Brane cosmology: an introduction
Abstract
These notes give an introductory review on brane cosmology. This subject deals with the cosmological behaviour of a braneuniverse, i.e. a threedimensional space, where ordinary matter is confined, embedded in a higher dimensional spacetime. In the tractable case of a fivedimensional bulk spacetime, the brane (modified) Friedmann equation is discussed in detail, and various other aspects are presented, such as cosmological perturbations, bulk scalar fields and systems with several branes.
February 4, 2021
1 Introduction
It has been recently suggested that there might exist some extra spatial dimensions, not in the traditional KaluzaKlein sense where the extradimensions are compactified on a small enough radius to evade detection in the form of KaluzaKlein modes, but in a setting where the extra dimensions could be large, under the assumption that ordinary matter is confined onto a threedimensional subspace, called brane (more precisely ‘3brane’, referring to the three spatial dimensions) embedded in a larger space, called bulk.
Altough the idea in itself is not completely new [1], the fact that it might be connected to recent string theory developments has suscitated a renewed interest. In this respect, an inspiring input has been the model suggested by Horava and Witten [2], sometimes dubbed Mtheory, which describes the low energy effective theory corresponding to the strong coupling limit of heterotic string theory. This model is associated with an elevendimensional bulk spacetime with 11dimensional supergravity, the eleventh dimension being compactified via a orbifold symmetry. The two fixed points of the orbifold symmetry define two 10dimensional spacetime boundaries, or 9branes, on which the gauge groups are defined. Starting from this configuration, one can distinguish three types of spatial dimensions: the orbifold dimension, three large dimensions corresponding to the ordinary spatial directions and finally six additional dimensions, which can be compactified in the usual KaluzaKlein way. It turns out that the orbifold dimension might be larger than the six KaluzaKlein extra dimensions, resulting in an intermediary picture with a fivedimensional spacetime, two boundary 3branes, one of which could be our universe, and a “large” extradimension. This model provides the motivating framework for many of the brane cosmological models.
This concept of a 3brane has also been used in a purely phenomenological way by ArkaniHamed, Dimopoulos and Dvali [3] (ADD) as a possible solution to the hierarchy problem in particle physics. Their setup is extremely simple since they consider a flat dimensional spacetime, thus with compact extra dimensions with, for simplicity, a torus topology and a common size . From the fundamental (Planck) dimensional mass , which embodies the coupling of gravity to matter from the dimensional point of view, one can deduce the effective fourdimensional Planck mass , either by integrating the EinsteinHilbert action over the extra dimensions, or by using directly Gauss’ theorem. One then finds
(1) 
On sizes much larger than , dimensional gravity behaves effectively as our usual fourdimensional gravity, and the two can be distinguished only on scales sufficiently small, of the order of or below. The simple but crucial remark of ADD is that a fundamental mass of the order of the electroweak scale can explain the huge fourdimensional Planck mass we observe, provided the volume in the extra dimensions is very large. Of course, such a large size for the usual extradimensions à la KaluzaKlein is forbidden by collider constraints, but is allowed when ordinary matter is confined to a threebrane. In contrast, the constraints on the behaviour of gravity are much weaker since the usual Newton’s law has been verified experimentally only down to a fraction of millimeter[4]. This leaves room for extra dimensions as large as this millimeter experimental bound.
The treatment of extradimensions can be refined by allowing the spacetime to be curved, by the presence of the brane and possibly by the bulk. In this spirit, Randall and Sundrum have proposed a very interesting model[5, 6] with an Anti de Sitter (AdS) fivedimensional spacetime (i.e. a maximally symmetric spacetime with a negative cosmological constant), and shown that, for an appropriate tension of the brane representing our universe, one recovers effectively fourdimensional gravity even with an infinite extradimension. Another model [7], which will not be discussed in this review, is characterized, in contrast with the previous ones, by a gravity which becomes fivedimensional at large scales and it could have interesting applications for the present cosmological acceleration[8].
The recent concept of extradimensions with branes has been explored in a lot of aspects of particle physics, gravity, astrophysics and cosmology [9, 10]. The purpose of this review is to present a very specific, although very active, facet of this vast domain, dealing with the cosmological behaviour of brane models when the curvature of spacetime along the extradimensions, and in particular the brane selfgravity, is explicitly taken into account. For technical reasons, this can be easily studied in the context of codimension one spacetimes and we will therefore restrict this review to the case of fivedimensional spacetimes. Even in this restricted area, the number of works during the last few years is so large that this introductory review is not intended to be comprehensive but will focus on some selected aspects. This also implies that the bibliography is far from exhaustive.
