420
C. Cosme
2.2 Radiation Era
After inflation and the reheating period (which we will assume to be instantaneous,
for simplicity), the Universe becomes radiation-dominated and R = 0. Above the
EWPT, the dominant term in the potential of the dark scalar is the quartic one
(see Eq. (1)), since the thermal effects can keep the Higgs field localized about
its origin. The dark scalar acquires an effective field mass m φ =
3 λ φ φ and, when
the condition m φ H is satisfied, it starts to oscillate about the origin with an
amplitude φ rad given by:
φ rad (T ) =
φ inf
T rad
T =
π 2 g ∗
270
1/4 φ inf
M P l
1/2 T
λ
1/4
φ
.
(7)
where the temperature at the onset of fields oscillations, T rad , reads
T rad = λ
1/4
φ
φ inf M P l
270
π 2 g ∗
1/4
,
(8)
g ∗ is the number of relativistic degrees of freedom and M P l is the reduced Planck
mass. Since the dark scalar’s amplitude decays as a −1 ∝ T and ρ φ ∼ a −4 , we
conclude that the field behaves like dark radiation during this period.
As soon as the temperature of the Universe drops below the electroweak scale,
both the Higgs and the dark scalar fields acquire a vacuum expectation value (vev)
and, consequently, the Higgs generate a mass for the dark scalar, as we will see in
the next section. The electroweak phase transition is completed when the leading
thermal contributions to the Higgs potential become Boltzmann suppressed, at
approximately T EW ∼ m W , where m W is the W boson mass.
3 Dynamics After the Electroweak Symmetry Breaking
At the EWPT, the relevant interaction potential is
V (φ, h) = −
g 2
4
φ
2 h
2
+
λ φ
4
φ
4
+
λ h
4
h
2
− ˜
v
2
2
,
(9)
where the Higgs self-coupling is λ h 0.13.
At this point, the Higgs and the dark scalar acquire a non-vanishing vev,
respectively:
h 0 =
1 −
g 4
4 λ φ λ h
−1/2
˜
v ≡ v,
φ 0 =
g v
2λ φ
,
(10)