[PPT] - Bubble Wall Velocities Jonathan Kozaczuk ACFI, UMass Amherst ACFI PowerPoint Presentation

SLIDE 1

Bubble Wall Velocities

Jonathan Kozaczuk ACFI, UMass Amherst ACFI EWPT Workshop, 4/7/17

SLIDE 2

2

Why should we care about bubble wall dynamics?

SLIDE 3

Electroweak baryogenesis

Bubble wall catches up with diffusing current, freezing in B

B / B 6= 0 ΓB = 0 / / ΓB 6= 0

f , ¯ f hφi 6= 0

3

f hφi 6= 0

B /

, ¯ f

SLIDE 4

Electroweak baryogenesis

4

Requiring sufficient B-violation during diffusion requires relatively slow bubble walls Wall velocities conventionally required to be subsonic: vw<0.58 Resulting asymmetry can strongly depend on vw, depending on the form of the primary CP-violating source

Huber et al, 2001 (subtle and there are exceptions; see e.g. No, 2011; Caprini + No, 2011; Katz+Riotto, 2016 )

1 2 3 4 5 6 7 –4 –3 –2 –1 10 20 30 40 50 60 –4 –3 –2 –1

ln10(vw) → ln10(vw) → H1 + H2 H1 − H2 η ↑

SLIDE 5

Gravitational waves

5

Spectrum, and prospects for detection, depend on the wall velocity

h2Ωenv(f) = 1.67 × 10−5 ✓H∗ β ◆2 ✓ κα 1 + α ◆2 ✓100 g∗ ◆ 1

3 ✓ 0.11 v3

w

0.42 + v2

w

◆ Senv(f) v ' ( ↵ (0.73 + 0.083p↵ + ↵)

−1

vw ⇠ 1 v6/5

w 6.9 ↵ (1.36 0.037p↵ + ↵) −1 ,

vw . 0.1

Caprini et al, 2015

SLIDE 6

Guiding questions

6

How do we compute the wall velocity? What are the theoretical challenges involved in these calculations?

SLIDE 7

Master equation

7

Scalar field EOM for one scalar D.O.F. at finite T:

T=0 effective potential Distribution functions of all particles coupled to Higgs

See e.g. Moore+Prokopec, 1996; Konstandin et al, 2014

⇤φ + V 0(φ) + X dm2 dφ Z d3k (2π)32E f(k, z) = 0

SLIDE 8

Master equation

8

Scalar field EOM for one scalar D.O.F. at finite T: Rewrite in terms of finite-T effective potential:

T=0 effective potential Distribution functions of all particles coupled to Higgs

See e.g. Moore+Prokopec, 1996; Konstandin et al, 2014

⇤φ + V 0(φ) + X dm2 dφ Z d3k (2π)32E f(k, z) = 0

⇤φ + V 0(φ, T) + X dm2 dφ Z d3k (2π)32E δf(k, z) = 0

SLIDE 9

Master equation

9

Scalar field EOM for one scalar D.O.F. at finite T: Rewrite in terms of finite-T effective potential:

T=0 effective potential Distribution functions of all particles coupled to Higgs

See e.g. Moore+Prokopec, 1996; Konstandin et al, 2014

⇤φ + V 0(φ) + X dm2 dφ Z d3k (2π)32E f(k, z) = 0

⇤φ + V 0(φ, T) + X dm2 dφ Z d3k (2π)32E δf(k, z) = 0 Z  ⇤φ + V 0(φ, T) + X dm2 dφ Z d3k (2π)32E δf(k, z)

φ0(z) dz = ∆p

SLIDE 10

Master equation

10

Scalar field EOM for one scalar D.O.F. at finite T: Rewrite in terms of finite-T effective potential: Non-accelerating wall:

T=0 effective potential Distribution functions of all particles coupled to Higgs

See e.g. Moore+Prokopec, 1996; Konstandin et al, 2014

⇤φ + V 0(φ) + X dm2 dφ Z d3k (2π)32E f(k, z) = 0

⇤φ + V 0(φ, T) + X dm2 dφ Z d3k (2π)32E δf(k, z) = 0 Z  ⇤φ + V 0(φ, T) + X dm2 dφ Z d3k (2π)32E δf(k, z)

φ0(z) dz = ∆p

∆V (T) = − Z X dm2 dφ Z d3k (2π)32E δf(k, z)φ0(z) dz

SLIDE 11

Master equation

11

Boils down to drawing a free body diagram Two questions:

hφi = 0 hφi 6= 0

Friction

∆VT =0 vw

enough friction to stop the wall from accelerating? If so, what is the terminal velocity and bubble profile satisfying the master equation above?