2 Prehistory of brane cosmology
Since our purpose is to describe (homogeneous) cosmology within the brane, we will model our fourdimensional universe as an infinitesimally thin wall of constant spatial curvature embedded in a fivedimensional spacetime. This means that we need to consider spacetimes with planar (or spherical/hyperboloidal) symmetry along three of their spatial directions, i.e. homogeneous and isotropic along three spatial dimensions. The general metric compatible with these symmetries can be written in appropriate coordinate systems in the form
(2) 
where is the maximally symmetric threedimensional metric, with either negative, vanishing or positive spatial curvature (respectively labelled by , or ), and is a twodimensional metric (with spacetime signature) depending only on the two coordinates , which span time and the extra spatial dimension. We are interested in the history of a spatially homogeneous and isotropic threebrane, which can be simply described as a point trajectory in the twodimensional spacetime, the ordinary spatial dimensions inside the brane corresponding to the coordinates . Now, once such a trajectory has been defined, it is always possible to introduce a socalled Gaussian Normal (GN) coordinate system so that
(3) 
This coordinate system can be constructed by introducing the proper time on the trajectory and by defining the coordinate as the proper distance on (spacelike) geodesics normal to the trajectory. The time coordinate defined only on the brane is then propagated off the brane along these normal geodesics. It is convenient to take the origin on the brane so that in this coordinate system represents the brane itself.
Summarizing, it is always possible to write the metric (2), in a GN coordinate system where the brane is located at , in the form
(4) 
This is the most convenient system of coordinates when one wishes to focus on the brane itself since the induced metric in the brane is immediately obtained in its FriedmannLemaîtreRobertsonWalker (FLRW) form
(5) 
If is the proper time on the brane then . The GN coordinates can however suffer from coordinate singularities off the brane, and it might be more convenient, as discussed in Section 5, to use more appropriate coordinates in order to describe the bulk spacetime structure.
Having specified the form of the metric, we now turn to the fivedimensional Einstein equations, which can be written in the condensed form
(6) 
where is the fivedimensional Ricci tensor and its trace. stands for a possible bulk cosmological constant, whereas is the total energy momentum tensor. It includes the energymomentum tensor of the brane, which is distributional if one assumes the brane infinitely thin (a few works [11, 12] have discussed the cosmology for thick branes, which can also be related to the abundant literature on thick domain walls[13]) and thus of the form
(7) 
where is the total brane energy density and is the total brane pressure. The total energymomentum tensor also includes a possible bulk contribution, which we first assume to vanish (bulk matter will be considered in Section 8).
In the coordinate system (4), the components of the Einstein tensor read[14]
(8)  
(9)  
(10)  
(11) 
where a dot stands for a derivative with respect to and a prime a derivative with respect to . One way to solve Einstein’s equations is to insert the full energymomentum tensor, including the brane energymomentum tensor with its delta distribution, on the righthand side of (6) and solve the full system of equations. An alternative procedure is to consider Einstein’s equations without the brane energymomentum tensor, i.e. valid strictly in the bulk, and then impose on the general solutions boundary conditions to take into account the physical presence of the brane. The Einstein tensor is made of the metric up to second derivatives, so formally the Einstein equations with the distributional source are of the form
(12) 
If the brane is located at , integrating this equation over across the brane immediately yields
(13) 
In other words, the boundary conditions due to the presence of the brane must take the form of a relation between the jump, across the brane, of the first derivative of the metric with respect to and the brane energymomentum tensor. In the case of Einstein’s equations, the exact equivalent of the formal relation (13) is the socalled junction conditions [15], which can be written in a covariant form as
(14) 
where the brackets here denote the jump at the brane, i.e. , and the extrinsic curvature tensor is defined by
(15) 
being the unit vector normal to the brane, and
(16) 
the induced metric on the brane.
A frequent assumption in the brane cosmology literature has been, for simplicity, to keep the orbifold nature of the extra dimension in the HoravaWitten model and thus impose a mirror symmetry across the brane, although some works have relaxed this assumption [16, 17]. This enables us to replace the jump in the extrinsic curvature by twice the value of the extrinsic curvature at the location of the brane. The junction conditions (14) then imply
(17) 
where is the trace of . If one uses the GN coordinate system with the metric (4) and substitutes the explicit form of the brane energymomentum tensor (7), the junction conditions reduce to the two relations
(18) 
Going back to the bulk Einstein equations, one can solve the component (see 10) to get
(19) 
and the integration of the component with respect to and of the component with respect to time yields the first integral
(20) 
where is an arbitrary integration constant. When one evaluates this first integral at , i.e. in our braneuniverse, substituting the junction conditions given above in (17), one ends up with the following equation[14, 18]
(21) 
where the subscript ‘’ means evaluation at . This equation relates the Hubble parameter to the energy density but it is different from the usual Friedmann equation []. The most remarkable feature of (21) is that the energy density of the brane enters quadratically on the right hand side in contrast with the standard fourdimensional Friedmann equation where the energy density enters linearly. As for the energy conservation equation, it is unchanged in this fivedimensional setup and still reads
(22) 
as a consequence of the component of Einstein’s equations combined with the junction conditions (18).
