Criterion for ultra-fast bubble walls: the impact of hydrodynamic obstruction

The Bödeker-Moore thermal friction [1] is usually used to determine whether or not a bubble wall can run away. However, the friction on the wall is not necessarily a monotonous function of the wall velocity and could have a maximum before it reaches the Bödeker-Moore limit. In this paper, we compare the maximal hydrodynamic obstruction, a frictional force that exists in local thermal equilibrium, and the Bödeker-Moore thermal friction. We study the former in a fully analytical way, clarifying its physical origin and providing a simple expression for its corresponding critical phase transition strength above which the driving force cannot be balanced out by the maximal hydrodynamic obstruction. We find that for large parameter space, the maximal hydrodynamic obstruction is larger than the Bödeker-Moore thermal friction, indicating that the conventional criterion for the runaway behavior of the bubble wall may have to be modified. We also explain how to apply efficiently the modified criterion to particle physics models and discuss possible limitations of the analysis carried out in this paper.

In the case of the ultrarelativistic regime, one may use the ballistic approximation [58,61].Mathematically, the ballistic approximation amounts to where L w is the width of the wall in the wall frame, L MFP is the mean free path of the incoming particles in the plasma frame.In this case, the particles do not (have time to) collide with each other when they pass through the wall.The pressure on the bubble wall is simply due to the transmission of the particles from the outside to the inside, and their reflection from the wall.It is generally a monotonically increasing function of the velocity.This pressure, at large γ w ≫ 1, asymptotes to [1] where c i = 1(1/2) for bosons (fermions), g i is the number of internal degrees of freedom, T n is the nucleation temperature, and ∆m i is the mass gain of the particle as it transits from the exterior to the interior of the bubble.The criterion P driving > P BM (Bödeker and Moore (BM) criterion), where P driving is the driving pressure, is frequently used to determine whether a wall runs away or not.Note that P BM in Eq. ( 2) is only the friction in the ultrarelativistic regime at the leading order in γ w .Other particle-number-changing processes typically induce additional friction that depends on γ w [82,83,85,86].There can also be additional friction due to heavy new physics [84] that is also independent of γ w .In this paper, we will not consider those types of contributions.Therefore, one needs to assume that there is no additional γ w -independent friction in the ultrarelativistic regime other than the BM thermal friction.And we interpret "runaway" in this paper as that the wall can enter into a regime with extremely large γ w for which other γ w -dependent (if present) pressures become comparable with P BM .However, the total friction on the wall does not need to be a monotonous function of the wall velocity and could have a maximum before it reaches the BM limit.Indeed, it is observed in Refs.[62,63,79] that there is a pressure barrier peaked at the Jouguet velocity ξ J .This peak is largely due to the hydrodynamic obstruction [74], a frictional force that exists in LTE, as shown in Ref. [63]. 2 The hydrodynamic obstruction will also be called the LTE force occasionally.
In this paper, we compare the maximal hydrodynamic obstruction with the BM thermal friction Eq (2).We show that the former can be larger than the latter in large regions of the parameter space and hence the runaway criterion must be reconsidered.This means that in many models while the BM thermal friction cannot stop the wall's acceleration, the hydrodynamic obstruction can however do this job.We work out the critical phase transition strengths, α BM/LTE n,crit , above which the driving force cannot be balanced out by the maximal hydrodynamic obstruction/the BM thermal friction, and compare them with each other.Defining the ratio of the number of degrees of freedom (DoFs) in the broken and symmetric phases, b ≡ a − /a + , we find that hydrodynamic obstruction becomes more relevant than the BM thermal friction in prohibiting runaway for b ≲ 0.85.Although the value of b in the standard model (SM) at the electroweak phase transition (EWPT) is close to one, b EWPT ∼ 0.9, many beyond-standard-model (BSM) phase transitions feature a much smaller b, like in minimal toy models of phase transition with two scalar fields [104], minimal models of B − L breaking [44,105], Peccei-Quinn symmetry breaking [106], or conformal symmetry breaking [105,107].Deconfinement-confinement phase transitions like in the composite Higgs also feature a large change in the number of DoFs [81,[108][109][110].
The maximal hydrodynamic obstruction has been partly studied in Ref. [79].There, however, no attention has been paid to the comparison with the BM thermal friction.Furthermore, we give a simpler derivation of the critical phase transition strength α LTE n,crit without solving the fluid equations.Therefore, our derivation can be very easily implemented in, e.g., Mathematica.We also provide a fit for the obtained α LTE n,crit (b) as a function of b.The paper is organized as follows.In the next section, we derive the general formulae for the driving and frictional pressures for general wall motions (not necessarily stationary) from the EoM of the scalar field.In Sec. 3, we derive a different formula for the frictional pressure based on the matching conditions.In Sec. 4, we study the maximal LTE friction in the detonation regime which however is not the maximal hydrodynamic obstruction.The latter occurs in the hybrid regime and is studied in Sec. 5.In Sec. 6, we compare the maximal hydrodynamic obstruction with the BM thermal friction.In Sec. 7, we comment on the validity of the approximations used in this work and give some caveats.We conclude in Sec. 8.In the Appendix, we also briefly discuss how to include the out-of-equilibrium effects in the analysis.

Bubble wall dynamics and the friction
Analysis of the friction on bubble walls is usually based on the following EoM for the background field [60,61], where □ = ∂ µ ∂ µ , f i (p, x) are the particle distribution functions, and E i = p 2 i + m 2 i the particle energies.Here x and p denote general four-dimensional position and momentum coordinates.Below, without loss of generality, we shall assume a planar wall expanding in the z direction.
