu}^{\mathcal{\mathcal{E}}}+{\sigma}^{\mu \nu}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}& =& 0,\end{array}$$ 
(115)

$$\begin{array}{ccc}{\dot{q}}_{\langle \mu \rangle}^{\mathcal{\mathcal{E}}}+\frac{4}{3}\Theta {q}_{\mu}^{\mathcal{\mathcal{E}}}+\frac{1}{3}{\stackrel{\u20d7}{\nabla}}_{\mu}{\rho}_{\mathcal{\mathcal{E}}}+\frac{4}{3}{\rho}_{\mathcal{\mathcal{E}}}{A}_{\mu}+{\stackrel{\u20d7}{\nabla}}^{\nu}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}+{A}^{\nu}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}+{{\sigma}_{\mu}}^{\nu}{q}_{\nu}^{\mathcal{\mathcal{E}}}{{\varepsilon}_{\mu}}^{\nu \alpha}{\omega}_{\nu}{q}_{\alpha}^{\mathcal{\mathcal{E}}}& =& \frac{(\rho +p)}{\lambda}{\stackrel{\u20d7}{\nabla}}_{\mu}\rho .\end{array}$$ 
(116)

Equation ( 116 ) shows that [
291]

∙
if
${\mathcal{\mathcal{E}}}_{\mu \nu}=0$
and the brane energymomentum tensor has perfect fluid form, then the density
$\rho $
must be homogeneous,
${\stackrel{\u20d7}{\nabla}}_{\mu}\rho =0$
;

∙
the converse does not hold, i.e., homogeneous density does not in general imply vanishing
${\mathcal{\mathcal{E}}}_{\mu \nu}$
.
A simple example of the latter point is the FRW case: Equation ( 116 ) is trivially satisfied, while Equation ( 115 ) becomes
$$\begin{array}{c}{\dot{\rho}}_{\mathcal{\mathcal{E}}}+4H{\rho}_{\mathcal{\mathcal{E}}}=0.\end{array}$$ 
(117)

This equation has the dark radiation solution
$$\begin{array}{c}{\rho}_{\mathcal{\mathcal{E}}}={\rho}_{\mathcal{\mathcal{E}}0}{\left(\frac{{a}_{0}}{a}\right)}^{4}.\end{array}$$ 
(118)

If
${\mathcal{\mathcal{E}}}_{\mu \nu}=0$
, then the field equations on the brane form a closed system. Thus for perfect fluid branes with homogeneous density and
${\mathcal{\mathcal{E}}}_{\mu \nu}=0$
, the brane field equations form a consistent closed system. However, this is unstable to perturbations, and there is also no guarantee that the resulting brane metric can be embedded in a regular bulk.
It also follows as a corollary that inhomogeneous density requires nonzero
${\mathcal{\mathcal{E}}}_{\mu \nu}$
:
$$\begin{array}{c}{\stackrel{\u20d7}{\nabla}}_{\mu}\rho \ne 0\Rightarrow {\mathcal{\mathcal{E}}}_{\mu \nu}\ne 0.\end{array}$$ 
(119)

For example, stellar solutions on the brane necessarily have
${\mathcal{\mathcal{E}}}_{\mu \nu}\ne 0$
in the stellar interior if it is nonuniform. Perturbed FRW models on the brane also must have
${\mathcal{\mathcal{E}}}_{\mu \nu}\ne 0$
. Thus a nonzero
${\mathcal{\mathcal{E}}}_{\mu \nu}$
, and in particular a nonzero
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
, is inevitable in realistic astrophysical and cosmological models.
3.5 Propagation and constraint equations on the brane
The propagation equations for the local and nonlocal energy density and momentum density are supplemented by further
$1+3$
covariant propagation and constraint equations for the kinematic quantities
$\Theta $
,
${A}_{\mu}$
,
${\omega}_{\mu}$
,
${\sigma}_{\mu \nu}$
, and for the free gravitational field on the brane. The kinematic quantities govern the relative motion of neighbouring fundamental worldlines. The free gravitational field on the brane is given by the brane Weyl tensor
${C}_{\mu \nu \alpha \beta}$
. This splits into the gravitoelectric and gravitomagnetic fields on the brane:
$$\begin{array}{c}{E}_{\mu \nu}={C}_{\mu \alpha \nu \beta}{u}^{\alpha}{u}^{\beta}={E}_{\langle \mu \nu \rangle},{H}_{\mu \nu}=\frac{1}{2}{\varepsilon}_{\mu \alpha \beta}{C}_{\nu \gamma}^{\alpha \beta}{u}^{\gamma}={H}_{\langle \mu \nu \rangle},\end{array}$$ 
(120)

where
${E}_{\mu \nu}$
is not to be confused with
${\mathcal{\mathcal{E}}}_{\mu \nu}$
. The Ricci identity for
${u}^{\mu}$
$$\begin{array}{c}{\nabla}_{[\mu}{\nabla}_{\nu ]}{u}_{\alpha}=\frac{1}{2}{R}_{\alpha \nu \mu \beta}{u}^{\beta},\end{array}$$ 
(121)

and the Bianchi identities
$$\begin{array}{c}{\nabla}^{\beta}{C}_{\mu \nu \alpha \beta}={\nabla}_{[\mu}\left({R}_{\nu ]\alpha}+\frac{1}{6}R{g}_{\nu ]\alpha}\right),\end{array}$$ 
(122)

produce the fundamental evolution and constraint equations governing the above covariant quantities. The field equations are incorporated via the algebraic replacement of the Ricci tensor
${R}_{\mu \nu}$
by the effective total energymomentum tensor, according to Equation ( 63 ). The brane equations are derived directly from the standard general relativity versions by simply replacing the energymomentum tensor terms
$\rho ,...$
by
${\rho}_{\text{tot}},...$
. For a general fluid source, the equations are given in [
218]
. In the case of a single perfect fluid or minimallycoupled scalar field, the equations reduce to the following nonlinear equations:

∙
Generalized Raychaudhuri equation (expansion propagation):
$$\begin{array}{c}\dot{\Theta}+\frac{1}{3}{\Theta}^{2}+{\sigma}_{\mu \nu}{\sigma}^{\mu \nu}2{\omega}_{\mu}{\omega}^{\mu}{\stackrel{\u20d7}{\nabla}}^{\mu}{A}_{\mu}+{A}_{\mu}{A}^{\mu}+\frac{{\kappa}^{2}}{2}(\rho +3p)\Lambda =\frac{{\kappa}^{2}}{2}(2\rho +3p)\frac{\rho}{\lambda}{\kappa}^{2}{\rho}_{\mathcal{\mathcal{E}}}.\end{array}$$ 
(123)


∙
Vorticity propagation:
$$\begin{array}{c}{\dot{\omega}}_{\langle \mu \rangle}+\frac{2}{3}\Theta {\omega}_{\mu}+\frac{1}{2}curl{A}_{\mu}{\sigma}_{\mu \nu}{\omega}^{\nu}=0.\end{array}$$ 
(124)


∙
Shear propagation:
$$\begin{array}{c}{\dot{\sigma}}_{\langle \mu \nu \rangle}+\frac{2}{3}\Theta {\sigma}_{\mu \nu}+{E}_{\mu \nu}{\stackrel{\u20d7}{\nabla}}_{\langle \mu}{A}_{\nu \rangle}+{\sigma}_{\alpha \langle \mu}{{\sigma}_{\nu \rangle}}^{\alpha}+{\omega}_{\langle \mu}{\omega}_{\nu \rangle}{A}_{\langle \mu}{A}_{\nu \rangle}=\frac{{\kappa}^{2}}{2}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}.\end{array}$$ 
(125)


∙
Gravitoelectric propagation (Maxwell–Weyl Edot equation):
$$\begin{array}{ccc}& & {\dot{E}}_{\langle \mu \nu \rangle}+\Theta {E}_{\mu \nu}curl{H}_{\mu \nu}+\frac{{\kappa}^{2}}{2}(\rho +p){\sigma}_{\mu \nu}2{A}^{\alpha}{\varepsilon}_{\alpha \beta (\mu}{{H}_{\nu )}}^{\beta}3{\sigma}_{\alpha \langle \mu}{{E}_{\nu \rangle}}^{\alpha}+{\omega}^{\alpha}{\varepsilon}_{\alpha \beta (\mu}{{E}_{\nu )}}^{\beta}=\end{array}$$  
$$\begin{array}{ccc}& & \frac{{\kappa}^{2}}{2}(\rho +p)\frac{\rho}{\lambda}{\sigma}_{\mu \nu}\end{array}$$  
$$\begin{array}{ccc}& & \frac{{\kappa}^{2}}{6}\left[4{\rho}_{\mathcal{\mathcal{E}}}{\sigma}_{\mu \nu}+3{\dot{\pi}}_{\langle \mu \nu \rangle}^{\mathcal{\mathcal{E}}}+\Theta {\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}+3{\stackrel{\u20d7}{\nabla}}_{\langle \mu}{q}_{\nu \rangle}^{\mathcal{\mathcal{E}}}+6{A}_{\langle \mu}{q}_{\nu \rangle}^{\mathcal{\mathcal{E}}}+3{\sigma}_{\langle \mu}^{\alpha}{\pi}_{\nu \rangle \alpha}^{\mathcal{\mathcal{E}}}+3{\omega}^{\alpha}{\varepsilon}_{\alpha \beta (\mu}{{\pi}_{\nu )}^{\mathcal{\mathcal{E}}}}^{\beta}\right].\end{array}$$ 
(126)


∙
Gravitomagnetic propagation (Maxwell–Weyl Hdot equation):
$$\begin{array}{ccc}& & {\dot{H}}_{\langle \mu \nu \rangle}+\Theta {H}_{\mu \nu}+curl{E}_{\mu \nu}3{\sigma}_{\alpha \langle \mu}{{H}_{\nu \rangle}}^{\alpha}+{\omega}^{\alpha}{\varepsilon}_{\alpha \beta (\mu}{{H}_{\nu )}}^{\beta}+2{A}^{\alpha}{\varepsilon}_{\alpha \beta (\mu}{{E}_{\nu )}}^{\beta}=\end{array}$$  
$$\begin{array}{ccc}& & \frac{{\kappa}^{2}}{2}\left[curl{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}3{\omega}_{\langle \mu}{q}_{\nu \rangle}^{\mathcal{\mathcal{E}}}+{\sigma}_{\alpha (\mu}{{\varepsilon}_{\nu )}}^{\alpha \beta}{q}_{\beta}^{\mathcal{\mathcal{E}}}\right].\end{array}$$ 
(127)


∙
Vorticity constraint:
$$\begin{array}{c}{\stackrel{\u20d7}{\nabla}}^{\mu}{\omega}_{\mu}{A}^{\mu}{\omega}_{\mu}=0.\end{array}$$ 
(128)


∙
Shear constraint:
$$\begin{array}{c}{\stackrel{\u20d7}{\nabla}}^{\nu}{\sigma}_{\mu \nu}curl{\omega}_{\mu}\frac{2}{3}{\stackrel{\u20d7}{\nabla}}_{\mu}\Theta +2{\varepsilon}_{\mu \nu \alpha}{\omega}^{\nu}{A}^{\alpha}={\kappa}^{2}{q}_{\mu}^{\mathcal{\mathcal{E}}}.\end{array}$$ 
(129)


∙
Gravitomagnetic constraint:
$$\begin{array}{c}curl{\sigma}_{\mu \nu}+{\stackrel{\u20d7}{\nabla}}_{\langle \mu}{\omega}_{\nu \rangle}{H}_{\mu \nu}+2{A}_{\langle \mu}{\omega}_{\nu \rangle}=0.\end{array}$$ 
(130)


∙
Gravitoelectric divergence (Maxwell–Weyl divE equation):
$$\begin{array}{ccc}& & {\stackrel{\u20d7}{\nabla}}^{\nu}{E}_{\mu \nu}\frac{{\kappa}^{2}}{3}{\stackrel{\u20d7}{\nabla}}_{\mu}\rho {\varepsilon}_{\mu \nu \alpha}{\sigma}_{\beta}^{\nu}{H}^{\alpha \beta}+3{H}_{\mu \nu}{\omega}^{\nu}=\end{array}$$  
$$\begin{array}{ccc}& & \frac{{\kappa}^{2}}{3}\frac{\rho}{\lambda}{\stackrel{\u20d7}{\nabla}}_{\mu}\rho +\frac{{\kappa}^{2}}{6}\left(2{\stackrel{\u20d7}{\nabla}}_{\mu}{\rho}_{\mathcal{\mathcal{E}}}2\Theta {q}_{\mu}^{\mathcal{\mathcal{E}}}3{\stackrel{\u20d7}{\nabla}}^{\nu}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}+3{{\sigma}_{\mu}}^{\nu}{q}_{\nu}^{\mathcal{\mathcal{E}}}9{{\varepsilon}_{\mu}}^{\nu \alpha}{\omega}_{\nu}{q}_{\alpha}^{\mathcal{\mathcal{E}}}\right).\end{array}$$ 
(131)


∙
Gravitomagnetic divergence (Maxwell–Weyl divH equation):
$$\begin{array}{ccc}& & {\stackrel{\u20d7}{\nabla}}^{\nu}{H}_{\mu \nu}{\kappa}^{2}(\rho +p){\omega}_{\mu}+{\varepsilon}_{\mu \nu \alpha}{\sigma}_{\beta}^{\nu}{E}^{\alpha \beta}3{E}_{\mu \nu}{\omega}^{\nu}=\end{array}$$  
$$\begin{array}{ccc}& & {\kappa}^{2}(\rho +p)\frac{\rho}{\lambda}{\omega}_{\mu}+\frac{{\kappa}^{2}}{6}\left(8{\rho}_{\mathcal{\mathcal{E}}}{\omega}_{\mu}3curl{q}_{\mu}^{\mathcal{\mathcal{E}}}3{{\varepsilon}_{\mu}}^{\nu \alpha}{{\sigma}_{\nu}}^{\beta}{\pi}_{\alpha \beta}^{\mathcal{\mathcal{E}}}3{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}{\omega}^{\nu}\right).\end{array}$$ 
(132)


∙
Gauss–Codazzi equations on the brane (with
${\omega}_{\mu}=0$
):
$$\begin{array}{ccc}{R}_{\langle \mu \nu \rangle}^{\perp}+{\dot{\sigma}}_{\langle \mu \nu \rangle}+\Theta {\sigma}_{\mu \nu}{\stackrel{\u20d7}{\nabla}}_{\langle \mu}{A}_{\nu \rangle}{A}_{\langle \mu}{A}_{\nu \rangle}& =& {\kappa}^{2}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}},\end{array}$$ 
(133)

$$\begin{array}{ccc}{R}^{\perp}+\frac{2}{3}{\Theta}^{2}{\sigma}_{\mu \nu}{\sigma}^{\mu \nu}2{\kappa}^{2}\rho 2\Lambda & =& {\kappa}^{2}\frac{{\rho}^{2}}{\lambda}+2{\kappa}^{2}{\rho}_{\mathcal{\mathcal{E}}},\end{array}$$ 
(134)

where
${R}_{\mu \nu}^{\perp}$
is the Ricci tensor for 3surfaces orthogonal to
${u}^{\mu}$
on the brane, and
${R}^{\perp}={h}^{\mu \nu}{R}_{\mu \nu}^{\perp}$
.
Figure 4
: The evolution of the dimensionless shear parameter
${\Omega}_{shear}={\sigma}^{2}/6{H}^{2}$
on a Bianchi I brane, for a
$V=\frac{1}{2}{m}^{2}{\phi}^{2}$
model. The early and latetime expansion of the universe is isotropic, but the shear dominates during an intermediate anisotropic stage. (Figure taken from [
221]
.)
The standard 4D general relativity results are regained when
${\lambda}^{1}\to 0$
and
${\mathcal{\mathcal{E}}}_{\mu \nu}=0$
, which sets all right hand sides to zero in Equations ( 123 , 124 , 125 , 126 , 127 , 128 , 129 , 130 , 131 , 132 , 133 , 134 ). Together with Equations ( 113 , 114 , 115 , 116 ), these equations govern the dynamics of the matter and gravitational fields on the brane, incorporating both the local, highenergy (quadratic energymomentum) and nonlocal, KK (projected 5D Weyl) effects from the bulk. Highenergy terms are proportional to
$\rho /\lambda $
, and are significant only when
$\rho >\lambda $
. The KK terms contain
${\rho}_{\mathcal{\mathcal{E}}}$
,
${q}_{\mu}^{\mathcal{\mathcal{E}}}$
, and
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
, with the latter two quantities introducing imperfect fluid effects, even when the matter has perfect fluid form.
Bulk effects give rise to important new driving and source terms in the propagation and constraint equations. The vorticity propagation and constraint, and the gravitomagnetic constraint have no direct bulk effects, but all other equations do. Highenergy and KK energy density terms are driving terms in the propagation of the expansion
$\Theta $
. The spatial gradients of these terms provide sources for the gravitoelectric field
${E}_{\mu \nu}$
. The KK anisotropic stress is a driving term in the propagation of shear
${\sigma}_{\mu \nu}$
and the gravitoelectric/gravitomagnetic fields,
$E$
and
${H}_{\mu \nu}$
respectively, and the KK momentum density is a source for shear and the gravitomagnetic field. The 4D Maxwell–Weyl equations show in detail the contribution to the 4D gravitoelectromagnetic field on the brane, i.e.,
$({E}_{\mu \nu},{H}_{\mu \nu})$
, from the 5D Weyl field in the bulk.
An interesting example of how highenergy effects can modify general relativistic dynamics arises in the analysis of isotropization of Bianchi spacetimes. For a Binachi type I brane, Equation (
134 ) becomes [
221]
$$\begin{array}{c}{H}^{2}=\frac{{\kappa}^{2}}{3}\rho \left(1+\frac{\rho}{2\lambda}\right)+\frac{{\Sigma}^{2}}{{a}^{6}},\end{array}$$ 
(135)

if we neglect the dark radiation, where
$a$
and
$H$
are the average scale factor and expansion rate, and
$\Sigma $
is the shear constant. In general relativity, the shear term dominates as
$a\to 0$
, but in the braneworld, the highenergy
${\rho}^{2}$
term will dominate if
$w>0$
, so that the matterdominated early universe is isotropic [
221,
48,
47,
308,
280,
18,
64]
. This is illustrated in Figure 4 .
Note that this conclusion is sensitive to the assumption that
${\rho}_{\mathcal{\mathcal{E}}}\approx 0$
, which by Equation ( 115 ) implies the restriction
$$\begin{array}{c}{\sigma}^{\mu \nu}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}\approx 0.\end{array}$$ 
(136)

