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 - empty universe

  

  • 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

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

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

    $$\Rightarrow \vec F_\Lambda=-\nabla V_\Lambda$$