In the simplest case where and , one can easily solve the cosmological equations (2122) for a perfect fluid with equation of state ( constant). One finds that the evolution of the scale factor is given by[14]
(23) 
In the most interesting cases for cosmology, radiation and pressureless matter, the evolution of the scale factor is thus given by, respectively, (instead of the usual ) and (instead of ). Such behaviour is problematic because it cannot be reconciled with nucleosynthesis. Indeed, the standard nucleosynthesis scenario depends crucially on the balance between the microphysical reaction rates and the expansion rate of the universe. Any drastic change in the evolution of the scale factor between nucleosynthesis and now would dramatically modify the predictions for the light element abundances. After discussing some particular solutions in the next section, the subsequent section will present a brane cosmological model with much nicer features.
3 Cosmological solutions for ‘domain walls’
It is instructive to consider the brane cosmological solutions for the simplest equation of state, , which also characterizes the socalled domain walls. With this particular equation of state, the cosmological equations can be explicitly integrated [19, 20, 21]. Indeed, the energy density is necessarily constant, as implied by (22), and the Friedmann equation (21) is of the form
(24) 
where
(25) 
is a constant. The case is referred to as a ‘critical’ brane (or domain wall), while and correspond to subcritical and supercritical branes respectively. Defining , the Friedmann equation (24) can be rewritten as
(26) 
which is analogous to the equation for a point particle with kinetic energy on the left hand side and (minus) the potential energy on the right hand side. For a critical brane (), one immediately finds the following three solutions, depending on the spatial curvature of the brane:
(27)  
(28)  
(29) 
which in fact correspond to the usual FLRW solutions with radiation, the term proportional to playing the effective rôle of radiation.
For a non critical brane, integration of (26) yields the following solutions,
(30)  
(31)  
(32) 
with
(33) 
The particular solutions for (and ) are
(34) 
In all the above equations the (additive) integration constant defining the origin of times is not written explicitly and one can always use this time shift to rewrite any solution in a more adequate form (for instance so that at ). Moreover, all solutions have their time reversed counterpart obtained by changing into .
The analytical expressions (31) or (32) cover very different cosmological behaviours depending on the value of their parameters. Using the analogy with the point particle mentioned before, one can see that the global cosmological behaviour depends on the number of positive roots of the quadratic ‘potential’ on the right hand side of (26). For , and , the potential has two positive roots and this leads to two solutions, one expanding from the singularity and recontracting to , the other contracting from infinity to a minimum scale factor and then expanding. These two types of solutions correspond to the two branches of (31). For and with , or with , there is no positive root and the corresponding cosmology starts from the singularity and then expands indefinitely. This behaviour is described by the solution (30) and the solution (31) for . For and , there is a single positive root, and the cosmological solution starts expanding from and then recollapses back to . This corresponds to the solution (32). For , and , the solution is confined between the two positive roots, which gives an oscillating cosmology between a minimum scale factor and a maximum scale factor. This is also described by the solution (32). Finally, for and , there is a single positive root and the solution corresponds to a contraction from infinity followed, after a bounce at a minimum scale factor, by an expansion back to infinity. This is described by the solution (31).
4 Simplest realistic brane cosmology
As explained earlier, the simplest model of selfgravitating brane cosmology, that of a brane embedded in an empty bulk with vanishing cosmological constant (in fact a Minkowski bulk because of the symmetries), does not appear compatible [14] with the standard landmarks of modern cosmology. It is thus necessary to consider more sophisticated models in order to get a viable scenario, at least as far as homogeneous cosmology is concerned.
An instructive exercise is to look for non trivial (i.e. with a non empty brane) static solutions. In the simplest case , one immediately sees from (21) that a static solution, corresponding to , can be obtained with a negative cosmological constant and an energy density satisfying
(35) 
One can check that this is compatible with the other equations, in particular the conservation equation in the brane (22), which imposes that the matter equation of state is that of a cosmological constant, i.e. . It turns out that this configuration is exactly the starting point of the two models due to Randall and Sundrum (RS) [5, 6]. From the point of view of brane cosmology, embodied in the unconventional Friedmann equation (21), the RS models thus appear as the simplest non trivial static brane configurations. The case of the single brane model[6], with a positive tension, is particularly interesting, because ordinary fourdimensional gravity, at least at linear order, is effectively recovered on large enough lengthscales [22], with
(36) 
where is the Anti de Sitter (AdS) lengthscale defined by the (negative) cosmological constant,
(37) 
Since usual gravity is recovered, the generalization of this model to cosmology seems a priori a good candidate for a viable brane cosmology.
