Simulations in Early-Universe Theory
Jan Smit
Institute for Theoretical Physics,
University of Amsterdam, the Netherlands
Lattice ’05, Dublin
1
end of inflation
?
?
LG?
EW
QCD
NS
EQ
DEC
now
BG?
LG: leptogenesis
EW & QCD transitions
BG: baryogenesis
NS: nucleosynthesis
EQ: equal energy in ‘radiation’ and ‘matter’ (non-rel.)
DEC: photon decoupling (↔ CMB)
This talk: EW and earlier, field theory
2
Plan:
1. Inflation, preheating, defects,
baryogenesis, thermalization
2. Classical approximation
3. Φ-derivable approximations
4. Outlook
3
examples, not a review
4
Inflation
inflaton field σ, FLRW scale factor a (flat),
Hubble rate H, energy density ρ
∂V (σ, · · · )
∇2 σ
0 = σ̈ + 3H σ̇ − 2 +
a
∂σ
s
ȧ
ρ
H≡
=
a
3m2
P
(∇σ)2
σ̇ 2
+
ρ =
+ V (σ, · · · ) + · · ·
2
2
2a
potential energy dominates, strong damping,
‘slow roll’, exponential expansion
5
V
V
sigma
‘small-field’ potential
σ↑
σ↓
sigma
‘large-field’ potential
small-field models, ‘new inflation’
large-field models, ‘chaotic inflation’
e.g. large-field potential (hybrid inflation):
V (σ, ϕ) = V0 +
1
2
µ2σ σ 2 + g 2σ 2ϕ† ϕ − µ2ϕ† ϕ + λ(ϕ† ϕ)2
(µσ µ, when g 2σ 2 − µ2 < 0, ϕ†ϕ → (µ2 − g 2σ 2)/λ)
6
Preheating
after inflation: universe is cold∗
σ-energy transferred to other d.o.f.
V (σ, · · · ) = V (σ) + (g 2σ 2 − µ2)ϕ† ϕ + · · ·
∗ different
in ‘warm inflation’ Berera & Ramos
7
time-dependent effective ϕ-mass
2
2 2
=
−µ
+
g
σ
µ2
eff
- hσi oscillates → ϕ resonates
- µ2
eff < 0 → tachyonic instability
σ-modes can also resonate and σ is also tachyonic during ∂ 2V /∂σ 2 < 0
⇒ large occupation numbers in σ and ϕ
in limited momentum range: preheating
8
Zoo of defects
domain walls, strings, textures, monopoles, Qballs, I-balls, oscillons, half knots, sphalerons,
Chern-Simons numbers, . . .
9
Baryogenesis
explain baryon to photon ratio
nB
' 6.5 10−10
nγ
from nB = 0 after inflation and B, C & CP
violation during particle physics processes out
of equilibrium
very many scenarios proposed
here:
10
Cold ElectroWeak Baryogenesis
anomaly in divergence of baryon current
1
µν
q=
trF
F̃
µν
16π 2
topological-charge density
µ
∂µjB = 3 q,
tachyonic EW transition with CP violation ⇒
baryon asymmetry
B(t) = 3
∗ Garcı́a-Bellido,
Z t
0
dt0
Z
d3 x hq(x, t0)i
Grigoriev, Kusenko, Shaposhnikov, PRD 60 (1999)
123504; Krauss, Trodden, PRL 83 (1999) 1502
11
Thermalization
redistribution of energies in d.o.f.
how long does it take?
(in the expanding universe)
(re?)heating temperature Trh marks beginning
of radiation-dominated universe
need Treh & 1 GeV for nucleo-synthesis
12
Classical Approximation
real time
hO(t)i = Trρ O(t) = Trρ U †(t) O U (t)
difficult in field theory
classical approximation
- reasonable for large occupation numbers
- depends on initial conditions
13
coherent-state∗ or Wigner represenation
hO(φ, π)i =
Z
Dφ Dπ ρc (φc , πc) O(φc , πc )
classical approximation∗∗
1.
2.
3.
