This document discusses the effective picture of bubble expansion during a cosmological phase transition. It describes how the friction term experienced by an expanding bubble wall can be separated into a gamma-independent and gamma-dependent component. The gamma-independent term arises from leading-order particle transmissions while the gamma-dependent term arises from next-to-leading order processes involving soft gauge boson emissions. Numerical simulations of the bubble wall equations of motion can take into account these friction terms as well as contributions from particle scattering off the wall.

1. Effective picture of bubble expansion
王少江
Based on 2011.11451 with Prof. Rong-Gen Cai
/101
2020引⼒和宇宙学研讨会 2020/12/20 08:55-09:15

2. GWs from FOPT
/102

3. GWs from FOPT
/102

4. GWs from FOPT
/102
TeV PeVMeV
BSMTPT
QCDPT EWPT PQPT

5. Phase dynamics
/103

6. Phase dynamics
/103
ϕ
Veff(ϕ)
ϕ−
ϕ+

7. Phase dynamics
/103
ϕ
Veff(ϕ)
ϕ−
ϕ+
α ∼
ΔVeff
ρRad
Γ(t) ∼ e−S(t)

8. Phase dynamics
/103
ϕ
Veff(ϕ)
ϕ−
ϕ+
α ∼
ΔVeff
ρRad
Γ(t) ∼ e−S(t)

9. Phase dynamics
/103
ϕ
Veff(ϕ)
ϕ−
ϕ+
α ∼
ΔVeff
ρRad
Γ(t) ∼ e−S(t)

10. Phase dynamics
/103
ϕ
Veff(ϕ)
ϕ−
ϕ+
α ∼
ΔVeff
ρRad
Γ(t) ∼ e−S(t)

11. Phase dynamics
/103
ϕ
Veff(ϕ)
ϕ−
ϕ+
α ∼
ΔVeff
ρRad
Γ(t) ∼ e−S(t) Γ(T*) ∼ H(T*)
R* ∼ ⟨Ri (T*)⟩

12. Phase dynamics
/103
ϕ
Veff(ϕ)
ϕ−
ϕ+
α ∼
ΔVeff
ρRad
Γ(t) ∼ e−S(t) Γ(T*) ∼ H(T*)
R* ∼ ⟨Ri (T*)⟩
fpeak ∼
1
R*

13. Bubble dynamics
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14. Bubble dynamics
/104
Bubble expansion in vacuum
vw → 1

15. Bubble dynamics
/104
Ωscalar
GW ∝ κ2
collision ∼
(
Ewall
kinetic
ρvac )
2
Bubble expansion in vacuum
vw → 1

16. Bubble dynamics
/104
Ωscalar
GW ∝ κ2
collision ∼
(
Ewall
kinetic
ρvac )
2
Bubble expansion in vacuum
vw → 1
Bubble expansion in plasma
vw ↑ → veq

17. Bubble dynamics
/104
Ωscalar
GW ∝ κ2
collision ∼
(
Ewall
kinetic
ρvac )
2
Bubble expansion in vacuum
vw → 1
Bubble expansion in plasma
vw ↑ → veq
Ωplasma
GW
∝ κ2
soundwave ∼
(
Efluid
kinetic
ρvac )
2

18. Numerical simulation
/105

19. Numerical simulation
/105
□ ϕ =
∂Veff
∂ϕ
+δf
∂μTμν
p + ∂ν
ϕ
∂VT
∂ϕ
= − ∂ν
ϕ ⋅ δf
δf =
∑
i=B,F
gi
dm2
i
dϕ ∫
d3 ⃗k
(2π)3
δfi
2Ei( ⃗k )

20. Numerical simulation
/105
□ ϕ =
∂Veff
∂ϕ
+δf
∂μTμν
p + ∂ν
ϕ
∂VT
∂ϕ
= − ∂ν
ϕ ⋅ δf
δf =
∑
i=B,F
gi
dm2
i
dϕ ∫
d3 ⃗k
(2π)3
δfi
2Ei( ⃗k )

21. Numerical simulation
/105
□ ϕ =
∂Veff
∂ϕ
+δf
∂μTμν
p + ∂ν
ϕ
∂VT
∂ϕ
= − ∂ν
ϕ ⋅ δf
δf =
∑
i=B,F
gi
dm2
i
dϕ ∫
d3 ⃗k
(2π)3
δfi
2Ei( ⃗k )
σγ3
w
··rw +
2σγw
rw
= Δpdr + Δpfr□ ϕ =
∂Veff
∂ϕ
+ηTuμ
∂μϕ
∫
+∞
−∞
dr′ϕ′(r′) EOM = 0
Effective EOM