Let us therefore consider a brane with the total energy density
(38) 
where is a tension, constant in time, and the energy density of ordinary cosmological matter. Substituting this decomposition into (21), one obtains [23, 18, 24]
(39) 
where is the AdS mass scale. If one finetunes the brane tension and the bulk cosmological cosmological constant like in (35) so that
(40) 
the first term on the right hand side of (39) vanishes and, because of (36), the tension is proportional to Newton’s constant,
(41) 
The second term in (39) then becomes the dominant term if is small enough and is exactly the linear term of the usual Friedmann equation, with the same coefficient of proportionality.
The third term on the right hand side of (39), quadratic in the energy density, provides a highenergy correction to the Friedmann equation which becomes significant when the value of the energy density approaches the value of the tension and dominates at still higher energy densities. In the very high energy regime, , one recovers the unconventional behaviour (23), not surprisingly since the bulk cosmological constant is then negligible.
Finally, the last term in (39) behaves like radiation and arises from the integration constant . This constant is analogous to the Schwarzschild mass, as will be shown in the next section, and it is related to the bulk Weyl tensor, which vanishes when . In a cosmological context, this term is constrained to be small enough at the time of nucleosynthesis in order to satisfy the constraints on the number of extra light degrees of freedom (this will be discussed quantitatively just below). In the matter era, this term then redshifts quickly and would be in principle negligible today.
To summarize the above results, the brane Friedmann equation (21), for a RS type brane (i.e. satisfying (38) and (40)), reduces to
(42) 
which shows that, at low energies, i.e. at late times, one recovers the usual Friedmann equation. Going backwards in time, the term becomes significant and makes brane cosmology deviate from the usual FLRW behaviour.
It is also useful, especially as a preparation to the equations for the cosmological perturbations, (see Section 7), to notice that the generalized Friedmann equation (42) can be seen as a particular case of the more general effective fourdimensional Einstein’s equations for the brane metric , obtained by projection on the brane. Using the Gauss equation and the junction conditions, and decomposing the total energymomentum tensor of the brane into a pure tension part and an ordinary matter part so that
(43) 
one arrives to effective fourdimensional Einstein equations, which read [25]
(44) 
with
(45) 
and where
(46) 
is the projection on the brane of the fivedimensional Weyl tensor. In this new form, the effective Einstein’s equations are analogous the standard fourdimensional equations with the replacement of the usual matter energymomentum tensor by the sum of an effective matter energymomentum tensor,
(47) 
which is constructed only with , and of an additional part that depends on the bulk, which defines what we will call the Weyl energymomentum tensor
(48) 
Although this formulation might appear very simple, the reader should be warned that this equation is in fact directly useful only in the cosmological case, where reduces to the arbitrary constant . In general, will hide a dependence on the brane content and only a detailed study of the bulk can in practice provide the true behaviour of gravity in the brane.
Let us now work out a few explicit cosmological solutions. For an equation of state , with constant, one can integrate explicitly the conservation equation (22) to obtain as usual
(49) 
Substituting in the Friedmann equation (39), one gets
(50) 
with
(51) 
In the case and , defining , the Friedmann equation (50) reduces to the form
(52) 
which is similar to (26). For a critical brane, i.e. , the corresponding cosmological evolution is given by
(53) 
It is clear from this analytical expression that there is a transition, at a typical time of the order of , i.e. of the AdS lengthscale , between an early high energy regime characterized by the behaviour and a late low energy regime characterized by the standard evolution .
For a non critical brane, the cosmological solutions are similar to the expressions (31) and (32) [the expression (30) does not apply here because of the relation between the coefficients , and ]. After choosing an appropriate origin of time (so that ), they can be rewritten in the form
(54) 
and
(55) 
The solution (54) describes a brane cosmology with a positive cosmological constant. This implies that, by finetuning adequately the brane tension, one can obtain a cosmology with, like in (53), an unconventional early phase followed by a conventional phase, itself followed by a period dominated by a cosmological constant.
In the present section, we have so far considered only the metric in the brane. But the metric outside the brane can be also determined explicitly by solving the full system of Einstein’s equations. [18]. The metric coefficients defined in (4) are given by
(56) 
and
(57) 
which, in the special case , reduce to the much simpler expressions
(58)  
(59) 
with
(60) 
In the limit , i.e. , which implies , one recovers the RS metric .
