Fate of Electroweak Vacuum during Preheating
Yohei Ema
Based on arXiv:1602.00483In collaboration with K. Mukaida and K. Nakayama
Progress in Particle Physics 2016.09.06
University of Tokyo
Introduction
Metastability
102 104 106 108 1010 1012 1014 1016 1018 1020
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
RGE scale m in GeV
Hig
gsqu
artic
coup
lingl
3s bands inMt = 173.3 ± 0.8 GeV HgrayLa3HMZL = 0.1184 ± 0.0007HredLMh = 125.1 ± 0.2 GeV HblueL
Mt = 171.1 GeV
asHMZL = 0.1163
asHMZL = 0.1205
Mt = 175.6 GeV
Electroweak (EW) vacuum may be metastable??[Buttazzo+ 13]
µ [GeV]
�
becomes negative for the center value of .� ⇠ 1010 GeV Mtop
Cosmology must be compatible with it.
hmax
⇠ 1010 GeV
hmax
Ve↵(h)
h
Metastability
107 108109
1010
1011
10121013
1014
1016
120 122 124 126 128 130 132168
170
172
174
176
178
180
Higgs pole mass Mh in GeVToppolemassM
tinGeV
1017
1018
1019
1,2,3 s
Instability
Stability
Meta-stability
102 104 106 108 1010 1012 1014 1016 1018 1020
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
RGE scale m in GeV
Hig
gsqu
artic
coup
lingl
3s bands inMt = 173.3 ± 0.8 GeV HgrayLa3HMZL = 0.1184 ± 0.0007HredLMh = 125.1 ± 0.2 GeV HblueL
Mt = 171.1 GeV
asHMZL = 0.1163
asHMZL = 0.1205
Mt = 175.6 GeV
Electroweak (EW) vacuum may be metastable??
Instability
Meta-stability
Stability
⌧EW > ⌧U
[Buttazzo+ 13]
µ [GeV]
� Mtop
Mhiggs
becomes negative for the center value of .� ⇠ 1010 GeV Mtop
Cosmology must be compatible with it.
hmax
⇠ 1010 GeV
Metastability vs. inflation• Light fields acquire fluctuations during inflation.
ph�h2i ' Hinf
2⇡
pN
[Starobinsky, Yokoyama 94]
(random walk with for each one e-folding)Hinf/2⇡
• Inflation scale must be low for EW vacuum to be stable.(if BSM does not affect much on the higgs potential)
[e.g. Hook+ 14; Espinosa+ 15]
(Potential and reheating dynamics are neglected.)
Hinf
. 4⇥ 108 GeV
✓60
N
◆✓hmax
1010 GeV
◆Roughly, or, P (|h| > h
max
) ' e�x
2
/p⇡x < e�3N , x =
p2⇡h
max
/pNH
inf
Stabilization during inflation
Induce effective mass for higgs during inflation.
• The following couplings are often considered:
[e.g. Espinosa+ 08; Lebedev+ 13]
• One way to avoid this constraint:
(BSM or low-scale inflation are of course other choices…)
m2e↵;h = c2�2, 12⇠H2
inf during inflation
c & O(Hinf/�inf), ⇠ & O(0.1)Suppress fluctuations if
�Lint =
(12c
2�2h2 · · · quartic12⇠Rh2 · · · curvature
Stabilization during inflation
Induce effective mass for higgs during inflation.
• The following couplings are often considered:
[e.g. Espinosa+ 08; Lebedev+ 13]
• One way to avoid this constraint:
(BSM or low-scale inflation are of course other choices…)
m2e↵;h = c2�2, 12⇠H2
inf during inflation
c & O(Hinf/�inf), ⇠ & O(0.1)Suppress fluctuations if
�Lint =
(12c
2�2h2 · · · quartic12⇠Rh2 · · · curvature
How about after inflation??
Motivation1. EW vacuum metastability (BSM negligible)
vs. high scale inflation
Motivation1. EW vacuum metastability (BSM negligible)
vs. high scale inflation
2. Stabilization mechanism during inflation:
�Lint =1
2c2�2h2,
1
2⇠Rh2.