For a bubble wall with a constant velocity, which we will refer to as a stationary wall, one can always work in the wall rest frame where all quantities are time independent.However, for a general motion, we cannot define a global rest frame of the wall.But we can still define an "instantaneous" wall rest frame which can help us find the expressions for the driving and frictional pressures.With fixed x (to be distinguished from the general four-dimensional coordinate) and y, the worldline of the wall can be parameterized by a proper time τ .Near an arbitrarily chosen proper time of the wall, we define an instantaneous comoving rest frame as follows.First, we assume a certain Ansatz for the wall in the plasma frame from which the wall velocity and acceleration can be read [111].For example, one can take (v b is the symmetry broken value of ϕ) where {t ′ , z ′ } are the coordinates in the plasma frame and L ′ w is the wall width observed in that frame.The function z ′ 0 (t ′ ) describes the trajectory of the wall.We can choose t ′ = 0 at the reference point of the worldline (corresponding to the chosen τ ).Then ).The velocity v w is used to define the (instantaneous) comoving coordinates In the instantaneous rest frame of the wall, one can choose z = 0 as the position of the wall and at the origin of coordinates t = 0.It is easy to show that the first time derivative of ϕ in the instantaneous comoving frame, ∂ t ϕ(0) ≡ φ(0), is vanishing.The second time derivative of ϕ, φ(0), depends on the acceleration a and vanishes also if a = 0 which corresponds to a stationary wall.
In the instantaneous comoving frame, multiplying the EoM by ∂ z ϕ and integrating over z, one gets where δ is a length much larger than the wall width, and effectively can be taken to be infinity in the present context.Above, we also have Taylor expanded quantities in t at t = 0 and kept the leading contributions.The second term is vanishing because the integrand (∂ z ϕ)∂ 2 z ϕ is a total derivative and ∂ z ϕ vanishes for large |z| (for any τ ).The term with φ(0)| τ is related to the acceleration of the wall at τ as discussed above.Thus we obtain where and The sum in the last expression is to be taken on the species changing mass across the wall.
Although the above expressions implicitly depend on τ , we expect that their dependence on τ from the τ -dependence in ϕ disappears or becomes very weak as long as δ is much larger than the wall width.Thus P driving can be identified as the zero-temperature potential difference between the symmetric and broken phases, and is therefore not a function of the temperature nor τ .P friction can still be τ -dependent due to the time-dependence in the distribution functions which should be determined by solving the Boltzmann equation.A stationary motion is reached when P driving = P friction , which leads to φ(0)| τ = 0, as expected.Looking at Eq. ( 10), it would seem that P friction only depends on bath particles directly coupling with the scalar field ϕ with a wall-dependent mass (particles that we will call active particles in what follows).This is not exactly the case, because the distribution functions depend on the interactions among all the particles in the bath.Writing f i = f eq i + δf i , the frictional pressure can be further decomposed into two parts where P dissipative is given by Eq. ( 10) with f i replaced by the out-of-equilibrium part δf i .In this paper, we are interested in P LTE which reads Here V T (ϕ, T ) is the finite-temperature corrections to the potential in the non-interacting-gas approximation [112] and T , which in general depends on z, is the temperature of all the particles being in local thermal equilibrium.Remembering the z-dependence in V T (ϕ(z), T (z)) and using the chain rule for derivatives, P LTE can be further written as Note that there is no clear criterion to distinguish between the driving and friction forces.Different authors may attribute the same force to either the driving force or the friction force.
Our definitions above are different from those used in Ref. [77] where ∆V T was identified as a part of the driving force. 3However, the quantity P friction − P driving is invariant under different identifications.A merit of the definitions we are using is that the driving pressure is constant, in agreement with the convention used in Refs.[1,82].
In this paper, we shall assume LTE for the plasma and study (P friction − P driving )(ξ w ) as a function of the wall velocity.In particular, we are interested in its maximum, (P LTE − P driving ) max ≡ (P friction − P driving ) ξw=ξ peak (see the left panel of Fig. 2 below).If (P LTE − P driving ) max > 0, then there should be a stationary solution for the wall in LTE.On the other hand, if (P LTE − P driving ) max < 0, a stationary solution is not possible and the wall runs away unless P dissipative becomes large enough to stop the acceleration of the wall.Since P driving is constant, (P LTE − P driving ) max corresponds to P max LTE , the maximal LTE frictional pressure.
In realistic situations, the plasma is typically out of equilibrium.However, if we assume that out-of-equilibrium effects would add more friction to the wall, which is confirmed in a simple model in Ref. [63] and also in the Appendix for general case, P max LTE > P driving implies P max friction > P driving and the former can serve as a conservative condition for non-runaway bubble walls.
It is still not clear how to compute P LTE using the formula (14) directly.In the next section, we derive another form of P LTE that can be conveniently used to determine P max LTE .