Relaxing this assumption can lead to nonisotropizing solutions [
1,
65,
46]
.
The system of propagation and constraint equations, i.e., Equations (
113 , 114 , 115 , 116 ) and ( 123 , 124 , 125 , 126 , 127 , 128 , 129 , 130 , 131 , 132 , 133 , 134 ), is exact and nonlinear, applicable to both cosmological and astrophysical modelling, including stronggravity effects. In general the system of equations is not closed: There is no evolution equation for the KK anisotropic stress
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
.
4 Gravitational Collapse and Black Holes on the Brane
The physics of braneworld compact objects and gravitational collapse is complicated by a number of factors, especially the confinement of matter to the brane, while the gravitational field can access the extra dimension, and the nonlocal (from the brane viewpoint) gravitational interaction between the brane and the bulk. Extradimensional effects mean that the 4D matching conditions on the brane, i.e., continuity of the induced metric and extrinsic curvature across the 2surface boundary, are much more complicated to implement [
111,
78,
314,
110]
. Highenergy corrections increase the effective density and pressure of stellar and collapsing matter. In particular this means that the effective pressure does not in general vanish at the boundary 2surface, changing the nature of the 4D matching conditions on the brane. The nonlocal KK effects further complicate the matching problem on the brane, since they in general contribute to the effective radial pressure at the boundary 2surface. Gravitational collapse inevitably produces energies high enough, i.e.,
$\rho \gg \lambda $
, to make these corrections significant.
We expect that extradimensional effects will be negligible outside the highenergy, shortrange regime. The corrections to the weakfield potential, Equation (
41 ), are at the second postNewtonian (2PN) level [
114,
150]
. However, modifications to Hawking radiation may bring significant corrections even for solarsized black holes, as discussed below.
A vacuum on the brane, outside a star or black hole, satisfies the brane field equations
$$\begin{array}{c}{R}_{\mu \nu}={\mathcal{\mathcal{E}}}_{\mu \nu},{R}_{\mu}^{\mu}=0={\mathcal{\mathcal{E}}}_{\mu}^{\mu},{\nabla}^{\nu}{\mathcal{\mathcal{E}}}_{\mu \nu}=0.\end{array}$$ 
(137)

The Weyl term
${\mathcal{\mathcal{E}}}_{\mu \nu}$
will carry an imprint of highenergy effects that source KK modes (as discussed above). This means that highenergy stars and the process of gravitational collapse will in general lead to deviations from the 4D general relativity problem. The weakfield limit for a static spherical source, Equation ( 41 ), shows that
${\mathcal{\mathcal{E}}}_{\mu \nu}$
must be nonzero, since this is the term responsible for the corrections to the Newtonian potential.
4.1 The black string
The projected Weyl term vanishes in the simplest candidate for a black hole solution. This is obtained by assuming the exact Schwarzschild form for the induced brane metric and “stacking” it into the extra dimension [
52]
,
$$\begin{array}{ccc}{}^{\left(5\right)}d{s}^{2}& =& {e}^{2\lefty\right/\ell}{\stackrel{~}{g}}_{\mu \nu}d{x}^{\mu}d{x}^{\nu}+d{y}^{2},\end{array}$$ 
(138)

$$\begin{array}{ccc}{\stackrel{~}{g}}_{\mu \nu}& =& {e}^{2\lefty\right/\ell}{g}_{\mu \nu}=(12GM/r)d{t}^{2}+\frac{d{r}^{2}}{12GM/r}+{r}^{2}d{\Omega}^{2}.\end{array}$$ 
(139)

(Note that Equation ( 138 ) is in fact a solution of the 5D field equations ( 22 ) if
${\stackrel{~}{g}}_{\mu \nu}$
is any 4D Einstein vacuum solution, i.e., if
${\stackrel{~}{R}}_{\mu \nu}=0$
, and this can be generalized to the case
${\stackrel{~}{R}}_{\mu \nu}=\stackrel{~}{\Lambda}{\stackrel{~}{g}}_{\mu \nu}$
[
7,
15]
.) Each
$\{y=const.\}$
surface is a 4D Schwarzschild spacetime, and there is a line singularity along
$r=0$
for all
$y$
. This solution is known as the Schwarzschild black string, which is clearly not localized on the brane
$y=0$
. Although
${}^{\left(5\right)}{C}_{ABCD}\ne 0$
, the projection of the bulk Weyl tensor along the brane is zero, since there is no correction to the 4D gravitational potential:
$$\begin{array}{c}V\left(r\right)=\frac{GM}{r}\Rightarrow {\mathcal{\mathcal{E}}}_{\mu \nu}=0.\end{array}$$ 
(140)

The violation of the perturbative corrections to the potential signals some kind of non
$Ad{S}_{5}$
pathology in the bulk. Indeed, the 5D curvature is unbounded at the Cauchy horizon, as
$y\to \infty $
[
52]
:
$$\begin{array}{c}{}^{\left(5\right)}{{R}_{ABCD}}^{\left(5\right)}{R}^{ABCD}=\frac{40}{{\ell}^{4}}+\frac{48{G}^{2}{M}^{2}}{{r}^{6}}{e}^{4\lefty\right/\ell}.\end{array}$$ 
(141)

Furthermore, the black string is unstable to largescale perturbations [
127]
.
Thus the “obvious” approach to finding a brane black hole fails. An alternative approach is to seek solutions of the brane field equations with nonzero
${\mathcal{\mathcal{E}}}_{\mu \nu}$
[
72]
. Brane solutions of static black hole exteriors with 5D corrections to the Schwarzschild metric have been found [
72,
111,
78,
314,
160,
49,
159]
, but the bulk metric for these solutions has not been found. Numerical integration into the bulk, starting from static black hole solutions on the brane, is plagued with difficulties [
292,
54]
.
4.2 Taylor expansion into the bulk
One can use a Taylor expansion, as in Equation ( 82 ), in order to probe properties of a static black hole on the brane [
73]
. (An alternative expansion scheme is discussed in [
50]
.) For a vacuum brane metric,
$$\begin{array}{ccc}{\stackrel{~}{g}}_{\mu \nu}(x,y)& =& {\stackrel{~}{g}}_{\mu \nu}(x,0){\mathcal{\mathcal{E}}}_{\mu \nu}(x,0+){y}^{2}\frac{2}{\ell}{\mathcal{\mathcal{E}}}_{\mu \nu}(x,0+)y{}^{3}\end{array}$$  
$$\begin{array}{ccc}& & +\frac{1}{12}{\left[\square {\mathcal{\mathcal{E}}}_{\mu \nu}\frac{32}{{\ell}^{2}}{\mathcal{\mathcal{E}}}_{\mu \nu}+2{R}_{\mu \alpha \nu \beta}{\mathcal{\mathcal{E}}}^{\alpha \beta}+6{{\mathcal{\mathcal{E}}}_{\mu}}^{\alpha}{\mathcal{\mathcal{E}}}_{\alpha \nu}\right]}_{y=0+}{y}^{4}+...\end{array}$$ 
(142)

This shows in particular that the propagating effect of 5D gravity arises only at the fourth order of the expansion. For a static spherical metric on the brane,
$$\begin{array}{c}{\stackrel{~}{g}}_{\mu \nu}d{x}^{\mu}d{x}^{\nu}=F\left(r\right)d{t}^{2}+\frac{d{r}^{2}}{H\left(r\right)}+{r}^{2}d{\Omega}^{2},\end{array}$$ 
(143)

the projected Weyl term on the brane is given by
$$\begin{array}{ccc}{\mathcal{\mathcal{E}}}_{00}& =& \frac{F}{r}\left[{H}^{\prime}\frac{1H}{r}\right],\end{array}$$ 
(144)

$$\begin{array}{ccc}{\mathcal{\mathcal{E}}}_{rr}& =& \frac{1}{rH}\left[\frac{{F}^{\prime}}{F}\frac{1H}{r}\right],\end{array}$$ 
(145)

$$\begin{array}{ccc}{\mathcal{\mathcal{E}}}_{\theta \theta}& =& 1+H+\frac{r}{2}H\left(\frac{{F}^{\prime}}{F}+\frac{{H}^{\prime}}{H}\right).\end{array}$$ 
(146)

These components allow one to evaluate the metric coefficients in Equation ( 142 ). For example, the area of the 5D horizon is determined by
${\stackrel{~}{g}}_{\theta \theta}$
; defining
$\psi \left(r\right)$
as the deviation from a Schwarzschild form for
$H$
, i.e.,
$$\begin{array}{c}H\left(r\right)=1\frac{2m}{r}+\psi \left(r\right),\end{array}$$ 
(147)

where
$m$
is constant, we find
$$\begin{array}{c}{\stackrel{~}{g}}_{\theta \theta}(r,y)={r}^{2}{\psi}^{\prime}\left(1+\frac{2}{\ell}\lefty\right\right){y}^{2}+\frac{1}{6{r}^{2}}\left[{\psi}^{\prime}+\frac{1}{2}(1+{\psi}^{\prime})(r{\psi}^{\prime}\psi {)}^{\prime}\right]{y}^{4}+...\end{array}$$ 
(148)

This shows how
$\psi $
and its
$r$
derivatives determine the change in area of the horizon along the extra dimension. For the black string
$\psi =0$
, and we have
${\stackrel{~}{g}}_{\theta \theta}(r,y)={r}^{2}$
. For a large black hole, with horizon scale
$\gg \ell $
, we have from Equation ( 41 ) that
$$\begin{array}{c}\psi \approx \frac{4m{\ell}^{2}}{3{r}^{3}}.\end{array}$$ 
(149)

This implies that
${\stackrel{~}{g}}_{\theta \theta}$
is decreasing as we move off the brane, consistent with a pancakelike shape of the horizon. However, note that the horizon shape is tubular in Gaussian normal coordinates [
113]
.
4.3 The “tidal charge” black hole
The equations ( 137 ) form a system of constraints on the brane in the stationary case, including the static spherical case, for which
$$\begin{array}{c}\Theta =0={\omega}_{\mu}={\sigma}_{\mu \nu},{\dot{\rho}}_{\mathcal{\mathcal{E}}}=0={q}_{\mu}^{\mathcal{\mathcal{E}}}={\dot{\pi}}_{\mu \nu}^{\mathcal{\mathcal{E}}}.\end{array}$$ 
(150)

The nonlocal conservation equations
${\nabla}^{\nu}{\mathcal{\mathcal{E}}}_{\mu \nu}=0$
reduce to
$$\begin{array}{c}\frac{1}{3}{\stackrel{\u20d7}{\nabla}}_{\mu}{\rho}_{\mathcal{\mathcal{E}}}+\frac{4}{3}{\rho}_{\mathcal{\mathcal{E}}}{A}_{\mu}+{\stackrel{\u20d7}{\nabla}}^{\nu}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}+{A}^{\nu}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}=0,\end{array}$$ 
(151)

where, by symmetry,
$$\begin{array}{c}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}={\Pi}_{\mathcal{\mathcal{E}}}\left(\frac{1}{3}{h}_{\mu \nu}{r}_{\mu}{r}_{\nu}\right),\end{array}$$ 
(152)

for some
${\Pi}_{\mathcal{\mathcal{E}}}\left(r\right)$
, with
${r}_{\mu}$
being the unit radial vector. The solution of the brane field equations requires the input of
${\mathcal{\mathcal{E}}}_{\mu \nu}$
from the 5D solution. In the absence of a 5D solution, one can make an assumption about
${\mathcal{\mathcal{E}}}_{\mu \nu}$
or
${g}_{\mu \nu}$
to close the 4D equations.
If we assume a metric on the brane of Schwarzschildlike form, i.e.,
$H=F$
in Equation ( 143 ), then the general solution of the brane field equations is [
72]
$$\begin{array}{ccc}F& =& 1\frac{2GM}{r}+\frac{2G\ell Q}{{r}^{2}},\end{array}$$ 
(153)

$$\begin{array}{ccc}{\mathcal{\mathcal{E}}}_{\mu \nu}& =& \frac{2G\ell Q}{{r}^{4}}\left[{u}_{\mu}{u}_{\nu}2{r}_{\mu}{r}_{\nu}+{h}_{\mu \nu}\right],\end{array}$$ 
(154)

where
$Q$
is a constant. It follows that the KK energy density and anisotropic stress scalar (defined via Equation ( 152 )) are given by
$$\begin{array}{c}{\rho}_{\mathcal{\mathcal{E}}}=\frac{\ell Q}{4\pi {r}^{4}}=\frac{1}{2}{\Pi}_{\mathcal{\mathcal{E}}}.\end{array}$$ 
(155)

The solution ( 153 ) has the form of the general relativity Reissner–Nordström solution, but there is no electric field on the brane. Instead, the nonlocal Coulomb effects imprinted by the bulk Weyl tensor have induced a “tidal” charge parameter
$Q$
, where
$Q=Q\left(M\right)$
, since
$M$
is the source of the bulk Weyl field. We can think of the gravitational field of
$M$
being “reflected back” on the brane by the negative bulk cosmological constant [
71]
. If we impose the smallscale perturbative limit (
$r\ll \ell $
) in Equation ( 40 ), we find that
$$\begin{array}{c}Q=2M.\end{array}$$ 
(156)

Negative
$Q$
is in accord with the intuitive idea that the tidal charge strengthens the gravitational field, since it arises from the source mass
$M$
on the brane. By contrast, in the Reissner–Nordström solution of general relativity,
$Q\propto +{q}^{2}$
, where
$q$
is the electric charge, and this weakens the gravitational field. Negative tidal charge also preserves the spacelike nature of the singularity, and it means that there is only one horizon on the brane, outside the Schwarzschild horizon:
$$\begin{array}{c}{r}_{h}=GM\left[1+\sqrt{1\frac{2\ell Q}{G{M}^{2}}}\right]=GM\left[1+\sqrt{1+\frac{4\ell}{GM}}\right].\end{array}$$ 
(157)

The tidalcharge black hole metric does not satisfy the farfield
${r}^{3}$
correction to the gravitational potential, as in Equation ( 41 ), and therefore cannot describe the endstate of collapse. However, Equation ( 153 ) shows the correct 5D behaviour of the potential (
$\propto {r}^{2}$
) at short distances, so that the tidalcharge metric could be a good approximation in the strongfield regime for small black holes.
4.4 Realistic black holes
Thus a simple branebased approach, while giving useful insights, does not lead to a realistic black hole solution. There is no known solution representing a realistic black hole localized on the brane, which is stable and without naked singularity. This remains a key open question of nonlinear braneworld gravity. (Note that an exact solution is known for a black hole on a
$1+2$
brane in a 4D bulk [
96]
, but this is a very special case.) Given the nonlocal nature of
${\mathcal{\mathcal{E}}}_{\mu \nu}$
, it is possible that the process of gravitational collapse itself leaves a signature in the black hole endstate, in contrast with general relativity and its nohair theorems. There are contradictory indications about the nature of the realistic black hole solution on the brane:

∙
Numerical simulations of highly relativistic static stars on the brane [319] indicate that general relativity remains a good approximation.