Motivation1. EW vacuum metastability (BSM negligible)
vs. high scale inflation
2. Stabilization mechanism during inflation:
�Lint =1
2c2�2h2,
1
2⇠Rh2.
3. How about after inflation?? [our work]- Mass term oscillates resonance.
- Even tachyonic during some period (curvature).
Outline
1.Introduction
2.Resonant particle production
3.Preheating dynamics of Higgs
4.Summary
Outline
1.Introduction
2.Resonant particle production
3.Preheating dynamics of Higgs
4.Summary
Dynamics of inflaton
1. Slow-roll during inflation.
2. Oscillate after inflation.
3. Finally decay, and reheating completes.
= beginning of hot big bang
if exponential particle production
preheating epoch
= accelerated expansion
Parametric resonance
hh h
h
inflaton
q ⌘ c2�2
4m2�
� 1If , adiabaticity only at around the origin.
Quartic coupling: �Lint =1
2c2�2h2
+ Bose enhancement [many times of oscillation].
Parametric resonance occurs.
µqtc ' O(0.1)nh ⇠ p3⇤ e2µqtcm�t,
where the typical momentum is p⇤ ⌘p
cm�� ⇠ m�q1/4.
[Kofman+ 94; 97]⇠m2e↵;h
m2�
Tachyonic resonanceCurvature coupling:�Lint =
1
2⇠Rh2
R =1
M2P
h4V (�)� �2
ifrom Friedmann equations.
Higgs is tachyonic for some region at around the origin.
inflaton
h h h
µcrv ' 2
* particle production is dominated by the first a few oscillations.
[Bassett+ 97; Tujikawa+ 99; Dufaux+ 06]
where the typical momentum is p(tac)⇤ ' m�q1/4, q ⌘ 3
4
✓⇠ � 1
4
◆�2
M2P
.
Tachyonic resonance occurs for q � 1 :
nh ⇠ p(tac)3
⇤ eµcrvp⇠
�iniMP ,
Outline
1.Introduction
2.Resonant particle production
3.Preheating dynamics of Higgs
4.Summary
Set up
� hc2�2h2, ⇠Rh2
Main idea: resonant Higgs production induces tachyonic mass.
• Consider inflaton oscillation epoch
• Inflaton potential: V (�) =1
2m2
��2
�m2self;h ' �3|�|hh2i
It may force Higgs to roll down to the true vacuum.
From now,
• Analytically derive the condition for EW vacuum decay. • Verify it by numerical simulations.
Higgs-inflaton coupling :Lint = �1
2c2�2h2 nh ⇠ 1
32⇡2
r⇡
2µqtcm�te2µqtcm�tp3⇤ µqtc ' O(0.1)with
• Cosmic expansion should kill resonance before that time.
at the end of the resonance, or���m2
self;h(t)���⇠0
. p2⇤
c . 10�4
0.1
µqtc
� h m�
1013 GeV
i
• Higgs tachyonic mass grows:
That time interval �t : c2�2(m��t)2 ⇠���m2
self;h
���⇠0
�m2self;h(t) ' 3�hh2(t)i ⇠ �3|�| nh(t)
!k⇤;h(t)
• dominates at around the origin.�m2self;h c2�2
EW vacuum decays.• If the growth rate exceeds unity, |�me↵;h|�t ⇠ |�m2self;h|/p2⇤ > 1,
Quartic: lattice simulationWe have verified our condition by classical lattice simulations.
(added 6th term for large field for convergence )
c = 1⇥ 10�4 c = 2⇥ 10�4
�ini =p2Mpl,m� = 1.5⇥ 1013 GeV, N = 1283, L = 10/m�, dt = 10�3/m�
10-8
10-6
10-4
10-2
100
102
0 20 40 60 80 100
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
10-8
10-6
10-4
10-2
100
102
0 10 20 30 40 50 60
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
Quartic: lattice simulationWe have verified our condition by classical lattice simulations.
