# Equations of State

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## Equations of State

Recall we had the following (Friedmann, Fluid, Acceleration) equations:

$\left({{\dot {a}} \over a}\right)^{2}={8\pi \over 3}G\epsilon -{k \over a^{2}}\,\!$ ${\dot {\epsilon }}=-3{{\dot {a}} \over a}(\epsilon +P)\,\!$ ${{\ddot {a}} \over a}=-{4\pi \over 3}G(\epsilon +3P)\,\!$ We introduced the Fluid and Acceleration equations in order to related $\epsilon$ and $a$ in the Friedmann equation. Unfortunately, in doing so, we introduced a new unknown: the pressure $P$ . To close the equations, we need to relate $P$ and $\epsilon$ with an equation of state. Equations of state generally have the form:

${P=w\epsilon }\,\!$ where pressure is proportional to energy density with some (dimensionless) constant of proportionality.

Using this generic equation of state (we haven’t decided what $w$ is yet) the Fluid equation becomes:

{\begin{aligned}{\dot {\epsilon }}&=-3{{\dot {a}} \over a}(1+w)\epsilon \\{{\dot {\epsilon }} \over \epsilon }&=-3(1+w){{\dot {a}} \over a}\\\epsilon &\propto a^{-3(1+w)}\\\end{aligned}}\,\! Note that we’ve assumed ${\dot {w}}=0$ , which is okay most of the time. We don’t have any evidence so far that $w$ changes.

In general, $\epsilon \leftrightarrow \Omega$ can consist of multiple components: $\Omega =\sum _{i}{\Omega _{i}}$ e.g.

$\Omega {\begin{cases}r&=radiation\\m&=matter\ (dark\ and\ luminous)\\b&=baryons\ (dark\ and\ luminous)\\\nu &=neutrinos\\\Lambda &=dark\ energy\end{cases}}\,\!$ To get density ($\epsilon$ ) as a function of time, want to solve for $w$ . Below we examine special cases of interest.

### 1 Matter (Non-relativistic particles)

Pressure-less “dust” $P=0$ , $w=0\Rightarrow \epsilon \propto a^{-3}$ because volume goes as $V\propto {1 \over a^{3}}$ .

Relativistic particles (photons, bosons): $w={1 \over 3}$ , $P={\epsilon \over 3}\Rightarrow \epsilon \propto a^{-4}$ because $VV\propto {1 \over a^{3}}$ , and energy is given by $E\propto {1 \over a}$ .
($\Lambda$ )/Dark Energy: $w=-1$ , $P=-\epsilon \Rightarrow \epsilon =$ constant in time.