JCAP11(2016)025
where
p
hh
2
i plays the role of the relevant energy scale at which the coupling should be
evaluated. Since we are only interested in the high energy regime, in our numerical analysis
we use the step-function approximation
λ
h
(µ) = 0.01 × sign
h
SM
c
−
p
hh
2
i
, (2.8)
where h
SM
c
∼ 10
10
GeV is the critical scale of the Standard Model at which λ
h
flips sign.
3 Pure parametric resonance
Let us first consider the case where the trilinear interaction is negligible, σ
hφ
≈ 0. The
Higgs-inflaton interaction is quartic so that we recover the well-known parametric resonance
setting [16].
The equations of motion for the Higgs field are quadratic in h apart from the quartic
self-interaction. During the parametric resonance regime, the effect of the latter can be
approximated as h
4
→ 6h
2
hh
2
i, which is known as the Hartree approximation. In that case,
the equations of motion for different momentum modes decouple. In terms of the rescaled
Higgs momentum modes X
k
≡ a
3/2
h
k
, where a is the scale factor, one has [16]
¨
X
k
+ ω
2
k
X
k
= 0 with ω
2
k
=
k
2
a
2
+
λ
hφ
2
Φ
2
cos
2
(mt) + 3λ
h
a
−3
hX
2
i +
3
2
wH
2
. (3.1)
In the last term, w = p/ρ = −
1 +
2
3
˙
H
H
2
is the equation of state parameter of the Universe,
which vanishes in the matter-like background. We thus neglect this term.
If the Higgs-inflaton coupling λ
hφ
is substantial, the Higgs modes experience amplifica-
tion due to broad parametric resonance. The parameter characterizing the strength of the
resonance is
q(t) =
λ
h
Φ
2
(t)
2m
2
(3.2)
such that q 1 corresponds to the broad resonance regime. In this case, the modes grow
exponentially leading to a large Higgs field variance hh
2
i. The fluctuations can be so signif-
icant that they exceed the size of the barrier separating the electroweak vacuum from the
catastrophic one at large field values. In this case, vacuum destabilization occurs. In what
follows, we will estimate the corresponding critical size of λ
hφ
.
As was shown in [16], in the broad resonance regime the Higgs modes evolve adiabatically
away from the inflaton zero-crossings and can be described by the WKB approximation
X
k
'
α
k
√
2ω
k
e
−i
R
ω
k
dt
+
β
k
√
2ω
k
e
i
R
ω
k
dt
, (3.3)
where α
k
, β
k
are some constants. Adiabaticity is broken for certain modes near the inflaton
zero-crossing, where the frequency ω
k
evolves very quickly. There the system can be treated
in analogy to the Schr¨odinger equation as a scattering of plane wave solutions. The adiabatic
constants α
k
and β
k
can be thought of as Bogolyubov coefficients. We assume a vacuum
initial condition for the Higgs modes with α
k
= 1 and β
k
= 0. The occupation number of
Higgs quanta after j ' mt/π zero crossings is then
n
j+1
k
= |β
j+1
k
|
2
(3.4)
– 4 –
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