∃

Z X dm2 dφ Z d3k (2π)3 2E δf(k, z)φ0(z)dz = −∆V (T)

SLIDE 12

"Runaway Bubbles"

Is there enough friction to stop the wall from accelerating once it’s moving ultra-relativistically?

Bodeker + Moore, 2009

∆Vvac = − Z X dm2 dφ Z d3k (2π)3 2E f(k, z)φ0(z)dz

12

SLIDE 13

"Runaway Bubbles"

13

Is there enough friction to stop the wall from accelerating once it’s moving ultra-relativistically? For g >> 1:

Bodeker + Moore, 2009

High-T expansion (m/T<<1)

Z X dm2 dφ Z d3k (2π)3 2E f(k, z)φ0(z)dz ' X ⇥ m2

i (h2) m2 i (h1)

⇤ Z dm2 dφ Z d3k (2π)3 2E f(k)|h1 ∆Vvac = − Z X dm2 dφ Z d3k (2π)3 2E f(k, z)φ0(z)dz ≈ X aiT 2 ⇥ m2

i (h2) − m2 i (h1)

⇤ + O(1/γ2)

Equilibrium distributions

SLIDE 14

"Runaway Bubbles"

14

Is there enough friction to stop the wall from accelerating once it’s moving ultra-relativistically? For g >> 1:

Bodeker + Moore, 2009

High-T expansion (m/T<<1)

Z X dm2 dφ Z d3k (2π)3 2E f(k, z)φ0(z)dz ' X ⇥ m2

i (h2) m2 i (h1)

⇤ Z dm2 dφ Z d3k (2π)3 2E f(k)|h1 ∆Vvac < − Z X dm2 dφ Z d3k (2π)3 2E f(k, z)φ0(z)dz vacuum energy difference

verwhelms the friction

≈ X aiT 2 ⇥ m2

i (h2) − m2 i (h1)

⇤ + O(1/γ2)

Equilibrium distributions

∆Vvac = − Z X dm2 dφ Z d3k (2π)3 2E f(k, z)φ0(z)dz

SLIDE 15

In the high-T approximation, the difference between vacua of the finite-T effective potential is given by Runaway condition can be interpreted in terms of finite-T effective potential with no cubic term: If, after dropping thermal cubic terms, it is energetically favorable to tunnel to the broken phase, the bubble can run away

"Runaway Bubbles"

15

Bodeker + Moore, 2009

∆Vvac + X aiT 2[m2

i (h2) − m2 i (h1)] < 0

Runaway ∆V (T) ≈ ∆Vvac + X aiT 2 ⇥ m2

i (h2) − m2 i (h1)

⇤ − X biT ⇥ m3

i (h2) − m3 i (h1)

⇤

SLIDE 16

If, after dropping thermal cubic terms, it is energetically favorable to tunnel to the broken phase, the bubble can run away

"Runaway Bubbles"

16

Bodeker + Moore, 2009

∆Vvac + X aiT 2[m2

i (h2) − m2 i (h1)] < 0

Runaway

From JK et al, 2014 for the NMSSM

SLIDE 17

Important (and simple) criterion to check when doing pheno studies Theories with tree-level cubic terms (e.g. singlet models) are especially susceptible to runaways Recent progress and outstanding theoretical questions associated with friction in the ultra-relativistic limit:

NLO (e.g. 1à2) processes enhance the friction, tend to prevent

runaway! Important for gravitational wave spectra

Effect is dominated by soft emission (in the wall frame). Full

calculation is challenging

Sphalerons behind the wall?!

Some comments

17

Bodeker+Moore, 2017

SLIDE 18

Beyond runaway

18

What about non-relativistic walls? How do we get a number out?