In summary, we have obtained a cosmological model, based on a braneworld scenario, which appears to be compatible with current observations, because it converges to the standard model at low enough energies. Let us now quantify the constraints on the parameters of this model. As mentioned above an essential constraint comes from nucleosynthesis: the evolution of the universe since nucleosynthesis must be approximately the same as in usual cosmology. This is the case if the energy scale associated with the tension is higher than the nucleosynthesis energy scale, i.e.
(61) 
Combining this with (41) this implies for the fundamental mass scale (defined by )
(62) 
However, one must not forget another constraint, not of cosmological nature: the requirement to recover ordinary gravity down to scales of the submillimeter order. This requires[4]
(63) 
which yields in turn the constraint
(64) 
Therefore the most stringent constraint comes, not from cosmology, but from gravity experiments in this particular model.
There are also constraints on the Weyl parameter [18, 26]. The ratio
(65) 
is constrained by the number of additional relativistic degrees of freedom allowed during nucleosynthesis [27], which is usually expressed as the number of additional light neutrino species . A typical bound implies at the time of nucleosynthesis. It is also important to stress that if has been considered here as an arbitrary constant, a more refined, and more realistic, treatment must take into account the fact that cosmology is only approximately homogeneous: the small inhomogeneities in the brane can produce bulk gravitons and the energy outflow carried by these gravitational waves can feed the asymptotic ‘Schwarzschild mass’, i.e. the Weyl parameter which would not be any longer a constant (in agreement with the fact that the bulk is then no longer vacuum). This process is at present under investigation and the first estimates give a Weyl parameters of the order of the nucleosynthesis limit [28, 29].
So far, we have thus been able to build a model, which reproduces all qualitative and quantitative features of ordinary cosmology in the domains that have been tested by observations. The obvious next question is whether this will still hold for a more realistic cosmology that includes perturbations from homogeneity, and more interestingly, whether brane cosmology is capable of providing predictions that deviate from usual cosmology and which might tested in the future. This question will be addressed, but unfortunately not answered, in the section following the next one which presents the brane cosmological solutions in a totally different way.
5 A different point of view
In the previous sections, we have from the start restricted ourselves to a particular system of coordinates, namely a GN coordinate system. This choice entails no physical restriction (at least locally) and is very useful for a ‘branebased’ point of view, but there also exists an alternative approach for deriving the brane cosmological solutions given earlier, which relies on a ‘bulkbased’ point of view [30, 31] and corresponds to a more appropriate choice of coordinates for the bulk. The link between the two points of view can be understood from a general analysis [21] which consists in solving the fivedimensional Einstein equations for a generic metric of the form (2). It turns out that simple coordinates emerge, expressing in a manifest way the underlying symmetries of the solutions. This approach leads to the following simple form for the solutions (with the required ‘cosmological symmetries’) of Einstein’s equations in the bulk (6) [with ]:
(66) 
where
(67) 
The above metric is known as the fivedimensional SchwarzschildAnti de Sitter (SchAdS) metric (AdS for , which is the case we are interested in; for , this the Schwarschildde Sitter metric). It is clear from (67) that is indeed the fivedimensional analog of the Schwarschild mass, as said before (the dependence instead of the usual is simply due to the different dimension of spacetime).
It is manifestly static (since the metric coefficients are timeindependent), which means that the solutions of Einstein’s equations have more symmetries than assumed a priori. In fact, this is quite analogous to what happens in fourdimensional general relativity when one looks for vacuum solutions with only spherical symmetry: one ends up with the Schwarschild geometry, which is static. The above result for empty fivedimensional spacetimes can thus be seen as a generalization of Birkhoff’s theorem, as it is known in the fourdimensional case.
Since the solutions of the bulk Einstein equations, with the required symmetries, are necessarily SchAdS, it is easy to infer that the solutions (5657), or (59), obtained in the particular GN coordinate system correspond to the same geometry as (66) but written in a more complicated coordinate system, as can be checked[32] by finding the explicit coordinate transformation going from (66) to (4).
If the coordinates in (66) are much simpler to describe the bulk spacetime, it is not so for the brane itself. Indeed, whereas the brane is ‘at rest’ (at ) in the GN coordinates, its position in the new coordinate system, will be in general timedependent. This means that the brane is moving in the manifestly static reference frame (66). The trajectory of the brane can be defined by its coordinates and given as functions of a parameter . Choosing to be the proper time imposes the condition
(68) 
where is the brane velocity and a dot stands for a derivative with respect to the parameter . The normalization condition (68) yields and the components of the unit normal vector (defined such that and ) are, up to a sign ambiguity,
(69) 
The fourdimensional metric induced in the brane worldsheet is then directly given by
(70) 
and it is clear that the scale factor of the brane, denoted previously, can be identified with the radial coordinate of the brane .