∙
Exact analysis of Oppenheimer–Snyder collapse on the brane shows that the exterior is nonstatic [110] , and this is extended to general collapse by arguments based on a generalized AdS/CFT correspondence [303, 94] .
The first result suggests that static black holes could exist as limits of increasingly compact static stars, but the second result and conjecture suggest otherwise. This remains an open question.
More recent numerical evidence is also not conclusive, and it introduces further possible subtleties to do with the size of the black hole [
183]
.
On very small scales relative to the
$Ad{S}_{5}$
curvature scale,
$r\ll \ell $
, the gravitational potential becomes 5D, as shown in Equation ( 40 ),
$$\begin{array}{c}V\left(r\right)\approx \frac{G\ell M}{{r}^{2}}=\frac{{G}_{5}M}{{r}^{2}}.\end{array}$$ 
(158)

In this regime, the black hole is so small that it does not “see” the brane, so that it is approximately a 5D Schwarzschild (static) solution. However, this is always an approximation because of the selfgravity of the brane (the situation is different in ADDtype braneworlds where there is no brane tension). As the black hole size increases, the approximation breaks down. Nevertheless, one might expect that static solutions exist on sufficiently small scales. Numerical investigations appear to confirm this [
183]
: Static metrics satisfying the asymptotic
$Ad{S}_{5}$
boundary conditions are found if the horizon is small compared to
$\ell $
, but no numerical convergence can be achieved close to
$\ell $
. The numerical instability that sets in may mask the fact that even the very small black holes are not strictly static. Or it may be that there is a transition from static to nonstatic behaviour.
Or it may be that static black holes do exist on all scales.
The 4D Schwarzschild metric cannot describe the final state of collapse, since it cannot incorporate the 5D behaviour of the gravitational potential in the strongfield regime (the metric is incompatible with massive KK modes). A nonperturbative exterior solution should have nonzero
${\mathcal{\mathcal{E}}}_{\mu \nu}$
in order to be compatible with massive KK modes in the strongfield regime. In the endstate of collapse, we expect an
${\mathcal{\mathcal{E}}}_{\mu \nu}$
which goes to zero at large distances, recovering the Schwarzschild weakfield limit, but which grows at short range. Furthermore,
${\mathcal{\mathcal{E}}}_{\mu \nu}$
may carry a Weyl “fossil record” of the collapse process.
4.5 Oppenheimer–Snyder collapse gives a nonstatic black hole
The simplest scenario in which to analyze gravitational collapse is the Oppenheimer–Snyder model, i.e., collapsing homogeneous and isotropic dust [
110]
. The collapse region on the brane has an FRW metric, while the exterior vacuum has an unknown metric. In 4D general relativity, the exterior is a Schwarzschild spacetime; the dynamics of collapse leaves no imprint on the exterior.
The collapse region has the metric
$$\begin{array}{c}d{s}^{2}=d{\tau}^{2}+\frac{a(\tau {)}^{2}\left[d{r}^{2}+{r}^{2}d{\Omega}^{2}\right]}{(1+k{r}^{2}/4{)}^{2}},\end{array}$$ 
(159)

where the scale factor satisfies the modified Friedmann equation (see below),
$$\begin{array}{c}\frac{{\dot{a}}^{2}}{{a}^{2}}=\frac{8\pi G}{3}\rho \left(1+\frac{\rho}{2\lambda}+\frac{{\rho}_{\mathcal{\mathcal{E}}}}{\rho}\right).\end{array}$$ 
(160)

The dust matter and the dark radiation evolve as
$$\begin{array}{c}\rho ={\rho}_{0}{\left(\frac{{a}_{0}}{\mu}\right)}^{3},{\rho}_{\mathcal{\mathcal{E}}}={\rho}_{\mathcal{\mathcal{E}}0}{\left(\frac{{a}_{0}}{\mu}\right)}^{4},\end{array}$$ 
(161)

where
${a}_{0}$
is the epoch when the cloud started to collapse. The proper radius from the centre of the cloud is
$R\left(\tau \right)=ra\left(\tau \right)/(1+\frac{1}{4}k{r}^{2})$
. The collapsing boundary surface
$\Sigma $
is given in the interior comoving coordinates as a freefall surface, i.e.
$r={r}_{0}=const.$
, so that
${R}_{\Sigma}\left(\tau \right)={r}_{0}a\left(\tau \right)/(1+\frac{1}{4}k{r}_{0}^{2})$
.
We can rewrite the modified Friedmann equation on the interior side of
$\Sigma $
as
$$\begin{array}{c}{\dot{R}}^{2}=\frac{2GM}{R}+\frac{3G{M}^{2}}{4\pi \lambda {R}^{4}}+\frac{Q}{{R}^{2}}+E,\end{array}$$ 
(162)

where the “physical mass”
$M$
(total energy per proper star volume), the total “tidal charge”
$Q$
, and the “energy” per unit mass
$E$
are given by
$$\begin{array}{ccc}M& =& \frac{4\pi {a}_{0}^{3}{r}_{0}^{3}{\rho}_{0}}{3(1+\frac{1}{4}k{r}_{0}^{2}{)}^{3}},\end{array}$$ 
(163)

$$\begin{array}{ccc}Q& =& \frac{{\rho}_{\mathcal{\mathcal{E}}0}{a}_{0}^{4}{r}_{0}^{4}}{(1+\frac{1}{4}k{r}_{0}^{2}{)}^{4}},\end{array}$$ 
(164)

$$\begin{array}{ccc}E& =& \frac{k{r}_{0}^{2}}{(1+\frac{1}{4}k{r}_{0}^{2}{)}^{2}}>1.\end{array}$$ 
(165)

Now we assume that the exterior is static, and satisfies the standard 4D junction conditions.
Then we check whether this exterior is physical by imposing the modified Einstein equations (
137 ).
We will find a contradiction.
The standard 4D Darmois–Israel matching conditions, which we assume hold on the brane, require that the metric and the extrinsic curvature of
$\Sigma $
be continuous (there are no intrinsic stresses on
$\Sigma $
). The extrinsic curvature is continuous if the metric is continuous and if
$\dot{R}$
is continuous. We therefore need to match the metrics and
$\dot{R}$
across
$\Sigma $
.
The most general static spherical metric that could match the interior metric on
$\Sigma $
is
$$\begin{array}{c}d{s}^{2}=F(R{)}^{2}\left[1\frac{2Gm\left(R\right)}{R}\right]d{t}^{2}+\frac{d{R}^{2}}{12Gm\left(R\right)/R}+{R}^{2}d{\Omega}^{2}.\end{array}$$ 
(166)

We need two conditions to determine the functions
$F\left(R\right)$
and
$m\left(R\right)$
. Now
$\Sigma $
is a freely falling surface in both metrics, and the radial geodesic equation for the exterior metric gives
${\dot{R}}^{2}=1+2Gm\left(R\right)/R+\stackrel{~}{E}/F(R{)}^{2},$
where
$\stackrel{~}{E}$
is a constant and the dot denotes a proper time derivative, as above. Comparing this with Equation ( 162 ) gives one condition. The second condition is easier to derive if we change to null coordinates. The exterior static metric, with
$$dv=dt+\frac{dR}{(12Gm/R)F},$$
$$\begin{array}{c}d{s}^{2}={F}^{2}\left(1\frac{2Gm}{R}\right)d{v}^{2}+2FdvdR+{R}^{2}d{\Omega}^{2}.\end{array}$$ 
(167)

The interior Robertson–Walker metric takes the form [
110]
$$\begin{array}{c}d{s}^{2}=\frac{{\tau}_{,v}^{2}\left[1(k+{\dot{a}}^{2}){R}^{2}/{a}^{2}\right]d{v}^{2}}{(1k{R}^{2}/{a}^{2})}+\frac{2{\tau}_{,v}dvdR}{\sqrt{1k{R}^{2}/{a}^{2}}}+{R}^{2}d{\Omega}^{2},\end{array}$$ 
(168)

where
$$d\tau ={\tau}_{,v}dv+\left(1+\frac{1}{4}k{r}^{2}\right)\frac{dR}{r\dot{a}1+\frac{1}{4}k{r}^{2}}.$$
Comparing Equations ( 167 ) and ( 168 ) on
$\Sigma $
gives the second condition. The two conditions together imply that
$F$
is a constant, which we can take as
$F\left(R\right)=1$
without loss of generality (choosing
$\stackrel{~}{E}=E+1$
), and that
$$\begin{array}{c}m\left(R\right)=M+\frac{3{M}^{2}}{8\pi \lambda {R}^{3}}+\frac{Q}{2GR}.\end{array}$$ 
(169)

In the limit
${\lambda}^{1}=0=Q$
, we recover the 4D Schwarzschild solution. In the general braneworld case, Equations ( 166 ) and ( 169 ) imply that the brane Ricci scalar is
$$\begin{array}{c}{R}_{\mu}^{\mu}=\frac{9G{M}^{2}}{2\pi \lambda {R}^{6}}.\end{array}$$ 
(170)

However, this contradicts the field equations ( 137 ), which require
$$\begin{array}{c}{R}_{\mu}^{\mu}=0.\end{array}$$ 
(171)

It follows that a static exterior is only possible if
$M/\lambda =0,$
which is the 4D general relativity limit. In the braneworld, collapsing homogeneous and isotropic dust leads to a nonstatic exterior.
Note that this nogo result does not require any assumptions on the nature of the bulk spacetime, which remains to be determined.
Although the exterior metric is not determined (see [
123]
for a toy model), we know that its nonstatic nature arises from

∙
5D bulk graviton stresses, which transmit effects nonlocally from the interior to the exterior, and

∙
the nonvanishing of the effective pressure at the boundary, which means that dynamical information from the interior can be conveyed outside via the 4D matching conditions.
The result suggests that gravitational collapse on the brane may leave a signature in the exterior, dependent upon the dynamics of collapse, so that astrophysical black holes on the brane may in principle have KK “hair”. It is possible that the nonstatic exterior will be transient, and will tend to a static geometry at late times, close to Schwarzschild at large distances.
4.6 AdS/CFT and black holes on 1brane RStype models
Oppenheimer–Snyder collapse is very special; in particular, it is homogeneous. One could argue that the nonstatic exterior arises because of the special nature of this model. However, the underlying reasons for nonstatic behaviour are not special to this model; on the contrary, the role of highenergy corrections and KK stresses will if anything be enhanced in a general, inhomogeneous collapse. There is in fact independent heuristic support for this possibility, arising from the AdS/CFT correspondence.
The basic idea of the correspondence is that the classical dynamics of the
$Ad{S}_{5}$
gravitational field correspond to the quantum dynamics of a 4D conformal field theory on the brane. This correspondence holds at linear perturbative order [
87]
, so that the RS 1brane infinite
$Ad{S}_{5}$
braneworld (without matter fields on the brane) is equivalently described by 4D general relativity coupled to conformal fields,
$$\begin{array}{c}{G}_{\mu \nu}=8\pi G{T}_{\mu \nu}^{(cft)}.\end{array}$$ 
(172)

According to a conjecture [
303]
, the correspondence holds also in the case where there is strong gravity on the brane, so that the classical dynamics of the bulk gravitational field of the brane black hole are equivalent to the dynamics of a quantumcorrected 4D black hole (in the dual CFTplusgravity description). In other words [
303,
94]
:

∙
Quantum backreaction due to Hawking radiation in the 4D picture is described as classical dynamics in the 5D picture.

∙
The black hole evaporates as a classical process in the 5D picture, and there is thus no stationary black hole solution in RS 1brane.
A further remarkable consequence of this conjecture is that Hawking evaporation is dramatically enhanced, due to the very large number of CFT modes of order
$(\ell /{\ell}_{p}{)}^{2}$
. The energy loss rate due to evaporation is
$$\begin{array}{c}\frac{\dot{M}}{M}\sim N\left(\frac{1}{{G}^{2}{M}^{3}}\right),\end{array}$$ 
(173)

where
$N$
is the number of light degrees of freedom. Using
$N\sim {\ell}^{2}/G$
, this gives an evaporation timescale [
303]
$$\begin{array}{c}{t}_{evap}\sim {\left(\frac{M}{{M}_{\odot}}\right)}^{3}{\left(\frac{1mm}{\ell}\right)}^{2}yr.\end{array}$$ 
(174)

A more detailed analysis [
95]
shows that this expression should be multiplied by a factor
$\approx 100$
. Then the existence of stellarmass black holes on long time scales places limits on the
$Ad{S}_{5}$
curvature scale that are more stringent than the tabletop limit, Equation ( 6 ). The existence of black hole Xray binaries implies
$$\begin{array}{c}\ell \lesssim {10}^{2}mm,\end{array}$$ 
(175)

already an order of magnitude improvement on the tabletop limit.
One can also relate the Oppenheimer–Snyder result to these considerations. In the AdS/CFT picture, the nonvanishing of the Ricci scalar, Equation (
170 ), arises from the trace of the Hawking CFT energymomentum tensor, as in Equation ( 172 ). If we evaluate the Ricci scalar at the black hole horizon,
$R\sim 2GM$
, using
$\lambda =6{M}_{5}^{6}/{M}_{p}^{2}$
, we find
$$\begin{array}{c}{R}_{\mu}^{\mu}\sim \frac{{{M}_{5}}^{12}{\ell}^{6}}{{M}^{4}}.\end{array}$$ 
(176)

The CFT trace on the other hand is given by
${T}^{(cft)}\sim N{T}_{h}^{4}/{M}_{p}^{2}$
, so that
$$\begin{array}{c}8\pi G{T}^{(cft)}\sim \frac{{{M}_{5}}^{12}{\ell}^{6}}{{M}^{4}}.\end{array}$$ 
(177)

Thus the Oppenheimer–Snyder result is qualitatively consistent with the AdS/CFT picture.
Clearly the black hole solution, and the collapse process that leads to it, have a far richer structure in the braneworld than in general relativity, and deserve further attention. In particular, two further topics are of interest:

∙
Primordial black holes in 1brane RStype cosmology have been investigated in [150, 61, 129, 227, 62, 287] .
Highenergy effects in the early universe (see the next Section
5 ) can significantly modify the evaporation and accretion processes, leading to a prolonged survival of these black holes.
Such black holes evade the enhanced Hawking evaporation described above when they are formed, because they are much smaller than
$\ell $
.

∙
Black holes will also be produced in particle collisions at energies
$\gtrsim {M}_{5}$
, possibly well below the Planck scale. In ADD braneworlds, where
${M}_{4+d}=\mathcal{O}(TeV)$
is not ruled out by current observations if
$d>1$
, this raises the exciting prospect of observing black hole production signatures in the nextgeneration colliders and cosmic ray detectors (see [51, 116, 93] ).
5 BraneWorld Cosmology: Dynamics
A
$1+4$
dimensional spacetime with spatial 4isotropy (4D spherical/plane/hyperbolic symmetry) has a natural foliation into the symmetry group orbits, which are
$1+3$
dimensional surfaces with 3isotropy and 3homogeneity, i.e., FRW surfaces. In particular, the
$Ad{S}_{5}$
bulk of the RS braneworld, which admits a foliation into Minkowski surfaces, also admits an FRW foliation since it is 4isotropic. Indeed this feature of 1brane RStype cosmological braneworlds underlies the importance of the AdS/CFT correspondence in braneworld cosmology [
254,
282,
136,
289,
293,
210,
259,
125]
.
The generalization of
$Ad{S}_{5}$
that preserves 4isotropy and solves the vacuum 5D Einstein equation ( 22 ) is Schwarzschild–
$Ad{S}_{5}$
, and this bulk therefore admits an FRW foliation. It follows that an FRW braneworld, the cosmological generalization of the RS braneworld, is a part of Schwarzschild–
$Ad{S}_{5}$
, with the
${Z}_{2}$
symmetric FRW brane at the boundary. (Note that FRW branes can also be embedded in nonvacuum generalizations, e.g., in Reissner–Nordström–
$Ad{S}_{5}$
and Vaidya–
$Ad{S}_{5}$
.) In natural static coordinates, the bulk metric is
$$\begin{array}{ccc}{}^{\left(5\right)}d{s}^{2}& =& F\left(R\right)d{T}^{2}+\frac{d{R}^{2}}{F\left(R\right)}+{R}^{2}\left(\frac{d{r}^{2}}{1K{r}^{2}}+{r}^{2}d{\Omega}^{2}\right),\end{array}$$ 
(178)

$$\begin{array}{ccc}F\left(R\right)& =& K+\frac{{R}^{2}}{{\ell}^{2}}\frac{m}{{R}^{2}},\end{array}$$ 
(179)

where
$K=0,\pm 1$
is the FRW curvature index, and
$m$
is the mass parameter of the black hole at
$R=0$
(recall that the 5D gravitational potential has
${R}^{2}$
behaviour). The bulk black hole gives rise to dark radiation on the brane via its Coulomb effect. The FRW brane moves radially along the 5th dimension, with
$R=a\left(T\right)$
, where
$a$
is the FRW scale factor, and the junction conditions determine the velocity via the Friedmann equation for
$a$
[
249,
32]
. Thus one can interpret the expansion of the universe as motion of the brane through the static bulk. In the special case
$m=0$
and
$da/dT=0$
, the brane is fixed and has Minkowski geometry, i.e., the original RS 1brane braneworld is recovered in different coordinates.
The velocity of the brane is coordinatedependent, and can be set to zero. We can use Gaussian normal coordinates, in which the brane is fixed but the bulk metric is not manifestly static [
27]
:
$$\begin{array}{c}{}^{\left(5\right)}d{s}^{2}={N}^{2}(t,y)d{t}^{2}+{A}^{2}(t,y)\left[\frac{d{r}^{2}}{1K{r}^{2}}+{r}^{2}d{\Omega}^{2}\right]+d{y}^{2}.\end{array}$$ 
(180)

Here
$a\left(t\right)=A(t,0)$
is the scale factor on the FRW brane at
$y=0$
, and
$t$
may be chosen as proper time on the brane, so that
$N(t,0)=1$
. In the case where there is no bulk black hole (
$m=0$
), the metric functions are
$$\begin{array}{ccc}N& =& \frac{\dot{A}(t,y)}{\dot{a}\left(t\right)},\end{array}$$ 
(181)

$$\begin{array}{ccc}A& =& a\left(t\right)\left[cosh\left(\frac{y}{\ell}\right)\left\{1+\frac{\rho \left(t\right)}{\lambda}\right\}sinh\left(\frac{\lefty\right}{\ell}\right)\right].\end{array}$$ 
(182)

Again, the junction conditions determine the Friedmann equation. The extrinsic curvature at the brane is
$$\begin{array}{c}{K}_{\nu}^{\mu}=diag{\left(\frac{{N}^{\prime}}{N},\frac{{A}^{\prime}}{A},\frac{{A}^{\prime}}{A},\frac{{A}^{\prime}}{A}\right)}_{brane}.\end{array}$$ 
(183)

Then, by Equation ( 62 ),
$$\begin{array}{ccc}\frac{{N}^{\prime}}{N}{}_{brane}& =& \frac{{\kappa}_{5}^{2}}{6}(2\rho +3p\lambda ),\end{array}$$ 
(184)

$$\begin{array}{ccc}\frac{{A}^{\prime}}{A}{}_{brane}& =& \frac{{\kappa}_{5}^{2}}{6}(\rho +\lambda ).\end{array}$$ 
(185)