22. Numerical simulation
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□ ϕ =
∂Veff
∂ϕ
+δf
∂μTμν
p + ∂ν
ϕ
∂VT
∂ϕ
= − ∂ν
ϕ ⋅ δf
δf =
∑
i=B,F
gi
dm2
i
dϕ ∫
d3 ⃗k
(2π)3
δfi
2Ei( ⃗k )
σγ3
w
··rw +
2σγw
rw
= Δpdr + Δpfr□ ϕ =
∂Veff
∂ϕ
+ηTuμ
∂μϕ
∫
+∞
−∞
dr′ϕ′(r′) EOM = 0
Effective EOM
Parameterization

23. Numerical simulation
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□ ϕ =
∂Veff
∂ϕ
+δf
∂μTμν
p + ∂ν
ϕ
∂VT
∂ϕ
= − ∂ν
ϕ ⋅ δf
δf =
∑
i=B,F
gi
dm2
i
dϕ ∫
d3 ⃗k
(2π)3
δfi
2Ei( ⃗k )
σγ3
w
··rw +
2σγw
rw
= Δpdr + Δpfr□ ϕ =
∂Veff
∂ϕ
+ηTuμ
∂μϕ
∫
+∞
−∞
dr′ϕ′(r′) EOM = 0
Effective EOM
Δpfr ∝ − γwParticle physicsParameterization

24. Numerical simulation
/105
□ ϕ =
∂Veff
∂ϕ
+δf
∂μTμν
p + ∂ν
ϕ
∂VT
∂ϕ
= − ∂ν
ϕ ⋅ δf
δf =
∑
i=B,F
gi
dm2
i
dϕ ∫
d3 ⃗k
(2π)3
δfi
2Ei( ⃗k )
σγ3
w
··rw +
2σγw
rw
= Δpdr + Δpfr□ ϕ =
∂Veff
∂ϕ
+ηTuμ
∂μϕ
∫
+∞
−∞
dr′ϕ′(r′) EOM = 0
Effective EOM
Δpfr ∝ − γwParticle physicsParameterization

25. Thermal friction
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26. Thermal friction
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P1→1 ≈
Δm2
T2
24
≡ ΔpLO
Bodeker & Moore 09 Particle transmission and reflection
Run-away

27. Thermal friction
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Transition splitting of a fermion
emitting a soft vector boson
Bodeker & Moore 17
P1→2 ≈ γwg2
ΔmVT3
≡ γwΔpNLO Non-run-away
P1→1 ≈
Δm2
T2
24
≡ ΔpLO
Bodeker & Moore 09 Particle transmission and reflection
Run-away

28. Thermal friction
/106
Transition splitting of a fermion
emitting a soft vector boson
Bodeker & Moore 17
P1→2 ≈ γwg2
ΔmVT3
≡ γwΔpNLO Non-run-away
P1→1 ≈
Δm2
T2
24
≡ ΔpLO
Bodeker & Moore 09 Particle transmission and reflection
Run-away
Re-summing multiple soft gauge
bosons scattering to all orders
Hoche et al 2007.10343
P1→N ≈ 0.005γ2
wg2
T4
≡ γ2
wΔpNLO
⋯
Non-run-away

29. Thermal friction
/106
Transition splitting of a fermion
emitting a soft vector boson
Bodeker & Moore 17
P1→2 ≈ γwg2
ΔmVT3
≡ γwΔpNLO Non-run-away
P1→1 ≈
Δm2
T2
24
≡ ΔpLO
Bodeker & Moore 09 Particle transmission and reflection
Run-away
Re-summing multiple soft gauge
bosons scattering to all orders
Hoche et al 2007.10343
P1→N ≈ 0.005γ2
wg2
T4
≡ γ2
wΔpNLO
⋯
Non-run-away
Thermal
Equilibrium
Thermal
Equilibrium
Mancha et al 2005.10875
Δp = − (γ2
w − 1)TΔs

