Cosmological models - Evolution of Inhomogenity

tosv

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 3-Ricci 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