4.
draw initial configuration from ρc (φ, π)
solve classical e.o.m. for φ and π
evaluate O(φ, π)
average over initial configurations
∗ Sallé,
JS, Vink, PRD 64 (2001) 025016
∗∗ perturbative
analysis: Aarts, JS, PLB 393 (1997) 395; NPB 511
(1998) 451 ; this meeting: Yavin
14
occupation numbers
1
(ω φ + iπk),
ak = √
2ωk k k
†
nk = hak aki
generalize to time-dependent (quasi)particle energies ωk and numbers nk:
1 1
†
hφk φk i ≡ nk +
,
2 ωk
i
†
hφk πki ≡ ñk +
2
1
†
ωk
hπkπki ≡ nk +
2
15
tachyonic transition at t = tc , µeff (tc ) = 0
gaussian approximation
2
φ̈k + (µ2
eff + k )φk = 0
3
µ2
eff = −M (t − tc),
M 3 = −2g 2 σcσ̇c
quench:
2, t < t ,
=
+µ
µ2
c
eff
= −µ2, t > tc
16
reproduce quantum h· · · i with classical distribution
"
!#
+ 2
− 2
X
|ξk |
|ξk |
1
0
exp −
+
2
nk + 1/2 + ñk
nk + 1/2 − ñk
k
where
1
ξk± = √
(ωk φk ± πk)
2ωk
nk and ñk grow faster than exponential∗ and
nk + 1/2 − ñk → 0, for k . kmax
q
M 3(t − tc)
kmax =
= µ, for the quench
P0
: k . kmax
p
∗ for the quench: n ∼ exp[ µ2 − k 2 (t − t )]
c
k
17
initial distribution for classical approximation:
gaussian ensemble
at time ti > tc when nk , ñk 1 and non-linear
terms in e.o.m. still small,
or ‘just the half’ at ti = tc
kmax cutoff, or kmax = cutoff
18
milestones in preheating:
‘re-scattering’ effects after parametric-resonance∗
and tachyonic∗∗ preheating lead to efficient mixing
∗ Khlebnikov,
∗∗ Felder,
Tkachev, PRL 77 (1996) 219
Garcı́a-Bellido, Greene, Kofman, Linde, Tkachev, PRL 87
(2001) 011601
19
Example∗: ‘new inflation’ with Coleman-Weinberg
potential∗∗
V (σ) =
!
|σ| 1
1
1 4
λσ ln
−
+
λv 4
4
v
4
16
λ = 10−12 , v = 10−3 mP , with expansion
initial hσi ≈ 0, it rolls into minimum of V and oscillates,
energy decays into σ-particles
tachyonic & parametric-resonance preheating
∗ Desroche,
Felder, Kratochvil, Linde, PRD71 (2005) 103516
∗∗ questionable
in real time!
20
10
nk
20
t=801
10
1010
t=839
1010
1
0.01 0.1
nk
1020
nk
20
1
10
k
1
0.01 0.1
nk
t=876
1020
1010
1
10
k
10
k
t=903
1010
1
0.01 0.1
1
10
k
1
0.01 0.1
1
occupation numbers nk , resonance visible at t = 876
t and k in λv-units
21
with φ, V = V (σ) +
1
2
g 2σ 2φ2 (non-tachyonic for φ)
number densities nσ , nφ
10
25
n
g2 =Λ
10
1015
105
800
25
n
g2 =ː10
1015
1400
t
2000
105
800
1400
t
2000
inefficient resonance, nφ nσ
σ → φφ perturbatively,
p
Trh ≈ mP Γσ→φφ (≈ 107 GeV for g 2 . λ)
22
Example∗: large-field inflaton-‘Higgs’ SU(2) model
1 µ2 σ 2 + g 2σ 2ϕ† ϕ + λ(ϕ† ϕ − 1 v 2)2
V (σ, ϕ) = 2
σ
2
fit k-dependence of initial distribution at ti to
gaussian, kmax cutoff, no expansion
next slide:
‘mixing’ through scattering of waves
λ = 0.11/4, g 2 = 2λ, M = 0.18 m, m =
√
λv
σ → χ, tc → 0, no expansion,
∗ Garcı́a-Bellido,
Garcı́a-Pérez, González-Arroyo, PRD67 (2003) 103501
23
mt = 23
mt = 24
φ1
1
40 0.5
0.5
30
0
10
30
0
y
20
40
10
20
x
20
20
10
30
10
30
