The lower the temperature of the universe, the easier life gets: once the temperature of the universe drops below the rest energy of a particle, it is no longer often created in particle/antiparticle pairs.
Thermodynamics of a Fermi/Bose Gas
Phase Space is a 6-dimensional space of directions and linear momenta. It has volume:
The Distribution Function describes the number of particles in a particular state at a given time. Thus, the number of particles in a phase space volume is:
where is used to account for degeneracy. Reminder of Fermi/Bose statistics:
Where is the chemical potential (usually 0 for us), + is for fermions, and - is for bosons. The chemical potential is usually 0 because we are talking about photons, and it can be shown that since photon number is not conserved, then .
- # density in thermal equilibrium (integrating above, dividing by volume, recall ):
- Energy density (, weighted by energy):
- Entropy density . is pressure. For :
Without proving, we’ll state . The proof comes from kinetic theory. Thus:
So we have ways to calculate 3 very useful quantities for the early universe. We can take two useful limits of these quantities:
Ultra-relativistic () Particles in the Early Universe
In this limit, particles are effectively massless (their energy is dominated by their motion, ).
- "A." Bosons (1) The # density of relativistic bosons is:
where . This integral is calculable, and is the definition of the Reimann-Zeta function:
So getting back to the # density of bosons:
(2) The energy density of relativistic bosons is:
For photons, . We can calculate a flux related to their energy density:
(3) The entropy density of relativistic bosons is:
Recall that for radiation, the energy density . Since . This is only true if is fixed.
- "B." Fermions: here’s a trick:
This is an identity. (1) # density. Since , then , then using our identity:
(2) Energy density:
(3) Entropy density:
When doing computations like this for the universe, we’ll have to keep track of the ’s of each particle’s contribution. To aid us in doing this, we’ll define an effective degeneracy:
Using this definition, the total energy density .
Non-Relativistic () Particles In the Early Universe
In this limit, a particle’s energy is dominated by its rest mass: . Then in general, the denominator inside the integral over momentum simplifies:
So dropping the “” in the denominator:
This the the Maxwell-Boltzmann distribution if we put , the chemical potential, back in the exponent with the energy (). Thus, the # density is exponentially suppressed for . For example, at , if we are in thermal equilibrium, then the neutron-to-proton ratio is:
This determines the Hydrogen/Helium ratio!