The field equations yield the first integral [
27]
$$\begin{array}{c}(A{A}^{\prime}{)}^{2}\frac{{A}^{2}}{{N}^{2}}{\dot{A}}^{2}+\frac{{\Lambda}_{5}}{6}{A}^{4}+m=0,\end{array}$$ 
(186)

where
$m$
is constant. Evaluating this at the brane, using Equation ( 185 ), gives the modified Friedmann equation ( 188 ).
The dark radiation carries the imprint on the brane of the bulk gravitational field. Thus we expect that
${\mathcal{\mathcal{E}}}_{\mu \nu}$
for the Friedmann brane contains bulk metric terms evaluated at the brane. In Gaussian normal coordinates (using the field equations to simplify),
$$\begin{array}{c}{\mathcal{\mathcal{E}}}_{0}^{0}=3\frac{{A}^{\prime \prime}}{A}{}_{brane}+\frac{{\Lambda}_{5}}{2},{\mathcal{\mathcal{E}}}_{j}^{i}=\left(\frac{1}{3}{\mathcal{\mathcal{E}}}_{0}^{0}\right){\delta}_{j}^{i}.\end{array}$$ 
(187)

Either form of the cosmological metric, Equation ( 178 ) or ( 180 ), may be used to show that 5D gravitational wave signals can take “shortcuts” through the bulk in travelling between points A and B on the brane [
59,
151,
45]
. The travel time for such a graviton signal is less than the time taken for a photon signal (which is stuck to the brane) from A to B. Instead of using the junction conditions, we can use the covariant 3D Gauss–Codazzi equation ( 134 ) to find the modified Friedmann equation:
$$\begin{array}{c}{H}^{2}=\frac{{\kappa}^{2}}{3}\rho \left(1+\frac{\rho}{2\lambda}\right)+\frac{m}{{a}^{4}}+\frac{1}{3}\Lambda \frac{K}{{a}^{2}},\end{array}$$ 
(188)

on using Equation ( 118 ), where
$$\begin{array}{c}m=\frac{{\kappa}^{2}}{3}{\rho}_{\mathcal{\mathcal{E}}0}{a}_{0}^{4}.\end{array}$$ 
(189)

The covariant Raychauhuri equation ( 123 ) yields
$$\begin{array}{c}\dot{H}=\frac{{\kappa}^{2}}{2}(\rho +p)\left(1+\frac{\rho}{\lambda}\right)2\frac{m}{{a}^{4}}+\frac{K}{{a}^{2}},\end{array}$$ 
(190)

which also follows from differentiating Equation ( 188 ) and using the energy conservation equation.
When the bulk black hole mass vanishes, the bulk geometry reduces to
$Ad{S}_{5}$
, and
${\rho}_{\mathcal{\mathcal{E}}}=0$
.
In order to avoid a naked singularity, we assume that the black hole mass is nonnegative, so that
${\rho}_{\mathcal{\mathcal{E}}0}\ge 0$
. (By Equation ( 179 ), it is possible to avoid a naked singularity with negative
$m$
when
$K=1$
, provided
$\leftm\right\le {\ell}^{2}/4$
.) This additional effective relativistic degree of freedom is constrained by nucleosynthesis and CMB observations to be no more than
$\sim 5\%$
of the radiation energy density [
191,
19,
148,
34]
:
$$\begin{array}{c}{\frac{{\rho}_{\mathcal{\mathcal{E}}}}{{\rho}_{rad}}}_{nuc}\lesssim 0.05.\end{array}$$ 
(191)

The other modification to the Hubble rate is via the highenergy correction
$\rho /\lambda $
. In order to recover the observational successes of general relativity, the highenergy regime where significant deviations occur must take place before nucleosynthesis, i.e., cosmological observations impose the lower limit
$$\begin{array}{c}\lambda >(1MeV{)}^{4}\Rightarrow {M}_{5}>{10}^{4}GeV.\end{array}$$ 
(192)

This is much weaker than the limit imposed by tabletop experiments, Equation ( 42 ). Since
${\rho}^{2}/\lambda $
decays as
${a}^{8}$
during the radiation era, it will rapidly become negligible after the end of the highenergy regime,
$\rho =\lambda $
.
If
${\rho}_{\mathcal{\mathcal{E}}}=0$
and
$K=0=\Lambda $
, then the exact solution of the Friedmann equations for
$w=p/\rho =const.$
is [
27]
$$\begin{array}{c}a=const.\times \left[t\right(t+{t}_{\lambda}){]}^{1/3(w+1)},{t}_{\lambda}=\frac{{M}_{p}}{\sqrt{3\pi \lambda}(1+w)}\lesssim (1+w{)}^{1}{10}^{9}s,\end{array}$$ 
(193)

where
$w>1$
. If
${\rho}_{\mathcal{\mathcal{E}}}\ne 0$
(but
$K=0=\Lambda $
), then the solution for the radiation era (
$w=\frac{1}{3}$
) is [
19]
$$\begin{array}{c}a=const.\times \left[t\right(t+{t}_{\lambda}){]}^{1/4},{t}_{\lambda}=\frac{\sqrt{3}{M}_{p}}{4\sqrt{\pi \lambda}(1+{\rho}_{\mathcal{\mathcal{E}}}/\rho )}.\end{array}$$ 
(194)

For
$t\gg {t}_{\lambda}$
we recover from Equations ( 193 ) and ( 194 ) the standard behaviour,
$a\propto {t}^{2/3(w+1)}$
, whereas for
$t\ll {t}_{\lambda}$
, we have the very different behaviour of the highenergy regime,
$$\begin{array}{c}\rho \gg \lambda \Rightarrow a\propto {t}^{1/3(w+1)}.\end{array}$$ 
(195)

When
$w=1$
we have
$\rho ={\rho}_{0}$
from the conservation equation. If
$K=0=\Lambda $
, we recover the de Sitter solution for
${\rho}_{\mathcal{\mathcal{E}}}=0$
and an asymptotically de Sitter solution for
${\rho}_{\mathcal{\mathcal{E}}}>0$
:
$$\begin{array}{ccc}a& =& {a}_{0}exp\left[{H}_{0}\right(t{t}_{0}\left)\right],{H}_{0}=\kappa \sqrt{\frac{{\rho}_{0}}{3}\left(1+\frac{{\rho}_{0}}{2\lambda}\right)},\text{for}{\rho}_{\mathcal{\mathcal{E}}}=0,\end{array}$$ 
(196)

$$\begin{array}{ccc}{a}^{2}& =& \sqrt{\frac{m}{{H}_{0}}}sinh\left[2{H}_{0}\right(t{t}_{0}\left)\right]\text{for}{\rho}_{\mathcal{\mathcal{E}}}>0.\end{array}$$ 
(197)

A qualitative analysis of the Friedmann equations is given in [
48,
47]
.
5.1 Braneworld inflation
In 1brane RStype braneworlds, where the bulk has only a vacuum energy, inflation on the brane must be driven by a 4D scalar field trapped on the brane. In more general braneworlds, where the bulk contains a 5D scalar field, it is possible that the 5D field induces inflation on the brane via its effective projection [
165,
138,
100,
276,
141,
140,
304,
318,
180,
195,
139,
33,
238,
158,
229,
103,
13]
.
More exotic possibilities arise from the interaction between two branes, including possible collision, which is mediated by a 5D scalar field and which can induce either inflation [
90,
157]
or a hot bigbang radiation era, as in the “ekpyrotic” or cyclic scenario [
163,
154,
251,
301,
193,
231,
307]
, or in colliding bubble scenarios [
29,
107,
108]
. (See also [
21,
69,
214]
for colliding branes in an M theory approach.) Here we discuss the simplest case of a 4D scalar field
$\phi $
with potential
$V\left(\phi \right)$
(see [
207]
for a review).
Highenergy braneworld modifications to the dynamics of inflation on the brane have been investigated [
222,
156,
63,
302,
234,
233,
74,
204,
23,
24,
25,
240,
135,
184,
270,
221]
. Essentially, the highenergy corrections provide increased Hubble damping, since
$\rho \gg \lambda $
implies that
$H$
is larger for a given energy than in 4D general relativity. This makes slowroll inflation possible even for potentials that would be too steep in standard cosmology [
222,
70,
226,
277,
258,
206,
145]
.
The field satisfies the Klein–Gordon equation
$$\begin{array}{c}\ddot{\phi}+3H\dot{\phi}+{V}^{\prime}\left(\phi \right)=0.\end{array}$$ 
(198)

In 4D general relativity, the condition for inflation,
$\ddot{a}>0$
, is
${\dot{\phi}}^{2}<V\left(\phi \right)$
, i.e.,
$p<\frac{1}{3}\rho $
, where
$\rho =\frac{1}{2}{\dot{\phi}}^{2}+V$
and
$p=\frac{1}{2}{\dot{\phi}}^{2}V$
. The modified Friedmann equation leads to a stronger condition for inflation: Using Equation ( 188 ), with
$m=0=\Lambda =K$
, and Equation ( 198 ), we find that
$$\begin{array}{c}\ddot{a}>0\Rightarrow w<\frac{1}{3}\left[\frac{1+2\rho /\lambda}{1+\rho /\lambda}\right],\end{array}$$ 
(199)

where the square brackets enclose the brane correction to the general relativity result. As
$\rho /\lambda \to 0$
, the 4D result
$w<\frac{1}{3}$
is recovered, but for
$\rho >\lambda $
,
$w$
must be more negative for inflation. In the very highenergy limit
$\rho /\lambda \to \infty $
, we have
$w<\frac{2}{3}$
. When the only matter in the universe is a selfinteracting scalar field, the condition for inflation becomes
$$\begin{array}{c}{\dot{\phi}}^{2}V+\left[\frac{\frac{1}{2}{\dot{\phi}}^{2}+V}{\lambda}\left(\frac{5}{4}{\dot{\phi}}^{2}\frac{1}{2}V\right)\right]<0,\end{array}$$ 
(200)

which reduces to
${\dot{\phi}}^{2}<V$
when
${\rho}_{\phi}=\frac{1}{2}{\dot{\phi}}^{2}+V\ll \lambda $
.
In the slowroll approximation, we get
$$\begin{array}{ccc}{H}^{2}& \approx & \frac{{\kappa}^{2}}{3}V\left[1+\frac{V}{2\lambda}\right],\end{array}$$ 
(201)

$$\begin{array}{ccc}\dot{\phi}& \approx & \frac{{V}^{\prime}}{3H}.\end{array}$$ 
(202)

The braneworld correction term
$V/\lambda $
in Equation ( 201 ) serves to enhance the Hubble rate for a given potential energy, relative to general relativity. Thus there is enhanced Hubble `friction' in Equation ( 202 ), and braneworld effects will reinforce slowroll at the same potential energy.
We can see this by defining slowroll parameters that reduce to the standard parameters in the lowenergy limit:
$$\begin{array}{ccc}\epsilon & \equiv & \frac{\dot{H}}{{H}^{2}}=\frac{{M}_{p}^{2}}{16\pi}{\left(\frac{{V}^{\prime}}{V}\right)}^{2}\left[\frac{1+V/\lambda}{(1+V/2\lambda {)}^{2}}\right],\end{array}$$ 
(203)

$$\begin{array}{ccc}\eta & \equiv & \frac{\ddot{\phi}}{H\dot{\phi}}=\frac{{M}_{p}^{2}}{8\pi}\left(\frac{{V}^{\prime \prime}}{V}\right)\left[\frac{1}{1+V/2\lambda}\right].\end{array}$$ 
(204)

Selfconsistency of the slowroll approximation then requires
$\epsilon ,\left\eta \right\ll 1$
. At low energies,
$V\ll \lambda $
, the slowroll parameters reduce to the standard form. However at high energies,
$V\gg \lambda $
, the extra contribution to the Hubble expansion helps damp the rolling of the scalar field, and the new factors in square brackets become
$\approx \lambda /V$
:
$$\begin{array}{c}\epsilon \approx {\epsilon}_{gr}\left[\frac{4\lambda}{V}\right],\eta \approx {\eta}_{gr}\left[\frac{2\lambda}{V}\right],\end{array}$$ 
(205)

where
${\epsilon}_{gr},{\eta}_{gr}$
are the standard general relativity slowroll parameters. In particular, this means that steep potentials which do not give inflation in general relativity, can inflate the braneworld at high energy and then naturally stop inflating when
$V$
drops below
$\lambda $
. These models can be constrained because they typically end inflation in a kineticdominated regime and thus generate a blue spectrum of gravitational waves, which can disturb nucleosynthesis [
70,
226,
277,
258,
206]
. They also allow for the novel possibility that the inflaton could act as dark matter or quintessence at low energies [
70,
226,
277,
258,
206,
4,
239,
208,
43,
285]
.
The number of efolds during inflation,
$N=\int Hdt$
, is, in the slowroll approximation,
$$\begin{array}{c}N\approx \frac{8\pi}{{M}_{p}^{2}}{\int}_{{\phi}_{i}}^{{\phi}_{f}}\frac{V}{{V}^{\prime}}\left[1+\frac{V}{2\lambda}\right]d\phi .\end{array}$$ 
(206)

Braneworld effects at high energies increase the Hubble rate by a factor
$V/2\lambda $
, yielding more inflation between any two values of
$\phi $
for a given potential. Thus we can obtain a given number of efolds for a smaller initial inflaton value
${\phi}_{i}$
. For
$V\gg \lambda $
, Equation ( 206 ) becomes
$$\begin{array}{c}N\approx \frac{128{\pi}^{3}}{3{M}_{5}^{6}}{\int}_{{\phi}_{i}}^{{\phi}_{f}}\frac{{V}^{2}}{{V}^{\prime}}d\phi .\end{array}$$ 
(207)

The key test of any modified gravity theory during inflation will be the spectrum of perturbations produced due to quantum fluctuations of the fields about their homogeneous background values.
We will discuss braneworld cosmological perturbations in the next Section
6 . In general, perturbations on the brane are coupled to bulk metric perturbations, and the problem is very complicated.
However, on large scales on the brane, the density perturbations decouple from the bulk metric perturbations [
218,
191,
122,
102]
. For 1brane RStype models, there is no scalar zeromode of the bulk graviton, and in the extreme slowroll (de Sitter) limit, the massive scalar modes are heavy and stay in their vacuum state during inflation [
102]
. Thus it seems a reasonable approximation in slowroll to neglect the KK effects carried by
${\mathcal{\mathcal{E}}}_{\mu \nu}$
when computing the density perturbations.
Figure 5
: The relation between the inflaton mass
$m/{M}_{4}$
(
${M}_{4}\equiv {M}_{p}$
) and the brane tension
$(\lambda /{M}_{4}^{4}{)}^{1/4}$
necessary to satisfy the COBE constraints. The straight line is the approximation used in Equation ( 214 ), which at high energies is in excellent agreement with the exact solution, evaluated numerically in slowroll. (Figure taken from [
222]
.)
To quantify the amplitude of scalar (density) perturbations we evaluate the usual gaugeinvariant quantity
$$\begin{array}{c}\zeta \equiv \mathcal{\mathcal{R}}\frac{H}{\dot{\rho}}\delta \rho ,\end{array}$$ 
(208)

which reduces to the curvature perturbation
$\mathcal{\mathcal{R}}$
on uniform density hypersurfaces (
$\delta \rho =0$
).
This is conserved on large scales for purely adiabatic perturbations as a consequence of energy conservation (independently of the field equations) [
317]
. The curvature perturbation on uniform density hypersurfaces is given in terms of the scalar field fluctuations on spatially flat hypersurfaces
$\delta \phi $
by
$$\begin{array}{c}\zeta =H\frac{\delta \phi}{\dot{\phi}}.\end{array}$$ 
(209)

The field fluctuations at Hubble crossing (
$k=aH$
) in the slowroll limit are given by
$\langle \delta {\phi}^{2}\rangle \approx {\left(H/2\pi \right)}^{2}$
, a result for a massless field in de Sitter space that is also independent of the gravity theory [
317]
. For a single scalar field the perturbations are adiabatic and hence the curvature perturbation
$\zeta $
can be related to the density perturbations when modes reenter the Hubble scale during the matter dominated era which is given by
${A}_{s}^{2}=4\langle {\zeta}^{2}\rangle /25$
. Using the slowroll equations and Equation ( 209 ), this gives
$$\begin{array}{c}{A}_{s}^{2}\approx {\left(\frac{512\pi}{75{M}_{p}^{6}}\frac{{V}^{3}}{{V}^{\prime 2}}\right){\left[\frac{2\lambda +V}{2\lambda}\right]}^{3}}_{k=aH}.\end{array}$$ 
(210)

Thus the amplitude of scalar perturbations is increased relative to the standard result at a fixed value of
$\phi $
for a given potential.
The scaledependence of the perturbations is described by the spectral tilt
$$\begin{array}{c}{n}_{s}1\equiv \frac{dln{A}_{s}^{2}}{dlnk}\approx 4\epsilon +2\eta ,\end{array}$$ 
(211)

where the slowroll parameters are given in Equations ( 203 ) and ( 204 ). Because these slowroll parameters are both suppressed by an extra factor
$\lambda /V$
at high energies, we see that the spectral index is driven towards the Harrison–Zel'dovich spectrum,
${n}_{s}\to 1$
, as
$V/\lambda \to \infty $
; however, as explained below, this does not necessarily mean that the braneworld case is closer to scaleinvariance than the general relativity case.
As an example, consider the simplest chaotic inflation model
$V=\frac{1}{2}{m}^{2}{\phi}^{2}$
. Equation ( 206 ) gives the integrated expansion from
${\phi}_{i}$
to
${\phi}_{f}$
as
$$\begin{array}{c}N\approx \frac{2\pi}{{M}_{p}^{2}}\left({\phi}_{i}^{2}{\phi}_{f}^{2}\right)+\frac{{\pi}^{2}{m}^{2}}{3{M}_{5}^{6}}\left({\phi}_{i}^{4}{\phi}_{f}^{4}\right).\end{array}$$ 
(212)