(added 6th term for large field for convergence ) EW vacuum decays!!
c = 1⇥ 10�4 c = 2⇥ 10�4
�ini =p2Mpl,m� = 1.5⇥ 1013 GeV, N = 1283, L = 10/m�, dt = 10�3/m�
10-8
10-6
10-4
10-2
100
102
0 20 40 60 80 100
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
10-8
10-6
10-4
10-2
100
102
0 10 20 30 40 50 60
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
Higgs-curvature coupling :Lint = �1
2⇠Rh2 nh ⇠ 1
16⇡2
r⇡
2eneffµcrv
p⇠�ini/Mplp(tac)
3
⇤ with µcrv ' 2, ne↵ & 1
⇠ . 10⇥
2
ne↵µcrv
�2 "p2Mpl
�ini
#2
We require at the end of the resonance, or|�m2self;h|⇠R⇠0 . ⇠R
• : effective number of times of oscillationne↵
ne↵ ' 1 for �ini =p2MP , ne↵ ' 1.5-2 for �ini =
p0.2MP
• Bound depends on the initial inflaton amplitude.
c . 10�4
0.1
µqtc
� h m�
1013 GeV
icf. : independent of the initial inflaton amplitude
We have verified our condition by classical lattice simulations.
(added 6th term for large field for convergence )
⇠ = 10 ⇠ = 20
Curvature: lattice simulation
10-10
10-8
10-6
10-4
10-2
100
102
0 5 10 15 20 25 30
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
10-10
10-8
10-6
10-4
10-2
100
102
0 5 10 15 20 25 30
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
�ini =p2Mpl,m� = 1.5⇥ 1013 GeV, N = 1283, L = 20/m�, dt = 10�3/m�
We have verified our condition by classical lattice simulations.
(added 6th term for large field for convergence )
⇠ = 10 ⇠ = 20
Curvature: lattice simulation
10-10
10-8
10-6
10-4
10-2
100
102
0 5 10 15 20 25 30
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
10-10
10-8
10-6
10-4
10-2
100
102
0 5 10 15 20 25 30
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
EW vacuum decays!!�ini =
p2Mpl,m� = 1.5⇥ 1013 GeV, N = 1283, L = 20/m�, dt = 10�3/m�
Outline
1.Introduction
2.Resonant particle production
3.Preheating dynamics of Higgs
4.Summary
Motivation1. EW vacuum metastability (BSM negligible)
vs. high scale inflation
2. Stabilization mechanism during inflation:
�Lint =1
2c2�2h2,
1
2⇠Rh2.
3. How about during preheating?? [our work]- Mass term oscillates resonance.
- Even tachyonic during some period (curvature).
Summary• Studied EW vacuum stability during the preheating epoch with:
Lint = �1
2c2�2h2, � 1
2⇠Rh2
• Obtained upper bounds on the couplings:
Unstable during inflation
Dependent on reheating
Unstable during preheating
�1
2c2�2h2 :
�1
2⇠Rh2 :
c�ini
⇠⇠ 0.2
⇠ Hinf ⇠ 10Hinf
[high scale inflation]
⇠ 10M2pl/�
2ini
Back up
Adiabaticity• Mode equation of Higgs:
• Number density of Higgs:
nk;h(t) =1
2!k;h(t)
���hk;h
���2+ !2
k;h |hk;h|2�� 1
2
: [violation of adiabaticity] or,: [tachyonic] is important for particle production.
Its time evolution is nk;h = O(|!k;h/!2k;h|)!k;hnk;h
hk +⇥!2k;h(t) +�2(t)
⇤hk = 0,
�2 = �9H2/4� 3H/2, !2k;h(t) = m2
e↵;h(t) + k2/a2 + �m2self;h.where
� = �(t) sinm�t, �(t) ' �i
a(t)3/2m2
e↵;h(t) = c2�2, or
⇠
M2P
h4V � ˙�2
i,
R =1
M2P
h4V � �2
i
(|!k;h/!2
k;h| & 1
!2k;h < 0
Mathieu equationAk
0 5 10
5
10
-2q/3
Ak= 2q/3
Ak= 2q
q
zeroth
first
second
third
Ak=
[Tsujikawa+ 99]
• Quartic coupling
• Curvature coupling
Ak =k2
a2m2�
+ 2q, q =c2�2
4m2�
Ak =k2
a2m2�
+⇠�2
2M2pl
,
q =3
4
✓⇠ � 1
4
◆�2
M2pl
expansionexpansion
[quartic]
[curvature]
d2hk
dz2+ (Ak � 2q cos z)hk = 0
z = m�t+ const.
