> For the complete documentation index, see [llms.txt](https://slowdiveptg.gitbook.io/notes/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://slowdiveptg.gitbook.io/notes/basics-of-cosmological-physics/chapter-3.md).

# Chapter 3 Relativistic Cosmology

## Robertson-Walker Metric

* Flat (Euclid) space

  $$
  (\text{d}l)^2=(\text{d}x)^2+(\text{d}y)^2+(\text{d}z)^2\\
  \Rightarrow \Delta l=\int\_1^2\sqrt{(\text{d}x)^2+(\text{d}y)^2+(\text{d}z)^2}
  $$
* Flat (Minkowski) spacetime

  $$
  (\text{d}l)^2=(c\text{d}t)^2-(\text{d}x)^2-(\text{d}y)^2-(\text{d}z)^2\\
  \Rightarrow \Delta l=\int\_1^2\sqrt{(c\text{d}t)^2-(\text{d}x)^2-(\text{d}y)^2-(\text{d}z)^2}
  $$
* Proper distance

  distance at $t\_A=t\_B$ (hypothesis on the existence of a well-defined universal time)
* Curvature

  $$
  K=\frac{3}{\pi}\lim\_{D\to 0}\frac{2\pi D-C}{D^3}
  $$

  where $D​$ is the radius, $C​$ is the perimeter

  Some **2-D​** examples

  * Plain - zero curvature
  * Sphere - positive curvature
  * Hyperbolic paraboloid (马鞍面；双曲抛物面) - negative curvature
* 2-D curvature

  Spherical polar coordinates $(r,\theta,\phi)$

  $$
  \begin{aligned}
  (\text{d}l)^2&=(\text{d}D)^2+(r\text{d}\phi)^2\\
  &=(R\text{d}\theta)^2+(r\text{d}\phi)^2\\
  &=\left(\frac{\text{d}r}{\cos\theta}\right)^2+(r\text{d}\phi)^2\\
  &=\left(\frac{\text{d}r}{\sqrt{1-r^2/R^2}}\right)^2+(r\text{d}\phi)^2
  \end{aligned}
  $$

  In general, 2-D curvature

  $$
  (\text{d}l)^2=\left(\frac{\text{d}r}{\sqrt{1-kr^2}}\right)^2+(r\text{d}\phi)^2
  $$
* 3-D curvature

  $$
  (\text{d}l)^2=\left(\frac{\text{d}r}{\sqrt{1-kr^2}}\right)^2+(r\text{d}\theta)^2+(r\sin\theta\text{d}\phi)^2
  $$
* R-W Metric

  $$
  (\text ds)^2=(c\text dt)^2-\left(\frac{\text{d}r}{\sqrt{1-kr^2}}\right)^2-(r\text{d}\theta)^2-(r\sin\theta\text{d}\phi)^2
  $$

  * Proper distance

    $$
    \Delta L=\sqrt{(-\Delta s)^2},\ \text dt=0
    $$
  * Comoving distance

    $$
    K(t)=\frac{k}{a^2(t)},\ r(t)=a(t)x\\
    \Rightarrow (\text ds)^2=(c\text dt)^2-a^2(t)\left\[\left(\frac{\text{d}x}{\sqrt{1-kx^2}}\right)^2-(x\text{d}\theta)^2-(x\sin\theta\text{d}\phi)^2\right]
    $$

    Traditionally, we use $r$ to represent **comoving radial distance**

    $$
    \Rightarrow (\text ds)^2=(c\text dt)^2-a^2(t)\left\[\frac{\text{d}r^2}{1-kr^2}-r^2(\text{d}\theta^2+\sin^2\theta\text{d}\phi^2)\right]
    $$
* SR model - [the Milne Model](https://en.wikipedia.org/wiki/Milne_model) - **empty universe**

  ![](https://1509032923-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MPMxe8Bu9WDT3p-DA_8%2Fsync%2F665ede152071e5a99ab8662aae0219df36c44216.png?generation=1608873186437555\&alt=media)