Some references: Moore + Prokopec, 1995 & 1996 John + Schmidt, 2000 Konstandin et al, 2014 JK, 2016

SLIDE 19

Beyond runaway

Ultimate goal: find a wall velocity and bubble profile that solves the scalar field EOMs Simplification: look for configurations satisfying constraint equations

19

(1 v2

w)φ00 i + ∂V (φi)

∂φi + X

j

∂m2

j(φi)

∂φi Z d3p (2π)32Ej fj(p, x) = 0

(in the wall frame, neglecting sphericity, assuming stationary solution)

Z 2 4(1 v2

w)~

00 + rφV (i) + X

j

rφm2

j(i)

Z d3p (2⇡)32Ej fj(p, x) 3 5 · d~

dx dx = 0

Z 2 4(1 v2

w)~

00 + rφV (i) + X

j

rφm2

j(i)

Z d3p (2⇡)32Ej fj(p, x) 3 5 · d2~

dx2 dx = 0

(Vanishing pressure) (Vanishing pressure gradient)

SLIDE 20

Wall velocities with singlets

Illustrate in the real singlet extension of SM First, write down Boltzmann equations for distributions Utilize effective kinetic theory for excitations: , Infrared modes with (e.g. for gauge bosons) dealt with separately

(1 v2

w)φ00 i + ∂V (φi)

∂φi + X

j

∂m2

j(φi)

∂φi Z d3p (2π)32Ej fj(p, x) = 0

d dtfi ⌘ ✓ ∂ ∂t + ˙ z ∂ ∂z + ˙ pz ∂ ∂pz ◆ fi = C[f]i C[f]i = 1 2Ni X

jmn

1 2Ep Z d3kd3p0d3k0 (2π)92Ek2Ep02Ek0

Mij!mn(p, k; p0, k0)
2 (2π)4δ(p + k p0 k0)

⇥ Pij!mn [fi(p), fj(k), fm(p0), fn(k0)] (4.3)

E 1 Lw the plasma ange p & gT, e find for th

ta p ⌧ T erly describ

20

SLIDE 21

Wall velocities with singlets

Classify particles depending on strength of interaction with the condensate: Top quarks, SU(2)L gauge bosons, Higgs and singlet fields “feel” the passage of the wall the most; dominate the contribution in EOM All other particles treated as in local thermal equilibrium at common (space-time dependent) temperature and fluid velocity (determined from bulk properties of the PT: see Jose Miguel’s talk!) Ansatz for relevant distributions: ,

fa = ⇣ e(E+δa)/T ± 1 ⌘1

δj = µj E T (δTj + δTbg) pz(δvj + vbg)

Tn vfluid r/t v+ T+ T− vw

is study, we will work to linear order in th all (µj/T, δTj/T, δTbg/T, δvj, vbg ⌧ 1).

ng phase transitions, and we verify the va

21

SLIDE 22

Wall velocities with singlets

Take moments of the Boltzmann equations Obtain collision terms from interactions of the various species in the plasma

ci

2

∂ ∂tµi + ci

3

∂ ∂t(δTi + δTbg) + ci

3T

3 ∂ ∂z (δvi + vbg) + Z d3p (2π)3T 2 C[f]i = ci

1

2T ∂m2

i

∂t ci

3

∂ ∂tµi + ci

4

∂ ∂t(δTi + δTbg) + ci

4T

3 ∂ ∂z (δvi + vbg) + Z Ed3p (2π)3T 3 C[f]i = ci

2

2T ∂m2

i

∂t ci

3

3 ∂ ∂z µi + ci

4

3 ∂ ∂z (δTi + δTbg) + ci

4T

3 ∂ ∂t(δvi + vbg) + Z pzd3p (2π)3T 3 C[f]i = 0 X c4 ✓ ∂ ∂tδTbg + c4T 3 ∂ ∂z vbg ◆ + Z Ed3p (2π)3T 3 C[f]bg = 0 X c4 3 ✓ ∂ ∂z δTbg + T ∂ ∂tvbg ◆ + Z pzd3p (2π)3T 3 C[f]bg = 0 Z d3p (2π)3T 2 C[f]i ≡ X

j

δµjΓi

µ1,j + δTiΓi T1,j

Z

d3p (2π)3T 3 EiC[f]i ≡ X

j

δµjΓi

µ2,j + δTiΓi T2,j

Z

d3p (2π)3T 4 pz,iC[f]i ≡ X

j

δvjΓi

v1,j

Z

d3p (2π)3T 3 EiC[f]bg ≡ − X

j

⇣ δµje Γµ2,j + δTie ΓT2,j ⌘ Z d3p (2π)3T 4 pz,iC[f]bg ≡ − X

j

⇣ δvje Γv1,j ⌘ 22

SLIDE 23

Interaction rates

For the top quarks and Higgs bosons, work in a formal leading log, small coupling/high-temperature expansion, including hard thermal loops in internal propagators

t/u-channel diagrams lead to IR divergences cut off by thermal masses; yield parametrically “large” logarithms . Only keep these contributions Systematically drop all terms of . Then calculate in the gauge basis and treat external modes as massless Resum the hard thermal loops for the t/u-channel propagators Keep only processes of ,