The new highenergy term on the right leads to more inflation for a given initial inflaton value
${\phi}_{i}$
.
The standard chaotic inflation scenario requires an inflaton mass
$m\sim {10}^{13}GeV$
to match the observed level of anisotropies in the cosmic microwave background (see below). This corresponds to an energy scale
$\sim {10}^{16}GeV$
when the relevant scales left the Hubble scale during inflation, and also to an inflaton field value of order
$3{M}_{p}$
. Chaotic inflation has been criticised for requiring superPlanckian field values, since these can lead to nonlinear quantum corrections in the potential.
If the brane tension
$\lambda $
is much below
${10}^{16}GeV$
, corresponding to
${M}_{5}<{10}^{17}GeV$
, then the terms quadratic in the energy density dominate the modified Friedmann equation. In particular the condition for the end of inflation given in Equation ( 200 ) becomes
${\dot{\phi}}^{2}<\frac{2}{5}V$
. In the slowroll approximation (using Equations ( 201 ) and ( 202 ))
$\dot{\phi}\approx {M}_{5}^{3}/2\pi \phi $
, and this yields
$$\begin{array}{c}{\phi}_{end}^{4}\approx \frac{5}{4{\pi}^{2}}{\left(\frac{{M}_{5}}{m}\right)}^{2}{M}_{5}^{4}.\end{array}$$ 
(213)

In order to estimate the value of
$\phi $
when scales corresponding to largeangle anisotropies on the microwave background sky left the Hubble scale during inflation, we take
${N}_{cobe}\approx 55$
in Equation ( 212 ) and
${\phi}_{f}={\phi}_{end}$
. The second term on the right of Equation ( 212 ) dominates, and we obtain
$$\begin{array}{c}{\phi}_{cobe}^{4}\approx \frac{165}{{\pi}^{2}}{\left(\frac{{M}_{5}}{m}\right)}^{2}{M}_{5}^{4}.\end{array}$$ 
(214)

Imposing the COBE normalization on the curvature perturbations given by Equation ( 210 ) requires
$$\begin{array}{c}{A}_{s}\approx \left(\frac{8{\pi}^{2}}{45}\right)\frac{{m}^{4}{\phi}_{cobe}^{5}}{{M}_{5}^{6}}\approx 2\times {10}^{5}.\end{array}$$ 
(215)

Substituting in the value of
${\phi}_{cobe}$
given by Equation ( 214 ) shows that in the limit of strong brane corrections, observations require
$$\begin{array}{c}m\approx 5\times {10}^{5}{M}_{5},{\phi}_{cobe}\approx 3\times {10}^{2}{M}_{5}.\end{array}$$ 
(216)

Thus for
${M}_{5}<{10}^{17}GeV$
, chaotic inflation can occur for field values below the 4D Planck scale,
${\phi}_{cobe}<{M}_{p}$
, although still above the 5D scale
${M}_{5}$
. The relation determined by COBE constraints for arbitrary brane tension is shown in Figure 5 , together with the highenergy approximation used above, which provides an excellent fit at low brane tension relative to
${M}_{4}$
.
It must be emphasized that in comparing the highenergy braneworld case to the standard 4D case, we implicitly require the same potential energy. However, precisely because of the highenergy effects, largescale perturbations will be generated at different values of
$V$
than in the standard case, specifically at lower values of
$V$
, closer to the reheating minimum. Thus there are two competing effects, and it turns out that the shape of the potential determines which is the dominant effect [
203]
.
For the quadratic potential, the lower location on
$V$
dominates, and the spectral tilt is slightly further from scale invariance than in the standard case. The same holds for the quartic potential.
Data from WMAP and 2dF can be used to constrain inflationary models via their deviation from scale invariance, and the highenergy braneworld versions of the quadratic and quartic potentials are thus under more pressure from data than their standard counterparts [
203]
, as shown in Figure 6 .
Figure 6
: Constraints from WMAP data on inflation models with quadratic and quartic potentials, where
$R$
is the ratio of tensor to scalar amplitudes and
$n$
is the scalar spectral index. The high energy (H.E.) and low energy (L.E.) limits are shown, with intermediate energies in between, and the 1
$\sigma $
and 2
$\sigma $
contours are also shown. (Figure taken from [
203]
.)
Other perturbation modes have also been investigated:

∙
Highenergy inflation on the brane also generates a zeromode (4D graviton mode) of tensor perturbations, and stretches it to superHubble scales, as will be discussed below. This zeromode has the same qualitative features as in general relativity, remaining frozen at constant amplitude while beyond the Hubble horizon. Its amplitude is enhanced at high energies, although the enhancement is much less than for scalar perturbations [192] :
$$\begin{array}{ccc}{A}_{t}^{2}& \approx & \left(\frac{32V}{75{M}_{p}^{2}}\right)\left[\frac{3{V}^{2}}{4{\lambda}^{2}}\right],\end{array}$$ 
(217)

$$\begin{array}{ccc}\frac{{A}_{t}^{2}}{{A}_{s}^{2}}& \approx & \left(\frac{{M}_{p}^{2}}{16\pi}\frac{{{V}^{\prime}}^{2}}{{V}^{2}}\right)\left[\frac{6\lambda}{V}\right].\end{array}$$ 
(218)

Equation ( 218 ) means that braneworld effects suppress the largescale tensor contribution to CMB anisotropies. The tensor spectral index at high energy has a smaller magnitude than in general relativity,
$$\begin{array}{c}{n}_{t}=3\epsilon ,\end{array}$$ 
(219)

but remarkably the same consistency relation as in general relativity holds [145] :
$$\begin{array}{c}{n}_{t}=2\frac{{A}_{t}^{2}}{{A}_{s}^{2}}.\end{array}$$ 
(220)

This consistency relation persists when
${Z}_{2}$
symmetry is dropped [146] (and in a twobrane model with stabilized radion [118] ). It holds only to lowest order in slowroll, as in general relativity, but the reason for this [286] and the nature of the corrections [44] are not settled.
The massive KK modes of tensor perturbations remain in the vacuum state during slowroll inflation [
192,
121]
. The evolution of the superHubble zero mode is the same as in general relativity, so that highenergy braneworld effects in the early universe serve only to rescale the amplitude. However, when the zero mode reenters the Hubble horizon, massive KK modes can be excited.

∙
Vector perturbations in the bulk metric can support vector metric perturbations on the brane, even in the absence of matter perturbations (see the next Section 6 ). However, there is no normalizable zero mode, and the massive KK modes stay in the vacuum state during braneworld inflation [39] . Therefore, as in general relativity, we can neglect vector perturbations in inflationary cosmology.
Braneworld effects on largescale isocurvature perturbations in 2field inflation have also been considered [
12]
. Braneworld (p)reheating after inflation is discussed in [
309,
321,
5,
310,
67]
.
5.2 Braneworld instanton
The creation of an inflating braneworld can be modelled as a de Sitter instanton in a way that closely follows the 4D instanton, as shown in [
105]
. The instanton consists of two identical patches of
$Ad{S}_{5}$
joined together along a de Sitter brane (
$d{S}_{4}$
) with compact spatial sections. The instanton describes the “birth” of both the inflating brane and the bulk spacetime, which are together “created from nothing”, i.e., the point at the south pole of the de Sitter 4sphere. The Euclidean
$Ad{S}_{5}$
metric is
$$\begin{array}{c}{}^{\left(5\right)}d{s}_{euclid}^{2}=d{r}^{2}+{\ell}^{2}{sinh}^{2}(r/\ell )\left[d{\chi}^{2}+{sin}^{2}\chi d{\Omega}_{\left(3\right)}^{2}\right],\end{array}$$ 
(221)

where
$d{\Omega}_{\left(3\right)}^{2}$
is a 3sphere, and
$r\le {r}_{0}$
. The Euclidean instanton interpolates between
$r=0$
(“nothing”) and
$r={r}_{0}$
(the created universe), which is a spherical brane of radius
$$\begin{array}{c}{H}_{0}^{1}\equiv \ell sinh({r}_{0}/\ell ).\end{array}$$ 
(222)

After creation, the braneworld evolves according to the Lorentzian continuation,
$\chi \to i{H}_{0}t+\pi /2$
,
$$\begin{array}{c}{}^{\left(5\right)}d{s}^{2}=d{r}^{2}+(\ell {H}_{0}{)}^{2}{sinh}^{2}(r/\ell )\left[d{t}^{2}+{H}_{0}^{2}{cosh}^{2}\left({H}_{0}t\right)d{\Omega}_{\left(3\right)}^{2}\right]\end{array}$$ 
(223)

(see Figure 7 ).
Figure 7
: Braneworld instanton. (Figure taken from [
105]
.)
5.3 Models with nonempty bulk
The singlebrane cosmological model can be generalized to include stresses other than
${\Lambda}_{5}$
in the bulk:

∙
The simplest example arises from considering a charged bulk black hole, leading to the Reissner–Nordström
$Ad{S}_{5}$
bulk metric [17] . This has the form of Equation ( 178 ), with
$$\begin{array}{c}F\left(R\right)=K+\frac{{R}^{2}}{{\ell}^{2}}\frac{m}{{R}^{2}}+\frac{{q}^{2}}{{R}^{4}},\end{array}$$ 
(224)

where
$q$
is the “electric” charge parameter of the bulk black hole. The metric is a solution of the 5D Einstein–Maxwell equations, so that
${}^{\left(5\right)}{T}_{AB}$
in Equation ( 44 ) is the energymomentum tensor of a radial static 5D “electric” field. In order for the field lines to terminate on the boundary brane, the brane should carry a charge
$q$
. Since the RN
$Ad{S}_{5}$
metric is 4isotropic, it is still possible to embed a FRW brane in it, which is moving in the coordinates of Equation ( 178 ). The effect of the black hole charge on the brane arises via the junction conditions and leads to the modified Friedmann equation [17] ,
$$\begin{array}{c}{H}^{2}=\frac{{\kappa}^{2}}{3}\rho \left(1+\frac{\rho}{2\lambda}\right)+\frac{m}{{a}^{4}}\frac{{q}^{2}}{{a}^{6}}+\frac{1}{3}\Lambda \frac{K}{{a}^{2}}.\end{array}$$ 
(225)

The field lines that terminate on the brane imprint on the brane an effective negative energy density
$3{q}^{2}/\left({\kappa}^{2}{a}^{6}\right)$
, which redshifts like stiff matter (
$w=1$
). The negativity of this term introduces the possibility that at high energies it can bring the expansion rate to zero and cause a turnaround or bounce (but see [144] for problems with such bounces).
Apart from negativity, the key difference between this “dark stiff matter” and the dark radiation term
$m/{a}^{4}$
is that the latter arises from the bulk Weyl curvature via the
${\mathcal{\mathcal{E}}}_{\mu \nu}$
tensor, while the former arises from nonvacuum stresses in the bulk via the
${\mathcal{\mathcal{F}}}_{\mu \nu}$
tensor in Equation ( 63 ). The dark stiff matter does not arise from massive KK modes of the graviton.

∙
Another example is provided by the Vaidya–
$Ad{S}_{5}$
metric, which can be written after transforming to a new coordinate
$v=T+\int dR/F$
in Equation ( 178 ), so that
$v=const.$
are null surfaces, and
$$\begin{array}{ccc}{}^{\left(5\right)}d{s}^{2}& =& F(R,v)d{v}^{2}+2dvdR+{R}^{2}\left(\frac{d{r}^{2}}{1K{r}^{2}}+{r}^{2}d{\Omega}^{2}\right),\end{array}$$ 
(226)

$$\begin{array}{ccc}F(R,v)& =& K+\frac{{R}^{2}}{{\ell}^{2}}\frac{m\left(v\right)}{{R}^{2}}.\end{array}$$ 
(227)

This model has a moving FRW brane in a 4isotropic bulk (which is not static), with either a radiating bulk black hole (
$dm/dv<0$
), or a radiating brane (
$dm/dv>0$
) [53, 198, 197, 199] . The metric satisfies the 5D field equations ( 44 ) with a nullradiation energymomentum tensor,
$$\begin{array}{c}{}^{\left(5\right)}{T}_{AB}=\psi {k}_{A}{k}_{B},{k}_{A}{k}^{A}=0,{k}_{A}{u}^{A}=1,\end{array}$$ 
(228)

where
$\psi \propto dm/dv$
. It follows that
$$\begin{array}{c}{\mathcal{\mathcal{F}}}_{\mu \nu}={\kappa}_{5}^{2}\psi {h}_{\mu \nu}.\end{array}$$ 
(229)

In this case, the same effect, i.e., a varying mass parameter
$m$
, contributes to both
${\mathcal{\mathcal{E}}}_{\mu \nu}$
and
${\mathcal{\mathcal{F}}}_{\mu \nu}$
in the brane field equations. The modified Friedmann equation has the standard 1brane RStype form, but with a dark radiation term that no longer behaves strictly like radiation:
$$\begin{array}{c}{H}^{2}=\frac{{\kappa}^{2}}{3}\rho \left(1+\frac{\rho}{2\lambda}\right)+\frac{m\left(t\right)}{{a}^{4}}+\frac{1}{3}\Lambda \frac{K}{{a}^{2}}.\end{array}$$ 
(230)

By Equations ( 68 ) and ( 228 ), we arrive at the matter conservation equations,
$$\begin{array}{c}{\nabla}^{\nu}{T}_{\mu \nu}=2\psi {u}_{\mu}.\end{array}$$ 
(231)

This shows how the brane loses (
$\psi >0$
) or gains (
$\psi <0$
) energy in exchange with the bulk black hole. For an FRW brane, this equation reduces to
$$\begin{array}{c}\dot{\rho}+3H(\rho +p)=2\psi .\end{array}$$ 
(232)

The evolution of
$m$
is governed by the 4D contracted Bianchi identity, using Equation ( 229 ):
$$\begin{array}{c}{\nabla}^{\mu}{\mathcal{\mathcal{E}}}_{\mu \nu}=\frac{6{\kappa}^{2}}{\lambda}{\nabla}^{\mu}{\mathcal{S}}_{\mu \nu}+\frac{2}{3}\left[{\kappa}_{5}^{2}\left(\dot{\psi}+\Theta \psi \right)3{\kappa}^{2}\psi \right]{u}_{\mu}+\frac{2}{3}{\kappa}_{5}^{2}{\stackrel{\u20d7}{\nabla}}_{\mu}\psi .\end{array}$$ 
(233)

For an FRW brane, this yields
$$\begin{array}{c}{\dot{\rho}}_{\mathcal{\mathcal{E}}}+4H{\rho}_{\mathcal{\mathcal{E}}}=2\psi \frac{2}{3}\frac{{\kappa}_{5}^{2}}{{\kappa}^{2}}\left(\dot{\psi}+3H\psi \right),\end{array}$$ 
(234)

where
${\rho}_{\mathcal{\mathcal{E}}}=3m\left(t\right)/\left({\kappa}^{2}{a}^{4}\right)$
.

