Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 61
Figure 14: Naive phase diagram of strongly interacting matter in the T − n
plane. Here n is the baryon density, n0 is the present value of n, Tc = 170 MeV,
and T0 =2.725 K is the CMB temperature. At the present time the mixed
phase occurs, in the universe, only at the level of atomic nuclei (green circle) or
within compact objects such as neutron stars (adapted from Kämpfer, 2000).
latent heat (e.g. Schmid et al., 1997)
l = Tc∆s. (112)
The value of latent heat which is avaliable only from quenched lattice QCD
(gluons only, no quarks) is given by (e.g. Schmid et al., 1999)
l ≈ 1.4T 4 c . (113)
The latent heat should be compared with the difference in entropy between an
ideal Hadron Gas (HG) and an ideal QGP. This defines the ratio (e.g. Schmid
et al., 1997)
Rl =
l
. (114)
(Tc∆s) ideal
A first–order phase transition is classified as strong if Rl ≈ 1. The Bag Model
(Section 2.3.1) gives Rl = 1 and from quenched lattice QCD we have, from
equation (113), Rl ≈ 0.2 (e.g. Schmid et al., 1999).
Without dirt (e.g. PBHs, axions) the bubbles nucleate due to thermal fluctuations
in a process called homogeneous nucleation (e.g. Schmid et al., 1999).
For homogeneous nucleation the period of supercolling is ∆tsc ∼ 10 −3 tH, with
tH being the Hubble time (cf. equation 27) at the beginning of the transition.
The typical bubble nucleation distance is dnuc ≈ 1cm ≈ 10 −6 RH with RH being
the Hubble radius (cf. equation 28) (e.g. Schmid et al., 1999).
The change in free energy of the system by creating a spherical bubble with
radius R is (e.g. Schmid et al., 1999)
∆F = 4π
3 (pQGP − pHG) R 3 +4πσR 2 . (115)