# Cosmological models - Evolution of Inhomogenity

• tosv
In summary, you have made progress in rewriting the equations in terms of known spatial gradients and you can continue by using the definition of the auxiliary variable and the given evolution equations.
tosv

## Homework Statement

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.

## The Attempt at a Solution

Briefly my attempt at a solution looks like this:
$\dot{S}_{a}=[a\tilde{\nabla}_{a}(\sigma^{bc}S_{bc})]^{\cdot}$
$=\dot{a}\tilde{\nabla}_{a}(\sigma^{bc}S_{bc})+a[\tilde{\nabla}_{a}(\sigma^{bc}S_{bc})]^{\cdot}$
$=-\sigma_{a}^{b}S_{b}+a[\tilde{\nabla}_{a}(\sigma^{bc}S_{bc})]^{\cdot}$
$=-\sigma_{a}^{b}S_{b}+a\tilde{\nabla}_{a}\left(\dot{\sigma}^{bc}S_{bc}+\sigma^{bc}\dot{S}_{bc}\right)$

$\dot{S}_{bc}=-\frac{1}{3}\left(-\frac{1}{3}\Theta^{2}-2\sigma^{2}-\frac{1}{2}\mu\right)\sigma_{bc}$
$=\frac{1}{9}\Theta^{2}\sigma_{bc}+\frac{2}{3}σ^{2}\sigma_{bc}-\frac{1}{3}\mu\sigma_{bc}-\frac{2}{3}\Theta S_{bc}+\Theta\sigma_{d\langle b}\sigma_{c\rangle}^{d}+3\sigma_{d\langle b}E_{c\rangle}^{d}+2\dot{\sigma}_{d\langle b}\sigma_{c\rangle}^{d}$

Here is my problem, I do not know how I can continue to rewrite $\dot{S}_{bc}$, does anyone has any advice?

Thank you for sharing your progress on your project. It seems like you have made some good progress in rewriting the equations in terms of known spatial gradients. One suggestion I have is to try using the definition of the auxiliary variable S_{bc} to rewrite \dot{S}_{bc} in terms of known spatial gradients. You can also try using the given evolution equations for \sigma_{ab}, E_{ab}, and \Theta to simplify the expression further. Additionally, you can try using the definitions of D_{a}, Z_{a}, and T_{a} to express the spatial gradients in terms of these variables. I hope this helps and good luck with your project!

## 1. What is the definition of a cosmological model?

A cosmological model is a mathematical framework that describes the evolution and structure of the universe on a large scale. It takes into account various factors such as expansion, matter distribution, and the influence of dark energy.

## 2. How do cosmological models explain the evolution of inhomogeneity in the universe?

Cosmological models use the principles of general relativity to explain how inhomogeneities in the early universe evolved over time. These models suggest that small fluctuations in the density of matter eventually led to the formation of large-scale structures such as galaxies and galaxy clusters.

## 3. What role do dark matter and dark energy play in cosmological models?

Dark matter and dark energy are two components that are necessary to explain the observations and predictions of cosmological models. Dark matter is thought to make up about 27% of the universe's total mass and helps to explain the observed gravitational effects on galaxies and galaxy clusters. Dark energy is believed to be responsible for the accelerated expansion of the universe and makes up about 68% of the universe's energy.

## 4. Are there different types of cosmological models?

Yes, there are several different types of cosmological models, including the Big Bang model, the Steady State model, and the Inflationary model. Each of these models has its own set of assumptions and predictions about the evolution of the universe.

## 5. How do scientists test and refine cosmological models?

Scientists use various methods to test and refine cosmological models, including observations of the cosmic microwave background radiation, the distribution of galaxies, and the measurement of the universe's expansion rate. By comparing these observations to the predictions of different models, scientists can determine which model best fits the data and make adjustments to improve their accuracy.

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