# Cosmology Lecture 02

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### The Friedmann Equations, continued

Recall we had the following equations:

$\left({{\dot {a}} \over a}\right)^{2}={8\pi \over 3}G\rho -{k \over a^{2}}\,\!$ ${\dot {\rho }}=-3{{\dot {a}} \over a}(\rho +P)\,\!$ ${{\ddot {a}} \over a}=-{4\pi \over 3}G(\rho +3P)\,\!$ To close the equations, we need to relate $P$ and $\rho$ with an equation of state:

${P=w\rho (c^{2})}\,\!$ Note that we will generally set $c=1$ in this class. Combined with (2), this gives us:

{\begin{aligned}{\dot {\rho }}&=-3{{\dot {a}} \over a}(1+w)\rho \\{{\dot {\rho }} \over \rho }&=-3(1+w){{\dot {a}} \over a}\\\rho &\propto a^{-3(1+w)}\\\end{aligned}}\,\! Note that we’ve assumed ${\dot {w}}=0$ , which is okay most of the time. Some special cases of interest are:

• Pressure-less “dust” $P=0$ , $w=0\Rightarrow \rho \propto a^{-3}$ because volume goes as $V\propto {1 \over a^{3}}$ .
• Relativistic particles (photons, bosons): $w={1 \over 3}$ , $P={\rho \over 3}\Rightarrow \rho \propto a^{-4}$ because $VV\propto {1 \over a^{3}}$ , and energy is given by $E\propto {1 \over a}$ .
• ($\Lambda$ )/Dark Energy: $w=-1$ , $P=-\rho \Rightarrow \rho =$ constant in time.

To get density ($\rho$ ) as a function of time, want to solve for $w$ .

### Critical Density, $\rho _{crit}$ We define $\rho _{crit}$ to be the critical density at which $k=0$ (and $E=0$ ):

${\rho _{crit}}={3H^{2} \over 8\pi G}\,\!$ Today, we measure: $\rho _{crit,0}={3H_{0}^{2} \over 8\pi G}={3 \over 8\pi }{\left(100h{km \over sMpc}\right)^{2} \over G}=2.78\cdot 10^{11}h^{2}{M_{\odot } \over Mpc^{3}}=1.88\cdot 10^{-29}h^{2}{g \over cm^{3}}$ . Note that the mass of the sun is $M_{\odot }=2\cdot 10^{33}g$ , and the mass of the proton is $M_{p}=1.67\cdot 10^{-24}g$ .

### Density Parameter, $\Omega (t)$ $\Omega$ measures the ratio of the density of the universe to the critical density:

$\Omega (t)\equiv {\rho (t) \over \rho _{c}(t)}={8\pi G\rho (t) \over 3H^{2}(t)}\,\!$ $\Omega {\begin{cases}\leq 1&\Rightarrow open\ (k\leq 0)\\=1&\Rightarrow flat\ (k=0)\\\geq 1&\Rightarrow closed\ (k\geq 0)\end{cases}}\,\!$ In general, $\Omega$ can consist of multiple components: $\Omega =\sum _{i}{\Omega _{i}}$ e.g.

$\Omega {\begin{cases}r&=radiation\\m&=matter\ (dark\ and\ luminous)\\b&=baryons\ (dark\ and\ luminous)\\\nu &=neutrinos\\\Lambda &=dark\ energy\end{cases}}\,\!$ $\Omega =1$ is an unstable equilibrium; any perturbation from $\Omega =1$ in the early universe ensures $\Omega$ is far from 1 today. That we measure $\Omega _{m}\approx 0.3$ today implies that the early universe must have been extremely finely tuned.

### Evolution of $H(t)$ Today (at $a=1$ ): $k={8\pi \over 3}G\rho _{0}-H_{0}^{2}=H_{0}^{2}(\Omega _{0}-1)$ . Using the $1^{st}$ Friedmann equation (1), we have:

${H^{2}=H_{0}^{2}({\Omega _{0} \over a^{3+3w}}+{1-\Omega _{0} \over a^{2}})}\,\!$ (THE Friedmann Equation) where $H_{0}\equiv {\sqrt {8\pi G\rho \over 3{\rho _{crit}}}}$ , and it is understood that $\Omega _{0}=\sum _{i}{\Omega _{i,0}}$ . Each $\Omega _{i}$ has it’s own $w_{i}$ , so really ${\Omega _{0} \over a^{3+3w}}=\sum _{i}{\Omega _{i,0} \over a^{3+3w_{i}}}$ .

### Evolution of $\Omega (a)$ We’ll show in PS#1 that for any single component:

${1-\Omega (a) \over \Omega (a)}={1-\Omega _{0} \over \Omega _{0}}a^{1+3w}\,\!$ Plotting $\Omega (a)$ , we will find that for early $a$ , $\Omega$ is extremely close to 1.

### Evolution of $a(t)$ : Solving the Friedmann Equation

The evolution of the expansion of the universe is governed by:

$\left({{\dot {a}} \over a}\right)^{2}={8\pi \over 3}G\rho -{k \over a^{2}}\,\!$ We can apply this to several models of the universe:

• The Einstein-deSitter (flat) Model: $k=0$ , $\Omega (a)=\Omega _{0}=1$ . Using that $\rho \propto a^{-3(1+w)}$ , we have:
{\begin{aligned}\left({{\dot {a}} \over a}\right)^{2}&\propto a^{-3(1+w)}\\a^{-1}a^{{3 \over 2}(1+w)}da&\propto dt\\a^{-{3 \over 2}(1+w)}&\propto t\\\end{aligned}}\,\! ${a(t)\propto t^{2 \over 3(1+w)}}\,\!$ Thus, the rate of expansion of the universe depends on $w$ : (a) The matter-dominated era: $\Omega \approx \Omega _{m}\Rightarrow w=0,P=0,a\propto t^{2 \over 3}$ . (b) The radiation-dominated era: $\Omega \approx \Omega _{r}\Rightarrow w={1 \over 3}\Rightarrow a(t)\propto t^{\frac {1}{2}}$ . (c) The $\Lambda$ -dominated era: $\Omega \approx \Omega _{\Lambda }\Rightarrow w=-1,P=-\rho ,\rho$ constant in time $\Rightarrow a(t)\propto e^{Ht}$ , where H is now actually a constant. This is exponential inflation. We used to think that this only happened early on (like $10^{-34}$ seconds), but now we think that this has also been happening recently. Next time, we will do the harder two cases: open and closed.