3 Matching conditions for stationary and non-stationary walls The hydrodynamics of the bubble growth is based on the total energy-momentum conservation.Usually, a stationary wall is assumed and one works in the rest frame of the wall.Here again, we do not assume a stationary motion of the wall.The energy-momentum tensors for the scalar background and the fluid read where u µ is the fluid four-velocity, e f and p f are the fluid energy density and pressure that vanish at zero temperature.However, one usually combines the fluid energy density and pressure with the tree-level scalar potential energy, e = e f + V (ϕ), p = p f − V (ϕ).The advantage of using them is that in terms of e and p the matching conditions for hydrodynamic quantities take the standard form that appears commonly in the literature.Note that the fluid enthalpy ω = e f + p f = e + p.In terms of e and p, the energy-momentum tensor for the fluid then takes the form as The matching conditions are obtained from the ν = 0 and ν = 3 components of ∇ µ T µν = 0. Working in the instantaneous comoving frame and writing u µ = γ(z)(1, 0, 0, −v(z)), one obtains where ω ≡ (e + p) is the enthalpy.Above, we have assumed that all the first derivatives in time of the hydrodynamic quantities are vanishing. 4Integrating over z, one obtains where a subscript "±" denotes quantities in front of/behind the bubble wall.To be explicit, ω + = ω s (T + ), ω − = ω b (T − ) (and similarly for p ± ), where the label "s/b" denotes the symmetric/broken phase, see Fig. 1.In the last equality of Eq. (18b) we have used the 4 Whether this assumption is true or not cannot be solely determined by the total energy-momentum conservation.It depends on the microscopic details of how the scalar field interacts with the plasma.If this assumption is not true, one would have additional terms in Eqs.(17) expressed by the first derivatives in time of the hydrodynamic quantities.But since for φ(0)| τ = 0 (i.e.vanishing acceleration of the wall) the time dependence disappears in all quantities in the comoving frame, the first time derivatives of the hydrodynamic quantities in the instantaneous comoving frame must depend on the acceleration of the wall a and vanish as a → 0. Since the critical phase transition strength α LTE n,crit is given ultimately by a stationary motion (a = 0, see below), our derivation of it would not be changed even if the present assumption is not fully true.
, and so it is valid only for LTE [77].Comparing Eq. (18b) with Eqs. ( 8) and ( 11), we thus obtain where with s = ω/T being the entropy density.Using Eq. (18a), one can rewrite it as In LTE, there is an additional matching condition due to the entropy conservation [77] 5 The above condition is a special situation of the non-negativity of entropy production [113].Using Eq. (18a), it can be also written as In Appendix A, we explain how to extend this picture if there is a large entropy injection and we argue that the LTE limit provides a lower bound on the resistance to the bubble expansion.
If we impose φ(0)| τ = 0, i.e., vanishing acceleration, we arrive at three matching conditions for three unknown hydrodynamic quantities {T − , v + , v − } (T + can be related to the nucleation temperature T n which is a given parameter characterizing the phase transition), i.e.Eqs.(18) (with φ(0)| τ = 0) and Eq. ( 23).We therefore can fully determine all these three quantities, and thus also the wall velocity ξ w (e.g., for detonations, ξ w = v + ).To study the frictional force in LTE as a function of ξ w , φ(0) must be allowed to take any value so that the wall velocity ξ w becomes a variable.In this case, one should in principle only use the matching conditions (18a), (23) and the definition of P LTE given in Eqs (19), (20).However, as we will see shortly, for the derivation of the critical phase transition strength α LTE n,crit , φ(0)| τ = 0 can be imposed.
It is argued in Ref. [79], based on numerical results and previous works [62,63], that the hydrodynamic obstruction is maximal when the wall velocity ξ w is very close to the Jouguet velocity ξ J .We will elaborate on this point in Sec. 5. We can however already anticipate the important points: the hydrodynamic obstruction originates mainly from the heating due to the shock wave existing in front of the bubble, which decreases the effective phase transition strength felt by the bubble wall.This heating becomes more efficient as the velocity of the shock-wave front increases ξ sw (see Eq. (62a) and the heating parameter Eq. ( 63)).The velocity of the shock-wave front has however to be smaller than the Jouguet velocity ξ sw ≤ ξ J , which implies that the maximal heating, and thus obstruction, will occur at the Jouguet velocity.For larger wall velocities, the LTE pressure decreases (when the motion becomes a detonation), because the shock wave disappears, and thus the heating vanishes.
The typical behaviour of the total pressure on the wall in LTE is shown in the left panel of Fig. 2. Since the Jouguet velocity is a connecting point between the hybrid and detonation regimes, one might think that P max LTE,hyb = P max LTE,det .However, this is generically not true as indicated by the very steep slope at ξ J .The reason behind this is the sudden disappearance of the shock wave when one moves from the hybrid regime to the detonation regime which typically leads to a lower P max LTE,det compared with P max LTE,hyb .Only when the ratio of the enthalpies in the broken and symmetric phases is very close to one, is the equality P max LTE,hyb = P max LTE,det satisfied [79], see Sec. 5 for more details.Therefore, we have As α n increases, one should expect that the peak of the red curve in the left panel of Fig. 2 would move down due to the increase of P driving .At a critical value, denoted as α LTE n,crit , the peak would touch exactly the zero line.For bigger α n , P LTE − P driving cannot equal zero for all ξ w ∈ [0, 1) and thus the bubble wall runs away assuming LTE.Therefore, this value α LTE n,crit provides a criterion for runaway walls in LTE; LTE runaway criterion : Note that the peak has a narrow width, which means that when α n < α LTE n,crit , the wall velocity is typically close to ξ J such that ξ w ≈ ξ J may be a good approximation when the stationary motion is caused by hydrodynamic obstruction [63].Since the peak decreases dramatically after ξ J , only deflagration/hybrid (or ultrarelativistic detonation) solutions are viable [79].This may also be true if out-of-equilibrium effects are included [63] (see, however, Ref. [80]).
