Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 50
it turns out that particle and antiparticle have opposite values of µ. As a result,
µ vanishes if the number density n of particles and the respective number density
¯n of antiparticles are equal. Otherwise, µ is determined by the imbalance
n − ¯n (e.g. Lyth, 1993).
If the charges are all zero then all of the chemical potentials are zero and
(95) turns out to be some sort of generalized blackbody distribution (e.g. Lyth,
1993)
f(p) =gi(T ) e E
−1 T ± 1 . (96)
The charge density of the Universe is zero to very high accuracy. If that was
not the case then the expansion of the Universe would be governed by electrical
repulsion instead of gravity. The net baryon number of the Universe is not zero
but it is small in the sence that (e.g. Lyth, 1993)
η = nB
nγ
≪ 1 (97)
where nB is the baryon density and nγ the photon density.
Assuming that the same goes for the three lepton numbers (although we
cannot measure them directly) it turns out that the generalized blackbody distribution
is valid to great accuracy for all the relativistic species in equilibrium.
Since there are (2π) −3d3pd3x states in a given volume of phase space, the particle
number density n and the energy density ρ of particles of a particular species
i are given by (e.g. Lyth, 1993)
ni = gi(T )
(2π) 3
∞
0
ρi = gi(T )
(2π) 3
∞
0
f(p)4πp 2 dp (98)
Ef(p)4πp 2 dp (99)
If the mass m of the species in question is such that T ≫ m then one is on the
relativistic regime and it is a good approximation to consider E = p. Taking
this into account and inserting (96) into equations (98) and (99) we obtain,
separately for fermions and bosons (e.g. Lyth, 1993)
ρB,i = π2
30 gi(T )T 4
(100)
ρF,i = 7 π
8
2
30 gi(T )T 4
(101)
nB,i = ζ(3)
π2 gi(T )T 3
(102)
nF,i = 3 ζ(3)
4 π2 gi(T )T 3
(103)
where ζ(3) ≈ 1.2021. According to the generalized blackbody distribution each
relativistic species contributes with ∼ T 4 to ρ and ∼ T 3 to n. When T < m