The fluid equation is the equation of motion for the energy density \(\rho\) as a function of time. It is given by an adiabatic expansion of the universe.,

\[\begin{split} \begin{gather} \boxed{\dot\mu + 3\frac{\dot a}{a}(\mu + P) = 0}\\ \dot\mu = -3\frac{\dot a}{a}(\mu + P)\\ \boxed{\dot \mu \frac{a}{\dot a} = -3(\mu + P)} \end{gather} \end{split}\]

Proof - Newtonian

You may prove exactly, the fluid equation for the adiabatic expansion of a sphere with total internal energy \(E = V(t)\mu(t)\) where \(V(t)\) is,

\[ \boxed{V(t) = \frac{4\pi}{3}r_s^3a(t)^3} \]

Equation of State

To solve for the fluid equation we need to know \(P(\mu)\) which is called the equation of state.

Linear Case

A very famous example of an equation of state is the ideal gas law

\[ P = \frac{N}{V}\tau = w\mu \]

This will be the one we will use for the simple universe.

\[\begin{split} \begin{equation} P = w\mu\\ w = \begin{cases} 0 & \text{matter}\\ 1/3 & \text{radiation}\\ -1 & \text{dark energy} \end{cases} \end{equation} \end{split}\]

• \(w\) : A constant relating to the type of energy. Notice that from the acceleration equation, the universe accelerates for \(w < -1/3\) so it motivates that the unknown energy (dark energy) is negative. Why we use exactly \(w = -1\) for dark energy is explained in the cosmological constant

For a universe of several energy components, \(\mu\) is a weighted sum of those energies,

\[ P = \sum_i w_i\mu_i \]

Now let’s apply each type of energy density to the fluid equation:

\[\begin{split} \begin{gather*} \dot \mu_i + 3 \frac{\dot a}{a}\left(\mu_i + P_i\right) = 0\\ \frac{d\mu_i}{\mu_i} = -3(1+w_i)\frac{da}{a}\\ \end{gather*} \end{split}\]

\[ \begin{equation} \boxed{\mu_i(a) = \mu_{i,0}a^{-3(1+w_i)}} \end{equation} \]

Properties

• This gives the useful proportional relations,

  \[\begin{split} \mu_m \propto a^{-3}\\ \mu_r \propto a^{-4}\\ \mu_\Lambda = \mu_{\Lambda,0} \end{split}\]

• Notice that the dominating energy density is determined by the size of the universe. For a large universe \(a \rightarrow \infty\), the energy with the smallest \(w\) survives (cosmological constant).