1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Cosmological models - Evolution of Inhomogenity

  1. Mar 9, 2012 #1
    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}+\sigma^{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
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Cosmological models - Evolution of Inhomogenity
Loading...