40
40
mt = 25
mt = 26
1
1
40
0.5
30
0
10
20
20
40
0.5
30
0
10
20
20
10
30
40
10
30
40
24
mt =27
mt =32
φ1
1
40 0.5
0.5
30
0
10
30
0
y
20
40
10
20
x
20
20
10
30
10
30
40
40
mt =36
mt =40
1
1
40
0.5
30
0
10
20
20
40
0.5
30
0
10
20
20
10
30
40
10
30
40
25
Example∗: equation of state (p/ρ) in large-field
inflaton-‘Higgs’ U(1) model
2
1
1
2
2
2
2
†
†
2
V = 2 µσ σ + g σ ϕ ϕ + λ ϕ ϕ − 2 v
ϕ = r eiα /
√
2, pressures:
2 − 1 (∇r/a)2 + 1 λ(r 2 − v 2)2
pH = 1
ṙ
2
6
4
2 α̇2 − 1 (∇α/a)2
pG = 1
r
2
6
1 σ̇ 2 − 1 (∇σ/a)2 + 1 (µ2 + g 2r 2 )σ 2
pσ = 2
σ
6
2
∗ Borsányi,
Patkos, Sexty, PRD 68 (2003) 063512
26
8
Goldstone
Higgs
Inflaton
ρ /3
7
6
pressure
5
t=140
4
3
t=160
t=85
2
1
0
t=85
-1
-2
0
5
10
15
20
25
ρ
g 2 = 10−2 , λ = g 2/2, µσ = 4 1011 , mH = 9 1014 GeV,
with expansion
27
Example∗: strings and hot spots
V (σ, ϕ) =
1
2
µ2σ σ 2 + g 2σ 2ϕ†ϕ + λ(ϕ† ϕ − v 2)2
√
g 2 = 10−4 , λ = 10−2 , m ≡ 2λ v = 3 1015 GeV
constant expansion rate H = 3.6 10−3 m
µ2eff = g 2σ 2 − m2 = 0 at σ = σc
all modes (kmax = cutoff) initialized with ‘just the half’
next two slides:
global strings decay, overshoots to σ < −σc ⇒ hot spots,
good mixing
∗ Copeland,
Pascoli, Rajantie, PRD65 (2002) 103517
28
transparant surface:
σ = −σc at time tm = 130; strings: |ϕ|2 = v 2 /10 at time
for which spatial average σ = 0 for the first time
29
transparant green surface: |ϕ|2 = v 2/10; blue: σ = σc; yellow: σ = −σc;
time tm = 270
30
Example∗:
tachyonic electroweak quench
SU(2)-Higgs model
1
µν +(D ϕ)† D µϕ−µ2ϕ†ϕ+λ(ϕ† ϕ)2
trF
F
−L =
µν
µ
2g 2
‘just the half’, kmax = µ,
Ainit = 0, A0 = 0, Gauss’ law
∗ Skullerud,
JS, Tranberg, JHEP 0308 (2003) 045
31
A
A
1
A
A
A
A
A
nk
0.01
A
A
A
A
A
A
A
A
A
A
A
A
A
A
0.0001
A
A
A
A
A
A
A
A
1e-06
t= 1
t= 2
t= 3
t= 4
t= 5
t= 6
t= 7
t= 8
t= 9
t = 10
A t = 11
t = 12
A
A
t =A 20
A
t = 30
t = 40
t = 50
t =100
A
A
1e-08
0
1
2
k/mH
3
4
5
Coulomb-gauge W-particle numbers as a function of
momentum, time in units of m−1
H
32
approx. gauge-independence of transverse nW
k (Coulomband unitary-gauge) after t = 50 m−1
H not much happens
in 50 . tmH < 100
temperature T ≈ 0.4 mH
chemical potential µch ≈ 0.7(0.9)mH for W(H)
longitudinal modes settle at slower rate
- suggests continuing with inclusion of lighter d.o.f. of
Standard Model, e.g. through Boltzmann eqns. & quantum scattering rates
33
Example: kinetic turbulence and self-similar
scaling∗ in electromagnetic field after tachyonic electroweak transition∗∗
inflaton + SU(2)×U(1) model
mW = 6.15 GeV, mH/mW = 4.65, M ti = 5,
g/g 0 as in SM
∗ Micha,
Tkachev, PRD 70 (2004) 043538
∗∗ Diaz-Gil,
Garcı́a-Bellido, Garcı́a-Pérez, González-Arroyo, poster
this meeting
34
Inflaton NS=48
NS=64
Higgs NS=48
NS=64
1.2
1
HIGGS
0.8
0.6
0.4
0.2
INFLATON
0
-0.2
0
50
100
150
200
Higgs hϕ†ϕi and inflaton hσi vs time
250
1
Inflaton NS=48
NS=64
Higgs NS=48
NS=64
-2/5
0.1
