4D gravity localized in non Bbb Z2 -symmetric thick branes

We present a comparative analysis of localization of 4D gravity on a non Bbb Z₂ -symmetric scalar thick brane in both a 5-dimensional riemannian space time and a pure geometric Weyl integrable manifold in which variations in the length of vectors during parallel transport are allowed and a geometric scalar field is involved in its formulation. This work was mainly motivated by the hypothesis which claims that Weyl geometries mimic quantum behaviour classically. We start by obtaining a classical 4-dimensional Poincaré invariant thick brane solution which does not respect Bbb Z₂ -symmetry along the (non-)compact extra dimension. This field configuration reproduces the Bbb Z₂ -symmetric solutions previously found in the literature, in both the Riemann and the Weyl frames, when the parameter k ₁ = 1. The scalar energy density of our field configuration represents several series of thick branes with positive and negative energy densities centered at y ₀ . Thus, our field configurations can be compared with the standard Randall-Sundrum thin brane case. The only qualitative difference we have encountered when comparing both frames is that the scalar curvature of the riemannian manifold turns out to be singular for the found solution, whereas its weylian counterpart presents a regular behaviour. By studying the transverse traceless modes of the fluctuations of the classical backgrounds, we recast their equations into a Schödinger's equation form with a volcano potential of finite bottom (in both frames). By solving the Schödinger equation for the massless zero mode m ² = 0 we obtain a single bound state which represents a stable 4-dimensional graviton in both frames. We also get a continuum gapless spectrum of KK states with positive m ²>0 that are suppressed at y ₀ , turning into continuum plane wave modes as y approaches spatial infinity. We show that for the considered solution to our setup, the potential is always bounded and cannot adopt the form of a well with infinite walls; thus, we do not get a discrete spectrum of KK states, and we conclude that the claim that weylian structures mimic, classically, quantum behaviour does not constitute a generic feature of these geometric manifolds.