Friedmann-Robertson-Walker Braneworlds

We study the cosmological evolution with nonsingular branes generated by a bulk scalar field coupled to gravity. The specific setup investigated leads to branes with a time-dependent warp factor. We calculate the effective Hubble parameter and the effective scale factor for the FRW branes obtained solutions. The spatially dependent branes solutions also were found.

apply the Gauss-Codazzi formalism. The reason is that it is not clear what (if any) are the Israel-Darmois junction conditions in a thick brane context. The junction conditions are at the heart of the projection procedure, and its lack makes the whole program fall apart. In this vein, the investigation of a five-dimensional thick braneworld setup whose four-dimensional part describes a Friedmann-Robertson-Walker (FRW) universe is, indeed, in order. There are, nevertheless, only a few works addressing this crucial point. The cumbersome algebraic task inevitably present in this endeavor can be attributed as a cause for such.
In this paper we address ourselves to this task, starting from a five dimensional scalar field whose solution describes a FRW brane, i. e., a braneworld whose four-dimensional part is given by a FRW universe. The main braneworld characteristic, the warp factor, is also take into account in the solution.
We organize this paper as follows. In Section II, we present the mathematical preliminaries supporting the metric ansatz with the four-dimensional background metric given by the Friedmann-Lemaître-Robertson-Walker. Then we derive gravitational equations and the expressions for the scalar field and its potential. In Section III, we find the time-dependent FRW brane solutions.
We solve the field equations for two cases with respect to spatial curvature, i.e, k = 0 and k = 0.
Going further we determine the spatial part of the set of equations in Section IV. In Section V we present the effective Hubble parameter as well as the effective scale factor for all the possibilities found previously. In the final section we conclude.

II. FIELD EQUATIONS
The assumption of isotropy and homogeneity implies the large scale geometry described by a metric of the form in a synchronized coordinate system (a suitable set of coordinates called comoving coordinates).
The comoving observers, also called Hubble observers, are the ones located at spacelike hypersurfaces accompanying the cosmic fluid, which is at rest with respect to such hypersurfaces. Here a(t) is an arbitrary function of the cosmic time called scale factor and k = 0, ±1, denotes the spatial curvature of the universe for Minkowski, Riemann and Lobachevsky geometry, respectively.
Let us consider 5D spacetimes for which the metric takes the following form where the background metric in 4D is given by the Friedmann-Lemaître-Robertson-Walker line element (1). The metric signature is given by (− + + + +). The function a(t, y) is an warp factor with time and extra dimension dependence, while u(t) performs the usual scale factor for an homogeneous and isotropic universe. The function b(t, y) shows the dynamics of the extra dimension at different times and positions in the bulk.
Let us consider the 5D action in the presence of a bulk scalar field with the potential V (φ) minimally coupled to the gravitational sector where M is the Planck mass and R is the five-dimensional Ricci scalar. In general we suppose that the scalar field φ depends only on time and the extra dimension y.

The Einstein equations read
and the energy momentum-tensor T M N for the scalar field φ(t, y) is The time-time component of the field equations for the space-time under consideration is given by whilest the the space components give The extra dimensional part contributes with Finally, the scalar equation of motion reads where a dot denotes a derivative with respect to t, and a prime represents a derivative with respect to the extra dimension y.
In what follows we shall depict several important cases resulting from some relevant particularizations.

III. TIME-DEPENDENT SOLUTIONS
Let us assume that φ(t, y) ≡ φ(y). Thus the G 0 5 component of the field equation becomes implying the possibility of spatial and temporal separation, i.e., a(y, t) ≡ α(y)β(t). Therefore, the equations (11)-(13) take the form At this point we need to consider separately some different regimes of the above equations. Afterwards, we solve the time-dependent part of the solutions and, then, the spatial part is analyzed.
Within the specifications outlined in this subsection epigraph, the equations read .. ..
It is interesting to notice that the set of equations (18)-(20) with k = 0 and u = 1, recover the result obtained by [4], whose solution is On the other hand, for the case in which k = 0 e u = 1 and using the following redefinition where f (t) is an arbitrary function, we have B. The Case ∆ = −Σ with k = 0 and k = 0 Now, for the case specified here, the set of equations to be solved is .. ..
All together, these equations implies ∆ = −Σ and therefore the solution is given by Note that as far as k = 0 and u(t) = 1, we recover result obtained in [4], i.e., as expected.
For k = 0, the solution is given by Making use of the usual trigonometric relation sec 2 x = 1 + tan 2 x, we have or simply .
Thus Eq. (29) can be rewritten as where use was made of Note that, similarly to the case of Eq. (22), one can reproduce the results obtained in [4], in which the solutions given in (32), for k = 1, k = −1 (both for u = 1) are respectively and We shall investigate the cosmological outputs of the obtained solutions in Section V.

