[Paper Review] Brane Cosmology for Vacuum and Cosmological Constant Bulk

Kh. Saaidi, Abolhassan Mohammadi|arXiv (Cornell University)|Jun 9, 2010

Black Holes and Theoretical Physics3 references3 citations

TL;DR

This paper investigates brane cosmology in a 5D bulk with vacuum or cosmological constant, focusing on a 3-brane filled with polytropic and modified polytropic gas. It derives scale factor evolution equations and shows that early-time deceleration and late-time acceleration emerge depending on bulk and brane parameters, with positive cosmological constant required for late-time acceleration in the modified polytropic model.

ABSTRACT

We consider the cosmology of a 3-brane universe, in a five dimensional space time (Bulk). We present some solutions to the five-dimensional Einstein equation, where a perfect fluid is confined to the 3-brane. We investigate the evolution of brane for two models of bulk. We choose ordinary and modified polytropic gas in brane and it is seen that these models have some new features.

Motivation & Objective

• To explore cosmological evolution in a 5D brane world scenario with a 3-brane containing a perfect fluid.
• To analyze the impact of different bulk conditions—vacuum and cosmological constant—on brane dynamics.
• To investigate the role of polytropic and modified polytropic equations of state for the brane fluid in shaping cosmic evolution.
• To determine the conditions under which the universe exhibits acceleration or deceleration at early and late times.

Proposed method

• Formulates the 5D Einstein equations with a metric including a fifth spatial coordinate and time, assuming a warped geometry.
• Decomposes the energy-momentum tensor into brane-localized matter and bulk contributions, with the brane matter modeled as a perfect fluid with polytropic equation of state.
• Derives the effective Friedmann equation on the brane by solving the 5D Einstein equations under the assumption of homogeneity and isotropy on the brane.
• Introduces a modified polytropic equation of state with parameter α, analyzing two cases: −1 < α < 0 and α = 1 + 1/n.
• Solves the dynamical equations for the scale factor a₀(t) in vacuum and cosmological constant bulk scenarios, using energy density expressions derived from the equation of state.
• Analyzes the behavior of ȧ₀² and ä₀ to determine acceleration/deceleration phases in early and late time limits.

Experimental results

Research questions

• RQ1 How does the inclusion of a polytropic gas on the brane affect the cosmological evolution in a 5D bulk?
• RQ2 What are the conditions under which the brane universe exhibits early-time deceleration and late-time acceleration?
• RQ3 How does the presence of a cosmological constant in the bulk influence the dynamics of the brane universe?
• RQ4 What constraints does the modified polytropic equation of state impose on the initial size and evolution of the brane?
• RQ5 Under what conditions is the cosmological constant in the bulk required to be positive for consistent late-time dynamics?

Key findings

• For the standard polytropic model with A < 0, the brane energy density is bounded at early times; for A > 0, the scale factor has a nonzero minimum value.
• In the vacuum bulk model, the universe decelerates at early times and approaches a state with ä₀ = 0 at late times, regardless of the sign of A.
• In the cosmological constant bulk model, early-time behavior mirrors the vacuum case, but late-time dynamics depend on the value of b₀², allowing for acceleration.
• The modified polytropic model with −1 ≤ α < 0 leads to a negative constant in the equation of state, rendering it physically unsuitable.
• For α = 1 + 1/n, the scale factor a₀ has a minimum initial value, and the model predicts early deceleration and late-time acceleration.
• In the modified polytropic model with a cosmological constant bulk, a positive Λ is required to ensure positive ȧ₀² at late times, confirming consistency with accelerated expansion.