Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 91
Figure 36: A schematic plot of the evolution of the scalar potential V for different
values of temperature. Also represented is the evolution of φ− and φ+.
Here T0 represents the temperature for which V ′′ (φ = 0) = 0, i.e., the lowest
temperature where the symmetric vacuum can exist (equation 167), T∗ is the
temperature for which a second local minimum of the potential first appears
(equation 168) and Tc is the temperature at which that second minimum becomes
degenerate with the origin (equation 171) (adapted from Anderson &
Hall, 1992).
the origin: V (φ+(Tc)) = 0. Hence, if we divide equation (161) by φ 2 , Tc occurs
where the resulting quadratic equations have two real equal roots. This gives
the relation 27 (Anderson & Hall, 1992)
Tc =
T0
1 − E2
λT 0 D
. (171)
At this critical temperature Tc the two minima become degenerate, and below
this temperature the stable minimum of V is at (e.g. Mégevand, 2000)
φ+ = 3ET
2λT0
1+ 1 − 8 λT0D
9 E2
1 − T 2 0
T 2
. (172)
When the temperature reaches T0 the barrier between minima disappears, and
φ = 0 becomes a maximum of the potential as it is clear from Figure 36 (e.g.
Mégevand, 2000).
27Some authors (e.g. Gynther, 2006) prefer to indicate λTc instead of λT0 . In fact, as we
shall see, the difference between Tc and T0 is very small and, hence, λTc ≈ λT0 . We prefer to
indicate λT0 because we will determine the value of Tc with the help of the value of T0.