Cosmology: Adiabatic Expansion of a Gas

For a quick reminder, an adiabatic process is one where there is no heat exchange between a system and its surroundings. This happens either because the process happens so quickly that the system "has no chance" to exchange heat, or the system is thermally insulated. Taking a look at the first law of thermodynamics for a gas,

\begin{eqnarray}
dU &=& dQ + dW \\
dU &=& -pdV
\end{eqnarray}

For an expanding universe, we can write the energy of our gas as being proportional to volume, which is proportional to the scale factor, cubed:

\begin{eqnarray}
d(\rho a^3) &=& -p d(a^3)
\end{eqnarray}

Putting in our equation of state $p=w \rho$, we find

\begin{eqnarray}
d(\rho a^3) &=& -w \rho d(a^3)\\
d\rho a^3 + \rho d(a^3) &=& -w \rho d(a^3)\\
d\rho a^3 &=& -(w+1) \rho d(a^3)\\
\frac{d\rho}{\rho} &=& \frac{d(a^3)}{a^3}
\end{eqnarray}

Integrating both sides, we get

\begin{eqnarray}
\rho &=& \rho_0 a^{-3(w+1)}
\end{eqnarray}