# Cosmological models - Evolution of Inhomogenity

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