Four-dimensional gravity on a thick domain wall

TL;DR

This work investigates how four-dimensional gravity emerges on a thick domain wall interpolating between two asymptotic $AdS_5$ spaces in gravity coupled to a scalar. The authors construct a closed-form thick-wall solution via a first-order formalism with a superpotential $W$, yielding a metric function $A(r)$ and scalar profile $\phi(r)$, with a single remaining parameter $c$ controlling the AdS curvature $bc$ and wall thickness. They analyze metric fluctuations in the transverse-traceless sector, which lead to a Schrödinger-type equation $( -\partial_z^2 + V_{QM}(z) - k^2 ) \psi(z) = 0$ with $V_{QM} = \frac{9}{4}A'(z)^2 + \frac{3}{2}A''(z)$, and find a single normalizable zero mode and no resonances in the continuum. Consequently, gravity is localized in the thick-wall setup in a manner qualitatively identical to the thin-wall Randall-Sundrum scenario, with heavy bulk modes decoupling and RS-like corrections vanishing in the high-curvature limit.

Abstract

We consider an especially simple version of a thick domain wall in $AdS$ space and investigate how four-dimensional gravity arises in this context. The model we consider has the advantage, that the equivalent quantum mechanics problem can be stated in closed form. The potential in this Schrödinger equation suggests that there could be resonances in the spectrum of the continuum modes. We demonstrate that there are no such resonances in the model we consider.

Figures (2)

• Figure 1: The potential $V_{QM}$.
• Figure 2: Wave functions for moderate $k^2$ (rapidly oscillating) and small $k^2$ (half period). The wave functions are shown in the unphysical normalization $\psi(0)=1$.