Wednesday, April 29, 2015

neutrinos everywhere!

please estimate how many cosmic neutrinos are in the observable Universe. are there more or less photons than neutrinos?

1. both photons and neutrinos are ultrarelativistic particles. integrating over the respective distributions gives for the density of photons $n_\gamma$

n_\gamma = g_\gamma\frac{\zeta(3)}{\pi^2}\left(\frac{kT}{\hbar c}\right)^3

and for the density of neutrinos $n_\nu$

n_\mu = \frac{3}{4}\frac{g_\nu}{g_\gamma} n_\gamma

then, the ratio between the densities is given by

\frac{n_\nu}{n_\gamma} = \frac{3}{4}\frac{g_\nu}{g_\gamma}\left(\frac{T_\nu}{T_\gamma}\right)^3

which can be evaluated using the relation of the two background temperatures

T_\nu = \left(\frac{4}{11}\right)^{1/3} T_\gamma

due to the fact that the photon background has a higher temperature because of $e^+e^-$-annihilation, and the ratio of the statistical weights $g_\nu/g_\gamma=2\times 3 / 2$ for two spin states of both the photon and the neutrino, as 3 neutrino families. putting everything together yields

\frac{n_\nu}{n_\gamma} = \frac{3}{4}\frac{g_\nu}{g_\gamma}\frac{4}{11} = \frac{9}{11}<1

i.e. about 20% fewer neutrinos than photons: the smaller temperature overcompensates the larger number of neutrino families.

2. the total number of neutrinos could be estimated like this: the number density $n_\nu$ is roughly $10^8$ per cubic metre, and the volume $4\pi/3\chi_H^3$ contains then $10^{83}$ neutrinos in total.

3. and $10^{83}$ is very close again to Eddington's number (the number of protons in the Universe), which is $10^{80}$!