Calculating energy density in the early Universe

Question

I've been told to calculate the energy density in the early Universe. It states that it is completely dominated by neutrinos (3 species), photons, electrons, and positrons.

Now, I've found an equation to calculate the energy density for bosons and fermions, with different $g$-factors (degree of freedom) for each type of particle. So that doesn't seem to complicated. But, in order to get the energy density of the early Universe, I need to incorporate all particles in this density - is my guess.

My thought is just to calculate and energy density for each particle, and then sum up, and divide by the number of different particles. But that just seems to trivial, so is that the way to go, or am I missing something in order to do this correctly ?

Lars · Answer

Normally you would use the effective degrees of freedom $g_star$, which sums up the particles. This varies from 106.75 at T>200 GeV to 3.36 (after electron-positron annihilation). The values for $g_star$ may be looked up in a table, e.g. https://arxiv.org/pdf/1609.04979.pdf.

Then the energy density is given as: $$ epsilon(T) = frac{pi^2}{30}g_star T^4 $$

Then the energy density is given as:
$$ epsilon(T) = frac{pi^2}{30}g_star T^4 $$

Calculating energy density in the early Universe

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