Similarly, there is also the runaway criterion based on the BM thermal friction on ultrarelativistic bubble walls [1]; Bödeker-Moore runaway criterion : In the right panel of Fig. 2, we show the frictional pressures in the ballistic approximation, obtained by computing the momentum exchange between the wall and particles that transmit from the outside to the inside of the wall P trans , and that bounce off from the wall P reflection .In this type of calculation, the plasma outside of the wall is assumed to be unperturbed, and thus the calculation is typically valid for a relativistic detonation regime.The total frictional pressure P sum approaches the BM limit for γ w ≫ 1 where P reflection approaches zero.It is expected that the red curve in the left panel of Fig. 2 is followed by the dark red curve (when −P driving is included) from the right panel when the same particle physics model is considered.Note that, however, in some specific models, one may still have a non-monotonic behaviour of the pressure as a function of ξ w (or γ w ) even in the relativistic regime.In Ref. [114], it was found that massive dark photons coupling with the order-parameter scalar can induce a non-monotonic dependence on γ w for the friction.The friction then has a peak at intermediate γ w that the authors dubbed maximum dynamic pressure.We however do not consider such a situation and assume that such an additional peak is absent.can be larger than α BM n,crit in large parameter space, indicating that the peak of the LTE force is larger than the BM thermal friction.For, α n ∈ (α BM n,crit , α LTE n,crit ], the wall acceleration would be stopped by the LTE force even though it cannot be stopped by the BM thermal friction that exists in the ultrarelativistic regime. Since α LTE n,crit is given by P max LTE = P driving , i.e., when the peak touches the zero line of Fig. 2 (left panel), we actually can use the additional matching condition (that is valid only for stationary walls) in deriving α LTE n,crit .With the conditions (18a) and ( 28), one has [77] where and + and − sectors correspond to deflagration/hybrid and detonation regimes, respectively.
4 Maximal LTE pressure in the detonation regime Before we discuss the maximal hydrodynamic obstruction in LTE, equivalently, α LTE n,crit , we first discuss the maximal LTE pressure in the detonation regime, α det n,max .This simpler analysis serves as a warm-up for the more complicated analysis given in the next section.Below, we carry out our analysis for the bag equation of state (EoS) and one of its generalizations where the sound speed can deviate from 1/ √ 3.

Bag model
To close the system of equations, we now need to introduce an EoS.We first consider the Bag model for which we have where a ± and ϵ are constants and we used the convention ϵ ≡ ∆V .With this EoS, α + = ϵ/(a + T 4 + ).In the detonation regime, Substituting these relations into Eqs.( 29) and ( 21), one obtains [77] where Since in deriving f (ξ w , α n ), one has used the matching condition (28), for given α n this expression is valid only for the particular values of ξ w that satisfy P LTE = P driving .Alternatively, one can view ξ w as a free variable but α n as an implicit function of ξ w defined by P LTE = P driving .From this perspective, α n (ξ J ) defines α det n,max .The Jouguet velocity ξ J corresponds to v − = 1/ √ 3 [112] ξ At ξ w = ξ J , the square-root term in Eq. ( 34) is vanishing.Substituting where we have replaced α n with α det n,max .One can approximate f (α det n,max ) with a linear function Substituting Eq. ( 37) into Eq.( 33), we obtain This approximated expression should work well for α det n,max ≳ 0.2.For very small α det n,max , one should use the exact result obtained from Eq. (36).
For −∆V T , we have where r ≡ γ w /γ − and b ≡ a − /a + .Above, we have used the LTE matching condition (23).
Therefore, −∆V T evaluated at ξ w = ξ J is Finally, the driving pressure is The equation For b → 1, this can be approximated by [79] α det n,max ≈ In Fig. 3 we show the comparison between the solution of Eq. ( 43) (blue solid line) and its approximation (44) (blue dashed line).

µν-model
The analysis for the bag EoS can be generalized to the µν-model [115], which is also called the ν-model in Ref. [116] and the template model in Refs.[79,81,117].The µν-model is a generalization of the bag model where the sound speed is allowed to deviate from 1/ √ 3. Explicitly, the EoS reads where the constants µ, ν are related to the sound speed in the symmetric and broken phases through In this case the two matching conditions (18a) and ( 28) have a more specific form [79] v where and Ψ is the ratio of the enthalpies in the broken and the symmetric phase such that For µ = ν = 4, Ψ + becomes the parameter b, and α + becomes ϵ/(a + T 4 + ).In the detonation regime for the µν-model, we have The Jouguet velocity is obtained by substituting v − = c b into Eq.(48a) and reads [79,116] For c b = 1/ √ 3, one recovers Eq. (35).Substituting the EoS into Eq.( 21) and using v − = c b , v + = ξ J , one obtains At the Jouguet point, −∆V T reads where we also used the LTE matching condition (23).The driving pressure reads Equaling P max LTE,det to P driving , one can obtain α det n,max .For Ψ n → 1, one has the approximation [79] For µ = 4.1, ν = 4.2, we show the comparison between the exact result (red solid) and the approximation (red dashed) in Fig. 3.In the numerical result, we also observe that, different from the approximation (56), P max LTE,det = P driving can have a detonation solution for α det n,max > 0 only when Ψ n is smaller than a critical value Ψ n,crit .For µ = 4.1, ν = 4.2, Ψ n,crit ≈ 0.97619.For µ = ν = 4, this critical value goes to one, as expected.56).The lines for the approximations almost overlap for the chosen value of ν.

