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Dynamics of radiating braneworlds

If the observable universe is a braneworld of Randall-Sundrum type, then particle interactions at high energies will produce 5-dimensional gravitons that escape into the bulk. As a result, the Weyl energy density on the brane does not behave like radiation in the early universe, but does so only later, in the low energy regime. Recently a simple model was proposed to describe this modification of the Randall-Sundrum cosmology. We investigate the dynamics of this model, and find the exact solution of the field equations. We use a dynamical systems approach to analyze global features of the phase space of solutions.


I. INTRODUCTION
String and M theory indicate that gravity may be a higher-dimensional interaction, while matter fields and gauge interactions are confined to a 1+3-brane moving in the higher-dimensional "bulk" spacetime. These ideas have led to relatively simple phenomenological models that allow one to explore some of the consequences of braneworld gravity. The second Randall-Sundrum (RS2) scenario is a five-dimensional Anti-de Sitter (AdS 5 ) spacetime with an embedded Minkowski brane where matter fields are confined and Newtonian gravity is effectively reproduced at low energies. The RS2 scenario is readily generalized to a Friedmann-Robertson-Walker (FRW) brane, and the bulk is Schwarzschild-AdS 5 . The modified Friedmann equation (for a flat universe without cosmological constant) is where λ is the brane tension and C is the mass parameter of the bulk black hole. The tidal field of this black hole on the brane is felt as an effective radiative energy density, the so-called "dark radiation". If the reheating and early radiation eras occur at sufficiently high energies, this simple picture is modified. High energy (ρ ≫ λ > ∼ (1 TeV) 4 ) particle interactions can produce 5D gravitons which are emitted into the bulk. This process will contribute to an increase in the dark radiation term, so that C will not be a constant in the early universe, but will grow until the energy scale drops below the threshold for 5D-graviton production. In the low-energy universe, C will have its asymptotic constant value, which is constrained by nucleosynthesis limits [1].
In [2], a simple model is constructed in which all the 5D gravitons are assumed to be emitted radially, and their backreaction on the bulk metric is assumed negligible, so that the bulk is a Vaidya-AdS 5 spacetime. (Earlier alternative models are developed in [3].) Subsequently, the corrections to this model arising from non-radial emission have been investigated [4]. These corrections are very complicated, and we will confine ourselves to the simple radial model. We find the exact solution of the dynamical equations, and analyze the asymptotic behaviour and the global phase space of solutions.

II. FIELD EQUATIONS
The 5D field equations are (5) The projected field equations on the brane [5,6] have three new terms from extra-dimensional gravity: where Λ = Λ 5 /2 + λ 2 κ 4 5 /12 and κ 2 = λκ 4 5 /6. The term S µν is quadratic in T µν and dominates at high energies (ρ > λ); this term gives rise to the ρ 2 /λ term in the modified Friedmann equation (1). The five-dimensional Weyl tensor is felt on the brane via its projection, E µν . This Weyl term is determined by the bulk metric, not by equations on the brane, and it gives rise to the C/a 4 term in the modified Friedmann equation (1).
The term F µν is also determined by the bulk metric, and arises from (5) T AB , i.e., from 5D sources in the bulk other than the vacuum energy Λ 5 , such as a bulk dilaton field. The presence of such sources modifies the energymomentum conservation equations on the brane [6] where n A is the unit normal to the brane. Radial radiation in the bulk is described by where k A is a radial null vector and ψ determines the energy density. A physical normalization of k A is k A u A = 1, where u A is the preferred 4-velocity, coinciding on the brane with the 4-velocity of comoving observers. Then Eq. (5) determines the form of F µν as When the null radiation describes 5D gravitons produced in particle interactions in the radiation era (or in reheating with radiative equation of state), then [2] ψ = κ 2 where α is a dimensionless constant. If all degrees of freedom in the standard model are relativistic, then α ∼ 0.02. The brane energy-momentum tensor on the brane is assumed to be of the perfect-fluid type with respect to For a FRW brane (which we assume to be spatially flat), the bulk metric has 4-dimensional plane symmetry, and then Eq. (2) leads to the 5-dimensional Vaidya-AdS 5 metric [7] When C is independent of v, the metric reduces to the Schwarzschild-AdS 5 metric. The coordinate v is a null coordinate (corresponding to ingoing radial light rays). The brane trajectory is given by where t is proper time for cosmic fluid particles, which move with 4-velocity u µ . The normalization of For an expanding (outgoing) brane this is infinite at the apparent horizon (F (v, r) = 0), so we assume that an expanding brane is always located in the region F (v, r) > 0 (for Schwarzschild-AdS 5 this means outside the horizon). See [8] for further discussion. The above equations lead to the modified Friedmann equation (1), and an equation governing the evolution of C [2] (valid outside the apparent horizon) The emission of 5D gravitons is at the expense of radiation energy density; by Eqs. (4)- (7),

