[Paper Review] Matter in the Bulk and its Consequences on the Brane: A Possible Source of Dark Energy

Subenoy Chakraborty, Asit Banerjee+1 more|ArXiv.org|Jul 2, 2007

Cosmology and Gravitation Theories1 references3 citations

TL;DR

This paper proposes that matter in the bulk of a higher-dimensional spacetime can generate an effective dark energy-like behavior on the brane through gravitational coupling, even when the bulk matter is non-exotic. By generalizing the Randall-Sundrum model to include a bulk perfect fluid or dust, the authors derive modified Einstein equations on the brane showing that bulk dust induces an effective equation of state consistent with dark energy, offering a geometric origin for cosmic acceleration without introducing new fields.

ABSTRACT

The usual brane world scenario with anti de Sitter bulk has been generalized by considering a general form of energy momentum tensor in the bulk. The modified Einstein equation on the brane has been constructed. Two examples have been cited of which, the first one shows the usual brane equations when matter in the bulk is a negative cosmological constant. In the second example, the bulk matter is in the form of perfect fluid and as a result, an effective perfect fluid is obtained in the brane. Also it is noted that the effect of the dust bulk on the brane shows a dark energy behaviour and may be a possible explanation of the dark energy from the present day observational point of view.

Motivation & Objective

• To generalize the standard brane world scenario by allowing arbitrary matter content in the bulk, beyond the usual cosmological constant.
• To investigate how bulk matter influences the effective dynamics on the brane through the projected energy-momentum tensor and Weyl curvature.
• To explore whether bulk dust or perfect fluid can naturally lead to dark energy-like behavior on the brane.
• To derive the modified Einstein equations on the brane under general bulk matter conditions, including energy-momentum exchange.

Proposed method

• Derive the effective Einstein equation on the brane using the Gauss and Codazzi equations, incorporating the bulk energy-momentum tensor $ T_{AB} $ and brane tension $ \lambda $.
• Apply $ Z_2 $ symmetry to simplify the junction conditions and express the projected bulk energy-momentum tensor $ T_{\mu\nu}^{(P)} $ in terms of bulk fluid variables.
• Use the Codazzi equation to relate the divergence of the brane energy-momentum tensor to the bulk matter flux, introducing energy exchange between bulk and brane.
• Analyze two cases: (1) bulk with negative cosmological constant (recovering standard RS model), and (2) bulk with perfect fluid, leading to effective fluid on the brane.
• Compute the effective energy density $ \rho_{\text{eff}} $ and pressure $ p_{\text{eff}} $ on the brane as functions of bulk density, pressure, and brane tension.
• Assess the energy conditions (e.g., strong energy condition) on the brane to determine if the induced matter exhibits dark energy behavior.

Experimental results

Research questions

• RQ1 Can bulk matter other than a cosmological constant induce dark energy-like effects on the brane?
• RQ2 How does the presence of a perfect fluid in the bulk modify the effective energy-momentum tensor on the brane?
• RQ3 Under what conditions does bulk dust lead to a violation of the strong energy condition on the brane?
• RQ4 What is the role of $ Z_2 $ symmetry and energy exchange in shaping the effective dynamics on the brane?
• RQ5 Can the induced effective fluid on the brane mimic dark energy without introducing new fields?

Key findings

• The effective energy density on the brane due to bulk dust is $ \rho_{\text{eff}} = \frac{3}{4}\rho + \frac{\kappa_5^2 \lambda^2}{8} $, which includes a positive contribution from brane tension.
• The effective pressure on the brane is $ p_{\text{eff}} = p + \frac{1}{4}\rho - \frac{\kappa_5^2 \lambda^2}{8} $, and for dust ($ p=0 $), this leads to $ p_{\text{eff}} = \frac{1}{4}\rho - \frac{\kappa_5^2 \lambda^2}{8} $.
• When $ \rho < \frac{\kappa_5^2 \lambda^2}{6} $, the strong energy condition is violated on the brane, indicating dark energy-like behavior.
• The induced matter on the brane behaves as a perfect fluid with $ \rho_{\text{eff}} + p_{\text{eff}} = \rho + p $, preserving the total fluid character.
• For a perfect fluid in the bulk satisfying the dominant energy condition, the induced fluid on the brane also satisfies it, ensuring physical consistency.
• The model provides a geometric origin for dark energy via bulk gravitational effects, without requiring exotic matter on the brane.