# Chapter 4 World Models

* Scalar factor: $a(t)$ natural units: $c=1$

  $$
  \left(\frac{\dot a}{a}\right)^2 + \frac{k}{a^2} = \frac{8\pi}{3}G\rho\\
  H(t)\equiv \frac{\dot a}{a},\quad \rho\_\Lambda=\frac{\Lambda}{8\pi G}\\
  H^2 + \frac{k}{a^2} = \frac{8\pi G}{3}\left(\sum\_i\rho\_i+\rho\_\Lambda\right)
  $$

  $k=0​$, flat

  $$
  \rho\_\text{tot} = \sum\_i\rho\_i+\rho\_\Lambda=\frac{3H^2}{8\pi G}\equiv\rho\_\text{crit}
  $$

  define $\Omega*i\equiv\rho\_i/\rho*\text{crit}​$

  $$
  \left{
  \begin{aligned}
  &\Omega\_m: \text{matter}\\
  &\Omega\_r: \text{radiation}\\
  &\Omega\_\Lambda: \text{dark energy}
  \end{aligned}
  \right.
  $$

  all are functions of time. Deriving

  $$
  \frac{k}{a^2H^2}=\sum\_i\Omega\_i + \Omega\_\Lambda - 1
  $$

  define

  $$
  \Omega\_k \equiv -k/a^2H^2\\
  \Rightarrow \sum\_i\Omega\_i+\Omega\_\Lambda+\Omega\_k=1
  $$

## Flat FRW Cosmologies

* Friedmann-Robertson-Walker: FRW models
* Assuming pressureless matter $ P=0$

  $$
  \rho\_m = \rho\_{m,0}\left(\frac{a}{a\_0}\right)^{-3}
  $$

  define $a\_0\equiv1$

  $$
  \left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho\_{m,0}a^{-3}+\frac{\Lambda}{3}\\
  \dot a^2=H\_0^2\Omega\_{m,0}a^{-1} + H\_0^2\Omega\_{\Lambda,0}a^2\\
  \Omega\_{m,0}+\Omega\_{\Lambda,0}=1
  $$

  * $\Lambda>0$

    $$
    u=\frac{2\Omega\_{\Lambda,0}}{\Omega\_{m,0}}a^3\\
    \dot u^2 = 9H\_0^2\Omega\_{\Lambda,0}(2u+u^2)=3\Lambda(2u+u^2)
    $$

    do integration to $u$, derive

    $$
    a^3=\frac{\Omega\_{m,0}}{2\Omega\_{\Lambda,0}}\[\cosh(3\Lambda)^{1/2}t-1]
    $$

    * For small $t$, $a\propto t^{2/3}$
      * For large $t$, $a\propto e^{\[(\Lambda)/3]^{1/2}t}$
  * $\Lambda<0$

    $$
    u=-\frac{2\Omega\_{\Lambda,0}}{\Omega\_{m,0}}a^3\\
    a^3=\frac{\Omega\_{m,0}}{2(-\Omega\_{\Lambda,0})}{1-\cosh\[3(-\Lambda)]^{1/2}t}
    $$

    * For small $t$, $a\propto t^{2/3}$
    * For large $t$, $a\propto -e^{\[(-\Lambda)/3]^{1/2}t}​$
  * $\Lambda=0​$ Einstein-de Sitter

    $$
    a = \left(\frac{t}{t\_0}\right)^{2/3}\\
    t\_0=\frac{2}{3H\_0}\\
    a = \left(\frac{9}{4}H\_0^2t^2\right)^{1/3}
    $$

    * A flat, pressureless universe with a small, but non-zero, cosmological constant **initially evolves as if it were Einstein-deSitter**

## Cosmologies with $k\neq0, \Lambda=0$

$$
\left(\frac{\dot a}{a}\right)^2+\frac{k}{a^2}=\frac{8\pi G}{3}\rho\\
\dot a^2=\Omega\_{m,0}H\_0^2a^{-1}-k=\Omega\_{m,0}H\_0^2a^{-1}+\Omega\_{k,0}H\_0^2\\
\Omega\_{m,0}+\Omega\_{k,0} = 1
$$

* $\Omega\_{k,0}>0, k<0$, negative curvature

  $$
  \dot a^2\sim\Omega\_{k,0}H\_0^2=-k>0, \ a\propto t
  $$

  $a$ grows linearly with time
* $\Omega\_{k,0}0​$, positive curvature

  let $\dot a=0$, derive $a\_{max}$

  $$
  a\_{\max }=\frac{\Omega\_{\mathrm{m}, 0}}{\left|\Omega\_{\mathrm{k}, 0}\right|}
  $$

![](https://1509032923-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MPMxe8Bu9WDT3p-DA_8%2Fsync%2F9ecc4669c5e1d18b6d3435cd65cf6b4725a2ed75.png?generation=1608876130109964\&alt=media)

|     | $\Omega\_{m,0}$ | $\Omega\_{\Lambda,0}$ | $\Omega\_{k,0}$ |             Concordance cosmology             |
| :-: | :-------------: | :-------------------: | :-------------: | :-------------------------------------------: |
|  1  |       0.3       |          0.7          |        0        |      $a\propto\exp\[(\Lambda/3)^{1/2}t]$      |
|  2  |       0.3       |           0           |       0.7       |                  $a\propto t$                 |
|  3  |        1        |           0           |        0        | $a = \left(\frac{9}{4}H\_0^2t^2\right)^{1/3}$ |
|  4  |        4        |           0           |        -3       |                 $a\_{max}=4/3$                |

* The age of universe can be an indirect proof for acceleration of universe expansion, for only universe with high $\Omega\_\Lambda$ can give a universe age old enough (global cluster)