# Equations of State

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## Equations of State

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

${\displaystyle \left({{\dot {a}} \over a}\right)^{2}={8\pi \over 3}G\epsilon -{k \over a^{2}}\,\!}$
${\displaystyle {\dot {\epsilon }}=-3{{\dot {a}} \over a}(\epsilon +P)\,\!}$
${\displaystyle {{\ddot {a}} \over a}=-{4\pi \over 3}G(\epsilon +3P)\,\!}$

We introduced the Fluid and Acceleration equations in order to related ${\displaystyle \epsilon }$ and ${\displaystyle a}$ in the Friedmann equation. Unfortunately, in doing so, we introduced a new unknown: the pressure ${\displaystyle P}$. To close the equations, we need to relate ${\displaystyle P}$ and ${\displaystyle \epsilon }$ with an equation of state. Equations of state generally have the form:

${\displaystyle {P=w\epsilon }\,\!}$

where pressure is proportional to energy density with some (dimensionless) constant of proportionality.

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

{\displaystyle {\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 ${\displaystyle {\dot {w}}=0}$, which is okay most of the time. We don’t have any evidence so far that ${\displaystyle w}$ changes.

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

${\displaystyle \Omega {\begin{cases}r&=radiation\\m&=matter\ (dark\ and\ luminous)\\b&=baryons\ (dark\ and\ luminous)\\\nu &=neutrinos\\\Lambda &=dark\ energy\end{cases}}\,\!}$

To get density (${\displaystyle \epsilon }$) as a function of time, want to solve for ${\displaystyle w}$. Below we examine special cases of interest.

### 1 Matter (Non-relativistic particles)

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

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