∼ log(#/g2)

f O(m2/T 2)
f O(α2

s)

wed processes. How

and O(αsαt), O(α2

t )

) (F) (G) (E) (J ) (I) (D) (H) (C) (B) (A

Leading-log in QCD-like gauge theory

Arnold, Moore, Yaffe, 2000-2003

23

SLIDE 24

Interaction rates

Leading log vacuum matrix elements

Process |M|2

tot

Internal Propagator O(g4

3):

t¯ t $ gg:

128 3 g4 3

u

t + t u

t

tg $ tg:

128

3 g4 3 s u + 96g4 3 s2+u2 t2

g, t tq(¯ q) $ tq(¯ q): 160g4

3 u2+s2 t2

g O(y2

t g2 3):

t¯ t $ hg, φ0g: 8y2

t g2 3

u

t + t u

t

t¯ b $ φ+g : 8y2

t g2 3

u

t + t u

t, b

tg $ th, tφ0: 8y2

t g2 3 s t

t tg $ bφ+: 8y2

t g2 3 s t

b tφ− $ bg: 8y2

t g2 3 s t

t O(y4

t ):

t¯ t $ hh, φ0φ0:

3 2y4 t

u

t + t u

t

t¯ t $ φ+φ−: 3y4

t u t

b t¯ t $ hφ0:

3 2y4 t

u

t + t u

t

t¯ b $ hφ+, φ0φ+:

3 2y4 t u t

t th, tφ0 $ ht, φ0t: 3

2y4 t s t

t tφ− $ hb, φ0b:

3 2y4 t u t

t tφ+ $ φ+t: 3y4

t u t

b

Hard thermal loop dressing:

u t ' s t ! 4 Re(p · e q k · e q∗ + se q · e q∗) |e q · e q|2 with e qµ = pµ p0µ Σµ(p p0) and 124, 125] Σ0(q) = m2

f(T)

2 |q| ✓ log |q| + q0 |q| q0 iπ ◆ Σ(q) = m2

f(T) b

q |q| ✓ 1 + iπ q0 2 |q| log |q| + q0 |q| q0 ◆ s2 + u2 t2 ! 1 2 ⇣ 1 +

Dµν(p p0)(p + p0)µ(k + k0)ν

2⌘ D00(q) = 1 |q|2 + Π00(q, T) Dij(q) = δij b qib qj q2 + ΠT (q, T) Di0(q) = Di0(q) = 0. Π00(q) = m2

D(T)

✓ 1 q0 2 |q| log |q| + q0 |q| q0 + iπq0 2 |q| ◆ ΠT (q) = m2

D(T)

 q0 2 |q| + q0q2 4 |q|3 ✓ log |q| + q0 |q| q0 iπ ◆ 24

SLIDE 25

Interaction rates

Results for the tops and Higgses …

Γh

µ1,h ' (1.1 ⇥ 103g2 3y2 t + 6.0 ⇥ 104y4 t )T

Γh

T1,h ' Γh µ2,h ' (2.5 ⇥ 103g2 3y2 t + 1.4 ⇥ 103y4 t )T

Γh

T2,h ' (8.6 ⇥ 103g2 3y2 t + 4.8 ⇥ 103y4 t )T

Γh

v,h ' (3.5 ⇥ 103g2 3y2 t + 1.8 ⇥ 103y4 t )T,

Γt

µ1,t ' (5.0 ⇥ 10−4g4 3 + 5.8 ⇥ 10−4g2 3y2 t + 1.5 ⇥ 10−4y4 t )T

Γt

T1,t ' Γt µ2,t ' (1.2 ⇥ 10−3g4 3 + 1.4 ⇥ 10−3g2 3y2 t + 3.6 ⇥ 10−4y4 t )T

Γt

T2,t ' (1.1 ⇥ 10−2g4 3 + 4.6 ⇥ 10−3g2 3y2 t + 1.1 ⇥ 10−3y4 t )T

Γt

v,t ' (2.0 ⇥ 10−2g4 3 + 1.7 ⇥ 10−3g2 3y2 t + 4.3 ⇥ 10−4y4 t )T,

These then enter the system of Boltzmann equations. Solved by Green’s function techniques for general profile