The dynamics of the brane is then obtained by writing the junction conditions for the brane in the new coordinate system[33, 30]. The ‘orthogonal’ components of the extrinsic curvature tensor are given by
(71) 
which, after insertion in the junction conditions (17), implies for a mirror symmetric brane (see Section 9.2 for an asymmetric brane)
(72) 
After taking the square of this expression, substituting (67) and rearranging, we get
(73) 
which, upon the identification between and , is exactly the Friedmann equation (21) obtained before. There is additional information in the ‘longitudinal’ part of the junction conditions. The ‘longitudinal’ component of the extrinsic curvature tensor is given by
(74) 
where is the acceleration, i.e. . Since , the acceleration vector is necessarily of the form , and . Finally, using the fact that is a Killing vector, one can easily show that . Substituting in the junction conditions, one finds
(75) 
Combining with the first relation (72), it is immediate to rewrite the above expression as the traditional cosmological conservation equation
(76) 
One has thus established the complete equivalence between the two pictures: in the first, the brane sits at a fixed position in a GN coordinate system and while it evolves cosmologically the metric components evolve with time accordingly; in the second, one has a manifestly static bulk spacetime in which the brane is moving, its cosmological evolution being simply a consequence of its displacement in the bulk (phenomenom sometimes called ‘mirage cosmology’[34]). Let us add that the metric (66) describes in principle only one side of the brane. In the case of a mirror symmetric brane, the complete spacetime is obtained by gluing,along the brane worldsheet, two copies of a portion of SchAdS spacetime. For an asymmetric brane, one can glue two (compatible) portions from different SchAdS spacetimes, as illustrated in Section 9.2.
6 Inflation in the brane
The archetypical scenario of nowadays early universe cosmology is inflation. Since the infancy of brane cosmology, brane inflation has attracted some attention [35, 36, 37]. The simplest way to get inflation in the brane is to detune the brane tension from its RandallSundrum value (35), and to take it bigger so that the net effective fourdimensional cosmological constant is positive. This leads to an exponential expansion in the brane, as illustrated before in (34) (we take here and for simplicity). In the GN coordinate system, the metric corresponding to this situation [19] is given by the specialization of the bulk metric (59) to the case . The metric components and are then separable, i.e. they can be written as
(77) 
with
(78) 
Of course, this model is too naive for realistic cosmology since brane inflation would never end. Like in standard cosmology, one can replace the cosmological constant, or here the deviation of the brane tension from its RS value, by a scalar field whose potential behaves, during slowroll motion, like an effective tension. The simplest scenario is to consider a fourdimensional scalar field confined to the brane[38]. The brane cosmology formulas established above then apply, provided one substitutes for the energy density and pressure the appropriate expressions for a scalar field, namely
(79) 
Like in the analysis for ordinary matter, the brane inflationary scenarios will be divided into two categories, according to the typical value of the energy density during inflation:

high energy brane inflation if

low energy brane inflation if , in which case the scenario is exactly similar to fourdimensional inflation.
New features appear for the high energy scenarios[38]: for example, it is easier to get inflation because the Hubble parameter is bigger than the standard one, producing a higher friction on the scalar field. It enables inflation to take place with potentials usually too steep to sustain it [39]. Because of the modified Friedmann equation, the slowroll conditions are also changed.
For high energy inflation, the predicted spectra for scalar and tensor perturbations are also modified. It has been argued [38] that, since the scalar field is intrinsically fourdimensional, the modification of the scalar spectrum is due simply to the change of the background equations of motion. However, the gravitational wave spectrum requires more care because the gravitational waves are fivedimensional objects. This question will be treated in detail in the next section. Let us also mention another type of inflation in brane models, based by a bulk fivedimensional scalar field that induces inflation within the brane [40].
One of the main reasons to invoke inflation in ordinary fourdimensional cosmology is the horizon problem of the standard big bang model, i.e. how to get a quasihomogeneous CMB sky when the Hubble radius size at the time of last scattering corresponds to an angular scale of one degree. Rather than adapt inflation in brane cosmology, one may wonder wether it is possible to solve the horizon problem altogether without using inflation. A promising feature of brane cosmology in this respect is the possibility for a signal to propagate more rapidly in the bulk than along the brane [41], thus modifying ordinary causality[42]. This property can also be found in non cosmological brane models, usually called Lorentz violation [43]. Unfortunately, in the context of the SchAdS brane models presented here, a quantitative analysis (for ) has shown that the difference between the gravitational wave horizon (i.e. for signals propagating in the bulk) and the photon horizon (i.e. for signals confined to the brane) is too small to be useful as an alternative solution to the horizon problem [44]. Note also that the homogeneity problem of standard cosmology, i.e. why the universe was so close to homogeneity in its infancy, might appear more severe in the braneworld context, where one must explain both the homogeneity along the ordinary spatial dimensions and the inhomogeneity along the fifth dimension [45].
