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Brane-worlds and Cosmology

The Friedmann equation for a FRW-brane-world in a flat bulk is derived and applied to the accelerated expansion of the universe.


Brane-Worlds
It has been recently noted that the predicted scale of 10 15 TeV for quantum gravity is only a conjecture without experimental support, and that the only experimentally verified scale of gauge interactions in four dimensions lies within the TeV scale. Therefore, the assumptions that gravitation becomes strong at the TeV scale, while the standard gauge interactions remain confined to the four dimensional space-time, does not conflict with the today's experimental data [1,2]. These ideas, aimed to solve the the hierarchy problem of the fundamental interactions, were originally inspired by the Horawa-Witten M-theory [3].
We may identify four basic principles of the brane-world program: The first one, of phenomenological nature, sets the fundamental scale of interaction, including gravity to the TeV scale. The other three principles are of theoretical nature, stating that the space-time (or, the brane-world) is embedded in a higher dimensional bulk space; that ordinary matter and the standard gauge interaction remain confined to the brane-world; and that the extra dimensions are probed by gravitons.
Under these assumptions, the extra dimensions can be large and sometimes non-compact. This follows from an simple argument using the Einstein-Hilbert action for the brane-world V 4 , assuming that the bulk has product where M * denotes the equivalent to Planck's mass in the higher dimensional space.
On the other hand, assuming that ordinary Einstein's gravity remains confined to the space-time V 4 , we find that where V is the volume of B N . Comparing the two expressions and denoting by ℓ the typical length of the extra dimensions, we obtain If M * ≈ 1TeV e M P l ≈ 10 15 TeV, then ℓ ≈ 10 13 cm when N = 1. This is approximately the diameter of the solar system, quite unsuitable as an internal dimension. On the other hand if N = 2, a six dimensional bulk, we obtain the sub millimeter size ℓ ≈ 10 −2 cm. Larger number of dimensions would produce even smaller values for ℓ.
One immediate consequence is the modification of Newton's gravitational theory at small distances. In fact, replacing M P l in the Newtonian gravitational potential, we find For large values of r as compared with ℓ, we obtain the usual Newtonian theory. However, when r ≈ ℓ we find so that for N = 2 the gravitational field would decay as 1/r 3 at distances of the order of 10 −2 cm. Interesting enough, Newton's theory has never been accurately verified at those small distances. A number of experiments to verify this are currently being assembled [4].
Other notable consequences are the possible generation of (high frequency) gravitational waves and the detection of black-holes (and worm-holes) in the space-time at the TeV scale, estimated to occur sometime around 2006 at the LHC [5]. Cosmology would also be a laboratory to observe these phenomena, where the presence of TeV gravitons at the core of galaxies could give a new estimate for the vacuum energy. Even at the classical level, we may be already observing the effect of the extra dimensions in the form of the accelerated expansion of the universe.

Some Geometrical Aspects
The brane-worlds program has been implemented mostly on particular models where the bulk has a fixed geometry and the space-time has a specific metric ansatz. Nonetheless, it is possible to give a fairly general description based on the four mentioned principles. Here we present only a brief sketch [6].
Consider that we have a D-dimensional bulk V D in which we have a local and isometric embedded brane-world V 4 given by the embedding map Z : V 4 → V D whose derivative map satisfies the embedding equations where G µν is the metric of V D in arbitrary coordinates, g AB denotes the metric of the space orthogonal to V 4 and g iA = s M A iAM , where The embedding associates an extrinsic curvature to V 4 , defined for each di- Contrarily to the Riemannian curvature, the extrinsic curvature gives a measure of the deviation from the brane-world and its tangent plane at any point, which we call the bending of the geometry. Thus, we may have a bent space even if it is flat in the Riemannian sense. We may define a mean curvature for each normal direction to V n by h A = g ij κ ijA . The total mean curvature is The integrability conditions for the embedding equations (1) are the Gauss-Codazzi-Ricci equations, respectively From (1), we obtain Applying this, the contractions of Gauss' equation gives where we have denoted ω 2 = b ijA b ijA [6]. The divergence term can be discarded under a volume integration in the extra dimensions and assuming that the boundary of the integration region is minimal (corresponding to h A = 0). After adding the Lagrangian for the confined matter in the right hand side, we obtain the brane-worlds Lagrangian The corresponding Einstein's equations for the brane-world can be calculated directly from the contractions of Gauss' equations where we have denoted the extrinsic term by and an additional term which also depends on the Riemann and Ricci curvatures of the bulk 9) Now, the confinement hypothesis implies that ordinary matter, is confined to the brane-world. Therefore, if the bulk is a solution of the high dimensional Einstein's equations, it is either a vacuum or cosmological constant solution Placing this on the left hand side of (7) we obtain To these equations we add the conservation law, obtained from the contracted Bianchi identity 8πGT ij ;j + S ij ;j = 0, Q ij ;j = 0 (11) We focus our attention to Q ij which depends essentially on the extrinsic curvature and does not depend on the Riemannian curvature. This is as far as we can go without being more specific. A full account of the brane-worlds equations in a general bulk, including the confinement and some quantum perspectives can be found on [6] and references therein.

FRW as a Brane-World in Flat Bulk
The standard FRW cosmological model is known to be embeddable in a flat 5-dimensional flat bulk with metric signature (4, 1) and g 55 = 1 [7]. We use the following parametrization where f (r) = senr, r, senhr corresponding to k = 1, 0, −1 respectively. The confined matter is the perfect fluid T ij = (p + ρ)U i U j + pg ij , where U i = δ 0 i in commoving coordinates. The embedding equations (4) become simply Codazzi's equation may be easily solved, with solution [8] b or, after denoting B =ḃ/b e H =ȧ/a, it follows that Therefore In this example we have S ij = 0 and the dynamical equations (7) for the FRW metric becomeä After eliminatingä, we obtain the modified Friedmann's equatioṅ Notice that b(t) = b 11 , measures how the universe bends from the tangent space along the radial direction. To this we add the conservation law and the state equation p = (γ − 1)ρ, which are combined iṅ It is possible to add an extra equation relating b(t) to the matter density. This follows from the sandwich conjecture in general relativity using junction conditions such as Israel's adapted to higher dimensions in the Randall-Sundrum moldels. However, these junction conditions are not unique and their use lead to instabilities [9]. The application of these conditions to homogeneous and isotropic cosmologies produces a modified Friedmann equation, with an added a term proportional to the square of the density [10].
On the other hand, the accelerated expansion of the universe may be seen as an evidence for equation (14), suggesting that the universe bends in proportion to its expansion: In fact, we easily see that the negative constant k ′2 on the left side of (14) acts as an accelerator for the expansion, even when k = 0.
Another interesting comment is that the signature of the bulk may vary as a response to the quantum fluctuations [6]. This means that the term k ′2 in (14) may change sign with the consequent shift from accelerated expansion to contraction.