Inflaton variance t
0.01
Higgs variance t-2/7
10
100
mt
turbulent scaling of Higgs hϕϕ†i and inflaton hσσi ∝ t−ν ,
ν = −2/(2m − 1), m = 3 (σ), m = 4 (H)
SU(2) electric
Scaled
0.45
0.12
0.4
0.35
0.08
0.3
3
|k| |E(k)|
0.04
0.25
0
0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0.15
0.1
mt=55
mt=105
mt=155
mt=205
mt=255
0.05
0
0
1
2
3
4
5
6
7
8
9
|k|/m
self-similar spectrum: n(k, t) = t−q n0(kt−p), q = (7/2)p,
p = 1/(2m − 1)
Example∗: cold electroweak baryogenesis
SU(2) Higgs model with effective CP violation
L = LSU(2)H −
k
ϕ† ϕ q,
2
mW
q=
1
µν
trF
F̃
µν
16π 2
√
quench, ‘just the half’, kmax = µ = mH / 2 = 0.25/as
R 3 0
µ
q = ∂µjCS, NCS = d x jCS, Chern–Simons number
B(t) = 3hNCS(t) − NCS(0)i
Nw : winding number in the Higgs field
NCS ≈ Nw , expected at low temperature
*Tranberg, JS, JHEP 0311 (2003) 016; Tranberg, this meeting
38
0.3
0.2
0.1
0
-0.1
-0.2
0
20
40
60
80
100
hϕ† ϕi(black), hNCSi(red), hNw i(green)
Higgs winding num√
ber, versus time; k = 3, mH = 2 mW
39
3
2e-05
1.5e-05
Nw/VmW
2.5e-05
1e-05
5e-06
0
0
0.5
1
1.5
k
2
2.5
3
3.5
dependence of nCS on k, mH = 2mW
40
resulting in
√
nB
−5
= (4 ± 1)10 k, mH = 2 mW
nγ
= −(4 ± 1)10−5 k, mH = 2 mW
41
Thermalization
classical: Rayleigh-Jeans effects at large times
- need quantum description
approach equilibrium through scattering and
damping
nonperturbative, e.g.
e−γt cos(ωt),
γ = cλ2 ,
e−γt = 1 − cλ2 t + · · ·
need summing infinite series of diagrams
42
e.g. use Dyson-Schwinger hiearchy
Phi-derivable∗
i
i δ 2S(φ)
Γ(φ, G) = S(φ)− Tr ln G+ Tr
G+Φ(φ, G)
2
2
δφδφ
i Φ=
+
+
+ ...
sum of all 2PI diagrams
∗ revived
simulation-wise by Berges, Cox, PLB 517 (2001) 369
43
equations of motion
δΓ(φ, G)
= 0,
δφ(x)
δΓ(φ, G)
=0
δG(x, y)
typical form
h
2
−2
2
2
∂ − µ − 3λφ − 3λG(x, x) G>(x, y) =
Z x0
0
Z y0
0
2
i
dz 0
Z
d3z Im[Σ>(x, z; φ, G)]G>(z, y)
dz 0
Z
d3z Σ>(x, z; φ, G) Im[G>(z, y)]
44
δΦ
selfenergy Σ(x, y) = 2i δG(x,y)
+
− iΣ =
+
+
+ ...
45
Φ-derivable:
- ‘thermodynamically consistent’
- global symmetries → WTIs
- renormalizable∗
∗ Van
Hees and Knoll, PRD 65 (2002) 025010 & 105005; PRD 66
(2002) 045008
Blaizot, Iancu, Reinosa, PLB 568 (2003) 160; NPA 736 (2004) 149
Cooper, Mihaila, Dawson, PRD 70 (2004) 105008; hep-ph/0502040
Berges, Borsányi, Reinosa, Serreau, hep-ph/0409123;hep-ph/0503240
46
- truncate Φ by order in coupling- or 1/N expansion
- put Γ on space-time lattice (at as), or
use spatial lattice and solve e.g. with RungeKutta
- initial conditions in terms of φ and G
results directly for average h· · · i
- numerically challenging ‘memory kernels’ Σ
47
Example∗: chiral quark-meson model
1 (∂ σ∂ µσ + ∂ π a∂ µπ a)
−L = ψ̄γ µ∂µψ + 2
µ
µ
2(σ 2 + π aπ a) + g ψ̄(σ + iγ τ aπ a)ψ
+1
µ
5
2
iΦ=
spectral function ρ, statistical function F
G(x, y) ∼ F (x, y) − iρ(x, y)/2
e.o.m. discretized in terms of components, e.g.
