[Paper Review] Scalar-Tensor Gravity on a Gauss-Bonnet Brane World

Stephen C. Davis|ArXiv.org|Feb 19, 2004

Black Holes and Theoretical Physics24 references3 citations

TL;DR

This paper investigates a five-dimensional brane world model with a bulk scalar field (dilaton/moduli) and Gauss-Bonnet curvature terms, showing that higher-order gravity terms lead to new 'Gauss-Bonnet branch' solutions that suppress scalar modes relative to tensor modes, enabling compatibility with solar system constraints while preserving four-dimensional gravity at large distances.

ABSTRACT

The effective four-dimensional, linearised gravity for a brane world model with higher order curvature terms and a bulk scalar field is analysed. Large and small distance gravitational laws are derived. The model has a single brane embedded in a five-dimensional bulk spacetime, and the scalar field represents the dilaton or a moduli field. The quadratic, Gauss-Bonnet curvature term (and corresponding higher kinetic terms for the scalar) is also included in the bulk action. It is particularly natural to include such terms in a brane world model. Boundary terms and junction conditions for the higher order terms are given. The extra terms allow additional solutions of the field equations, which give better agreement with observational constraints. Brans-Dicke gravity is obtained on the brane. The scalar and tensor perturbations are affected differently by the higher gravity terms, and this provides a way for the scalar modes to be suppressed relative to the tensor ones. Another new (but less useful) feature is the appearance of instabilities for some parameter ranges.

Motivation & Objective

• To investigate the gravitational implications of including a bulk scalar field (e.g., dilaton) and Gauss-Bonnet curvature terms in a five-dimensional brane world model.
• To derive and analyze the effective four-dimensional linearized gravity on the brane, particularly focusing on scalar and tensor mode behavior.
• To determine whether the inclusion of higher-order curvature terms (Gauss-Bonnet) can suppress scalar modes to satisfy solar system constraints while preserving agreement with large-scale gravity.
• To examine the stability of solutions, especially the emergence of instabilities in certain parameter regimes, and to explore stabilization via modified brane actions.

Proposed method

• Formulates a five-dimensional action with Einstein-Hilbert, Gauss-Bonnet, and scalar kinetic terms in the Jordan frame, including a conformally coupled bulk scalar field.
• Derives field equations and junction conditions for the brane, including boundary terms to ensure consistency, using the Gibbons-Hawking term and $Z_2$ symmetry.
• Solves the field equations for both 'Einstein branch' (standard gravity) and 'Gauss-Bonnet branch' (higher-order solutions), identifying new solutions due to the Gauss-Bonnet term.
• Performs linearized gravity analysis on the brane, computing the effective four-dimensional Newtonian potential and analyzing the behavior of scalar and tensor modes.
• Applies perturbation theory to the metric and scalar field, deriving the linearized Einstein and scalar equations on the brane to extract gravitational laws at large and small distances.
• Evaluates stability by analyzing the second-order variation of the action, identifying instabilities in certain parameter ranges and proposing stabilization via additional brane terms.

Experimental results

Research questions

• RQ1 How do Gauss-Bonnet terms in the bulk action affect the effective four-dimensional gravity on the brane, particularly the relative strength of scalar versus tensor modes?
• RQ2 Can the inclusion of higher-order curvature and scalar kinetic terms lead to solutions where scalar modes are suppressed, thus satisfying solar system constraints?
• RQ3 What are the junction conditions and boundary terms required for consistent dynamics when higher-order curvature invariants are present in the bulk?
• RQ4 Under what parameter ranges do the new 'Gauss-Bonnet branch' solutions become unstable, and can these instabilities be removed by modifying the brane action?
• RQ5 How do the large- and small-distance gravitational laws deviate from four-dimensional general relativity in this model?

Key findings

• The model yields a Brans-Dicke-type gravity on the brane, with the scalar field arising from the bulk dilaton, and the effective gravitational constant depending on the scalar field configuration.
• The Gauss-Bonnet branch solutions suppress scalar mode contributions relative to tensor modes, providing a mechanism to satisfy solar system constraints on fifth forces.
• Large-distance gravity on the brane asymptotically approaches four-dimensional Einstein gravity, with corrections suppressed by the brane tension and bulk parameters.
• Small-distance gravity exhibits deviations from Newton's law due to the presence of Kaluza-Klein modes and the modified bulk dynamics, with the form of the potential depending on the Gauss-Bonnet coupling.
• Several solutions, particularly in the Gauss-Bonnet branch, exhibit instabilities for certain parameter ranges, especially when the Gauss-Bonnet coupling is large or the scalar field gradient is strong.
• Stabilization of unstable solutions is possible by adding appropriate higher-order terms to the brane action, which modify the junction conditions and remove the tachyonic modes.