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Effective picture of bubble expansion

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2020 gravitation and cosmology workshop, Chongqing

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  1. 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. 2. GWs from FOPT /102
  3. 3. GWs from FOPT /102
  4. 4. GWs from FOPT /102 TeV PeVMeV BSMTPT QCDPT EWPT PQPT
  5. 5. Phase dynamics /103
  6. 6. Phase dynamics /103 ϕ Veff(ϕ) ϕ− ϕ+
  7. 7. Phase dynamics /103 ϕ Veff(ϕ) ϕ− ϕ+ α ∼ ΔVeff ρRad Γ(t) ∼ e−S(t)
  8. 8. Phase dynamics /103 ϕ Veff(ϕ) ϕ− ϕ+ α ∼ ΔVeff ρRad Γ(t) ∼ e−S(t)
  9. 9. Phase dynamics /103 ϕ Veff(ϕ) ϕ− ϕ+ α ∼ ΔVeff ρRad Γ(t) ∼ e−S(t)
  10. 10. Phase dynamics /103 ϕ Veff(ϕ) ϕ− ϕ+ α ∼ ΔVeff ρRad Γ(t) ∼ e−S(t)
  11. 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. 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. 13. Bubble dynamics /104
  14. 14. Bubble dynamics /104 Bubble expansion in vacuum vw → 1
  15. 15. Bubble dynamics /104 Ωscalar GW ∝ κ2 collision ∼ ( Ewall kinetic ρvac ) 2 Bubble expansion in vacuum vw → 1
  16. 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. 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. 18. Numerical simulation /105
  19. 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. 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. 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. 22. 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 Parameterization
  23. 23. 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
  24. 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. 25. Thermal friction /106
  26. 26. Thermal friction /106 P1→1 ≈ Δm2 T2 24 ≡ ΔpLO Bodeker & Moore 09 Particle transmission and reflection Run-away
  27. 27. 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
  28. 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. 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. 30. 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 δf = − ˜ηT(uμ ∂μϕ)2 Δpfr ∝ − (γ2 w − 1)
  31. 31. Effective picture /107
  32. 32. Effective picture /107 Δpfr = − ΔpLO − h(γw)ΔpNLO γ-independent friction term γ-dependent friction term
  33. 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. 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. 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. 36. Wall velocity /108
  37. 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. 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. 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. 40. Efficiency factor /109
  41. 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. 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. 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. 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. 45. Conclusion and discussions /1010
  46. 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. 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. 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

Description

2020 gravitation and cosmology workshop, Chongqing

Transcript

  1. 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. 2. GWs from FOPT /102
  3. 3. GWs from FOPT /102
  4. 4. GWs from FOPT /102 TeV PeVMeV BSMTPT QCDPT EWPT PQPT
  5. 5. Phase dynamics /103
  6. 6. Phase dynamics /103 ϕ Veff(ϕ) ϕ− ϕ+
  7. 7. Phase dynamics /103 ϕ Veff(ϕ) ϕ− ϕ+ α ∼ ΔVeff ρRad Γ(t) ∼ e−S(t)
  8. 8. Phase dynamics /103 ϕ Veff(ϕ) ϕ− ϕ+ α ∼ ΔVeff ρRad Γ(t) ∼ e−S(t)
  9. 9. Phase dynamics /103 ϕ Veff(ϕ) ϕ− ϕ+ α ∼ ΔVeff ρRad Γ(t) ∼ e−S(t)
  10. 10. Phase dynamics /103 ϕ Veff(ϕ) ϕ− ϕ+ α ∼ ΔVeff ρRad Γ(t) ∼ e−S(t)
  11. 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. 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. 13. Bubble dynamics /104
  14. 14. Bubble dynamics /104 Bubble expansion in vacuum vw → 1
  15. 15. Bubble dynamics /104 Ωscalar GW ∝ κ2 collision ∼ ( Ewall kinetic ρvac ) 2 Bubble expansion in vacuum vw → 1
  16. 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. 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. 18. Numerical simulation /105
  19. 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. 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. 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. 22. 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 Parameterization
  23. 23. 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
  24. 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. 25. Thermal friction /106
  26. 26. Thermal friction /106 P1→1 ≈ Δm2 T2 24 ≡ ΔpLO Bodeker & Moore 09 Particle transmission and reflection Run-away
  27. 27. 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
  28. 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. 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. 30. 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 δf = − ˜ηT(uμ ∂μϕ)2 Δpfr ∝ − (γ2 w − 1)
  31. 31. Effective picture /107
  32. 32. Effective picture /107 Δpfr = − ΔpLO − h(γw)ΔpNLO γ-independent friction term γ-dependent friction term
  33. 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. 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. 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. 36. Wall velocity /108
  37. 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. 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. 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. 40. Efficiency factor /109
  41. 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. 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. 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. 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. 45. Conclusion and discussions /1010
  46. 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. 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. 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

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