Warped geometry of brane worlds

Abstract

We study the dynamical equations for extra-dimensional dependence of a warp factor and a bulk scalar in 5d brane world scenarios with induced brane metric of constant curvature. These equations are similar to those for the time dependence of the scale factor and a scalar field in 4d cosmology, but with the sign of the scalar field potential reversed. Based on this analogy, we introduce novel methods for studying the warped geometry. We construct the full phase portraits of the warp factor/scalar system for several examples of the bulk potential. This allows us to view the global properties of the warped geometry. For flat branes, the phase portrait is two dimensional. Moving along typical phase trajectories, the warp factor is initially increasing and finally decreasing. All trajectories have timelike gradient-dominated singularities at one or both of their ends, which are reachable in a finite distance and must be screened by the branes. For curved branes, the phase portrait is three dimensional. However, as the warp factor increases the phase trajectories tend towards the two dimensional surface corresponding to flat branes. We discuss this property as a mechanism that may stretch the curved brane to be almost flat, with a small cosmological constant. Finally, we describe the embedding of branes in the 5d bulk using the phase space geometric methods developed here. In this language the boundary conditions at the branes can be described as a 1d curve in the phase space. We discuss the naturalness of tuning the brane potential to stabilize the brane world system.

Brane-World Gravity

“…For example, if there is a bulk scalar field, then there is energy-momentum exchange between the brane and bulk (in addition to the gravitational interaction) [225, 16, 236, 97, 194, 98, 35]. …”

“…The modified Friedmann equation has the standard 1-brane RS-type form, but with a dark radiation term that no longer behaves strictly like radiation: By Equations (68) and (228), we arrive at the matter conservation equations, This shows how the brane loses( ψ > 0) or gains ( ψ < 0) energy in exchange with the bulk black hole. For an FRW brane, this equation reduces to The evolution of m is governed by the 4D contracted Bianchi identity, using Equation (229): For an FRW brane, this yields where ρε = 3 m ( t )/( κ 2 a 4 ).A more complicated bulk metric arises when there is a self-interacting scalar field Φ in the bulk [225, 16, 236, 97, 194, 98, 35]. In the simplest case, when there is no coupling between the bulk field and brane matter, this gives where Φ ( x, y ) satisfies the 5D Klein-Gordon equation, The junction conditions on the field imply that then Equations (68) and (235) show that matter conservation continues to hold on the brane in this simple case: From Equation (235) one finds that where so that the modified Friedmann equation becomes When there is coupling between brane matter and the bulk scalar field, then the Friedmann and conservation equations are more complicated [225, 16, 236, 97, 194, 98, 35].…”

Brane inflation and the overshoot problem

“…For inflation to be insensitive to initial conditions one typically hopes that the inflaton field will naturally flow toward an attractor solution [26,27,28], on which the inflaton slow-rolls and the Universe inflates. However, should the energy of the inflaton be initially kinetic dominated, it is possible for the end of inflation to be reached before the solution has approached the attractor.…”

Brane cosmology and motion of test particles in five-dimensional warped product spacetimes

“…(6). Let us consider a bulk metric of the form: d S 2 = e 2 f (t,y) dt 2 − btdr 2 − btr 2 dθ 2 − btr 2 sin(θ ) 2 dφ 2 − p(t, y)dy 2 (16) where b is another constant and f is given by Eq. (5) as f (t, y) = at +l ln(cosh(cy)).…”

“…In this case, confinement is due purely to the classical gravitational effects, without requiring the presence of brane-type confinement mechanisms. This has led to the investigation of warped product spaces and their geometrical properties (see for example [16]. Randall and Sundrum [10,11], achieved it through the construction of five-dimensional warped product spaces using an exponential warp factor in a non-factorizable metric furnished with mirror symmetry, even when the fifth dimension was infinite.…”