Cosmology Lecture 08

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Thermal Equilibrium vs. Decoupling (“Freeze-Out”)

The rule of thumb here is to compare the interaction rate () of the particle we are interested in to the expansion rate of the universe. We’ll examine two extremes: , and .

  • "Ex:" Weak Interactions:

The cross-section for weak interaction goes as , where is the Fermi constant, . Thus, for (recall that in the relativistic limit):

Let’s compare this to the Hubble expansion rate of early universe, when , so curvature is negligible. Therefore:

where is Newton’s constant and is the effective degeneracy. In order for , we need:

We know , so the temperature requirement for decoupling of the weak interaction is:

In general we can say that particles decouple after the rest mass stops being much more that . We can compute the threshold temperature for particles based on their rest mass:

Relating Temperature and Time in the Radiation-Dominated Era

Recall that the energy density of relativistic bosons is given by:

We also have shown that in the radiation-dominated era, , so:

Now since :

A useful relation is that .

The First 30 Minutes (in 6 frames)

  • "Frame 1:" , , , . To put things in perspective, . The major players at this point in time are photons (), electrons and positrons (), neutrinos (), and protons and neutrons (). For , we’re assuming baryon asymmetry has occurred, and we should note that they only show up in small numbers. particles are all ultra-relativistic. Let’s figure out our :

Note that was computed as [3 species 2 particle/antiparticle pairs with 1 spin state each fermion factor]. We ignored because they were not relativistic. Their reactions are:

The last reaction is negligible because it has timescale minutes. Remember that we derived the neutron-to-proton ratio:

where . Thus:

The neutron-to-baryon ratio is:

  • "Frame 2:" ,, , drops by a factor of 100. Our major player are the same, so:
  • "Frame 3:" , , . Now the weak interaction rate falls below the Hubble time (). Therefore, neutrinos are decoupling from the thermal bath. Before this decoupling occurred, recall that were all in thermal equilibrium, . After decoupling, are in thermal equilibrium, so . For the neutrinos, their Fermi-Dirac distribution is “frozen” in place:
  • "Frame 4:" , , . Because , it’s hard to make new . That is, is a favored interaction over the reverse, so start to disappear. As they annihilate, entropy is transferred to photons, so the entropy density is conserved (see Kolb and Turner, 66).