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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.