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We know that energy density is given by this formula:

$$u = \frac{1}{V}\int_0^\infty\epsilon_pf(\epsilon_p) g_s V/h^3 4\pi p^2 dp$$

Where $\epsilon_p=pc$ for relativistic particles. also $f(\epsilon_p)= \frac{1}{\exp((\epsilon_p -\mu)/kT) \pm1}$ where minus is for bosons and plus is for fermions.

Now if we evaluate this for photons which have $\mu =0$ we get $u = aT^4$ where $ a= 4 \sigma /c$ is radiation constant.

If electrons and positrons are in equilibrium with photons (so $\mu = 0$ and $g_s =2$) We will get $u=(7/8) a T^4$

equilibrium equation: $\gamma + \gamma \iff e^+ + e^-$

Now think we have electron+ positron + three types of neutrinos + three types of anti neutrinos which are at all in equilibrium with photons. $g_s=1$ for neutrinos and anti neutrinos. now how can we get an equation for energy density? what we must use for $g_s$?

I am also not sure about $u = (7/8) aT^4$, Is it the energy density of electrons or positrons (so the total energy density is $2u$) or energy of both of them is in this equation?

Note: All of the particles are relativistic. These question conditions is for early universe.

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  • $\begingroup$ This should be migrated to the physics stack exchange as it is more concerned with physics than astronomy. $\endgroup$
    – zephyr
    Sep 5, 2017 at 19:43
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    $\begingroup$ @zephyr I found these in an astrophysics book. It was used for simulating early universe conditions. where all I said above was in equilibrium with photons in Temperature. But I don't know what I must use for $gs$ of a gas composite of all of them and also I don't know what actually $u= (7/8) a T^4$ shows. $\endgroup$
    – titansarus
    Sep 5, 2017 at 19:57
  • $\begingroup$ I understand that it is related to astronomy and used within astronomy, but these are fundamentally physics questions. $\endgroup$
    – zephyr
    Sep 5, 2017 at 19:59
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    $\begingroup$ This is fine as an astronomy question: after all is not astronomy an application of physics? $\endgroup$
    – adrianmcmenamin
    Sep 5, 2017 at 20:05

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