Calculating the mass of the neutrino for a relativistic case

[Arman777]

Homework Statement

What mass would a neutrino need to still be relativistic today (T = 2.37K) ?

Relevant Equations

$T_{\nu} = T_{\gamma}/1.40$

I came across a question that states

What mass would a neutrino need to still be relativistic today (T = 2.37K) ?

So for a particle to be relativistic we need $pc \gg mc^2$

Well Neutrino was relativistic in the early universe, so I took the time when the neutrino decoupled which is approximately $\approx 1 MeV$

So I did something like

$$\frac{E_{now}}{E_{dec}} = \frac{kT_{now}}{T_{dec}} = \frac{8.617\times 10^{-5} eV K^{-1} \times 2.73K}{1Mev} = 2.3 \times 10^{-10}$$

But I am kind of stuck here since we need some value for the neutrino mass I guess ? Or my approach is completely wrong (?)
In general, how can we solve this kind of problem? What makes the transition from Non-Relativistic to the relativistic case? The temperature of the universe right..? For instance when the temperature of the universe was larger than the $1 MeV$ we would call protons relativistic

[Moderator's note: Moved from a technical forum and thus no template.]

 

[Orodruin]

That makes no sense at all. The question is what momenta the neutrinos have now compared to their mass. The ratio of the energies now and then is not directly relevant. What is relevant is the CNB temperature relative to the CMB temperature as this sets the momentum scale, which you need to compare to the neutrino mass.

 

[Arman777]

What is relevant is the CNB temperature relative to the CMB temperature as this sets the momentum scale, which you need to compare to the

Oh do you mean CMB tempeture now vs CNB temperature right ? Thats what I did ?

Let us say we find a ratio $q$. Then what can I do ?
$q \gg \frac{m_{\tau}^{now}} {m_{\tau}^{CNB}}$

CNB happened when the neutrinos are decouples (I suppose) and the current CMB tempeture is 2.73 K ?

Did you mean the CMB tempeture at the photon decoupling ?

 

[Orodruin]

Oh do you mean CMB tempeture now vs CNB temperature right ? Thats what I did ?

Yes and no.

Even if neutrinos decouple before photons they would have the same temperature today if the Universe just expanded and nothing else happened. However, this is not the case. The photon gas has since been heated by electrons becoming non-relativistic. How this affects the ratio between photon and neutrino temperatures should be described in any introductory book on cosmology.

The only relevant quantity is the CNB temperature today as only this will relate to the neutrino momentum today.

 

[Arman777]

You are talking about reheating I see. So current CNB temperature is $1.95K$

$E_{CNB}^{now} = 1.68 \times 10^{-4} eV$. The earlier times $E_{CNB}^{dec}= 1 MeV$

so $$\frac{E_{CNB}^{now}}{E_{CNB}^{dec}} = \frac{1.68 \times 10^{-4} eV} {1MeV} = 1.68 \times 10^{10}$$

So the momentum of the neutrino decreased by a factor of $10^{10}$.

At this point we need some neutrino mass information to put a limit ?

 

[Orodruin]

Again, the temperature at decoupling is irrelevant and the only relevant temperature for the question is the temperature now.

The only relevant quantity is the CNB temperature today as only this will relate to the neutrino momentum today.

 

[Arman777]

So $$m_{\tau} << 1.68 \times 10^{-4} eV$$

Is this correct ?

 

[Orodruin]

Yes, although what you describe with the $\ll$ is what we would call an ultra-relativistic neutrino. For relativistic effects to be relevant, you only need $p \gtrsim m$ and this would qualify for us to call the neutrino relativistic.

 

[Arman777]

Yes, although what you describe with the $\ll$ is what we would call an ultra-relativistic neutrino. For relativistic effects to be relevant, you only need $p \gtrsim m$ and this would qualify for us to call the neutrino relativistic.

Thanks a lot.

The question is actually easy. I don't know why I couldn't think like this...