Cosmology Lecture 11

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Massive (Hot Dark Matter) continued

Last time we started talking about (1) and (2) below:

  • Lab upper bounds on .
  • Neutrino oscillations: :

Now suppose at time we have a pure produced. Then:

At time t:

Where , and , assuming a fixed momentum state. We assume this for simplicity, but a full wave-packet-based derivation is done in (Kayser (81) PRD 24,110). Anyway, the probability of finding a state at time t is:

For relativistic (we don’t really need to make this assumption, it just makes our lives easier), then:

Thus:

Where is the distance travels in meters, is in , and is in MeV Thus we have our expression for the probability of an neutrino turning into a neutrino in a vacuum:

  • Supernova : (SN1987A: Feb 23 1987) "(a)" Supernovae produce via: , which happens within the first (0.1 sec) of final star collapse (note that the n is what makes neutron stars. "(b)" They also produce via: , where . "(c)" Time for to travel from 1987A: If neutrinos are massless, , . However if , then , , so:

The difference in times of arrival for if neutrinos are massive or not is: