Bouncing Cosmology with 4D-EGB Gravity

Abstract

Not long ago, a novel gravitational scheme i.e, $4D-EGB$ (Einstein-Gauss-Bonnet) gravity have been proposed by Glavan and Lin 2020. They rescaled the coupling factor $\alpha$ with $\frac{\alpha }{D-4}$ and developed the field equations. The purpose of this paper is to workout the cosmic bounce with a cubic form of scale factor and workout the bouncing scenario under these assumptions. The flat FLRW metric is used along with the perfect fluid to study the energy conditions. The conditions are scrutinized by using different coupling factors $\alpha$ and cosmological constant $\Lambda$ values. The stability of the assumed scale factor model is in evidence of universal expansion and allows the universal bounce by developing the validations of energy conditions.

Appendices

Appendix A

For the flat FLRW metric (3), the non vanishing Christoffel symbols for this study are obtained as

$$\begin{aligned} \Gamma ^{0}_{ij}=\dot{a}(t)a(t)\delta _{ij}, \Gamma ^{i}_{0j}=\dot{a}(t)a(t)\delta ^{i}_{j}. \end{aligned}$$

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here, $\delta$ shows the Kronecker symbol. The non vanishing Ricci tensor components are

$$\begin{aligned} R_{00}=-\ddot{a}(t)/a(t)(D-1), R_{jj}=\ddot{a}(t)a(t)+\dot{a}^{2}(t) (D-2). \end{aligned}$$

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The Ricci scalar turns out to be

$$\begin{aligned} R=2\ddot{a}(t)/a(t)(D-1)+(\dot{a}(t)/a(t))(D-1)(D-2). \end{aligned}$$

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Hence the Einstein tensor becomes

$$\begin{aligned} G_{00}=\frac{1}{2}(\dot{a}(t)/a(t))^{2}(D-1)(D-2), \end{aligned}$$

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$$\begin{aligned} G_{jj}=-\ddot{a}(t)a(t)(D-2)-\frac{1}{2}\dot{a}(t)^{2}(D-3)(D-2). \end{aligned}$$

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Here, $i,j=1,2,3,...,D-1$. The mathematical forms of are as follows

$$\begin{aligned} \rho (t)=\frac{\alpha (D-1) t^2 (2 a_{1}+3 a_{2} t+...)^2}{\left( a_{0}+t^2 (a_{1}+a_{2} t)+...\right) ^2}-\Lambda \end{aligned}$$

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$$\begin{aligned} p(t)=-\frac{\alpha (D-3) t^2 (2 a_{1}+3 a_{2} t+...)^2}{\left( a_{0}+t^2 (a_{1}+a_{2} t)+...\right) ^2}-\frac{4 \alpha (a_{1}+3 a_{2} t+...)}{a_{0}+t^2 (a_{1}+a_{2} t+...)}+\Lambda . \end{aligned}$$

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$$\begin{aligned} \nonumber \rho (t)\pm p(t)= & {} \frac{\alpha (D-1) t^2 (2 a_{1}+3 a_{2} t+...)^2}{\left( a_{0}+t^2 (a_{1}+a_{2} t)+...\right) ^2} \pm \frac{-\alpha (D-3) t^2 (2 a_{1}+3 a_{2} t+...)^2}{\left( a_{0}+t^2 (a_{1}+a_{2} t)+...\right) ^2}\\+ & {} \frac{-4 \alpha (a_{1}+3 a_{2} t+...)}{a_{0}+t^2 (a_{1}+a_{2} t+...)}. \end{aligned}$$

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$$\begin{aligned} \nonumber \rho (t)\pm 3p(t)= & {} \frac{\alpha (D-1) t^2 (2 a_{1}+3 a_{2} t+...)^2}{\left( a_{0}+t^2 (a_{1}+a_{2} t)+...\right) ^2} \pm \frac{-3\alpha (D-3) t^2 (2 a_{1}+3 a_{2} t+...)^2}{\left( a_{0}+t^2 (a_{1}+a_{2} t)+...\right) ^2}\\+ & {} \frac{-12 \alpha (a_{1}+3 a_{2} t+...)}{a_{0}+t^2 (a_{1}+a_{2} t+...)}. \end{aligned}$$

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Appendix B

The energy conditions plots.