∙
A more complicated bulk metric arises when there is a selfinteracting scalar field
$\Phi $
in the bulk [225, 16, 236, 97, 194, 98, 35] . In the simplest case, when there is no coupling between the bulk field and brane matter, this gives
$$\begin{array}{c}{}^{\left(5\right)}{T}_{AB}={\Phi}_{,A}{\Phi}_{,B}{}^{\left(5\right)}{g}_{AB}\left[V(\Phi )+{\frac{1}{2}}^{\left(5\right)}{g}^{CD}{\Phi}_{,C}{\Phi}_{,D}\right],\end{array}$$ 
(235)

where
$\Phi (x,y)$
satisfies the 5D Klein–Gordon equation,
$$\begin{array}{c}{}^{\left(5\right)}\square \Phi {V}^{\prime}(\Phi )=0.\end{array}$$ 
(236)

The junction conditions on the field imply that
$$\begin{array}{c}{\partial}_{y}\Phi (x,0)=0.\end{array}$$ 
(237)

Then Equations ( 68 ) and ( 235 ) show that matter conservation continues to hold on the brane in this simple case:
$$\begin{array}{c}{\nabla}^{\nu}{T}_{\mu \nu}=0.\end{array}$$ 
(238)

From Equation ( 235 ) one finds that
$$\begin{array}{c}{\mathcal{\mathcal{F}}}_{\mu \nu}=\frac{1}{4{\kappa}_{5}^{2}}\left[4{\phi}_{,\mu}{\phi}_{,\nu}{g}_{\mu \nu}\left(3V\left(\phi \right)+\frac{5}{2}{g}^{\alpha \beta}{\phi}_{,\alpha}{\phi}_{,\beta}\right)\right],\end{array}$$ 
(239)

where
$$\begin{array}{c}\phi \left(x\right)=\Phi (x,0),\end{array}$$ 
(240)

so that the modified Friedmann equation becomes
$$\begin{array}{c}{H}^{2}=\frac{{\kappa}^{2}}{3}\rho \left(1+\frac{\rho}{2\lambda}\right)+\frac{m}{{a}^{4}}+\frac{{\kappa}_{5}^{2}}{6}\left[\frac{1}{2}{\dot{\phi}}^{2}+V\left(\phi \right)\right]+\frac{1}{3}\Lambda \frac{K}{{a}^{2}}.\end{array}$$ 
(241)

When there is coupling between brane matter and the bulk scalar field, then the Friedmann and conservation equations are more complicated [225, 16, 236, 97, 194, 98, 35] .
6 BraneWorld Cosmology: Perturbations
The background dynamics of braneworld cosmology are simple because the FRW symmetries simplify the bulk and rule out nonlocal effects. But perturbations on the brane immediately release the nonlocal KK modes. Then the 5D bulk perturbation equations must be solved in order to solve for perturbations on the brane. These 5D equations are partial differential equations for the 3D Fourier modes, with both initial and boundary conditions needed.
The theory of gaugeinvariant perturbations in braneworld cosmology has been extensively investigated and developed [
218,
191,
19,
222,
70,
226,
277,
258,
206,
122,
192,
121,
244,
246,
133,
169,
185,
311,
179,
245,
312,
166,
186,
134,
82,
109,
168,
37,
86,
252,
288,
275,
205,
57,
60,
75,
30,
41,
271,
40,
269,
201,
200]
and is qualitatively well understood. The key remaining task is integration of the coupled branebulk perturbation equations. Special cases have been solved, where these equations effectively decouple [
191,
19,
201,
200]
, and approximation schemes have recently been developed [
298,
320,
290,
299,
300,
177,
268,
38,
142,
91,
235,
237,
22]
for the more general cases where the coupled system must be solved. From the brane viewpoint, the bulk effects, i.e., the highenergy corrections and the KK modes, act as source terms for the brane perturbation equations. At the same time, perturbations of matter on the brane can generate KK modes (i.e., emit 5D gravitons into the bulk) which propagate in the bulk and can subsequently interact with the brane. This nonlocal interaction amongst the perturbations is at the core of the complexity of the problem. It can be elegantly expressed via integrodifferential equations [
244,
246]
, which take the form (assuming no incoming 5D gravitational waves)
$$\begin{array}{c}{A}_{k}\left(t\right)=\int d{t}^{\prime}\mathcal{G}(t,{t}^{\prime}){B}_{k}\left({t}^{\prime}\right),\end{array}$$ 
(242)

where
$\mathcal{G}$
is the bulk retarded Green's function evaluated on the brane, and
${A}_{k},{B}_{k}$
are made up of brane metric and matter perturbations and their (brane) derivatives, and include highenergy corrections to the background dynamics. Solving for the bulk Green's function, which then determines
$\mathcal{G}$
, is the core of the 5D problem.
We can isolate the KK anisotropic stress
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
as the term that must be determined from 5D equations. Once
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
is determined in this way, the perturbation equations on the brane form a closed system. The solution will be of the form (expressed in Fourier modes):
$$\begin{array}{c}{\pi}_{k}^{\mathcal{\mathcal{E}}}\left(t\right)\propto \int d{t}^{\prime}\mathcal{G}(t,{t}^{\prime}){F}_{k}\left({t}^{\prime}\right),\end{array}$$ 
(243)

where the functional
${F}_{k}$
will be determined by the covariant brane perturbation quantities and their derivatives. It is known in the case of a Minkowski background [
281]
, but not in the cosmological case.
The KK terms act as source terms modifying the standard general relativity perturbation equations, together with the highenergy corrections. For example, the linearization of the shear propagation equation (
125 ) yields
$$\begin{array}{c}{\dot{\sigma}}_{\mu \nu}+2H{\sigma}_{\mu \nu}+{E}_{\mu \nu}\frac{{\kappa}^{2}}{2}{\pi}_{\mu \nu}{\stackrel{\u20d7}{\nabla}}_{\langle \mu}{A}_{\nu \rangle}=\frac{{\kappa}^{2}}{2}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}\frac{{\kappa}^{2}}{4}(1+3w)\frac{\rho}{\lambda}{\pi}_{\mu \nu}.\end{array}$$ 
(244)

In 4D general relativity, the right hand side is zero. In the braneworld, the first source term on the right is the KK term, and the second term is the highenergy modification. The other modification is a straightforward highenergy correction of the background quantities
$H$
and
$\rho $
via the modified Friedmann equations.
As in 4D general relativity, there are various different, but essentially equivalent, ways to formulate linear cosmological perturbation theory. First I describe the covariant branebased approach.
6.1 1 + 3covariant perturbation equations on the brane
In the
$1+3$
covariant approach [
218,
201,
220]
, perturbative quantities are projected vectors,
${V}_{\mu}={V}_{\langle \mu \rangle}$
, and projected symmetric tracefree tensors,
${W}_{\mu \nu}={W}_{\langle \mu \nu \rangle}$
, which are gaugeinvariant since they vanish in the background. These are decomposed into (3D) scalar, vector, and tensor modes as
$$\begin{array}{ccc}{V}_{\mu}& =& {\stackrel{\u20d7}{\nabla}}_{\mu}V+{\overline{V}}_{\mu},\end{array}$$ 
(245)

$$\begin{array}{ccc}{W}_{\mu \nu}& =& {\stackrel{\u20d7}{\nabla}}_{\langle \mu}{\stackrel{\u20d7}{\nabla}}_{\nu \rangle}W+{\stackrel{\u20d7}{\nabla}}_{\langle \mu}{\overline{W}}_{\nu \rangle}+{\overline{W}}_{\mu \nu},\end{array}$$ 
(246)

where
${\overline{W}}_{\mu \nu}={\overline{W}}_{\langle \mu \nu \rangle}$
and an overbar denotes a (3D) transverse quantity,
$$\begin{array}{c}{\stackrel{\u20d7}{\nabla}}^{\mu}{\overline{V}}_{\mu}=0={\stackrel{\u20d7}{\nabla}}^{\nu}{\overline{W}}_{\mu \nu}.\end{array}$$ 
(247)

In a local inertial frame comoving with
${u}^{\mu}$
, i.e.,
${u}^{\mu}=(1,\stackrel{\u20d7}{0})$
, all time components may be set to zero:
${V}_{\mu}=(0,{V}_{i})$
,
${W}_{0\mu}=0$
,
${\stackrel{\u20d7}{\nabla}}_{\mu}=(0,{\stackrel{\u20d7}{\nabla}}_{i})$
.
Purely scalar perturbations are characterized by the fact that vectors and tensors are derived from scalar potentials, i.e.,
$$\begin{array}{c}{\overline{V}}_{\mu}={\overline{W}}_{\mu}={\overline{W}}_{\mu \nu}=0.\end{array}$$ 
(248)

Scalar perturbative quantities are formed from the potentials via the (3D) Laplacian, e.g.,
$\mathcal{V}={\stackrel{\u20d7}{\nabla}}^{\mu}{\stackrel{\u20d7}{\nabla}}_{\mu}V\equiv {\stackrel{\u20d7}{\nabla}}^{2}V$
. Purely vector perturbations are characterized by
$$\begin{array}{c}{V}_{\mu}={\overline{V}}_{\mu},{W}_{\mu \nu}={\stackrel{\u20d7}{\nabla}}_{\langle \mu}{\overline{W}}_{\nu \rangle},curl{\stackrel{\u20d7}{\nabla}}_{\mu}f=2\dot{f}{\omega}_{\mu},\end{array}$$ 
(249)

where
${\omega}_{\mu}$
is the vorticity, and purely tensor by
$$\begin{array}{c}{\stackrel{\u20d7}{\nabla}}_{\mu}f=0={V}_{\mu},{W}_{\mu \nu}={\overline{W}}_{\mu \nu}.\end{array}$$ 
(250)

The KK energy density produces a scalar mode
${\stackrel{\u20d7}{\nabla}}_{\mu}{\rho}_{\mathcal{\mathcal{E}}}$
(which is present even if
${\rho}_{\mathcal{\mathcal{E}}}=0$
in the background). The KK momentum density carries scalar and vector modes, and the KK anisotropic stress carries scalar, vector, and tensor modes:
$$\begin{array}{ccc}{q}_{\mu}^{\mathcal{\mathcal{E}}}& =& {\stackrel{\u20d7}{\nabla}}_{\mu}{q}^{\mathcal{\mathcal{E}}}+{\overline{q}}_{\mu}^{\mathcal{\mathcal{E}}},\end{array}$$ 
(251)

$$\begin{array}{ccc}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}& =& {\stackrel{\u20d7}{\nabla}}_{\langle \mu}{\stackrel{\u20d7}{\nabla}}_{\nu \rangle}{\pi}^{\mathcal{\mathcal{E}}}+{\stackrel{\u20d7}{\nabla}}_{\langle \mu}{\overline{\pi}}_{\nu \rangle}^{\mathcal{\mathcal{E}}}+{\overline{\pi}}_{\mu \nu}^{\mathcal{\mathcal{E}}}.\end{array}$$ 
(252)

Linearizing the conservation equations for a single adiabatic fluid, and the nonlocal conservation equations, we obtain
$$\begin{array}{ccc}\dot{\rho}+\Theta (\rho +p)& =& 0,\end{array}$$ 
(253)

$$\begin{array}{ccc}{c}_{s}^{2}{\stackrel{\u20d7}{\nabla}}_{\mu}+(\rho +p){A}_{\mu}& =& 0,\end{array}$$ 
(254)

$$\begin{array}{ccc}{\dot{\rho}}_{\mathcal{\mathcal{E}}}+\frac{4}{3}\Theta {\rho}_{\mathcal{\mathcal{E}}}+{\stackrel{\u20d7}{\nabla}}^{\mu}{q}_{\mu}^{\mathcal{\mathcal{E}}}& =& 0,\end{array}$$ 
(255)

$$\begin{array}{ccc}{\dot{q}}_{\mu}^{\mathcal{\mathcal{E}}}+4H{q}_{\mu}^{\mathcal{\mathcal{E}}}+\frac{1}{3}{\stackrel{\u20d7}{\nabla}}_{\mu}{\rho}_{\mathcal{\mathcal{E}}}+\frac{4}{3}{\rho}_{\mathcal{\mathcal{E}}}{A}_{\mu}+{\stackrel{\u20d7}{\nabla}}^{\nu}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}& =& \frac{(\rho +p)}{\lambda}{\stackrel{\u20d7}{\nabla}}_{\mu}\rho .\end{array}$$ 
(256)

Linearizing the remaining propagation and constraint equations leads to
$$\begin{array}{ccc}\dot{\Theta}+\frac{1}{3}{\Theta}^{2}{\stackrel{\u20d7}{\nabla}}^{\mu}{A}_{\mu}+\frac{1}{2}{\kappa}^{2}(\rho +3p)\Lambda & =& \frac{{\kappa}^{2}}{2}(2\rho +3p)\frac{\rho}{\lambda}{\kappa}^{2}{\rho}_{\mathcal{\mathcal{E}}},\end{array}$$ 
(257)

$$\begin{array}{ccc}{\dot{\omega}}_{\mu}+2H{\omega}_{\mu}+\frac{1}{2}curl{A}_{\mu}& =& 0,\end{array}$$ 
(258)

$$\begin{array}{ccc}{\dot{\sigma}}_{\mu \nu}+2H{\sigma}_{\mu \nu}+{E}_{\mu \nu}{\stackrel{\u20d7}{\nabla}}_{\langle \mu}{A}_{\nu \rangle}& =& \frac{{\kappa}^{2}}{2}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}},\end{array}$$ 
(259)

$$\begin{array}{ccc}{\dot{E}}_{\mu \nu}+3H{E}_{\mu \nu}curl{H}_{\mu \nu}+\frac{{\kappa}^{2}}{2}(\rho +p){\sigma}_{\mu \nu}& =& \frac{{\kappa}^{2}}{2}(\rho +p)\frac{\rho}{\lambda}{\sigma}_{\mu \nu}\end{array}$$  
$$\begin{array}{ccc}& & \frac{{\kappa}^{2}}{6}\left[4{\rho}_{\mathcal{\mathcal{E}}}{\sigma}_{\mu \nu}+3{\dot{\pi}}_{\mu \nu}^{\mathcal{\mathcal{E}}}+3H{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}+3{\stackrel{\u20d7}{\nabla}}_{\langle \mu}{q}_{\nu \rangle}^{\mathcal{\mathcal{E}}}\right],\end{array}$$ 
(260)

$$\begin{array}{ccc}{\dot{H}}_{\mu \nu}+3H{H}_{\mu \nu}+curl{E}_{\mu \nu}& =& \frac{{\kappa}^{2}}{2}curl{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}},\end{array}$$ 
(261)

$$\begin{array}{ccc}{\stackrel{\u20d7}{\nabla}}^{\mu}{\omega}_{\mu}& =& 0,\end{array}$$ 
(262)

$$\begin{array}{ccc}{\stackrel{\u20d7}{\nabla}}^{\nu}{\sigma}_{\mu \nu}curl{\omega}_{\mu}\frac{2}{3}{\stackrel{\u20d7}{\nabla}}_{\mu}\Theta & =& {q}_{\mu}^{\mathcal{\mathcal{E}}},\end{array}$$ 
(263)

$$\begin{array}{ccc}curl{\sigma}_{\mu \nu}+{\stackrel{\u20d7}{\nabla}}_{\langle \mu}{\omega}_{\nu \rangle}{H}_{\mu \nu}& =& 0,\end{array}$$ 
(264)

$$\begin{array}{ccc}{\stackrel{\u20d7}{\nabla}}^{\nu}{E}_{\mu \nu}\frac{{\kappa}^{2}}{3}{\stackrel{\u20d7}{\nabla}}_{\mu}\rho & =& \frac{{\kappa}^{2}}{3}\frac{\rho}{\lambda}{\stackrel{\u20d7}{\nabla}}_{\mu}\rho +\frac{{\kappa}^{2}}{6}\left[2{\stackrel{\u20d7}{\nabla}}_{\mu}{\rho}_{\mathcal{\mathcal{E}}}4H{q}_{\mu}^{\mathcal{\mathcal{E}}}3{\stackrel{\u20d7}{\nabla}}^{\nu}{\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}\right],\end{array}$$ 
(265)

$$\begin{array}{ccc}{\stackrel{\u20d7}{\nabla}}^{\nu}{H}_{\mu \nu}{\kappa}^{2}(\rho +p){\omega}_{\mu}& =& {\kappa}^{2}(\rho +p)\frac{\rho}{\lambda}{\omega}_{\mu}+\frac{{\kappa}^{2}}{6}\left[8{\rho}_{\mathcal{\mathcal{E}}}{\omega}_{\mu}3curl{q}_{\mu}^{\mathcal{\mathcal{E}}}\right].\end{array}$$ 
(266)

Equations ( 253 ), ( 255 ), and ( 257 ) do not provide gaugeinvariant equations for perturbed quantities, but their spatial gradients do.
These equations are the basis for a
$1+3$
covariant analysis of cosmological perturbations from the brane observer's viewpoint, following the approach developed in 4D general relativity (for a review, see [
92]
). The equations contain scalar, vector, and tensor modes, which can be separated out if desired. They are not a closed system of equations until
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
is determined by a 5D analysis of the bulk perturbations. An extension of the
$1+3$
covariant perturbation formalism to
$1+4$
dimensions would require a decomposition of the 5D geometric quantities along a timelike extension
${u}^{A}$
into the bulk of the brane 4velocity field
${u}^{\mu}$
, and this remains to be done. The
$1+3$
covariant perturbation formalism is incomplete until such a 5D extension is performed. The metricbased approach does not have this drawback.
6.2 Metricbased perturbations
An alternative approach to braneworld cosmological perturbations is an extension of the 4D metricbased gaugeinvariant theory [
170,
241]
. A review of this approach is given in [
40,
269]
. In an arbitrary gauge, and for a flat FRW background, the perturbed metric has the form
$$\begin{array}{c}{\delta}^{\left(5\right)}{g}_{AB}=\left[\begin{array}{ccccccc}& 2{N}^{2}\psi & & {A}^{2}({\partial}_{i}\mathcal{\mathcal{B}}{S}_{i})& & & N\alpha \\ & & {A}^{2}({\partial}_{j}\mathcal{\mathcal{B}}{S}_{j})& & {A}^{2}\left\{2\mathcal{\mathcal{R}}{\delta}_{ij}+2{\partial}_{i}{\partial}_{j}\mathcal{C}+2{\partial}_{(i}{F}_{j)}+{f}_{ij}\right\}& & & {A}^{2}({\partial}_{i}\beta {\chi}_{i})\\ & & N\alpha & & {A}^{2}({\partial}_{j}\beta {\chi}_{j})& & & 2\nu & \end{array}\right],\end{array}$$ 
(267)

where the background metric functions
$A,N$
are given by Equations ( 181 , 182 ). The scalars
$\psi ,\mathcal{\mathcal{R}},\mathcal{C},\alpha ,\beta ,\nu $
represent scalar perturbations. The vectors
${S}_{i}$
,
${F}_{i}$
, and
${\chi}_{i}$
are transverse, so that they represent 3D vector perturbations, and the tensor
${f}_{ij}$
is transverse traceless, representing 3D tensor perturbations.
In the Gaussian normal gauge, the brane coordinateposition remains fixed under perturbation,
$$\begin{array}{c}{}^{\left(5\right)}d{s}^{2}=\left[{g}_{\mu \nu}^{\left(0\right)}(x,y)+\delta {g}_{\mu \nu}(x,y)\right]d{x}^{\mu}d{x}^{\nu}+d{y}^{2},\end{array}$$ 
(268)

where
${g}_{\mu \nu}^{\left(0\right)}$
is the background metric, Equation ( 180 ). In this gauge, we have
$$\begin{array}{c}\alpha =\beta =\nu ={\chi}_{i}=0.\end{array}$$ 
(269)

In the 5D longitudinal gauge, one gets
$$\begin{array}{c}\mathcal{\mathcal{B}}+\dot{\mathcal{C}}=\prime =\beta +{\mathcal{C}}^{\prime}.\end{array}$$ 
(270)

