[Paper Review] Emergent Universe in Brane World Scenario

Asit Banerjee, Tanwi Bandyopadhyay+1 more|ArXiv.org|May 27, 2007

Cosmology and Gravitation Theories8 citations

TL;DR

This paper proposes a singularity-free emergent universe model in the brane world scenario, where the bulk contains a cosmological constant and the brane is filled with a modified Chaplygin gas with equation of state $p = \frac{1}{3}\rho - \frac{B}{\rho}$. The model exhibits a static, Einstein-like phase in the infinite past ($t \to -\infty$) with finite scale factor and vanishing energy density fluctuations, evolving into a $\Lambda$CDM-like phase in the future, offering a viable alternative to big bang cosmology without initial singularities.

ABSTRACT

A model of an emergent universe is obtained in brane world. Here the bulk energy is in the form of cosmological constant, while the brane consists of the Chaplygin gas with the modified equation of state such as $p=Aρ-B/ρ$. Initially the brane matter for the special choice $A=1/3$ may have negative or positive pressure depending on the relative magnitudes of the parameter $B$ and the cosmological constant of the bulk, while asymptotically in future the brane world approaches a $Λ$CDM model.

Motivation & Objective

• To construct a viable emergent universe model in the brane world scenario that avoids initial singularities.
• To explore the dynamics of a brane filled with modified Chaplygin gas under a 5D bulk with cosmological constant.
• To determine whether the brane can exhibit a static, Einstein-like state in the infinite past while asymptotically approaching $\Lambda$CDM behavior.
• To analyze the role of bulk cosmological constant and brane parameters in determining the initial pressure and evolution of the model.

Proposed method

• Assumes a 5D bulk metric with a cosmological constant $\Lambda_5$, and a flat 4D brane with modified Chaplygin gas equation of state $p = A\rho - B/\rho^\alpha$.
• Uses the generalized Friedmann equations on the brane derived from the induced gravity framework, incorporating energy conservation and curvature terms.
• Solves the dynamical equation for $a_0(t)$ analytically for $A = \frac{1}{3}$, yielding exact solutions in terms of hyperbolic functions depending on integration constants and parameters.
• Analyzes the behavior of the scale factor $a_0(t)$ in the limit $t \to -\infty$, identifying a non-singular, static initial state when $C < 0$ and $b > 0$.
• Evaluates the initial energy density and equation of state on the brane, showing dependence on $\Lambda_5$ and $B$, and determines conditions for positive or negative initial pressure.
• Compares the model's asymptotic behavior to $\Lambda$CDM, confirming convergence to a de Sitter-like phase in the far future.

Experimental results

Research questions

• RQ1 Can a brane world model with modified Chaplygin gas and a 5D cosmological constant exhibit a non-singular, static state in the infinite past?
• RQ2 What conditions on the bulk cosmological constant and brane parameters allow for a transition from initial positive or negative pressure to a $\Lambda$CDM-like phase?
• RQ3 How does the choice of $A = \frac{1}{3}$ in the equation of state enable analytical solutions and emergent behavior?
• RQ4 What is the role of the integration constant $C$ in determining whether the model starts with a big bang or a bounce, or exhibits emergent behavior?

Key findings

• The model exhibits a singularity-free, static state in the infinite past ($t \to -\infty$) with constant scale factor $a_0 = \left(\frac{|C|}{2b}\right)^{1/4}$ and finite energy density $\rho_{bi} = \left(\frac{3B}{2} + \frac{6\Lambda_5}{\kappa^2}\right)^{1/2}$.
• For $C < 0$ and $b > 0$, the solution is a bouncing model with a minimum at finite time, while for $C < 0$ and $b > 0$, the solution is emergent with no initial singularity.
• The initial pressure on the brane is $p_{bi} = -\frac{1}{3}\rho_{bi} + \frac{4\Lambda_5}{\kappa^2 \rho_{bi}}$, which can be positive or negative depending on the magnitude of $\Lambda_5$ and $B$.
• When $\Lambda_5 > 0$, the initial pressure is positive if $B < \frac{4\Lambda_5}{\kappa^2}$, allowing the model to start with positive pressure and evolve into $\Lambda$CDM.
• The model asymptotically approaches a $\Lambda$CDM phase in the future, with the scale factor growing exponentially and the equation of state converging to $p \to -\rho$.
• The spatial curvature of the brane is zero, distinguishing this model from earlier closed-universe emergent models and making it compatible with current observations of spatial flatness.