  * Time dilution

    $$
    \tau=\gamma^{-1}t\\
    (t,r)\Rightarrow(\tau,l)
    $$

## Friedmann Equation

* Einstein's field equation

  $$
  G\_{\alpha\beta}=\frac{8\pi G}{c^2}T\_{\alpha\beta}
  $$

  * Einstein tensor

    $$
    G\_{\alpha\beta}=R\_{\alpha\beta}-\frac{1}{2}g\_{\alpha\beta}R
    $$
  * Metric

    $$
    \mathrm ds^2=g\_{\alpha\beta}\mathrm dx^\alpha\mathrm dx^\beta
    $$
* Friedmann equation

  $$
  \left(\frac{\dot{a}}{a}\right)^2+\frac{kc^2}{a^2}=\frac{8\pi G}{3}\rho\\
  \frac{\ddot a}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3P}{c^2}\right)
  $$

  the second is known as acceleration equation
* At present time

  $$
  \Omega\_0=\frac{\rho\_0}{\rho\_c},\ \rho\_c=\frac{3H\_0^2}{8\pi G}
  $$

  rewrite Friedmann equation

  $$
  \dot a\_0^2=\frac{8\pi}{3}Ga\_0^2\rho\_0-kc^2\\
  \Rightarrow H\_0^2a\_0^2=H\_0^2a\_0^2\Omega\_0-kc^2\\
  \Rightarrow kc^2=H\_0^2a\_0^2(\Omega\_0-1),\ k=+1,0,-1
  $$

  * $\Omega\_0>1$, overcritical density, $k=+1$
  * $\Omega=1$, critical density, $k=0$
  * $0<\Omega\_0<1$, undercritical density, $k=-1$
* Difference between Newtonion cosmology
  * Newtonion

    $$
    kc^2=-\frac{2U}{mc^2}
    $$
  * GR, $k$ stands for curvature

## Cosmological constant

* Static universe
  * $a$ is a constant, $\dot a=\ddot a=0$
  * $H=0$
  * Age of the universe is infinite
  * Friedmann equation

    $$
    \frac{kc^2}{a^2}=\frac{8\pi G}{3}\rho=-\frac{8\pi G}{c^2}P\_0
    $$
  * To make sure that $\rho>0$, $P\_0<0$ , Einstein introduced a constant, Lorentz-invariant term $\Lambda$

    $$
    G\_{\alpha\beta}-\Lambda{g\_{\alpha\beta}}=\frac{8\pi G}{c^4}T\_{\alpha\beta}
    $$

    rewrite Friedmann equation

    $$
    \left(\frac{\dot{a}}{a}\right)^2+\frac{kc^2}{a^2}=\frac{8\pi G}{3}\rho+\frac{\Lambda c^2}{3}\\
    2\frac{\ddot a}{a}+\left(\frac{\dot{a}}{a}\right)^2+\frac{kc^2}{a^2}=-\frac{8\pi G}{c^2}P+\Lambda c^2
    $$
  * Define **vacuum energy density**

    $$
    \rho\_{vac}=\frac{\Lambda c^2}{8\pi G}
    $$

    we have

    $$
    2\frac{\ddot a}{a}+\left(\frac{\dot{a}}{a}\right)^2+\frac{kc^2}{a^2}=-\frac{8\pi G}{c^2}(P-\rho\_{vac})
    $$
  * For Newtonion cosmology, we have to introduce an additional potential energy term

    $$
    V\_\Lambda\equiv-\frac{1}{6}\Lambda mc^2r^2
    $$

    $$
    \begin{aligned}
    \Rightarrow U&=T+V+V\_\Lambda\\
    &=\frac{1}{2}m\dot{r}^2-\frac{4\pi}{3}G\rho r^2m-\frac{1}{6}\Lambda mc^2r^2
    \end{aligned}
    $$

    $$
    \Rightarrow \vec F\_\Lambda=-\nabla V\_\Lambda
    $$