30. Thermal friction
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Transition splitting of a fermion
emitting a soft vector boson
Bodeker & Moore 17
P1→2 ≈ γwg2
ΔmVT3
≡ γwΔpNLO Non-run-away
P1→1 ≈
Δm2
T2
24
≡ ΔpLO
Bodeker & Moore 09 Particle transmission and reflection
Run-away
Re-summing multiple soft gauge
bosons scattering to all orders
Hoche et al 2007.10343
P1→N ≈ 0.005γ2
wg2
T4
≡ γ2
wΔpNLO
⋯
Non-run-away
Thermal
Equilibrium
Thermal
Equilibrium
Mancha et al 2005.10875
Δp = − (γ2
w − 1)TΔs
δf = − ˜ηT(uμ
∂μϕ)2
Δpfr ∝ − (γ2
w − 1)

31. Effective picture
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32. Effective picture
/107
Δpfr = − ΔpLO − h(γw)ΔpNLO
γ-independent friction term γ-dependent friction term

33. Effective picture
/107
Δpfr = − ΔpLO − h(γw)ΔpNLO
γ-independent friction term γ-dependent friction term
σγ3
w
··rw +
2σγw
rw
= Δpdr + Δpfr
E = 4πσr2
wγw −
4
3
πr3
w(Δpdr + Δpfr)

34. Effective picture
/107
Δpfr = − ΔpLO − h(γw)ΔpNLO
γ-independent friction term γ-dependent friction term
σγ3
w
··rw +
2σγw
rw
= Δpdr + Δpfr
E = 4πσr2
wγw −
4
3
πr3
w(Δpdr + Δpfr)
(
σ+
rw
3
d|Δpfr |
dγw )
γ3
w
··rw +
2σγw
rw
= Δpdr + Δpfr

35. Effective picture
/107
Δpfr = − ΔpLO − h(γw)ΔpNLO
γ-independent friction term γ-dependent friction term
σγ3
w
··rw +
2σγw
rw
= Δpdr + Δpfr
E = 4πσr2
wγw −
4
3
πr3
w(Δpdr + Δpfr)
(
σ+
rw
3
d|Δpfr |
dγw )
γ3
w
··rw +
2σγw
rw
= Δpdr + Δpfr
h(γ) − h(1)
h(γeq) − h(1)
+
3γ
2R
= 1 +
1
2R3
, h(γeq) ≡
Δpdr − ΔpLO
ΔpNLO
General solution

36. Wall velocity
/108

37. Wall velocity
/108
1 10 100 1000 104
105
1
10
100
1000
104
105
R / R 0
(R)
eq = 101
eq = 102
eq = 103
eq = 104
eq =
γ(R) =
2γeqR3
+ γeq − 1
2R3 + 3(γeq − 1)R2
P1→2 ≡ h(γ)ΔpNLO, h(γ) = γ

38. Wall velocity
/108
1 10 100 1000 104
105
1
10
100
1000
104
105
R / R 0
(R)
eq = 101
eq = 102
eq = 103
eq = 104
eq =
γ(R) =
2γeqR3
+ γeq − 1
2R3 + 3(γeq − 1)R2
P1→2 ≡ h(γ)ΔpNLO, h(γ) = γ
1 10 100 1000 104
105
1
10
100
1000
104
105
R / R 0
(R)
eq = 101
eq = 102
eq = 103
eq = 104
eq =
γ(R) = γ2
eq +
9(γ2
eq − 1)2
16R2
+
γ2
eq − 1
2R3
−
3(γ2
eq − 1)
4R
P1→N ≡ h(γ)ΔpNLO, h(γ) = γ2

39. Wall velocity
/108
1 10 100 1000 104
105
1
10
100
1000
104
105
R / R 0
(R)
eq = 101
eq = 102
eq = 103
eq = 104
eq =
γ(R) =
2γeqR3
+ γeq − 1
2R3 + 3(γeq − 1)R2
P1→2 ≡ h(γ)ΔpNLO, h(γ) = γ
1 10 100 1000 104
105
1
10
100
1000
104
105
R / R 0
(R)
eq = 101
eq = 102
eq = 103
eq = 104
eq =
γ(R) = γ2
eq +
9(γ2
eq − 1)2
16R2
+
γ2
eq − 1
2R3
−
3(γ2
eq − 1)
4R
P1→N ≡ h(γ)ΔpNLO, h(γ) = γ2
lim
γeq→∞
γ(R) =
2
3
R +
1
3R2
lim
R→∞
γ(R) = γeq