III. EXACT SOLUTION
We introduce the dimensionless variables [2]: (Note that ℓ 2 = 6/κ 2 λ and κ 2 5 = κ 2 ℓ.) Then the high energy regime is given byρ,Ĥ > 1, and the dynamical equations (1), (11) and (12) becomê These equations imply the modified Raychaudhuri equation which we use to decouple the system: This can be integrated to givê wheret 0 is an integration constant. Introducing this into Eq. (15), we find This equation is of Bernoulli type, with solution (restoring units) where A is a dimensionless integration constant and Then, by Eq. (19) the Hubble parameter can be written as and then the scale factor follows as where a 0 is a constant (with dimension ℓ 1/2 ). Finally the expression for the dark radiation coefficient C is: When A > 0 (t 1 > t 0 ), the regions in which ρ ≥ 0 are: (i) t − t 1 ≤ 0 and t − t 0 ≤ 0, i.e. −∞ < t ≤ t 0 ; and (ii) t − t 1 ≥ 0 and t − t 0 ≥ 0, i.e. t 1 ≤ t < ∞. When A < 0 (t 1 < t 0 ), the ρ ≥ 0 region is: (iii) t − t 0 ≤ 0 and t − t 1 ≥ 0, i.e. t 1 ≤ t ≤ t 0 . By Eq. (23) it follows that region (i) corresponds to contracting models that end in a big crunch at t = t 0 . The allowed part (where F (v, a) > 0) of region (ii) corresponds to expanding models which start at t = t * ≥ t 1 , where t * is the time at which the brane is just outside the apparent horizon (or at an initial singularity). The allowed part of region (iii) corresponds to models that expand from the apparent horizon t = t * , then collapse to a big crunch at t = t 0 .
The ever-expanding models [region (ii)] are of most relevance. By Eq. (24), it follows that near the singularity (t → t 1 ), which corresponds to the high-energy limit, (we note that the solution is only valid near this limit if C is initially small). This behaviour is different from that in the RS model, a(t) ∼ t 1/4 , and the general relativistic one, a(t) ∼ t 1/2 . Whereas ρ and H are always positive, the sign of C depends on the initial data, that is on the constant A. If 0 < A < 8 then C is always negative. If 8 < A < 8 + 2α, then C is initially negative and changes sign at t = t 1 + 4ℓ[2(α + 4) − A]/A(A − 8), becoming positive forever. Finally, if A ≥ 8 + 2α, then C is always positive. Normally one requires C ≥ 0 to avoid a naked singularity in the bulk. An interesting feature is the behaviour of the dark radiation U = C/a 4 near the initial time t * . In general it has a minimum point at t * but grows quickly [this can be seen from Eqs. (14) and (11)]. For C > 0 (i.e. A > 8 + 2α) the general case describes the growth of the black hole mass from a minimum initial state, when the brane is at the horizon. The solid line in Fig. 1 illustrates this growth.
However, for the special case A = 2(4 + α) the behaviour is completely different: This special case is the one identified by perturbative analysis in Ref. [2], but the general case was not given, and nor were the exact general solutions, Eqs. (21), (24) and (25). The special case corresponds to C = 0 at a = 0, i.e., the black hole is created from nothing. However, the creation begins from a singularity (ρ(t 1 ) = ∞), so it is not clear what this means. The growth of C in this special case is illustrated by the dotted line in Fig. 1.
In the low energy limit, the standard RS2 behaviour is recovered, as expected:

IV. DYNAMICAL SYSTEMS ANALYSIS
Following [9], we use a dynamical systems approach to investigate the global properties of the phase space of solutions. Defining the modified Friedmann equation (14) takes the form In contrast to RS2 and general relativistic models, the phase space determined by the variables Ω = (Ω ρ , Ω U , ω) is not compact. This is because there are models in which Ω U can change sign. We could restrict the phase space by the physical argument that Ω U ≥ 0 because C represents the mass of a black hole in the bulk. In that case, in the restricted phase space, the models in which C changes sign will be represented by trajectories that in the past cross the boundary of the restricted phase space. In all cases, Eq. (30) implies that Ω U ≤ 1. Whenever Ω U ≥ 0, we must have 0 ≤ Ω ρ ≤ 1 and −1 ≤ ω ≤ 1.
In Fig. 4 we can see the line of equilibrium points Ω * 3 that extends to infinite values of Ω ρ and Ω U . (Its projection in the other two planes can be seen in Figs. 2 and 3). This plot describes the dynamics for contracting models only, where the point Ω * 2 appears. The corresponding one for expanding models can be obtaining by replacing Ω * 2 by Ω * 1 , that is, by moving the equilibrium point to the origin, and inverting the arrows. The three figures show the existence of trajectories on which Ω U , and hence C, changes sign, going from negative to positive values. Note that the contrary, to go from positive to negative values of C, is forbidden. Expanding models will always end up with a positive C.

V. CONCLUSION
We have investigated the dynamics of a RS2-type braneworld that is radiating 5D gravitons into the bulk during the high-energy radiation (or radiative reheating) era. The simplified model developed in [2] assumes that all gravitons are emitted radially, thus allowing the bulk spacetime to be described by the Vaidya-AdS 5 metric. We found the exact solution in this model to the field equations on the brane, as given in Eqs. (21)-(25). The Weyl radiation term C/a 4 on the brane, due to the tidal effect of the bulk black hole, is only purely radiative (i.e., C = const) asymptotically, in the low-energy universe. At early times, C evolves due to graviton emission, with two cases that depend on the choice of an integration constant, as illustrated in Fig. 1. Graviton emission leads to a loss of energy from the brane and a consequent asymmetry between expanding and collapsing models, that is not present in the non-radiating RS2 braneworld. This asymmetry, as well as the asymptotic behaviour of the models at early and late times, is illustrated in the phase planes that fully characterize the dynamics, shown in Figs. 2-4.