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4.2. Particles in Equilibrium

The various particles inhabiting the early universe can be usefully characterized according to three criteria: in equilibrium vs. out of equilibrium (decoupled), bosonic vs. fermionic, and relativistic (velocities near c) vs. non-relativistic. In this section we consider species which are in equilibrium with the surrounding thermal bath.

Let us begin by discussing the conditions under which a particle species will be in equilibrium with the surrounding thermal plasma. A given species remains in thermal equilibrium as long as its interaction rate is larger than the expansion rate of the universe. Roughly speaking, equilibrium requires it to be possible for the products of a given reaction have the opportunity to recombine in the reverse reaction and if the expansion of the universe is rapid enough this won't happen. A particle species for which the interaction rates have fallen below the expansion rate of the universe is said to have frozen out or decoupled. If the interaction rate of some particle with the background plasma is Gamma, it will be decoupled whenever

Equation 95 (95)

where the Hubble constant H sets the cosmological timescale.

As a good rule of thumb, the expansion rate in the early universe is "slow," and particles tend to be in thermal equilibrium (unless they are very weakly coupled). This can be seen from the Friedmann equation when the energy density is dominated by a plasma with rho ~ T4; we then have

Equation 96 (96)

Thus, the Hubble parameter is suppressed with respect to the temperature by a factor of T / Mp. At extremely early times (near the Planck era, for example), the universe may be expanding so quickly that no species are in equilibrium; as the expansion rate slows, equilibrium becomes possible. However, the interaction rate Gamma for a particle with cross-section sigma is typically of the form

Equation 97 (97)

where n is the number density and v a typical particle velocity. Since n propto a-3, the density of particles will eventually dip so low that equilibrium can once again no longer be maintained. In our current universe, no species are in equilibrium with the background plasma (represented by the CMB photons).

Now let us focus on particles in equilibrium. For a gas of weakly-interacting particles, we can describe the state in terms of a distribution function f (p), where the three-momentum p satisfies

Equation 98 (98)

The distribution function characterizes the density of particles in a given momentum bin. (In general it will also be a function of the spatial position x, but we suppress that here.) The number density, energy density, and pressure of some species labeled i are given by

Equation 99 (99)

where gi is the number of spin states of the particles. For massless photons we have ggamma = 2, while for a massive vector boson such as the Z we have gZ = 3. In the usual accounting, particles and antiparticles are treated as separate species; thus, for spin-1/2 electrons and positrons we have ge- = ge+ = 2. In thermal equilibrium at a temperature T the particles will be in either Fermi-Dirac or Bose-Einstein distributions,

Equation 100 (100)

where the plus sign is for fermions and the minus sign for bosons.

We can do the integrals over the distribution functions in two opposite limits: particles which are highly relativistic (T >> m) or highly non-relativistic (T << m). The results are shown in table 2, in which zeta is the Riemann zeta function, and zeta(3) approx 1.202.

Table 2. Number density, energy density, and pressure, for species in thermal equilibrium.
Table 2

From this table we can extract several pieces of relevant information. Relativistic particles, whether bosons or fermions, remain in approximately equal abundances in equilibrium. Once they become non-relativistic, however, their abundance plummets, and becomes exponentially suppressed with respect to the relativistic species. This is simply because it becomes progressively harder for massive particle-antiparticle pairs to be produced in a plasma with T << m.

It is interesting to note that, although matter is much more dominant than radiation in the universe today, since their energy densities scale differently the early universe was radiation-dominated. We can write the ratio of the density parameters in matter and radiation as

Equation 101 (101)

The redshift of matter-radiation equality is thus

Equation 102 (102)

This expression assumes that the particles that are non-relativistic today were also non-relativistic at zeq; this should be a safe assumption, with the possible exception of massive neutrinos, which make a minority contribution to the total density.

As we mentioned in our discussion of the CMB in the previous lecture, even decoupled photons maintain a thermal distribution; this is not because they are in equilibrium, but simply because the distribution function redshifts into a similar distribution with a lower temperature proportional to 1 / a. We can therefore speak of the "effective temperature" of a relativistic species that freezes out at a temperature Tf and scale factor af:

Equation 103 (103)

For example, neutrinos decouple at a temperature around 1 MeV; shortly thereafter, electrons and positrons annihilate into photons, dumping energy (and entropy) into the plasma but leaving the neutrinos unaffected. Consequently, we expect a neutrino background in the current universe with a temperature of approximately 2K, while the photon temperature is 3K.

A similar effect occurs for particles which are non-relativistic at decoupling, with one important difference. For non-relativistic particles the temperature is proportional to the kinetic energy 1/2 mv2, which redshifts as 1/a2. We therefore have

Equation 104 (104)

In either case we are imagining that the species freezes out while relativistic/non-relativistic and stays that way afterward; if it freezes out while relativistic and subsequently becomes non-relativistic, the distribution function will be distorted away from a thermal spectrum.

The notion of an effective temperature allows us to define a corresponding notion of an effective number of relativistic degrees of freedom, which in turn permits a compact expression for the total relativistic energy density. The effective number of relativistic degrees of freedom (as far as energy is concerned) can be defined as

Equation 105 (105)

(The temperature T is the actual temperature of the background plasma, assumed to be in equilibrium.) Then the total energy density in all relativistic species comes from adding the contributions of each species, to obtain the simple formula

Equation 106 (106)

We can do the same thing for the entropy density. From (94), the entropy density in relativistic particles goes as T3 rather than T4, so we define the effective number of relativistic degrees of freedom for entropy as

Equation 107 (107)

The entropy density in relativistic species is then

Equation 108 (108)

Numerically, g* and g*S will typically be very close to each other. In the Standard Model, we have

Equation 109 (109)

The events that change the effective number of relativistic degrees of freedom are the QCD phase transition at 300 MeV, and the annihilation of electron/positron pairs at 1 MeV.

Because of the release of energy into the background plasma when species annihilate, it is only an approximation to say that the temperature goes as T propto 1 / a. A better approximation is to say that the comoving entropy density is conserved,

Equation 110 (110)

This will hold under all forms of adiabatic evolution; entropy will only be produced at a process like a first-order phase transition or an out-of-equilibrium decay. (In fact, we expect that the entropy production from such processes is very small compared to the total entropy, and adiabatic evolution is an excellent approximation for almost the entire early universe. One exception is inflation, discussed in the next lecture.) Combining entropy conservation with the expression (108) for the entropy density in relativistic species, we obtain a better expression for the evolution of the temperature,

Equation 111 (111)

The temperature will consistently decrease under adiabatic evolution in an expanding universe, but it decreases more slowly when the effective number of relativistic degrees of freedom is diminished.

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