7 Cosmological perturbations
With the homogeneous scenario presented in section 4 as a starting point, one would like to explore the much richer, and much more difficult, question of cosmological perturbations and, in particular investigate whether brane cosmology leads to new effects that could be tested in the forthcoming cosmological observations, in particular of the anisotropies of the Cosmic Microwave Background (CMB). Brane cosmological perturbations is a difficult subject and although there are now many published works[46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62] with various formalisms on this question, no observational signature has yet been predicted. Rather than entering into the technicalities of the subject, for which the reader is invited to consult the original references, this section will try to summarize a few results concerning two different, but illustrative, aspects of perturbations: on one hand, the evolution of scalar type perturbations on the brane; on the other hand, the production of gravitational waves from quantum fluctuations during an inflationary phase in the brane.
7.1 Scalar cosmological perturbations
In a metricbased approach, there are various choices for the gauge in which the metric perturbations are defined. We will choose here a GN gauge, which has the advantage that the perturbed brane is still positioned at and that only the fourdimensional part of the metric is perturbed. One can then immediately identify the value at of the metric perturbations with the usual cosmological metric perturbations defined by a brane observer [49]. The most general metric with linear scalar perturbations about a FLRW brane is
(80) 
where is the scale factor and a vertical bar denotes the covariant derivative of the threedimensional metric .
The perturbed energymomentum tensor for matter on the brane, with background energy density and pressure , can be given as
(81) 
where 45), is is the tracefree anisotropic stress perturbation. The perturbed quadratic energymomentum tensor, defined in (
(82) 
The last term on the right hand side of the effective fourdimensional Einstein equations (44) is the projected Weyl tensor . Although it is due to the effect of bulk metric perturbations not defined on the brane, one can parametrize it as an effective energymomentum tensor (48)
(83) 
With these definitions, one can write explicitly the perturbed effective Einstein equations on the brane, which will look exactly as the fourdimensional ones for the geometrical part but with extra terms due to and for the matter part. In addition to the effective fourdimensional Einstein equations, the fivedimensional Einstein equations also provide three other equations. Two of them are equivalent to the conservation of the matter energy and momentum on the brane, i.e. of the tensor . The final one yields an equation of state for the Weyl fluid, which in the 4dimensional equations follows from the symmetry properties of the projected Weyl tensor, requiring in the background and at first order. The equations of motion for the effective energy and momentum of the projected Weyl tensor are provided by the 4dimensional contracted Bianchi identities, which are intrinsically fourdimensional, only being defined on the brane and not part of the fivedimensional field equations. The contracted Bianchi identities () and energymomentum conservation for matter on the brane () ensure, using Eq. (44), that
(84) 
In the background, this tells us that behaves like radiation, as we knew already, and for the firstorder perturbations we have
(85) 
This means that the effective energy of the projected Weyl tensor is conserved independently of the quadratic energymomentum tensor. The only interaction is a momentum transfer [25, 48], as shown by the perturbed momentum conservation equation
(87)  
where the right hand side represents the momentum transfer from the quadratic energymomentum tensor.
It is also possible to construct [56] gaugeinvariant variables corresponding to the curvature perturbation on hypersurfaces of uniform density, both for the brane matter energy density and for the total effective energy density (including the quadratic terms and the Weyl component). These quantities are extremely useful because their evolution on scales larger than the Hubble radius can be solved easily. However, their connection to the largeangle CMB anisotropies involves the knowledge of anisotropic stresses due to the bulk metric perturbations[56]. This means that for a quantitative prediction of the CMB anisotropies, even at large scales, one needs to determine the evolution of the bulk perturbations.
In summary, we have obtained a set of equations for the brane linear perturbations, where one recognizes the ordinary cosmological equations but modified by two types of corrections:

modification of the homogeneous background coefficients due to the additional terms in the Friedmann equation. These corrections are negligible in the low energy regime .

presence of source terms in the equations. These terms come from the bulk perturbations and cannot be determined solely from the evolution inside the brane. To determine them, one must solve the full problem in the bulk (which also means to specify some initial conditions in the bulk). In the effective fourdimensial perturbation equations, these terms from the fifth dimension appear like external source terms, in a way somewhat similar to the case of “active seeds” due to topological defects.
