Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 52
Table 12: The number of degrees of freedom for each kind of particle within
the SMPP: gi is the contribution due to a single particle, N is the number of
species of a particular particle and gN = Ngi is the total contribution for g(T )
of each kind of particle.
Particle gi N gN
quark 12 7
8 6 63.0
charged lepton 4 7
8 3 10.5
neutrino 2 7
8 3 5.25
photon 2 1 2
gluon 2 8 16
EW bosons 2 3 6
Higgs 4 1 4
contribute with 6 degrees of freedom corresponding to three species times two
possible helicity states. However at the EW transition (Section 3) the W and Z
contribution becomes 9. This is due to the Higgs mechanism during which the
W and Z bosons acquire mass and a third polarization degree of freedom (e.g.
Ignatius, 1993).
The meson π contributes with 3 degrees of freedom (one for each kind of
π meson: π − , π 0 and π + ). We may have to consider also the contribution of
kaons. This would be 4 degrees of freedom (e.g. Boyanovsky et al., 2006). On
Table 12 we have listed the contribution of each SMPP fundamental particle to
the total number of degrees of freedom.
As it was already mentioned it is a good approximation to treat all particles
with m ≪ 3T as though they were massless. The contribution of all other
particles can be neglected in the total energy density (e.g Schwarz, 2003). This is
why we did not consider the contribution of composite particles such as protons
and neutrons. For example, in the case of the proton we have mp ≈ 900 MeV.
Considering that protons form at the QCD epoch when the temperature of
the Universe was Tc = 170 MeV it turns out that in this case we do not have
m ≪ 3T and thus, we can safelly neglect the contribution of the proton to the
total number of degrees of freedom.
At very high temperatures (T > mt ∼ 172.5 GeV) all the particles of the
SMPP contribute to the effective number of degrees of freedom g(T ) (cf. equa-