Alk d dxδk + Γlkδk = Fl

δT ≡ (δµt, δTt, δvt, δµh, δTh, δvh) , F(x)T ≡ vw 2T ✓ ct

1

dm2

t (φh)

dx , ct

2

dm2

t (φh)

dx , 0, ch

1(x)dm2 h(φh, φs)

dx , ch

2

dm2

h(φh, φs)

dx , 0 ◆

A−1Γ
ij χjk = χikλk

Gi(x, y) = sgn(λi)e−λi(x−y)Θ [sgn(λi)(x − y)]

δi(x) = χij Z ∞

−∞

⇥ χ−1A−1F(y) ⇤

j Gj(x, y)dy. 25

SLIDE 26

HTL numerical result vs. Analytic result with thermal mass insertion Factor of ~4 difference, formally at the same order in the LL expansion Above result requires evaluating the angular integral Difference between dropping the constant piece or not accounts for most of the discrepancy

Beware the uncertainties! Different leading log prescriptions lead to factors of ~ 1– 10 difference in these rates à effects in vw Consider e.g. contribution to : Why? Because the logarithms are not numerically very large!

Interaction rates

tand the diff cess t¯ t ! gg.

ere we h to Γt

µ1,t,

endent sel

∆Γt

µ1,t ⇡ 1.1 ⇥ 10−3T,

∆Γt

µ1,t ' 16α2 s

9π3 ⇥ 9ζ2

2

16 log 9T 2 m2

q

T ⇡ 3.8 ⇥ 10−3T, Z d cos θ1 2 log ✓2 |p| |k| (1 cos θ) m2

t

◆ = 1 + log 4 |p| |k| m2

t

.

O(100%)

26

SLIDE 27

In the high-T approximation, the singlet scalar interaction rates are suppressed at leading log order relative to e.g. tops and Higgs Approximate collision term as small and drop from the Boltzmann

equations. They can then be solved exactly:

Full leading order result can go beyond this approximation. Likely

verestimates friction.

Singlet friction

Z d3p (2π)32E δfs(p, x) = vw Z d3p (2π)32E eEp/T

eEp/T ± 1

2 Q(pz) T Q Q(pz) = ( p p2

z + ms(φh, φs, T)2 − pz,

pz > − p m0

s(T)2 − ms(φh, φs, T)2

− p p2

z + ms(φh, φs, T)2 − m0 s(T)2 − pz,

pz < − p m0

s(T)2 − ms(φh, φs, T)2

(5.2)

27

SLIDE 28

Two prescriptions: Gauge invariant (approximate) treatment – drop thermal cubic term and gauge boson friction term Include both the cubic term and the gauge boson friction contribution Corresponding friction dominated by highly infrared modes; treat semi- classically

Gauge bosons

Moore, 2000

πm2

D,W (T)

8p d fW (p, T) dt = −[p2 + m2

W (φh)]fW (p, T) + N

dm2

W (φh)

dφh Z d3p (2π)32E δfW (p, x) = vw 3T 32πm2

D,W (T) φ0 h(x)

φh(x)2 Θ(x − x⇤) he quantity solves , with the SM-like Higgs wall w

ity x⇤ solves mW [φh(x⇤)] = 1/Lh, KB description used to derive E

,

Exact solution

28

SLIDE 29

Putting these pieces into the EOM (and dropping the gauge piece for now) we get: Need to find profile, , and such that the EOM or, more weakly,

ur constraints are satisfied:

How??

Putting it all together…

(1 v2

w)φ00 i + ∂V (φi, T)

∂φi + X

j

∂m2

j(φi)

∂φi T 2 h cj

1δµj + cj 2(δTj + δTbg)

i + ∂m2

s(φi)

∂φi Z d3p (2π)32E δfs(x, p) = 0 re φh,s(x ! ⌥1) = φh,s;±(T+) pically admit a solution for cert

f the above expressio

and φ0

h,s(x ! ±1) = 0.