Quartic: lattice simulationWe have verified our condition by classical lattice simulations.
Classical lattice simulation:
(1) Divide space coordinates into meshes.
(2) Introduce gaussian fluctuations (quantum fluctuation).
(3) Solve the discretized classical equations of motion.
[Khlebnikov, Tkachev 96]
�i,j �i+1,j · · ·
· · ·
�i,j�1
[Polarski, Starobinsky 96]
* Classical approximation is valid in the large occupation number limit.
Quartic: spectrumThe number density of higgs for each momentum:
(added 6th term for large field for convergence )
c = 1⇥ 10�4 c = 2⇥ 10�4
�ini =p2Mpl,m� = 1.5⇥ 1013 GeV, N = 1283, L = 10/m�, dt = 10�3/m�
10-1
100
101
102
103
104
1 10
Com
ovin
g nu
mbe
r den
sity
nk;
h +
1/2
Comoving momentum k/mφ
mφt = 05
1015202530
10-1
100
101
102
103
104
1 10
Com
ovin
g nu
mbe
r den
sity
nk;
h +
1/2
Comoving momentum k/mφ
mφt = 0102030405060708090
Curvature: spectrumThe number density of higgs for each momentum:
(added 6th term for large field for convergence )
�ini =p2Mpl,m� = 1.5⇥ 1013 GeV, N = 1283, L = 20/m�, dt = 10�3/m�
⇠ = 10 ⇠ = 20
10-1
100
101
102
103
104
1 10
Com
ovin
g nu
mbe
r den
sity
nk;
h +
1/2
Comoving momentum k/mφ
mφt/π = 012
10-1
100
101
102
103
104
1 10
Com
ovin
g nu
mbe
r den
sity
nk;
h +
1/2
Comoving momentum k/mφ
mφt/π = 0123456789
Coupling with SM particles
� hc2�2h2, ⇠Rh2So far…
Coupling with SM particles
� h
�
c2�2h2, ⇠Rh2
� : top quarks, EW gauge bosons
In reality…
Coupling with SM particles
� h
�
top Yukawa gauge
c2�2h2, ⇠Rh2
� : top quarks, EW gauge bosons
In reality…
Coupling with SM particles
� h
�
top Yukawa gauge
c2�2h2, ⇠Rh2
inflaton decay (reheating)
In reality…
� : top quarks, EW gauge bosons
Coupling with SM particles
� h
�
top Yukawa gauge
c2�2h2, ⇠Rh2In reality…
� : top quarks, EW gauge bosons
Coupling with SM particles
� h
�
top Yukawa gauge
c2�2h2, ⇠Rh2
� : top quarks, EW gauge bosons
• Higgs decays into top quarks via top Yukawa coupling.
• Higgs annihilates into gauge boson pairs.
Reduce Higgs resonance efficiency (instant preheating)
Induce effective potential for Higgs
We will see they are less significant during preheating.
Instant preheatingHiggs is massive for the large field value region of inflaton.
Higgs can decay via Yukawa at that region. [Felder+ 98]
There are two cases:
(1) [Resonant production rate] [Decay rate] ⌧
�(2) [Resonant production rate] [Decay rate]
• Early stage of quartic coupling
• Late stage of quartic coupling
• Curvature coupling
m2H;h = c2�2, ⇠R
* For curvature coupling, growth/decay rate , and hence (2) always holds./p
⇠
Instant preheating
In this case, the decay just slightly reduces the production rate.
(1) [Resonant production rate] [Decay rate] ⌧
(2) [Resonant production rate] [Decay rate] �
• Higgs decay rate: �h!tt ⇠ ↵tc�
Higgs completely decays for ↵tc� � m�/⇡.