Maximal hydrodynamic obstruction
Now we generalize the analysis given in the last section to the hybrid regime and derive the maximal phase transition strength (for a given value of b or Ψ n ) that allows for non-runaway hybrid motions, α hyb n,max .According to the discussion in Sec. 3, this also gives the critical phase transition strength above which the maximal LTE force can never balance out the driving force, i.e., α LTE n,crit = α hyb n,max .Below, we shall work with the µν-model directly and the results for the bag model are obtained by taking µ = ν = 4.With this EoS and in the hybrid regime, we have The common factor a + T µ + will be canceled out in the equation P max LTE = P driving .Compared with the detonation regime, in the hybrid regime, we now have an additional shock wave.As a consequence, we do not have the good properties v + = ξ w and T + = T n (⇒ α + = α n , Ψ + = Ψ n ) anymore.One then needs to find the corresponding modified relations with the help of the matching conditions across the shock-wave front.Since P max LTE occurs nearly at ξ J , α hyb n,max /α LTE n,crit can be derived by considering an infinitely thin shock wave.For larger ξ w , the shock wave disappears and the motion enters into the detonation regime.This transition is schematically shown in Fig. 4. Because of the sudden appearance/disappearance of the shock wave when ξ w crosses ξ J , v + and α + are discontinuous at the transition.For more details, see Ref. [112], in particular Figure 6 there.
We denote the hydrodynamic quantities on both sides of the shock-wave front in the rest frame of the latter as T sw,± , v sw,± .To understand the physics of the hydrodynamic obstruction, we first realize that the shock wave reheats the plasma to some temperature T sw,− > T n .In principle, the fluid equations [112] should be integrated from T sw,− to obtain the temperature right in front of the wall T + .However, as the velocity of the wall approaches the Jouguet velocity ξ w → ξ J , the distance between the shock-wave front and the wall diminishes to zero and T sw,− → T + .So, to study the pressure at the peak, we can set T sw,− = T + .We will see that T + becomes fixed as a function of α n .Similarly, the fluid velocity right behind the shock-wave front v sw,− , and that right in front of the wall v + , when transformed to the common plasma frame, are also equal to each other for ξ w → ξ J .Therefore, we have where ξ sw is the velocity of the shock-wave front, and a bar indicates that the quantity is transformed to the plasma frame: The last relation in Eq. ( 59) gives [77] With the µν-model, we have the following matching conditions at the shock-wave front [79] where Looking at Eq.(62a), we observe that the maximal heating (smallest r) is obtained for the largest ξ sw reachable.Furthermore, one can show that Combining Eqs. ( 61) and (62b) gives This relation can be used to show that the velocity of the shock-wave front is an increasing function of ξ w , which means in turn that r decreases (the heating increases) for a larger ξ w , until the shock wave disappears at the Jouguet velocity.We thus confirm here that the peak in the pressure (due to heating) is located at the Jouguet velocity.Finally, v + can be eliminated by substituting v − = c b into the matching conditions (18a) and (28) giving Substituting Eqs. ( 58), (62a), (64a) and (66) into Eq.( 65), one gets an algebraic equation for r and α n .After obtaining r(α n ), one also numerically obtains {v + , ξ w , ξ sw , α + } in terms of α n .Substituting them into Eq.(57a) and rewriting α n as α LTE n,crit , we then obtain P max LTE (α LTE n,crit )/(a + T µ + ).Together with Eq. (64b), we can also obtain (−∆V T )/(a + T µ + ), P driving /(a + T µ + ) in terms of α n .Finally, the equation Bag equation of state.We apply this numerical procedure for µ = ν = 4 (Ψ n = b in this case) and show the result in Fig. 5 (left panel) where we also compare the obtained α LTE n,crit with α det n,max and its approximation (44).One can see that α LTE n,crit increases very fast as b decreases.Unless when b is very close to one, α LTE n,crit significantly differs from α det n,max .We emphasize that α LTE n,crit (b) as a function of b is universal; it does not depend on the underlying particle physics model.Therefore, for the convenience of future applications, we provide the following fit, valid for µ = ν = 4, n,crit and the fit given in Eq. ( 67).
The comparison between the above fit and the numerical result is shown in the right panel of Fig. 5.Moreover, it seems that there is a critical value of b, around 0.77, where α LTE n,crit diverges.To clarify the physical significance of this divergence, we plot on the Left of Fig. 6 the value of the α + , as a function of α n , in the limit of ξ w → ξ J from the hybrid branch.This shows that within the LTE approximation, no matter how supercooled the phase transition is, in the limit of large α n , the effective strength that the phase boundary feels goes to a constant α + (α n → ∞) → 0.0668.This means that using the equations for the driving pressure in Eq. ( 57), the driving force saturates, even for α n going to infinity.On the Right of Fig. 6, we show that the driving force never reaches the friction for b < 0.77.It is expected that the upper bound on α + at the (hybrid) Jouguet point should be uplifted when the LTE condition is relaxed.However, it is well known that α + < 1/3 for any deflagration/hybrid solutions regardless of whether the LTE condition is satisfied at the wall or not.We observe a saturation at α + (α n → ∞) → 0.0668.This saturation indicates that from the point of view of the phase boundary, the effective phase transition strength saturates as we increase α n .Right: Comparison between the driving and maximal LTE pressure on the bubble wall: When the maximal LTE friction (blue line) is always higher than the driving force (Red line), the heating prevents the runaway, for any α n .This result agrees with the numerical analysis presented in Ref. [79].Since the peak in the pressure essentially comes from a heating effect in front of the bubble wall, which is absent in the limit of α n → ∞, this result must be unphysical.Our analysis may thus break down at some large value of α n and the unphysical conclusion should be mitigated when the hydrodynamics analysis is replaced with an analysis in terms of microphysical particle processes with an exchange of momentum between the wall and the plasma.How exactly the analysis may break down as α n increases is left for future work, but will be very briefly discussed in Sec. 7.
