Chapter 2 Newtonian Cosmology

$$\vec v_A=H_0 \vec r_A,\ \vec v_B=H_0 \vec r_B$$

• Recession velocity if B as seen by A is

  $$\vec v_{BA}=\vec v_B-\vec v_A=H_0(\vec r_B-\vec r_A)$$

  Hubble expansion is a natural property of an expanding universe that obeys the cosmological principle

Comoving coordinates

• coordinates that are carried along with the expansion

• comoving with space distance $\vec r$, comoving distance $\vec x$ (constant)

  $$\vec r_{BA} = a(t)\cdot\vec x_{BA}$$

  $a(t)$ is the scale factor - all we want to find out

• Birkhoof's theorem: the net gravitational effect of a uniform external medium on a spherical cavity is zero, the net gravitational effect comes from matter internal to radius $r$ (as a mass point)

  total energy of a particle of mass $m$ at A, B, C, D

  $$U=T+V=\frac{1}{2}m\dot{r}^2-\frac{GMm}{r}=\frac{1}{2}m\dot{r}^2-\frac{4\pi}{3}G\rho r^2m$$

Friedmann equation

$$\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2},\ \text{where }kc^2=-\frac{2U}{mx^2}$$

• $k$ must be independent of $x$, therefore $U\propto x^2$

• $k$ is a independent of $t$, for $U​$ is conserved

• Thus $k$ is just a constant

  • $k>0\Rightarrow UT$ , collapse

  • $k<0\Rightarrow U<0, V<T$ , expand forever

  • $k=0\Rightarrow U=0$ , expansion terminates when $t\to\infty$

• Hubble parameter

  $$\vec v=H_0\vec r,\ \vec r=a\vec x\\ \Rightarrow \vec v=|\dot{\vec r}|\hat{\vec r}=\frac{\dot a}{a}\vec r\\ \Rightarrow H\equiv \frac{\dot a}{a}$$

  which is a function of time, so called Hubble constant $H_0$ is the Hubble parameter at present

• Using Hubble parameter we can rewrite Friedmann equation

  $$H^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}$$

  if $k=0$ , the critical density $\rho$ is then

  $$\rho_c=\frac{3H_0^2}{8\pi G}$$
• In thermodynamics, the first law points that

  $$\text{d} E+P\text dV=T\text d S$$

  where $P$ is the pressure, $S$ is the entropy

• Expanding volume $V$ of unit comoving radius $x=1 (r=a)$, $E=mc^2$, energy within the volume

  $$E=\frac{4\pi}{3}a^3\rho c^2\\ \frac{\text dE}{\text{d} t}=4\pi a^2\rho c^2\frac{\text da}{\text{d} t}+\frac{4\pi}{3} a^3 c^2\frac{\text d\rho}{\text{d} t}\\ \frac{\text dV}{\text{d} t}=4\pi a^2\frac{\text da}{\text{d} t}$$
• Assuming a reservable expansion $\text dS=0​$ , we obtain the fluid equation

  $$\dot\rho+3\frac{\dot a}{a}\left(\rho+\frac{P}{c^2}\right)=0$$

  • The first term - the dilution in the density because the volume has increased

  • The second term - the loss in energy because the pressure of the material has done work as the universe’s volume increased.

• Equation of state

  There is a unique pressure associated with each density,

  $$P\equiv P(\rho)$$

  • For non-relativistic matter, $P=0$ , dust

  • For highly-relativistic matter , $P=\rho c^2/3$ , radiation pressure of light

• Time-derivation on both sides of the Friedmann equation

  $$2\frac{\dot a}{a}\frac{a\ddot a-\dot a^2}{a^2}=\frac{8\pi G}{3}\dot\rho+2\frac{kc^2}{a^3}\dot a\\ \frac{\ddot a}{a}-\left(\frac{\ddot a}{a}\right)^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^3}$$
• Acceleration equation

  $$\frac{\ddot a}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3P}{c^2}\right)$$
• Analytical solution

  • $k=0$

    • Dust, $P=0$

      $$\dot\rho+3\frac{\dot a}{a}\rho=0\Rightarrow\frac{\text{d}}{\text{d}t}(\rho a^3)=0$$

      $$\rho=\frac{\rho_0}{a^3}\Rightarrow\dot a^2=\frac{8\pi G\rho_0}{3}\frac{1}{a}\\\Rightarrow a(t)=\left(\frac{t}{t_0}\right)^{2/3},\ \rho(t)=\frac{\rho_0t_0^2}{t^2}\\ \Rightarrow H(t)=\frac{\dot a}{a}=\frac{2}{3t},\ t_0=\frac{2}{3H_0}$$

      However, the current age of universe $t_0​$ obtained in this model is wrong

    • Radiation, $P=\rho c^2/3$

      $$\dot\rho+4\frac{\dot a}{a}\rho=0\Rightarrow\frac{\text{d}}{\text{d}t}(\rho a^4)=0\\ \Rightarrow a(t)=\left(\frac{t}{t_0}\right)^{1/2},\ \rho(t)=\frac{\rho_0t_0^2}{t^2}\\ \Rightarrow H(t)=\frac{\dot a}{a}=\frac{1}{2t}<\frac{2}{3t},\ t_0=\frac{1}{2H_0}$$
    • Radiation dominated universe

      $$a(t)\propto t^{1/2},\ \rho_{rad}\propto t^{-2},\ \rho_{dust}\propto a^{-3}\propto t^{-3/2}$$

      Unstable, the radiation density decreases faster than the dust density, the universe will turn to matter dominated on day

    • Matter dominated universe

      $$a(t)\propto t^{2/3},\ \rho_{dust}\propto t^{-2},\ \rho_{dust}\propto a^{-4}\propto t^{-8/3}$$

      Stable

Olber's Paradox

• In a shell from $r​$ to $r+\mathrm d r​$, the total number of stars is $4\pi r^2n_0\mathrm d r​$, where $n_0​$ is the number density

• Total intensity of the sky

  $$\mu=\int_0^{r_{max}}4\pi r^2n_0\left(\frac{L_0}{4\pi r^2}\right)\mathrm d r=n_0L_0r_{max}\\ r_{max}\to\infty\Rightarrow \mu\to\infty$$

  which is $10^{13}$ times brighter than it actually is

• Thermodynamic problem

  • Infinite volume

  • Infinite time to reach thermodynamic equilibrium

  • BUT the universe is not homogeneous (not the temperature of CMB)