In this gauge, and for an
$Ad{S}_{5}$
background, the metric perturbation quantities can all be expressed in terms of a “master variable”
$\Omega $
which obeys a wave equation [
244,
246]
. In the case of scalar perturbations, we have for example
$$\begin{array}{c}\mathcal{\mathcal{R}}=\frac{1}{6A}\left({\Omega}^{\prime \prime}\frac{1}{{N}^{2}}\ddot{\Omega}\frac{{\Lambda}_{5}}{3}\Omega \right),\end{array}$$ 
(271)

with similar expressions for the other quantities. All of the metric perturbation quantities are determined once a solution is found for the wave equation
$$\begin{array}{c}{\left(\frac{1}{N{A}^{3}}\dot{\Omega}\right)}^{\cdot}+\left(\frac{{\Lambda}_{5}}{6}+\frac{{k}^{2}}{{A}^{2}}\right)\frac{N}{{A}^{3}}\Omega ={\left(\frac{N}{{A}^{3}}{\Omega}^{\prime}\right)}^{\prime}.\end{array}$$ 
(272)

The junction conditions ( 62 ) relate the offbrane derivatives of metric perturbations to the matter perturbations:
$$\begin{array}{c}{\partial}_{y}\delta {g}_{\mu \nu}={\kappa}_{5}^{2}\left[\delta {T}_{\mu \nu}+\frac{1}{3}\left(\lambda {T}^{\left(0\right)}\right)\delta {g}_{\mu \nu}\frac{1}{3}{g}_{\mu \nu}^{\left(0\right)}\delta T\right],\end{array}$$ 
(273)

where
$$\begin{array}{ccc}\delta {T}_{0}^{0}& =& \delta \rho ,\end{array}$$ 
(274)

$$\begin{array}{ccc}\delta {T}_{i}^{0}& =& {a}^{2}{q}_{i},\end{array}$$ 
(275)

$$\begin{array}{ccc}\delta {T}_{j}^{i}& =& \delta p{\delta}_{j}^{i}+\delta {\pi}_{j}^{i}.\end{array}$$ 
(276)

For scalar perturbations in the Gaussian normal gauge, this gives
$$\begin{array}{ccc}{\partial}_{y}\psi (x,0)& =& \frac{{\kappa}_{5}^{2}}{6}(2\delta \rho +3\delta p),\end{array}$$ 
(277)

$$\begin{array}{ccc}{\partial}_{y}\mathcal{\mathcal{B}}(x,0)& =& {\kappa}_{5}^{2}\delta p,\end{array}$$ 
(278)

$$\begin{array}{ccc}{\partial}_{y}\mathcal{C}(x,0)& =& \frac{{\kappa}_{5}^{2}}{2}\delta \pi ,\end{array}$$ 
(279)

$$\begin{array}{ccc}{\partial}_{y}\mathcal{\mathcal{R}}(x,0)& =& \frac{{\kappa}_{5}^{2}}{6}\delta \rho {\partial}_{i}{\partial}^{i}\mathcal{C}(x,0),\end{array}$$ 
(280)

where
$\delta \pi $
is the scalar potential for the matter anisotropic stress,
$$\begin{array}{c}\delta {\pi}_{ij}={\partial}_{i}{\partial}_{j}\delta \pi \frac{1}{3}{\delta}_{ij}{\partial}_{k}{\partial}^{k}\delta \pi .\end{array}$$ 
(281)

The perturbed KK energymomentum tensor on the brane is given by
$$\begin{array}{ccc}\delta {\mathcal{\mathcal{E}}}_{0}^{0}& =& {\kappa}^{2}\delta {\rho}_{\mathcal{\mathcal{E}}},\end{array}$$ 
(282)

$$\begin{array}{ccc}\delta {\mathcal{\mathcal{E}}}_{i}^{0}& =& {\kappa}^{2}{a}^{2}{q}_{i}^{\mathcal{\mathcal{E}}},\end{array}$$ 
(283)

$$\begin{array}{ccc}\delta {\mathcal{\mathcal{E}}}_{j}^{i}& =& \frac{{\kappa}^{2}}{3}\delta {\rho}_{\mathcal{\mathcal{E}}}{\delta}_{j}^{i}\delta {\pi}_{j}^{\mathcal{\mathcal{E}}i}.\end{array}$$ 
(284)

The evolution of the bulk metric perturbations is determined by the perturbed 5D field equations in the vacuum bulk,
$$\begin{array}{c}{\delta}^{\left(5\right)}{G}_{B}^{A}=0.\end{array}$$ 
(285)

Then the matter perturbations on the brane enter via the perturbed junction conditions ( 273 ).
For example, for scalar perturbations in Gaussian normal gauge, we have
$$\begin{array}{c}{\delta}^{\left(5\right)}{G}_{i}^{y}={\partial}_{i}\left\{{\psi}^{\prime}+\left(\frac{{A}^{\prime}}{A}\frac{{N}^{\prime}}{N}\right)\psi 2{\mathcal{\mathcal{R}}}^{\prime}\frac{{A}^{2}}{2{N}^{2}}\left[{\dot{\mathcal{\mathcal{B}}}}^{\prime}+\left(\u25bd\frac{\dot{\mathcal{A}}}{\mathcal{A}}\frac{\dot{\mathcal{N}}}{\mathcal{N}}\right){\mathcal{\mathcal{B}}}^{\prime}\right]\right\}.\end{array}$$ 
(286)

For tensor perturbations (in any gauge), the only nonzero components of the perturbed Einstein tensor are
$$\begin{array}{c}{\delta}^{\left(5\right)}{G}_{j}^{i}=\frac{1}{2}\left\{\frac{1}{{N}^{2}}{\ddot{f}}_{j}^{i}+{{f}^{\prime \prime}}_{j}^{i}\frac{{k}^{2}}{{A}^{2}}{f}_{j}^{i}+\frac{1}{{N}^{2}}\left(\frac{\dot{N}}{N}3\frac{\dot{A}}{A}\right){\dot{f}}_{j}^{i}+\left(\frac{{N}^{\prime}}{N}+3\frac{{A}^{\prime}}{A}\right){{f}^{\prime}}_{j}^{i}\right\}.\end{array}$$ 
(287)

In the following, I will discuss various perturbation problems, using either a
$1+3$
covariant or a metricbased approach.
6.3 Density perturbations on large scales
In the covariant approach, we define matter density and expansion (velocity) perturbation scalars, as in 4D general relativity,
$$\begin{array}{c}\Delta =\frac{{a}^{2}}{\rho}{\stackrel{\u20d7}{\nabla}}^{2}\rho ,Z={a}^{2}{\stackrel{\u20d7}{\nabla}}^{2}\Theta .\end{array}$$ 
(288)

Then we can define dimensionless KK perturbation scalars [
218]
,
$$\begin{array}{c}U=\frac{{a}^{2}}{\rho}{\stackrel{\u20d7}{\nabla}}^{2}{\rho}_{\mathcal{\mathcal{E}}},Q=\frac{a}{\rho}{\stackrel{\u20d7}{\nabla}}^{2}{q}^{\mathcal{\mathcal{E}}},\Pi =\frac{1}{\rho}{\stackrel{\u20d7}{\nabla}}^{2}{\pi}^{\mathcal{\mathcal{E}}},\end{array}$$ 
(289)

where the scalar potentials
${q}^{\mathcal{\mathcal{E}}}$
and
${\pi}^{\mathcal{\mathcal{E}}}$
are defined by Equations ( 251 , 252 ). The KK energy density (dark radiation) produces a scalar fluctuation
$U$
which is present even if
${\rho}_{\mathcal{\mathcal{E}}}=0$
in the background, and which leads to a nonadiabatic (or isocurvature) mode, even when the matter perturbations are assumed adiabatic [
122]
. We define the total effective dimensionless entropy
${S}_{\text{tot}}$
via
$$\begin{array}{c}{p}_{\text{tot}}{S}_{\text{tot}}={a}^{2}{\stackrel{\u20d7}{\nabla}}^{2}{p}_{\text{tot}}{c}_{\text{tot}}^{2}{a}^{2}{\stackrel{\u20d7}{\nabla}}^{2}{\rho}_{\text{tot}},\end{array}$$ 
(290)

where
${c}_{\text{tot}}^{2}={\dot{p}}_{\text{tot}}/{\dot{\rho}}_{\text{tot}}$
is given in Equation ( 103 ). Then
$$\begin{array}{c}{S}_{\text{tot}}=\frac{9\left[{c}_{s}^{2}\frac{1}{3}+\left(\frac{2}{3}+w+{c}_{s}^{2}\right)\rho /\lambda \right]}{\left[3\right(1+w\left)\right(1+\rho /\lambda )+4{\rho}_{\mathcal{\mathcal{E}}}/\rho ][3w+3(1+2w)\rho /2\lambda +{\rho}_{\mathcal{\mathcal{E}}}/\rho ]}\left[\frac{4}{3}\frac{{\rho}_{\mathcal{\mathcal{E}}}}{\rho}\Delta (1+w)U\right].\end{array}$$ 
(291)

If
${\rho}_{\mathcal{\mathcal{E}}}=0$
in the background, then
$U$
is an isocurvature mode:
${S}_{\text{tot}}\propto (1+w)U$
. This isocurvature mode is suppressed during slowroll inflation, when
$1+w\approx 0$
.
If
${\rho}_{\mathcal{\mathcal{E}}}\ne 0$
in the background, then the weighted difference between
$U$
and
$\Delta $
determines the isocurvature mode:
${S}_{\text{tot}}\propto (4{\rho}_{\mathcal{\mathcal{E}}}/3\rho )\Delta (1+w)U$
. At very high energies,
$\rho \gg \lambda $
, the entropy is suppressed by the factor
$\lambda /\rho $
.
The density perturbation equations on the brane are derived by taking the spatial gradients of Equations (
253 ), ( 255 ), and ( 257 ), and using Equations ( 254 ) and ( 256 ). This leads to [
122]
$$\begin{array}{ccc}\dot{\Delta}& =& 3wH\Delta (1+w)Z,\end{array}$$ 
(292)

$$\begin{array}{ccc}\dot{Z}& =& 2HZ\left(\frac{{c}_{s}^{2}}{1+w}\right){\stackrel{\u20d7}{\nabla}}^{2}\Delta {\kappa}^{2}\rho U\frac{1}{2}{\kappa}^{2}\rho \left[1+(4+3w)\frac{\rho}{\lambda}\left(\frac{4{c}_{s}^{2}}{1+w}\right)\frac{{\rho}_{\mathcal{\mathcal{E}}}}{\rho}\right]\Delta ,\end{array}$$ 
(293)

$$\begin{array}{ccc}\dot{U}& =& (3w1)HU+\left(\frac{4{c}_{s}^{2}}{1+w}\right)\left(\frac{{\rho}_{\mathcal{\mathcal{E}}}}{\rho}\right)H\Delta \left(\frac{4{\rho}_{\mathcal{\mathcal{E}}}}{3\rho}\right)Za{\stackrel{\u20d7}{\nabla}}^{2}Q,\end{array}$$ 
(294)

$$\begin{array}{ccc}\dot{Q}& =& (3w1)HQ\frac{1}{3a}U\frac{2}{3}a{\stackrel{\u20d7}{\nabla}}^{2}\Pi +\frac{1}{3\mu}\left[\left(\frac{4{c}_{s}^{2}}{1+w}\right)\frac{{\rho}_{\mathcal{\mathcal{E}}}}{\rho}3(1+w)\frac{\rho}{\lambda}\right]\Delta .\end{array}$$ 
(295)

The KK anisotropic stress term
$\Pi $
occurs only via its Laplacian,
${\stackrel{\u20d7}{\nabla}}^{2}\Pi $
. If we can neglect this term on large scales, then the system of density perturbation equations closes on superHubble scales [
218]
. An equivalent statement applies to the largescale curvature perturbations [
191]
. KK effects then introduce two new isocurvature modes on large scales (associated with
$U$
and
$Q$
), and they modify the evolution of the adiabatic modes as well [
122,
201]
.
Thus on large scales the system of brane equations is closed, and we can determine the density perturbations without solving for the bulk metric perturbations.
Figure 8
: The evolution of the covariant variable
$\Phi $
, defined in Equation ( 298 ) (and not to be confused with the Bardeen potential), along a fundamental worldline. This is a mode that is well beyond the Hubble horizon at
$N=0$
, about 50 efolds before inflation ends, and remains superHubble through the radiation era. A smooth transition from inflation to radiation is modelled by
$w=\frac{1}{3}\left[\right(2\frac{3}{2}\epsilon )tanh(N50)(1\frac{3}{2}\epsilon \left)\right]$
, where
$\epsilon $
is a small positive parameter (chosen as
$\epsilon =0.1$
in the plot). Labels on the curves indicate the value of
${\rho}_{0}/\lambda $
, so that the general relativistic solution is the dashed curve (
${\rho}_{0}/\lambda =0$
). (Figure taken from [
122]
.)
We can simplify the system as follows. The 3Ricci tensor defined in Equation ( 134 ) leads to a scalar covariant curvature perturbation variable,
$$\begin{array}{c}C\equiv {a}^{4}{\stackrel{\u20d7}{\nabla}}^{2}{R}^{\perp}=4{a}^{2}HZ+2{\kappa}^{2}{a}^{2}\rho \left(1+\frac{\rho}{2\lambda}\right)\Delta +2{\kappa}^{2}{a}^{2}\rho U.\end{array}$$ 
(296)

It follows that
$C$
is locally conserved (along
${u}^{\mu}$
flow lines):
$$\begin{array}{c}C={C}_{0},{\dot{C}}_{0}=0.\end{array}$$ 
(297)

We can further simplify the system of equations via the variable
$$\begin{array}{c}\Phi ={\kappa}^{2}{a}^{2}\rho \Delta .\end{array}$$ 
(298)

This should not be confused with the Bardeen metric perturbation variable
${\Phi}_{H}$
, although it is the covariant analogue of
${\Phi}_{H}$
in the general relativity limit. In the braneworld, highenergy and KK effects mean that
${\Phi}_{H}$
is a complicated generalization of this expression [
201]
involving
$\Pi $
, but the simple
$\Phi $
above is still useful to simplify the system of equations. Using these new variables, we find the closed system for largescale perturbations:
$$\begin{array}{ccc}\dot{\Phi}& =& H\left[1+(1+w)\frac{{\kappa}^{2}\rho}{2{H}^{2}}\left(1+\frac{\rho}{\lambda}\right)\right]\Phi \left[(1+w)\frac{{a}^{2}{\kappa}^{4}{\rho}^{2}}{2H}\right]U+\left[(1+w)\frac{{\kappa}^{2}\rho}{4H}\right]{C}_{0},\end{array}$$ 
(299)

$$\begin{array}{ccc}\dot{U}& =& H\left[13w+\frac{2{\kappa}^{2}{\rho}_{\mathcal{\mathcal{E}}}}{3{H}^{2}}\right]U\frac{2{\rho}_{\mathcal{\mathcal{E}}}}{3{a}^{2}H\rho}\left[1+\frac{\rho}{\lambda}\frac{6{c}_{s}^{2}{H}^{2}}{(1+w){\kappa}^{2}\rho}\right]\Phi +\left[\frac{{\rho}_{\mathcal{\mathcal{E}}}}{3{a}^{2}H\rho}\right]{C}_{0}.\end{array}$$ 
(300)

If there is no dark radiation in the background,
${\rho}_{\mathcal{\mathcal{E}}}=0$
, then
$$\begin{array}{c}U={U}_{0}exp\left(\int (13w)dN\right),\end{array}$$ 
(301)

and the above system reduces to a single equation for
$\Phi $
. At low energies, and for constant
$w$
, the nondecaying attractor is the general relativity solution,
$$\begin{array}{c}{\Phi}_{low}\approx \frac{3(1+w)}{2(5+3w)}{C}_{0}.\end{array}$$ 
(302)

At very high energies, for
$w\ge \frac{1}{3}$
, we get
$$\begin{array}{c}{\Phi}_{high}\to \frac{3}{2}\frac{\lambda}{{\rho}_{0}}(1+w)\left[\frac{{C}_{0}}{7+6w}\frac{2{\stackrel{~}{U}}_{0}}{5+6w}\right],\end{array}$$ 
(303)

where
${\stackrel{~}{U}}_{0}={\kappa}^{2}{a}_{0}^{2}{\rho}_{0}{U}_{0}$
, so that the isocurvature mode has an influence on
$\Phi $
. Initially,
$\Phi $
is suppressed by the factor
$\lambda /{\rho}_{0}$
, but then it grows, eventually reaching the attractor value in Equation ( 302 ). For slowroll inflation, when
$1+w\sim \epsilon $
, with
$0<\epsilon \ll 1$
and
${H}^{1}\left\dot{\epsilon}\right=\left{\epsilon}^{\prime}\right\ll 1$
, we get
$$\begin{array}{c}{\Phi}_{high}\sim \frac{3}{2}\epsilon \frac{\lambda}{{\rho}_{0}}{C}_{0}{e}^{3\epsilon N},\end{array}$$ 
(304)

where
$N=ln(a/{a}_{0})$
, so that
$\Phi $
has a growingmode in the early universe. This is different from general relativity, where
$\Phi $
is constant during slowroll inflation. Thus more amplification of
$\Phi $
can be achieved than in general relativity, as discussed above. This is illustrated for a toy model of inflationtoradiation in Figure 8 . The early (growing) and late time (constant) attractor solutions are seen explicitly in the plots.
Figure 9
: The evolution of
$\Phi $
in the radiation era, with dark radiation present in the background. (Figure taken from [
131]
.)
The presence of dark radiation in the background introduces new features. In the radiation era (
$w=\frac{1}{3}$
), the nondecaying lowenergy attractor becomes [
131]
$$\begin{array}{ccc}{\Phi}_{low}& \approx & \frac{{C}_{0}}{3}(1\alpha ),\end{array}$$ 
(305)

$$\begin{array}{ccc}\alpha & =& \frac{{\rho}_{\mathcal{\mathcal{E}}}}{\rho}\lesssim 0.05.\end{array}$$ 
(306)

