Calculating the Maximum total neutrino mass by using cosmological boun

[Arman777]

In an article its written,

$$\Omega_{\nu} = \frac{\rho_{\nu}}{\rho_{crit}}=\frac{\sum m_{i,\nu}n_{i,\nu}}{\rho_{crit}} = \frac{\sum m_{\nu}}{93.14h^2eV}$$

Now I am trying to derive this for myself but I could not. Can someone help me ?

So the values are,

$\rho_{crit} = 1.053 75 \times 10^{-5}h^2 GeV/c^2~~cm^{-3}$

Total neutrino average number density today : $n_{\nu} = 339.5~cm^{-3}$

I tried to write it like,

$$\frac{n_{\nu}\sum m_{\nu}}{\rho_{crit}} = \frac{\sum m_{\nu}}{93.14h^2eV}$$

$$\frac{n_{\nu}}{\rho_{cric}} = \frac{339.5cm^{-3}}{1.05375 \times 10^{-5}h^2 GeV/c^2~~cm^{-3}} = \frac{1}{3.103h^2 \times 10^{-8} GeV} = \frac{1}{31.0382916 h^2eV}$$

Which I am missing additional $1/3$.

I guess its a simple question but I couldn't see the answer.

 

[Vanadium 50]

In an article its written

By now you should know this is not an acceptable PF reference. Point us to where you are having trouble and don't make us guess!

31.0382916

Is innumerate nonsense. Please use proper rounding.

I strongly suspect, given the factor of 3, it's a difference between one neutrino flavor and all neutrino flavors, but since you didn't say what your sources are I can't say for sure.

 

[Arman777]

By now you should know this is not an acceptable PF reference. Point us to where you are having trouble and don't make us guess!
Is innumerate nonsense. Please use proper rounding.

I strongly suspect, given the factor of 3, it's a difference between one neutrino flavor and all neutrino flavors, but since you didn't say what your sources are I can't say for sure.

Yes sorry
https://www.google.com/url?sa=t&sou...Vaw0oNgNCVyC98o9jzCu867SY&cshid=1577529872455

I am thinking kind of the same buts its strange.. Umm equation 25.2

 

[Vanadium 50]

Since $\sum m_\nu = m_1 +m_2 + m_3$ (bottom of page 3), why do you want $n_\nu \sum m_\nu$ ?

 

[Arman777]

Since $\sum m_\nu = m_1 +m_2 + m_3$ (bottom of page 3), why do you want $n_\nu \sum m_\nu$ ?

I thought I can pull out the number density from the sum and obtain the similar format w.r.t to the paper.

 

[Vanadium 50]

Getting any information out of you is a slow and painful process. If you want people to find your mistake, you need to clearly explain what you are doing, and not let it slowly ooze out.

I also think you are in way over your head trying to study cosmology without having the prerequisites down cold. That's why you need to start a new thread every 3-4 days. Nobody is benefiting from this - you don't really learn cosmology, you just think you do, and people who are trying to help just get cross. You have to start at the beginning, not the middle.

If n is a number density, n * the average mass is a mass density. True or false?
If there are three species of neutrinos, m1 + m2 + m3 is:
A. The average mass
B. Three times the average mass.
C. One third the average mass.
D. None of the above.

 

[Arman777]

It's B... I understand it now. I should try to read things more carefully.

I also think you are in way over your head trying to study cosmology without having the prerequisites down cold.

you are right yes, but I cannot re-take physics courses. I am trying my best to learn.