Going away from the Jouguet point.Previously we have observed that the hydrodynamic obstruction occurs mostly due to a dramatic heating close to the Jouguet point.To justify our picture and to see how the heating grows when ξ w approaches ξ J from below, we also solve the fluid equations and search for the self-similar fluid temperature and velocity profiles for any ξ w approaching ξ J .On the left panel of Fig. 7, we present the evolution of v + (black) and α + /α n (blue) for α n = 0.1 (solid lines), α n = 1 (dashed lines) and α n = 5 (dot-dashed lines).We observe that the maximal heating occurs indeed at the Jouguet velocity, where both α + (ξ w ) and v + (ξ w ) are discontinuous.On the right panel, we show the evolution of α + (ξ w ).Its values on the hybrid side of the discontinuity confirm the saturation predicted and observed on the left panel of Fig. 6.

Comparison with the Bödeker-Moore criterion
The BM thermal frictional pressure is given in Eq. ( 2).This pressure is model dependent.Defining a characteristic energy scale M where the sum goes over the species that acquire a through the phase transition, Eq. ( 2) becomes where ∆g eff denotes the change in the number of relativistic DoFs in the plasma from the outside to the inside of the wall.We have assumed that the particles gaining mass have m i ≫ T − inside the wall such that they do not contribute to the relativistic DoFs after entering the bubble.To compare it with P max LTE , we consider the bag EoS where (a + − a − ) = π 2 ∆g eff /30 and we have Therefore, we have the BM runaway criterion where we have used the expression of the driving pressure in the bag EoS.The parameter M/T n can be calculated once the particle physics model is given.
In the left panel of Fig. 8, we compare α LTE n,crit with α BM n,crit for M/T n = 2, 5, 10.In the right panel of Fig. 8, taking M/T n = 5 as an example, we show the modified runaway criterion.The dashed and dotted lines represent the BM and LTE runaway criteria, respectively.The solid line, which is obtained from max{α LTE n,crit , α BM n,crit }, represents the modified runaway criterion.The shaded region gives a conservative estimate for the parameter space allowing non-runaway walls.The large red region would be (incorrectly) estimated to give runaway walls if hydrodynamic obstruction is not taken into account, even in the non-supercooled regime α n < 1.  n,crit , α BM n,crit }, above which the bubble wall's acceleration can be stopped by neither the hydrodynamic obstruction nor the leadingorder BM thermal friction.All the shaded regions (red and green) give non-runaway walls by the modified criterion, while the red region would give runaway walls if hydrodynamic obstruction is not taken into account.Recall our comment below Eq. (3) on the term "runaway" used in this paper.
Note that for α n < α LTE n,crit , the wall velocity is bounded by the Jouguet velocity, ξ w ≤ ξ J .For a phenomenologically large α n , e.g., α n = 10, γ(ξ J ) ≈ 7.64, which is much smaller than the typical value of γ w for the ultrarelativistic regime considered in Refs.[1,82], γ w ∼ 10 9 .Actually, the phase transition strength cannot be arbitrarily large, otherwise, the phase transition cannot complete [118].This ensures that the LTE force becomes maximal typically before the wall enters the ultrarelativistic regime for practical phase transition strengths.Therefore, α LTE n,crit alone can serve as a criterion for whether the wall can enter into the ultrarelativistic regime or not, and thus is important for phenomenological studies involving ultrarelativistic walls.Let us now illustrate within an actual particle physics model, how to compute the hydrodynamic obstruction.
How to compute b in BSM models.We have seen that b ≡ a − /a + is a crucial quantity for the determination of the hydrodynamic obstruction.Let us now discuss how to work out its value for BSM particle physics models, especially if the BSM sector is weakly coupled with the SM (as considered in, e.g., Ref. [119]).First of all, the matching conditions are based on the total energy-momentum conservation between the order-parameter scalar and the plasma.If some particle species do not respond to the passage of the wall quickly enough, they can be taken as completely "invisible" to the wall and they should not be taken into account when analyzing the total energy-momentum conservation.Therefore, a + should contain all the particles that couple directly/strongly to the bubble wall (that we call active) which give a contribution a +,active .However, there are some particles that couple indirectly to the wall via the active particles (that we call passive) such that the exchange of energy/momentum between the active and the passive particles occurs on a length scale much smaller than the size of the perturbed fluid around the wall, which can be approximated by the bubble radius. 8These passive particles also need to be counted and give a contribution a +,passive .Mathematically, we have where L therm is the length of thermalisation of the passive particles due to the exchange of energy/momentum between active and passive particles.In the opposite limit, particles coupling very weakly to active ones such that R bubble ≪ L therm can be fully ignored in the analysis.As an example, in the EWPT, the top quark would be an active particle, the electron would be a passive particle 9 and the neutrino would be discarded in counting the DoFs.The quantity a − is computed in the same way, with the further requirement that m − < T − .Since the passive particles do not couple to the order parameter, their DoFs do not change across the wall as long as the change in the temperature across the wall is not too big: Relevance for particle physics models and phenomenology.Let us quickly comment on a typical model where our discussion might be relevant.Spontaneous B − L symmetry breaking is a physically motivated phase transition with a Lagrangian of the following form where L α are the SM lepton doublets, N I are the three families of heavy right-handed (RH) neutrinos, Y D,αI are the Dirac Yukawa couplings between N I and L α , and y I are Majorana Yukawa couplings.After the phase transition, ⟨Φ⟩ = v ϕ / √ 2 and the type-I seesaw Lagrangian is recovered with The phase transition ⟨Φ⟩ = 0 → v ϕ / √ 2 can be strongly first order if Φ couples to another scalar s via a coupling of the form −∆L = λ sϕ 2 s 2 |Φ| 2 , inducing a mass M 2 s (ϕ) = λ sϕ ϕ 2 .It has been shown that the bubble wall induces an enhancement of the leptogenesis efficiency [44], which however strongly depends on the terminal velocity of the bubble wall.b ≡ a − /a + can be computed in the following way: (30/π 2 )a +,active = 6 + 2 + 1 = 9 (assuming three families of RH neutrinos, one complex Φ and a singlet s), and (30/π 2 )a −,active ≈ 1. a +,passive is more delicate to compute and depends on the value of Y D,αI .The N I couple to the SM via the couplings Y D,αI and the typical length scale for thermalisation between N I (active particles) and SM particles, typically via Higgs-mediated N L → N L, is given by [120] On the other hand, the largest bubble radius can be estimated as where βH ≡ β ≡ HT d dT S 3 /T is the timescale of the transition in Hubble units.Within our model for M I ∼ T n ∼ 10 10 GeV and β ∼ 10, we have So under perturbations, the SM particles never thermalise within a scale shorter than the bubble scale.Therefore, we have a +,passive = a −,passive = 0. We obtain finally b ∼ 0.1, which is a value that, according to the modified criterion, would not allow for runaway.In this case, we could approximate ξ w ≈ ξ J .