IV. SPATIAL-DEPENDENT SOLUTIONS
Now we shall consider the spatial part of the equations (11), (12) and (13): There are two interesting cases, concerning the separation constants which we are going to investigate in detail. The first one is given by the vanishing of both.
A. The case ∆ = Σ = 0 For better dealing with the system of equations we use the re-definition, In the light of (39), the Equations (36)-(38) become and Assuming, for ulterior convenience, that and substituting the equations (42) in (41) we find s = −1/3. Therefore the potential acquires the form The derivative of Eq. (43) with respect to φ, V φ , reads Therefore the potential itself can be expressed as and the Eqs. (42) become with Now we turn ourselves to a different arrangement of the separation constants. and Combining the equations (48), we obtain By assuming [5] that and substituting Eqs. (51) and (52) into (50), we have Taking the derivative of (53) with respect to φ one gets and inserting (54) in the scalar field equation (49), one arrives at the following consistence equation for the superpotential where a = 1 and b = 1/3. Now we define the following quantities from which one can see that some terms can be written as a total derivative, and the equation (55) takes the form d dφ In order to deal with a concrete and exact case, we assume that For simplicity, some authors consider Z(φ) = W (φ) [5,6]. Here we shall consider a more general case, substituting (58) into Eq. (57). This procedure leads to In order to keep some similarity with the well known literature, we look for a solution as the one presented in Ref. [7], for instance. We choose thereof In the light of Eq. (59), gathering Z, Z 2 φ and Z 2 terms together, we have By taking the Z coefficient equal to zero we get as the solution for v, the following expression where C 1 = ± 3/2C 4 and C 2 = C 3 + λ − 2σ if v = 1, in order to full accomplish the consistence constraint. In this vein, the expression (58) for W , becomes with the choice Z 0 = 1 and s = 0 in equation (60).
By means of Eqs. (67) and (68) for the expression (50), we obtain the following shape for the potential In the figures (1) and (2) we depict the profiles of φ(y) and A(y) in the relevant range where the scalar field is also varying. Before to delve into the effective quantities study, we remark by passing that, despite the rather non-trivial functional form of the obtained solutions, the resulting spacetime is after all well behaved. In fact, all the Kretschmann scalars associated to the solutions are finite.

V. EFFECTIVE HUBBLE PARAMETER AND SCALE FACTOR
In this section, from the found solutions for β(t), we obtain the effective Hubble parameter as well as the effective scale factor. The Hubble parameter, H = .
a/a, is used to measure the expansion rate of the universe. The time elapsed in this scenario is the so called proper time or cosmic time.
In view of the transformation it is possible to construct the effective Hubble parameter where A. The case ∆ = Σ = 0 Consider for a while the case that k = 0, so that and therefore H ef f (τ ) = 0 and a ef e = constant.
This case leads, then, to the typical Hubble parameter describing a static universe, sometimes called Einstein's universe.
Within this case (∆ = Σ = 0) but now with k = 0, the solution is given by Hence we have where we defined that dz = dt/ f (t). Thus, we find the expression for the cosmic time τ as given by Choosing, for simplicity, C = 0 and using f (t) ≡ e −2at , we have z = (1/a) e at and, consequently Therefore and it is possible see that returning to Eq. (75) one arrives at Finally, substituting the expressions (80) and (79) into (71), one can verify that the effective Hubble parameter decays with the inverse of τ ,while that scale factor is linearly growing and real as far as It is reasonable that the behavior of H ef f (τ ) is the one expected both for the matter and radiation dominated phase of the universe. Unfortunately, however, the corresponding scale factor grows much faster than it should in a realistic scenario.
and from Eq. (66), where b = Σ/3 and z = t 0 dt ′ /u(t ′ ). Inserting Eqs. (82) and (83) into the expression (71), one gets and therefore whose integration leads to We can note that the above solution to a ef e (τ ) with k = 0 is similar to the solution found for an usual universe (without any brane), and dominated by the vacuum energy, i.e., where Λ > 0 is the cosmological constant. This fact is more important that it may sound. In fact, in the context of thick braneworlds, as discussed in the Introduction, there is no how to directly relate the four-dimensional cosmological constant with some counterpart quantity in five dimensions, or even some property of the brane modeling. In this approach, however, we see that the separation constant (necessary to solve the gravitational system endowed with a brane) takes the place of the four-dimension cosmological constant. The solution given in (86) could, indeed, represent the current phase of accelerated expansion of our universe in a ΛCDM (Lambda Cold Dark Matter) model.
Finally, for the case in which k = 0, and remembering that , it can be seen that By taking dz = dt/u(t) and integrating the above expression, we obtain Now, by inverting the Eq. (90) for z as a function of τ , one gets In order to reach Eq. (92) we have used dτ = β(t)dt in the first equality. Substituting Eq. (90) in (91) one arrives at and therefore By means of (94) the effective scale factor is given by The above Hubble parameter, despite the fact that it presents an expected behavior at large values of τ , is again too fast at lower values of τ .

VI. CONCLUDING REMARKS
We have investigated exact solutions for a FRW braneworld, whose brane is performed by a bulk scalar field. The general idea was to find out explicit solutions which could, at least in some regime, to perform the large scale dynamics of the universe. In the course of our analysis a plenty of possibilities had appeared. Among them, we believe we pay attention to the most physically appealing cases.
In some aspects, the physical outputs can model a specific era of the known universe, as in the case represented by Eq. (81) in which the matter and radiation phases can be reached. By the same token, in the specific k = 0, ∆ = −Σ case, the separation constant Σ can mimic a four-dimensional cosmological constant for a de-Sitter-like universe. Therefore, the late-time acceleration can be modeled without regarding to any type of dark energy.
Currently we are delving into the possibility to describe more aspects of the cosmic evolution.
To accomplish that, more bulk scalar fields, as well as different potentials may be in order. We shall postpone these generalizations for a future work.