# Neutrinos as Dark Matter

Neutrinos are very weakly interacting, electrically neutral particles that are involved in nuclear interactions where protons and changed into neutrons or vice versa, and in other reactions as well. An example of a weak nuclear interaction involving a neutrino is the free neutron decay

neutron --> proton + electron + anti-neutrino

This decay has a mean life of 887 seconds or a half life of 10.25 minutes. The cross section for a typical interaction involving a neutrino is 5*10-44 (E/[1 MeV])2 cm2 which is very small when compared to the Thomson scattering cross section of 7*10-25 cm2. Thus a 1 MeV neutrino could travel through about 35 light years of water before interacting. Even with this very small probability of interactions, neutrinos have been detected coming from nuclear reactors, the Sun, and from supernova 1987A in the Large Magellanic Cloud. Experiments show that the neutrinos produced in muon interactions are different from the neutrinos involved in interactions with electrons. A third kind of particle, the tau, appears to be a heavier version of the muon which is itself a heavier version of the electron. It is assumed to have its own kind of neutrino as well. Thus there are 3 kinds of neutrinos: the electron neutrino, the muon neutrino, and the tau neutrino. Each kind has an anti-particle as well. The electron neutrino is known to have a mass at least 50,000 times smaller than the mass of the electron, and neutrinos are often assumed to be massless - which means zero rest mass.

Because the Universe was once so hot and dense that even neutrinos interacted many times during the Hubble time 1/H, there once was a thermal background of neutrinos in equilibrium with the thermal background of photons that is the CMBR. But since neutrino interactions are so weak, this thermal equilibrium only survived until 1 second after the Big Bang. But the neutrino background is still present today, with about 56 electron neutrinos, 56 electron anti-neutrinos, 56 muon neutrinos, etc., per cubic centimeter, for a total of 337 neutrinos per cubic centimeter in the Universe. The photons of the CMBR are slightly more numerous with 411 photons/cc.

## Measuring the Neutrino Mass

Because the number of neutrinos in the Universe is so large, even a very small neutrino mass can have drastic consequences for cosmology. So experiments to measure the neutrino mass are obviously very significant. There are three ways to detect a neutrino mass:

The cosmological effects of the neutrino mass would be obvious if the sum of the masses of the three types were larger than 40 eV, so this gives a limit on all three types.

## Technical Discussion of the Neutrino Background

At the time of weak decoupling, about 1 second after the Big Bang, the neutrinos and the photon-electron-positron plasma had the same temperature, which I will call Tn. All these particles were relativistic since k*Tn > 1 MeV, where k is the Boltzmann constant. The energy of a relativistic plasma occupying a volume a3 ("a" will be the scale factor of the Universe) is

Q = (2\sigma/c)(gb+(7/8)gf)a3 T4

where \sigma is the Stefan-Boltzmann constant, c is the speed of light, and gb and gf are the number of spin degrees of freedom for bosons (integral spin particles) and fermions (half-integral spin particles). For photons gb = 2, since even though the spin is 1, there are only 2 spin states instead of 2*Spin+1, because the longitudinal EM mode doesn't propagate. For neutrinos gf = 1, even though the spin is 1/2, because one of the helicity states doesn't exist. Finally for electrons with spin of 1/2, gf is 2 and for positrons gf is 2.

Thus the photon-electron-positron plasma has

Q = (4\sigma/c)(1 + 7/4) a3 T4

As the Universe expands and cools adiabatically the entropy in the volume a3 is conserved. Since the photon-electron-positron plasma has decoupled from the neutrinos its entropy is separately conserved. The entropy can be found by a thought experiment of heating the volume from 0 to T and using
dS = dQ/T

or
S = (4\sigma/c)(1 + 7/4) a3 \int 4 T2 dT

giving
S = (4/3)(4\sigma/c)(1 + 7/4) a3 T3

In the absence of annihilation conservation of entropy gives aT = constant.