7.2 Inflationary production of gravitational waves
Let us turn now to another facet of the brane cosmological perturbations: their production during a brane inflationary phase[57]. We concentrate on the tensor perturbations, which are subtler than the scalar perturbations, because gravitational waves have an extension in the fifth dimension. The brane gravitational waves can be defined by a perturbed metric of the form
(88) 
where the ‘TT’ stands for transverse traceless. The linearized Einstein equations for the metric perturbations give a wave equation, which reads in Fourier space ()
(89) 
where is defined in (78). This equation being separable, one looks for solutions , where the timedependent part obeys an ordinary FLRW KleinGordon equation while the dependent part must satisfy the Schrödinger type equation
(90) 
after introducing the new variable (with ) and the new function . The potential is given by
(91) 
The nonzero value of the Hubble parameter signals the presence of a gap[63] between the zero mode () and the continuum of KaluzaKlein modes (). The zero mode corresponds to and the constant is determined by the normalization . One finds
(92) 
Asymptotically, at low energies, i.e. for , and at very high energies, i.e. for . One can then evaluate the vacuum quantum fluctuations of the zero mode by using the standard canonical quantization. To do this explicitly, one writes the fivedimensional action for gravity at second order in the perturbations. Keeping only the zero mode and integrating over the fifth dimension, one obtains
(93) 
This has the standard form of a massless graviton in fourdimensional cosmology, apart from the overall factor instead of . It follows that quantum fluctuations in each polarization, , have an amplitude of on superhorizon scales. Quantum fluctuations on the brane at , where , thus have the typical amplitude[57]
(94) 
The same result can be obtained in a bulkbased approach [59].
At low energies, and one recovers exactly the usual fourdimensional result but at higher energies the multiplicative factor provides an enhancement of the gravitational wave spectrum amplitude with respect to the fourdimensional result. However, comparing this with the amplitude for the scalar spectrum[38], one finds that, at high energies (), the tensor over scalar ratio is in fact suppressed with respect to the fourdimensional ratio. An open question is how the gravitational waves will evolve during the subsequent cosmological phases, the radiation and matter eras.
8 Nonempty bulk
Out of simplicity, a majority of works in brane cosmology have focused on a braneuniverse embedded in a fivedimensional empty bulk, i.e. with only gravity propagating in the bulk. But many works have also considered extra fields in the bulk, either motivated by M/string theory [64, 65, 66], or simply in a purely phenomenomogical approach [67, 68], in some cases with the objective to solve some standing problems such as the cosmological constant problem [69] or more specific problems like the question of the radion stabilization[70, 71]. The most common extra field considered in the literature is not surprisingly the scalar field although one can find other generalizations such as including gauge fields [72, 73]. We will consider here only models with a bulk scalar field and start from the action
(95) 
where it is assumed that the fourdimensional metric which is minimally coupled to the fourdimensional matter fields in the brane, is conformally related to the induced metric , i.e.
(96) 
In the terminology of scalartensor theories, one would say that corresponds to the Einstein frame while corresponds to the Jordan frame. One can also define two brane matter energymomentum tensors, and , respectively with respect to and ,
(97) 
and they are related by . The corresponding energy densities, and , and pressures, and , will be related by the same factor . Variation of the action (95) with respect to the metric yields the fivedimensional Einstein equations (6), where, in addition to the (distributional) brane energymomentum, there is now the scalar field energymomentum tensor
(98) 
Variation of (95) with respect to yields the equation of motion for the scalar field, which is of the KleinGordon type, with a distributional source term since the scalar field is coupled to the brane via . This implies that there is an additional junction condition, now involving the scalar field at the brane location and which is of the form
(99) 
where is the trace of the energymomentum tensor. In summary, we now have a much more complicated system than for the empty bulk, since, in addition to the brane dynamics, one must solve for the dynamics of the scalar field. Although a full treatment would require a numerical analysis, a few interesting analytical solutions have been derived in the literature [74, 75, 76, 77, 78, 79], out of which we give below two examples.
8.1 Moving brane in a static bulk
Taking example on the empty bulk case, where the cosmology in the brane is induced by the motion of the brane in a static bulk, it seems natural to look for brane cosmological solutions corresponding to a moving brane in a static bulk geometry with a static scalar field. In the case of an exponential potential,
(100) 
it turns out that there exists a simple class of static solutions[80] (the full set of static solutions is larger[78]), which correspond to a metric of the form
(101) 
with
(102) 
where is an arbitrary constant, and a scalar field given by
(103) 
The next step is then to insert a moving (mirrorsymmetric) brane in the above bulk configuration[33, 76]. This is possible if the three junction conditions, two for the metric and one for the scalar field, are satisfied. The two metric junction conditions are easily derived by repeating the procedure given in section 5 for the metric (101) and will be generalizations of (72) and (75). The condition on the scalar field comes from (99).
These three junction conditions can be rearranged to give the following three relations:

a generalized Friedmann equation,
(104) 
a (non) conservation equation for the energy density,