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Cosmological models  Evolution of Inhomogenity 
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#1
Mar912, 09:55 AM

P: 13

1. The problem statement, all variables and given/known data
I'm working on a project to find evolution equations for a cosmological model, where the following propagations equations are known, [itex]\dot{\mu}=\Theta\mu[/itex] [itex]\dot{\Theta}=\frac{1}{3}\Theta^{2}2\sigma^{2}\frac{1}{2}\mu[/itex] [itex]\dot{\sigma}_{ab}=\frac{2}{3}\Theta\sigma_{ab}\sigma_{c\langle a}\sigma_{b\rangle}^{c}E_{ab}[/itex] [itex]\dot{E}_{ab}=\Theta E_{ab}+3\sigma_{c\langle a}E_{b\rangle}^{c}\frac{1}{2}\mu\sigma_{ab}[/itex] Particular spatial gradients are defined as [itex]D_{a}\equiv a\frac{a\tilde{\nabla}_{a}\mu}{\mu}[/itex] [itex]Z_{a}\equiv a\tilde{\nabla}_{a}\Theta[/itex] [itex]T_{a}\equiv a\tilde{\nabla}\sigma^{2}[/itex] From the traceless part of the 3Ricci tensor following definition of the auxilary variable are stated, [itex]S_{a}\equiv a\tilde{\nabla}_{a}\left(\sigma^{bc}S_{bc}\right)[/itex] where [itex]S_{bc}=\frac{1}{3}\Theta\sigma_{bc}+\sigma_{d\langle b}\sigma_{c\rangle}^{d}+E_{bc}[/itex] My goal is to determine [itex]\dot{S}_{a}[/itex] in terms of known spatial gradients. 2. Relevant equations 3. The attempt at a solution Briefly my attempt at a solution looks like this: [itex]\dot{S}_{a}=[a\tilde{\nabla}_{a}(\sigma^{bc}S_{bc})]^{\cdot}[/itex] [itex]=\dot{a}\tilde{\nabla}_{a}(\sigma^{bc}S_{bc})+a[\tilde{\nabla}_{a}(\sigma^{bc}S_{bc})]^{\cdot}[/itex] [itex]=\sigma_{a}^{b}S_{b}+a[\tilde{\nabla}_{a}(\sigma^{bc}S_{bc})]^{\cdot}[/itex] [itex]=\sigma_{a}^{b}S_{b}+a\tilde{\nabla}_{a}\left(\dot{\sigma}^{bc}S_{bc}+\s igma^{bc}\dot{S}_{bc}\right)[/itex] [itex]\dot{S}_{bc}=\frac{1}{3}\left(\frac{1}{3}\Theta^{2}2\sigma^{2}\frac{1}{2}\mu\right)\sigma_{bc}[/itex] [itex]=\frac{1}{9}\Theta^{2}\sigma_{bc}+\frac{2}{3}σ^{2}\sigma_{bc}\frac{1}{3}\mu\sigma_{bc}\frac{2}{3}\Theta S_{bc}+\Theta\sigma_{d\langle b}\sigma_{c\rangle}^{d}+3\sigma_{d\langle b}E_{c\rangle}^{d}+2\dot{\sigma}_{d\langle b}\sigma_{c\rangle}^{d}[/itex] Here is my problem, I do not know how I can continue to rewrite [itex]\dot{S}_{bc}[/itex], does anyone has any advice? 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 


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