Bulk-Brane Matching in Bianchi-Types Brane World
We discuss a comprehensive description of the geometry of the brane-world cosmologies, and present bulk and brane structure and matching between brane and bulk metrics. It is clear that the possibility of the matching condition is not always obvious and therefore it requires a separate analysis. In this study we have shown, under the assumption of consideration of the anisotropic metric except Kasner-AdS like, matching procedure is not achieved for Bianchi-types bulk metrics. Examples of this result are presented by the illustrations of the Bianchi-types II and V bulk metrics.
Randall and Sundrum (RS) made an intriguing alternative suggestion in which we reside in a universe, 3+1-dimensional surface (the “brane”), of more than four noncompact dimensions. They examine a brane in a space of higher dimension, which is called bulk and it is a slice of anti de Sitter spacetime (AdS) . In these models, five-dimensional Einstein field equations, where is the five-dimensional Einstein tensor, is the five-dimensional coupling constant, and is the energy-momentum tensor. It can be written as, where, , , and are the metric, tension, and energy-momentum tensors of the brane, respectively.
The effective four-dimensional gravitational equations on the brane take the form [2, 3]: where, where is the local quadratic energy-momentum correction and is nonlocal effect from the free bulk gravitational field. Thus, it is not possible to fully understand brane solutions without explicitly knowing the bulk solution.
In the literature, if we take bulk metrics as AdS-like and brane metrics as FRW-like, we make for an exact solution of (1.1), in the isotropic brane-world cosmology [4–9]. For instance, in FRW brane world, the bulk is Schwarzchild AdS and reduce to simple Coulomb term that gives a dark radiation term on the brane [10–12].
In anisotropic brane-world scenarios, the suitable bulk and brane metrics matching each other were first discovered by frolov . It is clear that Kasner-type brane-world model can be viewed as the generalization of an isotropic model. The five-dimensional Kasner anti de Sitter metric described by where is 3-dimentional spatial metric varying with time Here the exponents must satisfy the familiar Kasner restrictions, Thus, the brane also has a tension and matter-density given, respectively, as which is the the Randall-Sundrum like fine tuning between the brane tension and the bulk cosmological constant. Since the brane does not include matter, that is to say, it becomes a vacuum, it makes for a poor cosmological model. But the important point here is that it introduces anisotropy into the brane world models.
Some authors have analyzed anisotropic brane worlds including matter content [14, 15]. Particularly, dynamical systems techniques are used by Campos and Sopuerta to look into homogeneous and anisotropic Bianchi-type branes . For a summary of dynamical systems in the context of cosmology, including Bianchi-type cosmologies, refer to . However, in the these early studies many assumptions were made about the Weyl term, , due to the absence of an exact anisotropic bulk solution. This was addressed in  for the FLRW and Bianchi I case and shortly after Campos et al.  found a family of exact, anisotropic solution to the five-dimensional field equation. Therefore they were able to be explicitly see the relationship between the bulk Weyl curvature and the anisotropy on the brane. They found that it is not possible to have a perfect fluid or scalar field compatible with the anisotropic brane since the junction condition requires anisotropic stress on the brane. Fabbri et al. found more exact bulk solutions and agreed that an anisotropic brane cannot support a perfect fluid in the case where the bulk is static . Harko and Mak investigated Bianchi-type brane-world behaviour near the singularity and at late times and found that they tend to isotropize for certain matter content . Also, they found general solution of the field equations for Bianchi-type I and V in the brane .
Up to now, no complete solution for the brane and bulk metrics have been found for cosmological Bianchi brane worlds. The key difficulty is to find anisotropic generalization of AdS that can incorporate anisotropy on a cosmological brane, and that is necessarily nonconformally flat.
In this study we have shown, under the assumption of consideration of the anisotropic metric except Kasner-AdS-like, matching procedure is not achieved for Bianchi-types metrics . Examples of this result are shown by the illustrations of the Bianchi-types II and V metrics. Throughout this paper we will use the following notation: latin letters denote coordinate indices in the bulk spacetime and in the brane , and also tilde and upper “5” mean 5-dimensional quantities.
2. Bianchi-Type II and V Space Time
2.1. Bianchi-Type II
We consider the 5-dimensional metric: where the 3-dimensional spatial part of the metric can be expressed in diagonal form as We assume that the metric coefficients , and depend on both and .
The one-form have the relationship where, the are the structure constants corresponding to the particular Bianchi-type. In the case of type-II, the nonzero structure constants are The exact solution of the 5-dimentional Einstein Equation for vacuum case was obtained by Halpern . Following his paper, the metric coefficients can be expressed in the following manner: where
This set of solutions is purely exponential in character, with monotonic behavior similar to the Kasner (type I) solution. Note, however, that the relationship amongst these exponents is more complex than in the Kasner case.
