Energy Density

Critical Density

The critical (energy) density corresponds to the case where \(k = 0\),

\[\begin{split} \begin{gather} \left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\mu\\ \boxed{\mu_c \equiv \frac{3}{8\pi G}H(t)^2} \end{gather} \end{split}\]

Density Parameter

With the critical density value we motivate the use of normalizing with \(\mu_c\) such that we define the density parameter as,

\[ \begin{equation} \boxed{\Omega(t) \equiv \frac{\mu(t)}{\mu_c(t)}} \end{equation} \]

The universe today is about \(\Omega(t_0) \approx 1\) as expected since we live in a flat universe.

Using the density parameter, the Friedmann equation becomes,

\[ \begin{equation} \left(\frac{\dot a}{a}\right)^2 = \frac{k}{a^2}\left[1 - \Omega(t)\right]^{-1} \end{equation} \]

Radiation Energy Density

The energy density of radiation consist of all relativistic particles. Those include:

Cosmic Microwave Background : The photon emitted since the early universe with the energy density given by the CMB temperature \(T_0\),

\[\begin{split} \mu_{\text{CMB},0} = \alpha T_0^4 \\ \Omega_{\text{CMB},0} = 5.35 \times 10^{-5} \end{split}\]

Neutrino Background : Theory suggest that there should exist the quasi-relativistic particle, the neutrinos, that fills the sky very similar to the CMB with an energy density:

\[ \mu_v = 3\cdot\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\mu_\text{CMB} \approx 0.681 \mu_\text{CMB} \]

Starlight : The starlight (any radiation by massive bodies) is negligible

\[ \frac{\mu_\text{starlight}}{\mu_\text{CMB}} \approx 0.1 \]

In total we only include CMB and neutrino background into the radiation density which is found to be:

\[ \Omega_{r,0} = \frac{\mu_\text{CMB,0} + \mu_{v,0}}{\mu_c} \approx 9 \times 10^{-5} \]