B.C.’s: ,

~ (x) vw

Z dx (E.O.M.) · d⇥

dx = 0

Z dx (E.O.M.) · d2⇥

dx2 = 0

29

SLIDE 30

This is a set of integro-differential equations for : perturbations at a given point determined by integral involving profile Strategy: consider simpler case of constant friction term Much simpler: looks almost like the Euclidean EOMs for the bounce! Can solve this via path deformations

Solving for the profile

~ (x) ∝ d~ /dx

(1 v2

w)d2Φ

dx2 + rφV (Φ, T) + F dΦ dx = 0 Starting from initial guess, deform the path to minimize "normal force”, Corresponds to friction only parallel to the path in field space Result well fit by kink:

N(x) = JK et al, 2014

N(x) =

Wainwright, 2011 φi(x) = φ0

i

2 ✓ 1 + tanh x δi Li ◆ 30

SLIDE 31

How do things change when including the full friction term? Friction is no longer purely parallel to the trajectory satisfying the constant friction EOM Fortunate simplification: perpendicular friction typically perturbatively small! If we neglect this subdominant piece, only effect of going to full friction term is rescaling of profile:

Solving for the profile

rφV (Φ, T = 0) · dΦ(s) ds ⇠ X dm(Φ)2 dΦ(s) Z d3p (2π)32E f0(p, T) · dΦ(s) ds ⇢ (1 v2

w)d2Φ

dx2 + X dm(Φ)2 dΦ Z d3p (2π)32E δf(p, T) + O(δf2)

· dΦ(s)

ds ⇠ 0

Parallel: Perpendicular:

⇢ (1 v2

w)d2Φ

dx2 + rφV (T = 0) + X dm(Φ)2 dΦ Z d3p (2π)32E f0(p, T) + O(δf)

· d2Φ(s)

ds2 ⇠ 0 (5.20) Friction contribution dominates

Lh,s ! aLh,s, δs ! aδs

(Can check validity a posteriori) Wainwright, 2011

31

SLIDE 32

Suggests the following prescription:

1. Compute nucleation temperature and initial profile
2. Solve for constant friction profile. Fit to tanh
3. Solve hydrodynamic equations to determine T near the wall

(see Jose Miguel’s talk)

4. Vary values of vw , a. For each pair, solve for perturbations
5. Inset perturbations and profile into EOM and impose

constraints Values of vw , a approximately solving full set of EOMs and Boltzmann equations correspond to those satisfying the constraints.

A prescription

32

SLIDE 33

Walls move quickly For strong enough phase transitions, no subsonic solutions Likely optimistic friction estimate (rates can be larger, singlet friction neglected)

Results

vw φh(Tn)/Tn cs Set 1 Set 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.7 0.8 0.9 1 1.1

33

SLIDE 34

Walls move quickly and they’re thin

Results

δs/T φh(Tn)/Tn Set 1 Set 2

4
2

2 4 0.6 0.7 0.8 0.9 1 1.1 Lh/T φh(Tn)/Tn Set 1 Set 2 5 10 15 20 0.6 0.7 0.8 0.9 1 1.1 Ls/T φh(Tn)/Tn Set 1 Set 2 5 10 15 20 0.6 0.7 0.8 0.9 1 1.1

φi(x) = φ0

i

2 ✓ 1 + tanh x δi Li ◆ 34

SLIDE 35

Don’t always have to compute wall velocity from scratch. Can match

nto existing results for models with similar features

This is fortunate, but results only as good as the underlying microphysical calculation you are matching onto

Extending to other models

00

i − @V (, T)

@i = ⌘ivw 2

i

T 0

i

Phenomenological friction terms; matched to microphysical

calculation. Can have more complicated parametric

dependence

See e.g. Huber + Sopena, 2013; Megevand, 2013l; Konstandin et al, 2014…

35

SLIDE 36

Lots of theoretical uncertainties associated with approximations in microphysical calculation of wall velocities

Simplified hydro
Fluid approximation
Interaction rates (kinetic theory, high-T/leading log expansion)
Free passage for singlet

These lead to large (~100%) uncertainties on vw Many opportunities for improving these calculations

Closing thoughts

20 40 60 80 0.0 0.2 0.4 0.6 0.8 1.0 mh @GeVD vw

Konstandin et al, 2014 Moore+Prokopec, 1996 No HTL, analytic HTL, numerical

36

SLIDE 37

Zeroth order determination of whether to expect fast/”runaway” bubbles can be obtained using Bodeker-Moore criterion. Probably sufficient for gravitational wave spectrum, but no analog for checking whether vw<cs. Phenomenological approaches allow one to bootstrap using existing results (if model doesn’t look too different). Still, this approach is at most as accurate as the underlying microphysical calculation. Good idea to check robustness of baryon asymmetry calculation WRT to vw (and don’t use the MSSM predicted value!). Ultimately, sharp predictions for the baryon asymmetry will likely require improving these microphysical calculations.

Closing thoughts

37