2.2. Bianchi-Type V
We now consider 5-dimentional Bianchi-type V spatial geometry. We write the metric in the same manner as (2.1) with the nonzero structure constant of the Lie algebra of one-forms equal to Exact solution of the 5-dimentional Einstein equation was obtained by Halpern . Following his paper, the metric coefficients can be expressed in the following manner: where and are independent parameters with to ensure that all scale factors are real.
3. Brane in Anisotropic Bulk
In this section, we consider what will happen if the 3-brane is embedded in the Bianchi types II and V derived above. Following [13, 25], we describe some useful identities for suitable embedding, where, represents derivative with respect to .
For generalization of (2.1), we can take its components Then, 5-dimentional metric takes the form
Induced metric on the brane is If we chose and get proper time, we can write the local frame or It is not difficult to show these equations implying that the timelike vector is given by Also, using and , where, is normal vector, then we obtain some useful relations, Finally, we find that unit normal vector to the brane is or where, .
Now, After defining (3.12), we obtain useful form of extrinsic curvature tensor for the brane embedded in the spacetime defined as has the following nonvanishing components where we use overdots to represent partial derivatives with respect to , asterisks to represent partial derivatives with respect to , and overcommas to represent partial derivatives with respect to .
The Israel's junction condition is given by where is energy-momentum tensor of the brane and is induced metric on the brane. , and its trace are defined as where is brane matter-energi density, is matter pressure, is brane tension, and is four vector. The metric inhered by the brane and other hypersurfaces of the foliation is the first fundamental form, and its components are Using (3.16) and (3.18), we obtain energy-momentum tensor components Finally, if substituting the last equations into (3.15), one gets the brane equation of motions in the bulk
Because the anisotropic bulk coefficients , and depend on both and differently, the only way to satisfy (3.20) to (3.23) simultaneously without introducing anisotropic matter content on the brane, is to have the anisotropic term vanish in these equations. This is succeeded when , that is, when brane is not moving. Then we obtain . Finally these equations are reduced, respectively, in the following: From last tree equations, we infer that
Case I (for the Bianchi-type II). If we compare (3.28) with (2.5), we infer that , that is, complex values for the metric coefficients.
Case II (for the Bianchi-type V). If we compare (3.28) with (2.8), we infer that which is contrary to ensuring all scale factors are real.
Up till now, studies related to the isotropic brane-world models, because of the existence of suitable selections of bulk and brane metrics, there exists many models in the literature such as propose solutions to the current cosmological problems. Brane cosmology with anisotropy has not been clearly understood yet. Apart from Frolov's Kasner-AdS model, there are no additional anisotropic brane-world models which contain bulk-brane matching. The simplest generalizations of FRW brane worlds are Bianchi brane worlds. In this study, by using Frolov's method, after having obtained the equations of motion of brane, we investigated the bulk-brane matching of the Bianchi-type II and Bianchi-type V models whose exact bulk solutions are known. In the result, we have found that the coefficients of bulk and brane metrics are not matching each other since they are imaginary.
Just to finish we would like to mention some current and future work in the line of the present one. First, since Bianchi-types cosmology has large anisotropy, it would be interesting to suppose the matter on the brane possess some anisotropy, then take into account other Bianchi-type bulk solutions. In this sense, a good starting point would be to consider scenarios like those introduced in .
