Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 51
we are in the non–relativistic regime and we have ρ and n falling rapidly. The
reason is that the energy available in a collision is now insufficient to create the
species (e.g. Lyth, 1993).
As the temperature T falls below the mass m of a given species, particle–
antiparticle pairs rapidly annihilate (according to the generalized blackbody
distribution) and only one kind of particle of that species survives. The imbalance
n − ¯n becomes significant and µ no longer vanishes. However, even if
the surviving particles do not decay, their contribution to ρ and n during the
radiation–dominated era are negligible (e.g. Lyth, 1993).
If we are interested in the total energy density (i.e., in the energy density due
to all the particle species for which m ≪ 3T ) then it may be useful to introduce
the effective number of helicity degrees of freedom at a particular epoch (i.e.,
characterized by a given temperature T ) defined as (e.g. Liddle & Lyth, 1993)
g(T )=
bosons
gi(T )+ 7
8
fermions
gi(T ) (104)
where the sum goes over all particle species with m ≪ 3T . Notice that the
fermionic degrees of freedom are suppressed by a factor of 7/8 with respect to
bosonic degrees of freedom. This is due to the difference between Fermi–Dirac
statistics and Bose–Einstein statistics (e.g. Hands, 2001).
We may write, with the help of equation (104), the total energy density for
a radiation–dominated Universe as (e.g. Schwarz, 2003)
ρ = π2
30 g(T )T 4 . (105)
In particle physics helicity h is the projection of the angular momentum of the
particle to the direction of motion. Because angular momentum with respect
to an axis has discrete values, helicity is discrete too. For a relativistic particle
(m ≪ 3T ) there are two possible helicity eigenstates usually referred to as left–
handed and right–handed states 24 (e.g. Hands, 2001).
For each quark flavour we have to count two electric charges (quark +
anti–quark), two helicity states and three colour states. This gives a total of
2 × 2 × 3 = 12 degrees of freedom per quark. In the case of gluons we have to
consider that each one of the eight colour charges could have one of two helicity
states. Thus, gluons contribute with 2 × 8 = 16 degrees of freedom.
Each neutral lepton (i.e. neutrino) contributes with two degrees of freedom
corresponding to two possible helicity states. On the other hand each charged
lepton contibutes with four degrees of freedom corresponding to two helicity
states × two charges (lepton and anti–lepton). The photon contributes with
two degrees of freedom corresponding to two possible helicity states.
The Higgs boson contributes with 4 degrees of freedom corresponding to
the two possible helicity states of the scalar doublet. The W ± and Z 0 bosons
24 The antineutrinos observed so far all have right–handed helicity, while the neutrinos are
left–handed.