[Paper Review] Linearized gravity on branes: from Newton's law to cosmological perturbations

Nathalie Deruelle|ArXiv.org|Jan 10, 2003

Black Holes and Theoretical Physics3 citations

TL;DR

This paper investigates linearized gravity on a 4D brane embedded in a 5D anti-de Sitter bulk, demonstrating how Newton's law emerges with small $1/r^2$ corrections. It derives junction conditions for cosmological perturbations using a scalar field on the brane, showing that only specific graviton polarizations couple to the brane, and provides explicit expressions for metric perturbations in terms of bulk graviton modes, though full quantization remains incomplete.

ABSTRACT

We review here how Newton's law can be approximately recovered in the simple, "paradigmatic", case of a flat $Z_2$-symmetric brane in a 5-D anti-de Sitter bulk. We then comment on the difficulties encountered so far in extending this analysis to cosmological perturbations on a Robertson-Walker brane.

Motivation & Objective

• To extend the Randall-Sundrum mechanism from static Newtonian gravity to cosmological perturbations on a Robertson-Walker brane.
• To clarify the role of bulk gravitons in mediating gravitational interactions on the brane, particularly in the context of cosmological evolution.
• To derive and analyze the junction conditions linking bulk metric perturbations to brane-induced metric and scalar field perturbations.
• To identify which bulk graviton polarizations couple to matter on the brane and how they affect cosmological observables.
• To lay the groundwork for comparing brane-world cosmological perturbations with standard 4D Einstein gravity predictions.

Proposed method

• Uses a 5D perturbed anti-de Sitter spacetime with a Z2-symmetric brane at $w = \infty$, employing conformally Minkowskian coordinates.
• Applies linearized Einstein equations in the bulk, reducing them to ten independent components $\gamma_{\mu\nu}$ after gauge fixing $\gamma_{Aw} = 0$.
• Imposes junction conditions at the brane to relate bulk metric perturbations to induced metric and scalar field perturbations on the brane.
• Introduces a spatial tensor $F^i_j$ to decompose junction conditions into traceless and trace parts, isolating the coupling of specific graviton polarizations.
• Analyzes the system under the assumption of only zero-mode bulk gravitons, simplifying the dynamics to a single mode $e(k)$ for each wavevector.
• Derives explicit expressions for $\zeta$, $\chi$, and $\gamma_{\mu\nu}|_{\Sigma}$ in terms of $e(k)$, using Fourier modes and a time-dependent phase shift $T(\eta)$.

Experimental results

Research questions

• RQ1 How do the junction conditions for linearized gravity on a brane in 5D anti-de Sitter space reduce to Newton's law in the static limit?
• RQ2 Which bulk graviton polarizations couple to matter on the brane, and how do they affect the induced metric and scalar field perturbations?
• RQ3 What is the form of the metric perturbations on the brane when only zero-mode bulk gravitons are present?
• RQ4 How do the junction conditions constrain the relationship between the brane scalar field $\chi$ and the bulk graviton amplitude $e(k)$?
• RQ5 What is the correct quantization condition for bulk gravitons and brane fields, and how does it affect the power spectrum of cosmological perturbations?

Key findings

• The junction conditions reduce to a system of equations (5.7–5.10) that fully describe brane perturbations when matter is modeled as a scalar field.
• Only the $e_{33} = -e_{03} = e_{00} \equiv e(k)$ polarization couples to the brane scalar field, while $e_{13}, e_{23}$ correspond to free 4D gravitational waves.
• The brane metric perturbation $\gamma_{\mu\nu}|_{\Sigma}$ is explicitly expressed in terms of $e(k)$ via equation (5.15), with a time-dependent phase shift $T(\eta) = \int d\eta \sqrt{1 + \mathcal{L}^2 H^2}$.
• The scalar field perturbation $\chi$ and the brane metric potential $\zeta$ are both proportional to $e(k)/k$, with $\chi \propto \dot{\Phi} a e(k)/k$ as shown in (5.14).
• The induced metric perturbations are fully determined by the bulk graviton amplitude $e(k)$, assuming only zero modes are excited.
• The paper identifies the key obstacle to full comparison with 4D cosmology: the lack of a consistent quantization of the action $\int_{\text{bulk}} \sqrt{-g_5}(\mathcal{R}+\Lambda) d^5X + \kappa \int_{\text{brane}} \sqrt{-g_4} \mathcal{L}_m d^4x$.