Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 62
Hadronic bubbles grow very fast, within ∆tnuc ∼ 10 −6 tH until the released
heat has reheated the Universe to Tc and prohibits further bubble formation
(e.g. Schmid et al., 1999). By that time, only a small fraction of the volume of
the observable universe has gone through the transition (e.g. Boyanovsky et al.,
2006). For the remaining 99% of the transition, both phases (QGP and HG)
coexist at constant pressure (e.g. Schmid et al., 1999):
pc = pQGP (Tc) =pHG(Tc). (116)
Bubbles can grow only if they are created with radii greater than the critical
bubble radius Rcrit. Smaller bubbles disappear again due to the fact that the
energy gained from the bulk of the bubble is more than compensated by the
surface energy in the bubble wall. The value of Rcrit is given by the maximum
value of ∆F (e.g. Schmid et al., 1999)
Rcrit =
2σ
, (117)
pHG(T ) − pQGP (T )
which diverges at T = Tc meaning that bubble formation should stop after
reheating. The probability of forming a critical bubble per unit volume and
unit time can be written as (e.g. Schmid et al., 1999)
I ≈ T 4
c exp − A
η2
(118)
where
and
A = 16π
3
σ3 l2Tc (119)
η =1− T
. (120)
Tc
Using the results obtained from quenched lattice QCD we have A =3× 10 −5
(e.g. Boyanovsky et al., 2006).
During the period of coexistence hadronic bubbles grow slowly (due to the
expansion of the Universe only) causing a continuous growth of the volume fraction
occupied by the hadron phase, at the expense of the quark–gluon phase.
The latent heat released from the bubbles is distributed into the surrounding
QGP (by a supersonic shock wave and by neutrino radiation) keeping the Universe
at constant temperature Tc. This reheats the QGP to Tc and prohibits
further bubble formation. Since the amplitude of the shock is very small, on
scales smaller than the neutrino mean free path (which is 10 −6 RH at Tc), heat
transport by neutrinos is the most efficient (e.g. Boyanovsky et al., 2006).
The transition is completed when all space is occupied by the hadron phase
(e.g. Jedamzik, 1998; Schmid et al., 1999; Boyanovsky et al., 2006). A sketch
of homogeneous bubble nucleation is shown in Figure 15. In Figure 16 it is