40. Efficiency factor
/109

41. Efficiency factor
/109
1 10 100 1000 104
105
0.0
0.2
0.4
0.6
0.8
1.0
Rcol / R0
col(Rcol;/,eq)
eq = 101
eq = 102
eq = 103
eq = 104
/ =0.1
1 10 100 1000 104
105
0.0
0.2
0.4
0.6
0.8
1.0
Rcol / R0
col(Rcol;/,eq)
eq = 101
eq = 102
eq = 103
eq = 104
/ =0.1

42. Efficiency factor
/109
1 10 100 1000 104
105
0.0
0.2
0.4
0.6
0.8
1.0
Rcol / R0
col(Rcol;/,eq)
eq = 101
eq = 102
eq = 103
eq = 104
/ =0.1
1 10 100 1000 104
105
0.0
0.2
0.4
0.6
0.8
1.0
Rcol / R0
col(Rcol;/,eq)
eq = 101
eq = 102
eq = 103
eq = 104
/ =0.1
Rcol ≳ 𝒪(103
)Rσ
1. Previous estimations are the late-time limit of our estimation when
bubbles collide with radius much larger than the transition radius

43. Efficiency factor
/109
1 10 100 1000 104
105
0.0
0.2
0.4
0.6
0.8
1.0
Rcol / R0
col(Rcol;/,eq)
eq = 101
eq = 102
eq = 103
eq = 104
/ =0.1
1 10 100 1000 104
105
0.0
0.2
0.4
0.6
0.8
1.0
Rcol / R0
col(Rcol;/,eq)
eq = 101
eq = 102
eq = 103
eq = 104
/ =0.1
2. The difference between previous estimations and our estimation
is most pronounced in the intermediate regime Rσ ≲ Rcol ≲ 𝒪(102
)Rσ
Rcol ≳ 𝒪(103
)Rσ
1. Previous estimations are the late-time limit of our estimation when
bubbles collide with radius much larger than the transition radius

44. Efficiency factor
/109
1 10 100 1000 104
105
0.0
0.2
0.4
0.6
0.8
1.0
Rcol / R0
col(Rcol;/,eq)
eq = 101
eq = 102
eq = 103
eq = 104
/ =0.1
1 10 100 1000 104
105
0.0
0.2
0.4
0.6
0.8
1.0
Rcol / R0
col(Rcol;/,eq)
eq = 101
eq = 102
eq = 103
eq = 104
/ =0.1
2. The difference between previous estimations and our estimation
is most pronounced in the intermediate regime Rσ ≲ Rcol ≲ 𝒪(102
)Rσ
Rcol ≳ 𝒪(103
)Rσ
1. Previous estimations are the late-time limit of our estimation when
bubbles collide with radius much larger than the transition radius
3. Our estimation could evaluate in the regime when bubbles collide
before they start to approach the terminal velocity Rcol ≲ Rσ

45. Conclusion and discussions
/1010

46. Conclusion and discussions
/1010
(
σ +
R
3
dP1→N
dγ )
dγ
dR
+
2σγ
R
= Δpdr − P1→1 − P1→N
Derive an effective EOM for an expanding bubble wall in thermal plasma out-of equilibrium

47. Conclusion and discussions
/1010
(
σ +
R
3
dP1→N
dγ )
dγ
dR
+
2σγ
R
= Δpdr − P1→1 − P1→N
Derive an effective EOM for an expanding bubble wall in thermal plasma out-of equilibrium
Solve for bubble expansion with arbitrary -scaling friction P1→N ≡ h(γ)ΔpNLOγ
h(γ) − h(1)
h(γeq) − h(1)
+
3γ
2R
= 1 +
1
2R3

48. Conclusion and discussions
/1010
(
σ +
R
3
dP1→N
dγ )
dγ
dR
+
2σγ
R
= Δpdr − P1→1 − P1→N
Derive an effective EOM for an expanding bubble wall in thermal plasma out-of equilibrium
Solve for bubble expansion with arbitrary -scaling friction P1→N ≡ h(γ)ΔpNLOγ
h(γ) − h(1)
h(γeq) − h(1)
+
3γ
2R
= 1 +
1
2R3
□ ϕ =
∂Veff
∂ϕ
+
∑
i=B,F
gi
dm2
i
dϕ ∫
d3 ⃗k
(2π)3
δfi
2Ei( ⃗k )
=
∂Veff
∂ϕ
− ˜ηT(uμ
∂μϕ)2
? What kind of parametrization for the out-of equilibrium term could reproduce our
effective EOM of an expanding bubble wall with friction P1→N ≡ h(γ)ΔpNLO, h(γ) = γ2