The dark radiation serves to reduce the final value of
$\Phi $
, leaving an imprint on
$\Phi $
, unlike the
${\rho}_{\mathcal{\mathcal{E}}}=0$
case, Equation ( 302 ). In the very high energy limit,
$$\begin{array}{c}{\Phi}_{high}\to \frac{\lambda}{{\rho}_{0}}\left[\frac{2}{9}{C}_{0}\frac{4}{7}{\stackrel{~}{U}}_{0}\right]+16\alpha {\left(\frac{\lambda}{{\rho}_{0}}\right)}^{2}\left[\frac{{C}_{0}}{273}\frac{4{\stackrel{~}{U}}_{0}}{539}\right].\end{array}$$ 
(307)

Thus
$\Phi $
is initially suppressed, then begins to grow, as in the nodarkradiation case, eventually reaching an attractor which is less than the nodarkradiation attractor. This is confirmed by the numerical integration shown in Figure 9 .
6.4 Curvature perturbations and the Sachs–Wolfe effect
The curvature perturbation
$\mathcal{\mathcal{R}}$
on uniform density surfaces is defined in Equation ( 267 ). The associated gaugeinvariant quantity
$$\begin{array}{c}\zeta =\mathcal{\mathcal{R}}+\frac{\delta \rho}{3(\rho +p)}\end{array}$$ 
(308)

may be defined for matter on the brane. Similarly, for the Weyl “fluid” if
${\rho}_{\mathcal{\mathcal{E}}}\ne 0$
in the background, the curvature perturbation on hypersurfaces of uniform dark energy density is
$$\begin{array}{c}{\zeta}_{\mathcal{\mathcal{E}}}=\mathcal{\mathcal{R}}+\frac{\delta {\rho}_{\mathcal{\mathcal{E}}}}{4{\rho}_{\mathcal{\mathcal{E}}}}.\end{array}$$ 
(309)

On large scales, the perturbed dark energy conservation equation is [
191]
$$\begin{array}{c}(\delta {\rho}_{\mathcal{\mathcal{E}}}{)}^{\cdot}+4H\delta {\rho}_{\mathcal{\mathcal{E}}}+4{\rho}_{\mathcal{\mathcal{E}}}\dot{\mathcal{\mathcal{R}}}=\prime ,\end{array}$$ 
(310)

which leads to
$$\begin{array}{c}{\dot{\zeta}}_{\mathcal{\mathcal{E}}}=0.\end{array}$$ 
(311)

For adiabatic matter perturbations, by the perturbed matter energy conservation equation,
$$\begin{array}{c}(\delta \rho {)}^{\cdot}+3H(\delta \rho +\delta p)+3(\rho +p)\dot{\mathcal{\mathcal{R}}}=\prime ,\end{array}$$ 
(312)

we find
$$\begin{array}{c}\dot{\zeta}=0.\end{array}$$ 
(313)

This is independent of braneworld modifications to the field equations, since it depends on energy conservation only. For the total, effective fluid, the curvature perturbation is defined as follows [
191]
:
If
${\rho}_{\mathcal{\mathcal{E}}}\ne 0$
in the background, we have
$$\begin{array}{c}{\zeta}_{\text{tot}}=\zeta +\left[\frac{4{\rho}_{\mathcal{\mathcal{E}}}}{3(\rho +p)(1+\rho /\lambda )+4{\rho}_{\mathcal{\mathcal{E}}}}\right]\left({\zeta}_{\mathcal{\mathcal{E}}}\zeta \right),\end{array}$$ 
(314)

and if
${\rho}_{\mathcal{\mathcal{E}}}=0$
in the background, we get
$$\begin{array}{ccc}{\zeta}_{\text{tot}}& =& \zeta +\frac{\delta {\rho}_{\mathcal{\mathcal{E}}}}{3(\rho +p)(1+\rho /\lambda )}\end{array}$$ 
(315)

$$\begin{array}{ccc}\delta {\rho}_{\mathcal{\mathcal{E}}}& =& \frac{\delta {C}_{\mathcal{\mathcal{E}}}}{{a}^{4}},\end{array}$$ 
(316)

where
$\delta {C}_{\mathcal{\mathcal{E}}}$
is a constant. It follows that the curvature perturbations on large scales, like the density perturbations, can be found on the brane without solving for the bulk metric perturbations.
Note that
${\dot{\zeta}}_{tot}\ne 0$
even for adiabatic matter perturbations; for example, if
${\rho}_{\mathcal{\mathcal{E}}}=0$
in the background, then
$$\begin{array}{c}{\dot{\zeta}}_{tot}=H\left({c}_{\text{tot}}^{2}\frac{1}{3}\right)\frac{\delta {\rho}_{\mathcal{\mathcal{E}}}}{(\rho +p)(1+\rho /\lambda )}.\end{array}$$ 
(317)

The KK effects on the brane contribute a nonadiabatic mode, although
${\dot{\zeta}}_{tot}\to 0$
at low energies.
Although the density and curvature perturbations can be found on superHubble scales, the Sachs–Wolfe effect requires
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
in order to translate from density/curvature to metric perturbations.
In the 4D longitudinal gauge of the metric perturbation formalism, the gaugeinvariant curvature and metric perturbations on large scales are related by
$$\begin{array}{ccc}{\zeta}_{\text{tot}}& =& \mathcal{\mathcal{R}}\frac{H}{\dot{H}}\left(\frac{\dot{\mathcal{\mathcal{R}}}}{H}\psi \right),\end{array}$$ 
(318)

$$\begin{array}{ccc}\mathcal{\mathcal{R}}+\psi & =& {\kappa}^{2}{a}^{2}\delta {\pi}_{\mathcal{\mathcal{E}}},\end{array}$$ 
(319)

where the radiation anisotropic stress on large scales is neglected, as in general relativity, and
$\delta {\pi}_{\mathcal{\mathcal{E}}}$
is the scalar potential for
${\pi}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
, equivalent to the covariant quantity
$\Pi $
defined in Equation ( 289 ).
In 4D general relativity, the right hand side of Equation (
319 ) is zero. The (nonintegrated) Sachs–Wolfe formula has the same form as in general relativity:
$$\begin{array}{c}\frac{\delta T}{T}{}_{now}=({\zeta}_{rad}+\psi \mathcal{\mathcal{R}}){}_{dec}.\end{array}$$ 
(320)

The braneworld corrections to the general relativistic Sachs–Wolfe effect are then given by [
191]
$$\begin{array}{c}\frac{\delta T}{T}={\left(\frac{\delta T}{T}\right)}_{gr}\frac{8}{3}\left(\frac{{\rho}_{rad}}{{\rho}_{cdm}}\right){S}_{\mathcal{\mathcal{E}}}{\kappa}^{2}{a}^{2}\delta {\pi}_{\mathcal{\mathcal{E}}}+\frac{2{\kappa}^{2}}{{a}^{5/2}}\int da{a}^{7/2}\delta {\pi}_{\mathcal{\mathcal{E}}},\end{array}$$ 
(321)

where
${S}_{\mathcal{\mathcal{E}}}$
is the KK entropy perturbation (determined by
$\delta {\rho}_{\mathcal{\mathcal{E}}}$
). The KK term
$\delta {\pi}_{\mathcal{\mathcal{E}}}$
cannot be determined by the 4D brane equations, so that
$\delta T/T$
cannot be evaluated on large scales without solving the 5D equations. (Equation ( 321 ) has been generalized to a 2brane model, in which the radion makes a contribution to the Sachs–Wolfe effect [
176]
.) The presence of the KK (Weyl, dark) component has essentially two possible effects:

∙
A contribution from the KK entropy perturbation
${S}_{\mathcal{\mathcal{E}}}$
that is similar to an extra isocurvature contribution.

∙
The KK anisotropic stress
$\delta {\pi}_{\mathcal{\mathcal{E}}}$
also contributes to the CMB anisotropies. In the absence of anisotropic stresses, the curvature perturbation
${\zeta}_{\text{tot}}$
would be sufficient to determine the metric perturbation
$\mathcal{\mathcal{R}}$
and hence the largeangle CMB anisotropies via Equations ( 318 , 319 , 320 ). However, bulk gravitons generate anisotropic stresses which, although they do not affect the largescale curvature perturbation
${\zeta}_{\text{tot}}$
, can affect the relation between
${\zeta}_{\text{tot}}$
,
$\mathcal{\mathcal{R}}$
, and
$\psi $
, and hence can affect the CMB anisotropies at large angles.
A simple phenomenological approximation to
$\delta {\pi}_{\mathcal{\mathcal{E}}}$
on large scales is discussed in [
19]
, and the Sachs–Wolfe effect is estimated as
$$\begin{array}{c}\frac{\delta T}{T}\sim {\left(\frac{\delta {\pi}_{\mathcal{\mathcal{E}}}}{\rho}\right)}_{in}{\left(\frac{{t}_{eq}}{{t}_{dec}}\right)}^{2/3}\left[\frac{ln({t}_{in}/{t}_{4})}{ln({t}_{eq}/{t}_{4})}\right],\end{array}$$ 
(322)

where
${t}_{4}$
is the 4D Planck time, and
${t}_{in}$
is the time when the KK anisotropic stress is induced on the brane, which is expected to be of the order of the 5D Planck time.
A selfconsistent approximation is developed in [
177]
, using the lowenergy 2brane approximation [
298,
320,
290,
299,
300]
to find an effective 4D form for
${\mathcal{\mathcal{E}}}_{\mu \nu}$
and hence for
$\delta {\pi}_{\mathcal{\mathcal{E}}}$
. This is discussed below.
6.5 Vector perturbations
The vorticity propagation equation on the brane is the same as in general relativity,
$$\begin{array}{c}{\dot{\omega}}_{\mu}+2H{\omega}_{\mu}=\frac{1}{2}curl{A}_{\mu}.\end{array}$$ 
(323)

Taking the
$curl$
of the conservation equation ( 106 ) (for the case of a perfect fluid,
${q}_{\mu}=0={\pi}_{\mu \nu}$
), and using the identity in Equation ( 249 ), one obtains
$$\begin{array}{c}curl{A}_{\mu}=6H{c}_{s}^{2}{\omega}_{\mu}\end{array}$$ 
(324)

as in general relativity, so that Equation ( 323 ) becomes
$$\begin{array}{c}{\dot{\omega}}_{\mu}+\left(23{c}_{s}^{2}\right)H{\omega}_{\mu}=0,\end{array}$$ 
(325)

which expresses the conservation of angular momentum. In general relativity, vector perturbations vanish when the vorticity is zero. By contrast, in braneworld cosmology, bulk KK effects can source vector perturbations even in the absence of vorticity [
219]
. This can be seen via the divergence equation for the magnetic part
${H}_{\mu \nu}$
of the 4D Weyl tensor on the brane,
$$\begin{array}{c}{\stackrel{\u20d7}{\nabla}}^{2}{\overline{H}}_{\mu}=2{\kappa}^{2}(\rho +p)\left[1+\frac{\rho}{\lambda}\right]{\omega}_{\mu}+\frac{4}{3}{\kappa}^{2}{\rho}_{\mathcal{\mathcal{E}}}{\omega}_{\mu}\frac{1}{2}{\kappa}^{2}curl{\overline{q}}_{\mu}^{\mathcal{\mathcal{E}}},\end{array}$$ 
(326)

where
${H}_{\mu \nu}={\stackrel{\u20d7}{\nabla}}_{\langle \mu}{\overline{H}}_{\nu \rangle}$
. Even when
${\omega}_{\mu}=0$
, there is a source for gravimagnetic terms on the brane from the KK quantity
$curl{\overline{q}}_{\mu}^{\mathcal{\mathcal{E}}}$
.
We define covariant dimensionless vector perturbation quantities for the vorticity and the KK gravivector term:
$$\begin{array}{c}{\overline{\alpha}}_{\mu}=a{\omega}_{\mu},{\overline{\beta}}_{\mu}=\frac{a}{\rho}curl{\overline{q}}_{\mu}^{\mathcal{\mathcal{E}}}.\end{array}$$ 
(327)

On large scales, we can find a closed system for these vector perturbations on the brane [
219]
:
$$\begin{array}{ccc}{\dot{\overline{\alpha}}}_{\mu}+\left(13{c}_{s}^{2}\right)H{\overline{\alpha}}_{\mu}& =& 0,\end{array}$$ 
(328)

$$\begin{array}{ccc}{\dot{\overline{\beta}}}_{\mu}+(13w)H{\overline{\beta}}_{\mu}& =& \frac{2}{3}H\left[4\left(3{c}_{s}^{2}1\right)\frac{{\rho}_{\mathcal{\mathcal{E}}}}{\rho}9(1+w{)}^{2}\frac{\rho}{\lambda}\right]{\overline{\alpha}}_{\mu}.\end{array}$$ 
(329)

Thus we can solve for
${\overline{\alpha}}_{\mu}$
and
${\overline{\beta}}_{\mu}$
on superHubble scales, as for density perturbations. Vorticity in the brane matter is a source for the KK vector perturbation
${\overline{\beta}}_{\mu}$
on large scales. Vorticity decays unless the matter is ultrarelativistic or stiffer (
$w\ge \frac{1}{3}$
), and this source term typically provides a decaying mode. There is another pure KK mode, independent of vorticity, but this mode decays like vorticity. For
$w\equiv p/\rho =const.$
, the solutions are
$$\begin{array}{ccc}{\overline{\alpha}}_{\mu}& =& {b}_{\mu}{\left(\frac{a}{{a}_{0}}\right)}^{3w1},\end{array}$$ 
(330)

$$\begin{array}{ccc}{\overline{\beta}}_{\mu}& =& {c}_{\mu}{\left(\frac{a}{{a}_{0}}\right)}^{3w1}+{b}_{\mu}\left[\frac{8{\rho}_{\mathcal{\mathcal{E}}0}}{3{\rho}_{0}}{\left(\frac{a}{{a}_{0}}\right)}^{2(3w1)}+2(1+w)\frac{{\rho}_{0}}{\lambda}{\left(\frac{a}{{a}_{0}}\right)}^{4}\right],\end{array}$$ 
(331)

where
${\dot{b}}_{\mu}=0={\dot{c}}_{\mu}$
.
Inflation will redshift away the vorticity and the KK mode. Indeed, the massive KK vector modes are not excited during slowroll inflation [
40,
269]
.
6.6 Tensor perturbations
The covariant description of tensor modes on the brane is via the shear, which satisfies the wave equation [
219]
$$\begin{array}{c}{\stackrel{\u20d7}{\nabla}}^{2}{\overline{\sigma}}_{\mu \nu}{\ddot{\overline{\sigma}}}_{\mu \nu}5H{\dot{\overline{\sigma}}}_{\mu \nu}\left[2\Lambda +\frac{1}{2}{\kappa}^{2}\left(\rho 3p(\rho +3p)\frac{\rho}{\lambda}\right)\right]{\overline{\sigma}}_{\mu \nu}{\kappa}^{2}\left({\dot{\overline{\pi}}}_{\mu \nu}^{\mathcal{\mathcal{E}}}+2H{\overline{\pi}}_{\mu \nu}^{\mathcal{\mathcal{E}}}\right).\end{array}$$ 
(332)

Unlike the density and vector perturbations, there is no closed system on the brane for large scales. The KK anisotropic stress
${\overline{\pi}}_{\mu \nu}^{\mathcal{\mathcal{E}}}$
is an unavoidable source for tensor modes on the brane.
Thus it is necessary to use the 5D metricbased formalism. This is the subject of the next Section
7 .
7 Gravitational Wave Perturbations in BraneWorld Cosmology
The tensor perturbations are given by Equation ( 267 ), i.e., (for a flat background brane),
$$\begin{array}{c}{}^{\left(5\right)}d{s}^{2}={N}^{2}(t,y)d{t}^{2}+{A}^{2}(t,y)\left[{\delta}_{ij}+{f}_{ij}\right]d{x}^{i}d{x}^{j}+d{y}^{2}.\end{array}$$ 
(333)

The transverse traceless
${f}_{ij}$
satisfies Equation ( 287 ), which implies, on splitting
${f}_{ij}$
into Fourier modes with amplitude
$f(t,y)$
,
$$\begin{array}{c}\frac{1}{{N}^{2}}\left[\ddot{f}+\left(3\frac{\dot{A}}{A}\frac{\dot{N}}{N}\right)\dot{f}\right]+\frac{{k}^{2}}{{A}^{2}}f={f}^{\prime \prime}+\left(3\frac{{A}^{\prime}}{A}+\frac{{N}^{\prime}}{N}\right){f}^{\prime}.\end{array}$$ 
(334)

By the transverse traceless part of Equation ( 273 ), the boundary condition is
$$\begin{array}{c}{f}_{ij}^{\prime}{}_{brane}={\overline{\pi}}_{ij},\end{array}$$ 
(335)

where
${\overline{\pi}}_{ij}$
is the tensor part of the anisotropic stress of matterradiation on the brane.
The wave equation (
334 ) cannot be solved analytically except if the background metric functions are separable, and this only happens for maximally symmetric branes, i.e., branes with constant Hubble rate
${H}_{0}$
. This includes the RS case
${H}_{0}=0$
already treated in Section 2 . The cosmologically relevant case is the de Sitter brane,
${H}_{0}>0$
. We can calculate the spectrum of gravitational waves generated during brane inflation [
192,
121,
102,
167]
, if we approximate slowroll inflation by a succession of de Sitter phases. The metric for a de Sitter brane
$d{S}_{4}$
in
$Ad{S}_{5}$
is given by Equations ( 180 , 181 , 182 ) with