• Inflaton decay rate at that epoch: �inst ⇠c2
4⇡4p3↵t/2
m�
For c & O(0.1), inflaton completely decays via this process.
Otherwise, (1) is eventually violated and the situation reduces to (2).
↵t = y2t /4⇡
AnnihilationConsider the following simplified Lagrangian:
�Lint =1
2g2h�h
2�2 +1
4g2���
4,
with � : light scalar field (mimicking gauge bosons).
10-10
10-8
10-6
10-4
10-2
100
102
104
0 10 20 30 40 50 60
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>a3<χ2>
10-10
10-8
10-6
10-4
10-2
100
102
104
0 5 10 15 20 25 30
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>a3<χ2>
gh� = g�� = 0.5
Quartic Curvature
AnnihilationConsider the following simplified Lagrangian:
�Lint =1
2g2h�h
2�2 +1
4g2���
4,
with � : light scalar field (mimicking gauge bosons).
10-10
10-8
10-6
10-4
10-2
100
102
104
0 10 20 30 40 50 60
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>a3<χ2>
10-10
10-8
10-6
10-4
10-2
100
102
104
0 5 10 15 20 25 30
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>a3<χ2>
gh� = g�� = 0.5
Quartic Curvature
10-8
10-6
10-4
10-2
100
102
0 10 20 30 40 50 60N
orm
aliz
ed b
y th
e in
itial
infla
ton
ampl
itude
Φin
i
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
Quartic
AnnihilationConsider the following simplified Lagrangian:
�Lint =1
2g2h�h
2�2 +1
4g2���
4,
with � : light scalar field (mimicking gauge bosons).
10-10
10-8
10-6
10-4
10-2
100
102
104
0 10 20 30 40 50 60
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>a3<χ2>
10-10
10-8
10-6
10-4
10-2
100
102
104
0 5 10 15 20 25 30
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>a3<χ2>
gh� = g�� = 0.5
Quartic Curvature
10-10
10-8
10-6
10-4
10-2
100
102
0 5 10 15 20 25 30
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
Curvature
Annihilation
10-1
100
101
102
103
104
1 10
Com
ovin
g nu
mbe
r den
sity
nk;
h +
1/2
Comoving momentum k/mφ
mφt = 05
1015202530
10-1
100
101
102
103
104
1 10
Com
ovin
g nu
mbe
r den
sity
nk;χ
+ 1/
2
Comoving momentum k/mφ
mφt = 05
1015202530
Higgs is resonantly produced, while is not efficiently produced.�
Spectrum of Higgs Spectrum of �
(Effective mass might suppress the production.) m2e↵;� ' g2h�hh2i
Cosmic expansion• Assume EW vacuum survives the preheating epoch.
• For illustration, neglect top Yukawa and gauge couplings.
Just after preheating, Higgs is stabilized by m2H;h = c2�2, ⇠R.
However, while m2H;h / a�3, �m2
self;h / hh2i / a�2.
�m2self;h eventually dominates over m2
H;h.
For before ,hh2i < h2
max
m2H;h < |�m2
self;h|
m� = 1.5⇥ 1013 GeV, hmax
= 1010 GeV
c . 3⇥ 10�5
0.1
µqtc
�, ⇠ . 0.5
2
ne↵µcrv
�2 "p2MP
�ini
#2
.
* It means most parameters for stabilization during inflation cause catastrophe.
Cosmic expansion• Assume EW vacuum survives the preheating epoch.
• For illustration, neglect top Yukawa and gauge couplings.
Just after preheating, Higgs is stabilized by m2H;h = c2�2, ⇠R.
However, while m2H;h / a�3, �m2
self;h / hh2i / a�2.
�m2self;h eventually dominates over m2
H;h.
For before ,hh2i < h2
max
m2H;h < |�m2
self;h|
m� = 1.5⇥ 1013 GeV, hmax
= 1010 GeV
c . 3⇥ 10�5
0.1
µqtc
�, ⇠ . 0.5
2
ne↵µcrv
�2 "p2MP
�ini
#2
.Dynamics after preheating
is also non-trivial.
