Number of Neutrino Families

Expansion of the Universe

This constraint follows from the fact that the dynamics (expansion) of the early Universe is dominated by relativistic particles, the photons and leptons. Suppose the Universe is made up of

• photons + electron neutrinos + muon neutrinos

Now calculate how much energy is contained in the Universe. This will tell you how fast the Universe expands. Next, suppose that the Universe is made up of

• photons + electron neutrinos.

In this case, the amount of energy contained in the Universe is less and the Universe will expand more slowly. If there is another type of neutrino in addition to the electron, muon, and tau neutrinos, say the x-neutrino, then the energy contained in the Universe would be greater and the Universe would expand more quickly.

Expansion of the Universe and Big Bang Nucleosynthesis

How does the expansion rate of the Universe affect nucleosynthesis yields?

• Recall that in the production of helium, the following sequence of events occurs:
  • At high T: p + e <--> n + nu, n + positron <--> p + anti-nu, n <--> p + e + anti-nu

    At early times, these processes are in balance (equilibrium) and we expect to get a well defined number for the ratio of neutrons/protons which depends only on the temperature. What is this ratio?

  • since m(neutron) > m(proton) ---> we expect to get more protons than neutrons (based on an energy argument) in any given situation. Further, since neutrons are unstable (they decay in 10.3 minutes in vacuum), free neutrons left to their own devices will also turn into protons.
    • In equilibrium (the state of rapid collisions), we can show that
      n/p = exp[-{m(neutron)-m(proton)}c**2/{kT}]

      ---> 1 if T >> [m(neutron)-m(proton)]c**2 / k ~ 1.5 x 10**10 K and

      ---> 0 if T << 1.5 x 10**10 K

      and is somewhere in-between 0 and 1 for temperatures on the order of ~ 10**10 K.

    • We can get helium from the chain:
      • p + n ---> deuterium + photon
      • deuterium + deuterium ---> tritium + p
      • tritium + deuterium ---> helium(4) + n

      Note that to make helium(4), no beta-decays occurred. This means that no protons converted to neutrons and no neutrons converted to protons. The number of helium nuclei produced is 1/2 of the number of available neutrons (since each helium nucleus contains 2 neutrons). The helium abundance in the Universe is thus determined by the ratio n/p. What determines n/p at the time of nucleosynthesis?

    • As the Universe expands, the weak interactions (listed above) keep the neutrons and protons in equilibrium and the ratio n/p decreases as the Universe expands and cools. At a certain density and temperature the interaction rate becomes slower than the expansion rate of the Universe and the ratio n/p freezes out. What we mean is that since there are no more interactions, the numbers of neutrons and protons stops changing. The helium abundance is determined by when this freeze-out occurs.

    • If the Universe expands very quickly, the weak interactions can only keep up with the expansion of the Universe at early times and the n/p ratio freezes-out at high temperatures and the ratio n/p is large.

    • If the Universe expands very slowly, the weak interactions can keep up with the expansion for longer times (to lower T) and the freeze-out ratio of n/p is very small. This leads to a small helium abundance.

    The detailed calculations yield the following result on the number of neutrino families.

    Big Bang nucleosynthesis, the helium abundance, and a reasonable range for the baryon density suggests that we have 3 types of neutrinos. Theory, in principle, does not predict this result. Work by experimental particle physicists using large accelerators have produced results consistent with this suggestion. In an odd sense, one could argue that the Terrestrial accelerator results confirm the existence of the Big Bang.