µ
µ
µν
ρ = ρS + ρP iγ5 + ρV γµ + ρAγµ γ5 + ρT σµν /2
∗ Berges,
Borsányi, Serreau, NPB 660 (2003) 51
48
0.3
initial fermion distributions
0.8
nf0
A
0.25
B
0.4
0.2
0
FV(t,t,p)
0
p [m]
2
0
p [m]
2
p=0.16, A
B
p=0.48, A
B
p=0.78, A
B
0.15
0.1
0.05
0
-0.05
0
5
10
15
20
25
30 50 100
500
-1
t [m ]
memory loss 1/γFdamp ≈ 15 m−1 , m = thermal mass
49
3.5
mt = 4
mt = 6
mt = 12
mt = 24
mt = 60
mt = 120
mt = 240
mt = 600
3
log (1/nf - 1)
2.5
2
1.5
1
0.5
0
-0.5
0
0.5
1
1.5
2
2.5
p [m]
thermalization: ln(−1 + 1/np) → p/T , Fermi-Dirac
1/γFtherm ≈ 95 m−1 , g = 1
50
thermalization, relatively fast
no fermion doubling seen∗
∗
differs from Aarts &JS, NPB555 (1999) 355; PRD61 (2000)
025002, who used lattice fermion techniques in real time
51
Example∗: electroweak quench
−L = (1/2)∂µφa ∂ µφa−(µ2/2)φa φa+(λ/4)(φa φa)2
a = 1, . . . , N , λ ∝ 1/N , approximate Φ to Nextto-Leading-Order in 1/N expansion∗∗ , N = 4
next two slides:
comparison of nk in NLO with two ‘just the half’ classical approximations:
(a) only unstable modes initialized (kmax = µ = 0.7/as)
(b) all modes initialized (kmax = cutoff)
∗ Arrizabalaga,
∗∗ Berges,
JS, Tranberg, JHEP 0410 (2004) 017
NPA 699 (2002) 847
52
1000
Classical, unstable initialized
Quantum, NLO
Classical, all initialized
100
10
nk
1
0.1
0.01
0
1
k/µ
2
3
particle numbers at time tµ = 14; λ = 1/24, Lµ = 22.4
53
1000
1000
Quantum, NLO
Classical, unstable initialized
Classical, all initialized
100
100
10
10
nk
nk
1
1
0.1
0.1
0.01
0
Quantum, NLO
Classical, all initialized
Classical, unstable initialized
1
k/µ
2
3
0.01
0
1
2
k/µ
particle numbers at time tµ = 100 (Left) and 700 (Right);
λ = 1/4 (mH = 174 GeV), Lµ = 11.2
54
3
furthermore
Kibble-Zurek vs Hindmarsch-Rajantie∗ scaling in formation of topological defects in gauge theories
...
∗ Hindmarsch,
Rajantie, PRL 85 (2000) 4660
55
Outlook
• lots to do in the Big Bang
• quantum fields out of equilibrium: approximations
unavoidable
• classical approximation can be well motivated and
is very useful up to intermediate times
in tachyonic preheating, topological-like terms in
e.o.m. and -observables appear feasible with
kmax 1/as
need more experience with non-abelian gauge fields∗
∗ Moore,
JHEP 0111 (2001) 017
56
• Φ-derivable approximations good for later times
three-point interactions (non-zero range) are efficient for thermalization (broken symmetry phase,
Yukawa, gauge fields), but difficult to incorporate
in Φ beyond two loops
Nambu-Goldstone boson mass is still an issue
gauge fields not simulated yet (?)
dependence on gauge fixing parameter admissable∗
as part of controlled approximation?
fermion doubling?
need to gauge fix: a blessing for the chiral case?
real-time approach∗∗ to finite density?
∗ Arrizabalaga,
JS, PRD 66 (2002) 065014
∗∗ Sallé, JS, Vink, NP Proc.Suppl. 94 (2001) 427
57
Example∗: development of winding and ChernSimons number densities in tachyonic EW transition
same parameters as before
next two slides:
nW , tmH = 1, . . . , 15
nCS, tmH = 7, . . . , 15
∗ Van
der Meulen, Sexty, JS, Tranberg, to be pub.
58
59
60