[Paper Review] Stability of the Einstein Static Universe in $4 D$ Gauss-Bonnet Gravity

Shou-Long Li, Puxun Wu+1 more|arXiv (Cornell University)|Apr 5, 2020

Cosmology and Gravitation Theories103 references21 citations

TL;DR

This paper investigates the stability of the Einstein static universe in four-dimensional Gauss-Bonnet gravity, derived via a rescaling of the GB coupling constant and taking the D→4 limit. It demonstrates that both spatially closed and open universes filled with a perfect fluid can be simultaneously stable against homogeneous and inhomogeneous scalar perturbations under specific fluid equation-of-state conditions, supporting the viability of the emergent universe scenario in this modified gravity framework.

ABSTRACT

By rescaling the Gauss-Bonnet (GB) coupling constant $α ightarrow α/(D-4)$ and considering the $D ightarrow 4$ limit, the GB gravity gives rise to nontrivial modification of general relativity in four dimensions. In this work, we investigate the realization of the emergent universe scenario in the $4 D$ GB gravity. First, we obtain the Einstein static universe filled with a perfect fluid. Then, we show that both spatially closed and open universes can be stable against both homogeneous and inhomogeneous scalar perturbations simultaneously.

Motivation & Objective

• To assess the viability of the emergent universe scenario in four-dimensional Gauss-Bonnet gravity.
• To determine whether the Einstein static universe can be stable against both homogeneous and inhomogeneous scalar perturbations simultaneously in 4D GB gravity.
• To analyze the stability conditions for spatially closed (k=1) and open (k=-1) universes filled with a perfect fluid.
• To explore the implications of the D→4 limit of Gauss-Bonnet gravity for cosmological models without initial singularities.

Proposed method

• Derive the effective 4D Gauss-Bonnet gravity action by rescaling the GB coupling constant α→α/(D−4) and taking the D→4 limit.
• Obtain the Einstein static universe solution by solving the modified Friedmann equations in 4D GB gravity with a perfect fluid.
• Perform a linear perturbation analysis on the metric and fluid variables, decomposing into scalar, vector, and tensor modes.
• Focus on scalar perturbations by introducing Fourier modes and deriving a second-order differential equation for the perturbation amplitude Φn.
• Analyze the stability condition by requiring the oscillatory solution of the perturbation equation, which depends on the sign of the effective frequency squared Z.
• Use the condition Z>0, along with positivity of ρ, a₀, and α̃, to derive constraints on the fluid's equation of state parameter w.

Experimental results

Research questions

• RQ1 Can the Einstein static universe in 4D Gauss-Bonnet gravity be stable against both homogeneous and inhomogeneous scalar perturbations simultaneously?
• RQ2 What are the conditions on the fluid's equation of state parameter w for the stability of a spatially closed (k=1) Einstein static universe in 4D GB gravity?
• RQ3 What are the conditions on w for the stability of a spatially open (k=-1) Einstein static universe in 4D GB gravity?
• RQ4 Does the emergent universe scenario remain viable in 4D GB gravity after the D→4 limit and coupling constant rescaling?

Key findings

• The Einstein static closed universe (k=1) is stable against both homogeneous (k²=0) and inhomogeneous (k²≥8) scalar perturbations when 0≤w<1/3.
• For k²≥8, stability requires w>f(k²), where f(k²) is a monotonically increasing function that approaches 0 as k²→∞, ensuring f(k²≥8)<0.
• The closed universe remains stable for non-relativistic matter (w≈0) and radiation (w=1/3), which are physically relevant fluids.
• The Einstein static open universe (k=-1) is stable when w>1/3, indicating a requirement for stiff fluids.
• The stability analysis confirms that the 4D GB gravity model supports a stable Einstein static universe for both closed and open spatial geometries under appropriate fluid conditions.
• The results suggest that the emergent universe scenario can be realized in 4D GB gravity, as the initial static state is stable against scalar perturbations.