Comments on the validity of our approximations
We have seen that the hydrodynamic obstruction is essentially a consequence of the strong heating of the plasma due to the shock wave in front of the wall.For this heating to occur, we however have to fulfil some conditions on hydrodynamics and exchange of energy.In this section, we would like to discuss a few possible weaknesses of our results, for the purpose of further studies.
• The hydrodynamic regime: We would like to emphasize that in this paper, our computation relied on hydrodynamics, which is by definition valid for scales much larger than the mean free path of particles in the plasma.For example, for a gauge theory with coupling g, the mean free path of the particles coupling to the gauge field would be [120] which we evaluated at the nucleation temperature.As we have seen, in the LTE picture, the pressure as a function of the velocity is not a monotonic function, but features a peak in the pressure at ξ peak ∼ ξ J , which is related to α n via Eq.( 35) for the bag EoS.This means that for α n ≫ 1, γ(ξ J ) ∼ O(1)α 1/2 n .In order for the hydrodynamics analysis to be valid, one needs to require two conditions.First, the length scale of the perturbed fluid should be larger than the mean free path of the plasma.The former can be (very roughly) estimated as the bubble radius R w and hence we require This issue can be understood in another way, from the particle physics perspective.We have seen that the hydrodynamic obstruction disappears if the bubble wall manages to become a detonation.On the other hand, the hydrodynamic obstruction originates from the heating of the plasma due to the particles bouncing off the wall.For the hydrodynamic obstruction to be relevant in the early stage of the bubble wall, the time scale of the heating, t heating , needs to be smaller than the time scale for the bubble wall to accelerate to the Jouguet velocity, t acceleration : Second, one may also require the thickness of the shock wave to be larger than L MFP in order for the hydrodynamic description on the shock wave to be valid 10 , where R sh is the radial position of the shock-wave front.The equation R sh (ξ ⋆ w , α n ) − R w (ξ ⋆ w , α n ) = L MFP then defines a function ξ ⋆ w (α n ).For ξ ⋆ w (α n ) < ξ w < ξ J (α n ), the thickness of the shock wave would be smaller than L MFP .Strong heating α + /α n ≪ 1 require typically a short distance R sh (ξ w , α n ) − R w (ξ w , α n ), so we expect this condition to be more stringent than Eq.(78).Such a detailed analysis is beyond the scope of this paper and will be presented in another work [121].
• The thin-wall approximation: In hydrodynamics, one has implicitly assumed a thin wall whose size can be neglected compared to the hydrodynamic scale.Therefore, the hydrodynamics analysis may be valid only when The equation ), the thickness of the shock wave would be smaller than L w /γ w .
It is then possible that one should evaluate the peak height at min{ξ ⋆ w (α n ), ξ † w (α n )} instead of ξ J (α n ).This should reduce the height of the pressure peak.A detailed analysis of all the three conditions (78), (80) (81) and a study on the pressure when these conditions are violated go beyond this work and are left for future work.
• The self-similar solution Even in the regime of validity of hydrodynamics, there is no guarantee that the peak in the pressure will manifest.Although in our analysis, we do not need to solve the fluid equations, the classification of the fluid motions and the particular Jouguet point may still be based on self-similar solutions for the temperature and bulk velocity profiles.It is known that such self-similarity profiles are reached asymptotically when R w ≫ R initial .However, in the early stage of the bubble wall, the temperature profile might differ substantially from the self-similar solution.This would occur if the time scales of adaptation of the profile is much longer than the timescale of acceleration of the bubble. 11Verifying the validity of the self-similar profile or the real "existence" of the pressure peak, requires dynamical simulation, a study that we will perform in a forthcoming study [121].

Conclusion
In this paper, we have studied the criterion for the runaway behaviour of bubble walls within two opposite approximations: the ballistic approximation and the LTE approximation.These two approximations are usually applied to the ultrarelativistic regime and the relatively lower velocity regime, respectively.The former corresponds to the BM thermal friction and the latter corresponds to the hydrodynamic obstruction.These frictions thus typically exist in different regions of the wall velocity space.For both approximations, we formulate a different runaway criterion in terms of a critical phase transition strength, α , the acceleration of the wall cannot be stopped by the friction under study.The BM criterion for runaway walls α BM n,crit < α n , valid within the ballistic approximation, has been widely used in the literature.
