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 $\Omegai\equiv\rho_i/\rho\text{crit}​$

  $$\left\{ \begin{aligned} &\Omega_m: \text{matter}\\ &\Omega_r: \text{radiation}\\ &\Omega_\Lambda: \text{dark energy} \end{aligned} \right.$$

  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 $a_{max}$

  $$a_{\max }=\frac{\Omega_{\mathrm{m}, 0}}{\left|\Omega_{\mathrm{k}, 0}\right|}$$

	$\Omega_{m,0}$	$\Omega_{\Lambda,0}$	$\Omega_{k,0}$	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)