* It means most parameters for stabilization during inflation cause catastrophe.
Cosmic expansion• Assume EW vacuum survives the preheating epoch.
• For illustration, neglect top Yukawa and gauge couplings.
Just after preheating, Higgs is stabilized by m2H;h = c2�2, ⇠R.
However, while m2H;h / a�3, �m2
self;h / hh2i / a�2.
�m2self;h eventually dominates over m2
H;h.
For before ,hh2i < h2
max
m2H;h < |�m2
self;h|
m� = 1.5⇥ 1013 GeV, hmax
= 1010 GeV
c . 3⇥ 10�5
0.1
µqtc
�, ⇠ . 0.5
2
ne↵µcrv
�2 "p2MP
�ini
#2
.Dynamics after preheating
is also non-trivial.
Thermalization is crucial
for EW vacuum stability.* It means most parameters for stabilization during inflation cause catastrophe.
Complete reheating
� h
�
top Yukawa gauge
c2�2h2, ⇠Rh2
inflaton decay (reheating)
� : top quarks, EW gauge bosons
Inflaton must decay to complete the reheating process.
Characterized by the reheating temperature .TRH
Complete reheating
10-10
10-8
10-6
10-4
10-2
100
102
104
0 20 40 60 80 100
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>a3<χ2>
prelim
inary
• For 105 GeV . TRH . 1010 GeV,
radiation from inflaton decay may stabilize Higgs after preheating.
Thermal bounce is valid after that.
TRH � 1010 GeV,• For
It might also induce EW vacuum decay??
�ini =p2Mpl,m� = 1.5⇥ 1013 GeV, N = 1283,
L = 10/m�, dt = 10�3/m�, g�� = 5⇥ 10�4
parametric resonance might occur
in other sectors during preheating.
Quartic: lattice simulation
(added 6th term for large field for convergence )
c = 1⇥ 10�4 c = 2⇥ 10�4
We have changed the initial inflaton amplitude as .�ini =p0.2Mpl
10-8
10-6
10-4
10-2
100
102
0 20 40 60 80 100
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
10-8
10-6
10-4
10-2
100
102
0 10 20 30 40 50 60
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
�ini =p0.2Mpl,m� = 1.5⇥ 1013 GeV, N = 1283, L = 40/m�, dt = 10�3/m�
Quartic: lattice simulation
(added 6th term for large field for convergence )
c = 1⇥ 10�4 c = 2⇥ 10�4
We have changed the initial inflaton amplitude as .�ini =p0.2Mpl
10-8
10-6
10-4
10-2
100
102
0 20 40 60 80 100
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
10-8
10-6
10-4
10-2
100
102
0 10 20 30 40 50 60
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
EW vacuum decays!!�ini =
p0.2Mpl,m� = 1.5⇥ 1013 GeV, N = 1283, L = 40/m�, dt = 10�3/m�
Curvature: lattice simulation
(added 6th term for large field for convergence )
We have changed the initial inflaton amplitude as .�ini =p0.2Mpl
⇠ = 30
�ini =p0.2Mpl,m� = 1.5⇥ 1013 GeV, N = 1283, L = 40/m�, dt = 10�3/m�
⇠ = 20
10-10
10-8
10-6
10-4
10-2
100
102
0 5 10 15 20 25 30
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
10-10
10-8
10-6
10-4
10-2
100
102
0 5 10 15 20 25 30
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
Curvature: lattice simulation
(added 6th term for large field for convergence )
We have changed the initial inflaton amplitude as .�ini =p0.2Mpl
⇠ = 30
�ini =p0.2Mpl,m� = 1.5⇥ 1013 GeV, N = 1283, L = 40/m�, dt = 10�3/m�
⇠ = 20
10-10
10-8
10-6
10-4
10-2
100
102
0 5 10 15 20 25 30
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
10-10
10-8
10-6
10-4
10-2
100
102
0 5 10 15 20 25 30
Nor
mal
ized
by
the
initi
al in
flato
n am
plitu
de Φ
ini
mφt
a3<φ>2
a3(<φ2> - <φ>2)a3<h2>
EW vacuum decays!!