L. Randall and R. Sundrum, “An alternative to compactification,” Physical Review Letters, vol. 83, no. 23, pp. 4690–4693, 1999.View at: Publisher Site | Google Scholar | MathSciNet
A. N. Aliev and A. E. Gumrukcuoglu, “Gravitational field equations on and off a 3-brane world,” Classical and Quantum Gravity, vol. 21, no. 22, pp. 5081–5095, 2004.View at: Publisher Site | Google Scholar | MathSciNet
T. Shiromizu, K. I. Maeda, and M. Sasaki, “The Einstein equations on the 3-brane world,” Physical Review D, vol. 62, no. 2, Article ID 024012, 6 pages, 2000.View at: Publisher Site | Google Scholar | MathSciNet
P. Binetruy, C. Deffayet, and D. Langlois, “Non-conventional cosmology from a brane universe,” Nuclear Physics B, vol. 565, no. 1-2, pp. 269–287, 2000.View at: Publisher Site | Google Scholar | MathSciNet
P. BinEtruy, C. Deffayet, U. Ellwanger, and D. Langlois, “Brane cosmological evolution in a bulk with cosmological constant,” Physics Letters B, vol. 477, no. 1–3, pp. 285–291, 2000.View at: Publisher Site | Google Scholar
D. Langlois, “Brane cosmology,” Progress of Theoretical Physics, no. 148, pp. 181–212, 2002.View at: Publisher Site | Google Scholar | MathSciNet
D. Langlois, “Cosmology in a brane-universe,” Astrophysics and Space Science, vol. 283, pp. 469–479, 2003.View at: Google Scholar
C. Germani and R. Maartens, “Stars in the braneworld,” Physical Review D, vol. 64, no. 12, Article ID 124010, 6 pages, 2001.View at: Publisher Site | Google Scholar | MathSciNet
P. Binetruy, C. Deffayet, and D. Langlois, “Non-conventional cosmology from a brane universe,” Nuclear Physics B, vol. 565, pp. 269–287, 2000.View at: Google Scholar
P. Bowcock, C. Charmousis, and R. Gregory, “General brane cosmologies and their global spacetime structure,” Classical and Quantum Gravity, vol. 17, no. 22, pp. 4745–4763, 2000.View at: Publisher Site | Google Scholar | MathSciNet
P. Kraus, “Dynamics of anti-de Sitter domain walls,” The Journal of High Energy Physics, vol. 3, no. 12, article 011, 1999.View at: Publisher Site | Google Scholar | MathSciNet
D. Ida, “Brane-world cosmology,” The Journal of High Energy Physics, vol. 4, no. 9, 2000.View at: Publisher Site | Google Scholar | MathSciNet
A. V. Frolov, “Kasner-AdS spacetime and anisotropic brane-world cosmology,” Physics Letters B, vol. 514, no. 3-4, pp. 213–216, 2001.View at: Publisher Site | Google Scholar | MathSciNet
J. M. Aguirregabiria, L. P. Chimento, and R. Lazkoz, “Anisotropy and inflation in Bianchi I braneworlds,” Classical and Quantum Gravity, vol. 21, no. 4, pp. 823–829, 2004.View at: Publisher Site | Google Scholar | MathSciNet
J. D. Barrow and S. Hervik, “Magnetic brane-worlds,” Classical and Quantum Gravity, vol. 19, no. 1, pp. 155–172, 2002.View at: Publisher Site | Google Scholar | MathSciNet
A. Campos and C. F. Sopuerta, “Evolution of cosmological models in the brane-world scenario,” Physical Review D, vol. 63, no. 10, Article ID 104012, 14 pages, 2001.View at: Publisher Site | Google Scholar | MathSciNet
J. Wainwright and G. F. R. Ellis, Dynamical Systems in Cosmology, Cambridge University Press, Cambridge, UK, 1997.View at: Publisher Site
A. Campos and C. F. Sopuerta, “Bulk effects in the cosmological dynamics of brane-world scenarios,” Physical Review D, vol. 64, no. 10, Article ID 104011, 13 pages, 2001.View at: Publisher Site | Google Scholar | MathSciNet
A. Campos, R. Maartens, D. Matravers, and C. F. Sopuerta, “Braneworld cosmological models with anisotropy,” Physical Review D, vol. 68, no. 10, Article ID 103520, 9 pages, 2003.View at: Publisher Site | Google Scholar | MathSciNet
A. Fabbri, D. Langlois, D. A. Steer, and R. Zegers, “Brane cosmology with an anisotropic bulk,” Journal of High Energy Physics, no. 9, 2004.View at: Publisher Site | Google Scholar | MathSciNet
T. Harko and M. K. Mak, “Anisotropy in Bianchi-type brane cosmologies,” Classical and Quantum Gravity, vol. 21, no. 6, pp. 1489–1503, 2004.View at: Publisher Site | Google Scholar | MathSciNet
C.-M. Chen, T. Harko, and M. K. Mak, “Exact anisotropic brane cosmologies,” Physical Review D, vol. 63, no. 4, Article ID 044013, 12 pages, 2001.View at: Publisher Site | Google Scholar | MathSciNet
D. Giang and C. C. Dyer, “Velocity dominated singularities in the cheese slice universe,” International Journal of Modern Physics D, vol. 18, no. 1, pp. 13–23, 2009.View at: Publisher Site | Google Scholar
P. Halpern, “Exact solutions of five dimensional anisotropic cosmologies,” Physical Review D, vol. 66, no. 2, Article ID 027503, 4 pages, 2002.View at: Publisher Site | Google Scholar | MathSciNet
O. Sevinc, “Bianchi-type II spacetime and anisotropic brane-world cosmology,” Balkan Physical Letter, vol. 18, no. 181003, pp. 16–22, 2009.View at: Google Scholar