<p>We know that energy density is given by this formula:</p>

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

<p>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.</p>

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

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

<p>equilibrium equation:
$\gamma + \gamma \iff e^+ + e^-$ </p>

<p>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 $g_s$?</p>

<p>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?</p>

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

CC BY-SA 3.0
2017-09-05T19:43:09.903Z
This should be migrated to the physics stack exchange as it is more concerned with physics than astronomy.
zephyr
CC BY-SA 3.0
2017-09-05T19:57:20.200Z
@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.
titansarus
CC BY-SA 3.0
2017-09-05T19:59:08.007Z
I understand that it is related to astronomy and used within astronomy, but these are fundamentally physics questions.
zephyr
CC BY-SA 3.0
2017-09-05T20:05:19.543Z
This is fine as an astronomy question: after all is not astronomy an application of physics?
adrianmcmenamin
<astrophysics><early-universe><photons>
Question about finding energy density of a fermionic-bosonic gas
titansarus
# Question about finding energy density of a fermionic-bosonic gas

(Asked by: username_0 on 2017-09-05)

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.

**Tags:** astrophysics, early-universe, photons

> **Comments:**
> **username_1**: This should be migrated to the physics stack exchange as it is more concerned with physics than astronomy.  
> **username_0**: @username_1 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.  
> **username_1**: I understand that it is related to astronomy and used within astronomy, but these are fundamentally physics questions.  
> **username_2**: This is fine as an astronomy question: after all is not astronomy an application of physics?  

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## 0 answers

