# Limiting the mass of the neutrino for a relativistic case

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

$$\frac{E_{now}}{E_{dec}} = \frac{kT_{now}}{T_{dec}} = \frac{8.617\times 10^{-5} eV K^{-1} \times 2.73K}{1Mev} = 2.35 \times 10^{-10}$$

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

• Couple of comments. On the matter of typesetting, for "much greater than" use \gg ($\gg$) rather than >> ($>>$). Second I would not necessarily take "relativistic" to imply "energy much greater than mass"; depending on the context just being comparable to mass would be sufficient. If I really meant energy much greater than mass I would say "highly relativistic". – dmckee --- ex-moderator kitten Dec 10 '19 at 21:21
• Hmm then what can I do .. – Reign Dec 10 '19 at 21:28
• I've added the homework-and-exercises tag. In the future, please add this tag to this type of problem. This is one of the things that we ask you to do in our homework policy: physics.meta.stackexchange.com/questions/714/… – user4552 Dec 11 '19 at 6:06
• Please reference the source of this homework question. This is one of the things that we ask you to do in our homework policy: physics.meta.stackexchange.com/questions/714/… – user4552 Dec 11 '19 at 6:06
• To me this doesn't look like it has anything to do with cosmology or particle physics. The question seems to be telling you to assume that this neutrino is thermalized with the CMB. – user4552 Dec 11 '19 at 6:08

So for relativistis case $$pc > mc^2$$ We also know that at the reheating thr temperature of the neutrinos and photons evolved differently.

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

We know that $$T_{CMB}^{now}= 2.73 K$$ so

$$T_{CNB}^{now}= 1.95 K$$

By using $$E = kT$$

$E_{\tau} = 1.68 \times 10^{-4} eV$\$

So particle will be relativistic today if

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