During the period from 1 second after the Big Bang until 3 minutes after the Big Bang the temperature falls to well below the rest mass of the electron. Thus the electron-positron plasma annihilates and transfers its energy and entropy to the photons. This leaves the photons with a temperature Tp that is larger than the neutrino temperature Tn. Energy is not conserved in an adiabatically expanding gas because the pressure of the gas does external work, but the entropy is conserved. Thus the entropy before from photons and electron-positron pairs,

(4/3)(4\sigma/c)(1 + 7/4) a3 Tn3,

is equal to the entropy afterward just from photons,
(4/3)(4\sigma/c) a3 Tp3,

so
Tn/Tp = (4/11)1/3.

The photon temperature now is Tp = 2.728 K and the neutrino temperature now is 1.947 K.

If neutrinos are massless then we can compute their equivalent mass density using

rho = [Q/a3]/c2 = Nn (2\sigma/c)(7/8)gf Tn4

where Nn is the number of neutrino species: Nn = 6 for 3 types of neutrinos and 3 types of anti-neutrinos. This density works out to be very small compared to the critical density. It is 0.5*[Nn*7/8](4/11)4/3 = 0.68 times the equivalent mass density of the photons and only 3*10-34 grams per cubic centimeter.

Even though this density is negligible now, it was significant during the time that helium was formed during Big Bang Nucleosynthesis. The increased density due to the neutrino background during helium synthesis caused the universe to expand faster, and this reduced the time required for the temperature to fall to the point where deuterium could survive. As a result the helium abundance is a few percent larger than it would have been without the neutrino background.

If the neutrinos are not massless, then they could have a larger mass density now consisting of their number density times their rest mass. Each neutrino species has a number density of

n = (3/4)(4\pi)\Gamma(3)\zeta(3) (kTn/hc)3

where \Gamma is the gamma function (\Gamma(n+1) = n! so \Gamma(3) = 2), \zeta is the zeta function
\zeta(s) = 1 + 1/2s + 1/3s + ... and \zeta(3) = 1.202...

With Tn = 1.947 K now the number density of each neutrino species is 56 per cubic cm. This is just (3/22) times the number density of photons at Tp. With Nn = 6 species the total number density of neutrinos is 336 per cc. The mass density for massive neutrinos is then
rho = 112*(mn-e + mn-mu + mn-tau)

where mn-e is the mass of the electron neutrino (and electron anti-neutrino since anti-particles have the same mass as the corresponding particle), mn-mu is mass of the muon neutrino, and mn-tau is the mass of the tau neutrino. For masses in electron Volts this is
rho = 2*10-31(mn-e + mn-mu + mn-tau) gm/cc

compared to the critical density of 8*10-30 gm/cc for Ho = 65.

Thus a neutrino rest mass of 40 eV for one type would give the critical density in neutrinos, and a rest mass of 10 eV for one type or a sum of rest masses of 10 eV would be a significant factor in the formation of large scale structures in the Universe such as clusters and superclusters of galaxies.

For these masses the neutrinos are traveling slowly now but their thermal velocities were large in the past. The typical momentum of a relativistic particle in a thermal distribution is p = 3kT/c, and the product of the scale factor and the momentum, ap, is a constant. Thus neutrinos with rest mass m will be moving at redshift z with a typical velocity

v = pc/sqrt[p2 + (mc)2] = 3(1+z)k(1.947 K)/mc - ...

and the distance traveled, measured now (the comoving distance), is
D = \int (1+z) v dt = 2(c/Ho) sqrt(3*k*(1.947 K))/mc2) + ...

which gives D = 100 Mpc for mc2 = 5 eV in a critical density model with Ho = 65. Thus neutrinos can free stream out of the perturbations that make galaxies and cluster of galaxies, but will remain in the perturbations that make superclusters. Because this behavior is caused by the thermal velocity of the neutrinos, this form of dark matter is called Hot Dark Matter.