How Does Temperature Affect Particle Density and Expansion in the Universe?

[Arman777]

For $m>T$ we can write
$$n\propto T^{3/2}\exp[{-(m-\mu)/T}].\tag{2}$$

For $T>m$ we can write $n\propto T^{3}$ where $n$ is number density of particles with mass m. We can derive this relationship by using $n\propto a^{-3}$ and we also know that $a\propto T^{-1}$.

Is there a similar derivation for the first equation by using $n\propto a^{-3}$

 

[Phylosopher]

Can you elaborate?

 

[Arman777]

I edited my post

 

[mfb]

I'm guessing the question is about particle densities in the early universe? Then a is the scale factor.

For $T>m$ we can write $n\propto T^{-3}$ where $n$ is number density of particles with mass m. We can derive this relationship by using $n\propto a^{-3}$ and we also know that $a\propto T^{-1}$.

Shouldn't this be $n\propto T^{3}$? $n\propto a^{-3}$ assumes no particles can be produced or destroyed. Is that really an assumption you want to make - and if you do so, why don't you make it for m>T?

 

[Arman777]

I'm guessing the question is about particle densities in the early universe? Then a is the scale factor.Shouldn't this be $n\propto T^{3}$? $n\propto a^{-3}$ assumes no particles can be produced or destroyed. Is that really an assumption you want to make - and if you do so, why don't you make it for m>T?

If $n\propto a^{-3}$ assumes no pair production then we cannot say that at $m>T$, since at that time there are pair-production, So we cannot derive it by using $n\propto a^{-3}$.