Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 92
The number E 2 /λT0D is, in general, small, and the difference between Tc
and T0 is ∆T 10 −2 Tc. However, things change rapidly as the temperature
falls from Tc to T0 (e.g. Mégevand, 2000).
The exact temperature of the transition Tt depends on the evolution of the
bubbles after they are nucleated, which, in turn, depends on the viscosity of the
plasma (e.g. Mégevand, 2000).
Considering the present known values for mW , mZ, mt and mH (see Section
1.8) we obtain D ≈ 0.179, B ≈ −0.00523, E ≈ 0.0101, λ ≈ 0.1393,
T0 ≈ 137.8 GeV, λT0 ≈ 0.1321 and Tc ≈ 138.1 GeV. The value obtained for Tc
agrees with the value indicated in the literature which is Tc ∼ 100 GeV. Thus,
we consider
Tc = 100 GeV. (173)
In the case of the MSSM we have to consider two scalars φ1 and φ2 corresponding
to the two complex Higgs doublets H1 and H2 (cf. Section 1.8). The potential
can now be written in the standard form (e.g. Trodden, 1999)
V (φ1,φ2) =λ1(φ †
1 φ1 − v 2 1 )2 + λ2(φ †
2 φ2 − v 2 2 )2 +
+ λ3
+ λ4
(φ †
1φ1 − v 2 1 ) + (φ† 2φ2 − v 2 2 )
2 +
(φ †
1
†
φ1)(φ 2φ2) − (φ †
1
†
φ2)(φ 2φ1)
(174)
where v1 ≡〈H0 1 〉 and v2 ≡〈H0 2 〉 are the respective vacuum expectation values
of the two doublets, † represents the Hermitian conjugate, and the λi’s are
coupling constants. Notice that it is not restrictive to assume that the only
non–vanishing vacuum expectation values are v1 and v2, which are both real
and positive (e.g. Espinosa et al., 1993).
Even in the MSSM we may continue to apply for the free energy the SMPP–
like form given by equation (161), as long as we are in the limit in which the
pseudoscalar particle of the Higgs sector is heavy (mA ≫ Tc)(e.g. Mégevand,
2000). In that case the potential φ in equation (161) is given by (e.g. Moreno
et al., 1997)
φ = √ 2 φ 0 1 cos β + φ 0 2 sin β
(175)
where (e.g. Espinosa et al., 1993)
β = v2
v1
= 〈H0 2 〉
〈H 0 1 〉.
(176)
Although the one–loop approximation can be used to calculate the characteristics
of the phase transition, it is not guaranteed to be a reliable method. For an
improved analysis (making use of two–loop corrections to the effective potential)