General Properties of the Self-tuning Domain Wall Approach to the Cosmological Constant Problem

TL;DR

This work investigates five-dimensional brane–bulk systems with a bulk scalar, showing that self-tuning of the 4D cosmological constant is a generic feature for setups with at most two branes or a single brane with orbifold boundary conditions. A central finding is that localized gravity with an infinitely large extra dimension typically imposes fine-tuning on the brane tension, and in oscillatory bulk potentials the allowed tensions are quantized; singularities accompany most self-tuned solutions that localize gravity. The authors develop a superpotential-based first-order framework, derive exact solutions for integrable bulk potentials, and introduce perturbative and numerical methods to explore general cases, including a no-go theorem for non-singular localization. They further analyze attempts to resolve singularities (e.g., de Alwis–Nilles scenarios), showing these resolutions generically reintroduce fine-tuning unless additional fields provide extra degrees of freedom. Altogether, the paper highlights the tension between self-tuning and non-singular gravity localization, and suggests that extra fields might be required to fully realize self-tuning in realistic models.

Abstract

We study the dynamics of brane worlds coupled to a scalar field and gravity, and find that self-tuning of the cosmological constant is generic in theories with at most two branes or a single brane with orbifold boundary conditions. We demonstrate that singularities are generic in the self-tuned solutions compatible with localized gravity on the brane: we show that localized gravity with an infinitely large extra dimension is only consistent with particular fine-tuned values of the brane tension. The number of allowed brane tension values is related to the number of negative stationary points of the scalar bulk potential and, in the case of an oscillatory potential, the brane tension for which gravity is localized without singularities is quantized. We also examine a resolution of the singularities, and find that fine-tuning is generically re-introduced at the singularities in order to retain a static solution. However, we speculate that the presence of additional fields may restore self-tuning.

Figures (8)

• Figure 1: Shapes of the warp factor for singular and non-singular solutions to the equations of motion with an exponential bulk potential. We have drawn $\mathbb{Z}_2$ symmetric solutions for different values of the tension on the brane: when the tension increases, the horizon becomes closer and closer to the brane for the singular solutions while the warp factor, $e^{-2A/(D-1)}$, blows up faster and faster for the the singularity-free solutions. In both cases, the jump conditions require that the scalar coupling to the brane satisfies: $|(df/d\phi)/f|<1$ on the brane.
• Figure 2: The exact solution versus the perturbed solution for the case of a vanishing bulk potential. The curve that blows up at smaller values of $y$ corresponds to the exact solution. The initial unperturbed solution is given by the dashed curve. The singularity of the perturbed solution appears at the same distance from the brane as the singularity of the initial solution; the shift of the singularity with a variation of the tension is missing.
• Figure 3: The numerical solution overlayed on the exact solution for the case of a vanishing bulk potential. The fact that the two curves are indistinguishable shows that the numerical method works extremely well even close to the singularities.
• Figure 4: Asymptotic behaviors of $W[\phi]$ leading to a singularity free bulk geometry localizing gravity on the brane. The absence of singularities is equivalent to the conditions: ${W^{'\,}}_c^- = {W^{'\,}}_c^+=0$; the dynamics of the equations of motion require ${W^{"\,}}_c^- >0$ and ${W^{"\,}}_c^+<0$, while the localization of gravity requires $W_c^- <0$ and $W_c^+ >0$.
• Figure 5: The scalar bulk potential \ref{['V1']}. The physically interesting region, near the two negative stationary points, is emphasized.