However, the BM criterion implicitly assumes that the friction is a monotonous function of the wall velocity.This is however not true and the hydrodynamic obstruction gives a pressure barrier before the wall enters into the ultrarelativistic regime [62,63,79].The modified criterion on runaway when the hydrodynamic obstruction is taken into account should be α n > max{α BM n,crit , α LTE n,crit }.While both α BM n,crit and α LTE n,crit depend on the ratio of the number of DoFs in the symmetry and broken phases b, α BM n,crit further depends on a particle physics model-dependent parameter, the ratio between the averaged mass gain and the nucleation temperature M/T n .Therefore, max{α BM n,crit , α LTE n,crit } depends on the interplay of these two parameters.We have compared α BM n,crit with α LTE n,crit for some typical values of the model-dependent parameter M/T n (cf.Fig. 8).We have argued that the BM is not always a sufficient criterion for the runaway (even in the absence of other sources of friction due to, e.g., 1 → 2 processes).In Sec. 5, Eq. ( 67), we provide easily implementable expressions for the maximal LTE pressure and we emphasize the fact that α LTE n,crit (b) does not depend on the particularities of the underlying particle physics model.It appears that within the LTE approximation, theories with large enough change in DoF, b < 0.77, have typically non-runaway walls due to the hydrodynamic obstruction.This would reduce drastically the region of parameter space allowing for runaway.Furthermore, α LTE n,crit is the lower bound for the phase transition strength that the wall can enter the ultrarelativistic regime and thus is a very important quantity for phenomenological studies involving ultrarelativistic walls.
We note that in this paper we have ignored other sources of friction that may exist.For the estimate of the pressure peak, we have ignored out-of-equilibrium effects which were shown to be not small in Refs.[122][123][124], in contrast to Ref. [63].For the BM criterion, we have ignored other sources of friction due to other particle physics processes.As a consequence, both α LTE n,crit and α BM n,crit serve as a most conservative estimate of the maximal phase transition strength under which non-runaway stationary walls can exist.
We also attract the attention to several caveats that may change our conclusions.Specifically, the velocity at which the hydrodynamic obstruction is maximal, ξ J , is relatively low and can be reached at the early stage of the bubble expansion.Also, at the Jouguet velocity, the shock wave becomes infinitely thin.Whether the approximations or assumptions used in this paper, i.e., the LTE approximation, hydrodynamics, and the self-similarity used for the plasma temperature and velocity profiles, are valid or not in such an early stage, or when the shock wave becomes infinitely thin needs further investigation.We hope that our work can serve as a motivation to study the dynamical evolution of the bubble wall, not only its steady states, an investigation that we will initiate in Ref. [121].
A Discussion on the departure from LTE In the main text, we have always assumed LTE and neglected the increase of entropy at the phase boundary.In this appendix, we try to broaden this picture and discuss what the increase of entropy would change.
Modification in the matching conditions.The first modification would appear in the relation between the pressure and the potential [77] where δp encodes the out-of-equilibrium effects.We have introduced p ≡ −V (ϕ) − V T (ϕ, T ) to denote the equilibrium part of the pressure.In this notation, the p in the main text should be replaced with p.With this additional δp, the matching condition in Eq. ( 18b) is modified to (assuming stationary motion for simplicity for the present discussion) Then one can identify the friction as where P LTE is defined as previously.Comparing it with Eq. ( 12), one then has The above equation can be used to compute δp from δf i , the out-of-equilibrium part of the distribution functions.
Allowing for out-of-equilibrium, thermodynamic identities may be spoiled as there is no well-defined temperature for δf i .We will assume a close-to-equilibrium system such that δf i ≡ f − f eq i ≪ f eq i so that we still use f eq i to define most thermodynamic quantities except the additional δp in the pressure.In this sense, only the very direct effects on the friction of the wall are considered.Then, e(ϕ, T ) = −p(ϕ, T ) + T ∂ p(ϕ, T ) ∂T , as defined in LTE, and the entropy density is defined as s = ω/T = (e + p)/T .Although the entropy density is defined as previously, the presence of the deviation from equilibrium can induce the entropy production such that the matching condition Eq. ( 22) is modified by the introduction of a parameter ∆s: Using Eq. (18a), it can also be written as It is clear that the parameters ∆s and δp are in principle computable within a particle physics model and are related to each other.However, the computation of those parameters from a specific particle physics model is beyond the scope of this work.We leave it for future work.
Consequence of the entropy increase.For −∆V T , we have and thus the different contribution of the pressure in Eq. ( 57) becomes The consequence of the entropy injection is to lower the effective value of the parameter b by b → b (1 + ∆s/s + ) 4 , This effectively increases the frictional pressure on the bubble wall, since (1 + ∆s/s + ) > 1.
Assuming that P dissipative , i.e., ∆δp, is positive, one would confirm the idea that the LTE limit δp → 0, ∆s → 0, gives a lower bound on the pressure.Once δp (or ∆δp) is expressed in terms of ∆s/s + , then ∆s/s + can be an additional parameter for hydrodynamic simulations.This is similar to that although α n is computed from specific particle physics models, it is treated as a parameter in fluid dynamics [125].∆δp in the ultrarelativistic regime.One may get a feeling about ∆δp by considering the ultrarelativistic regime γ w ≫ 1.In this regime, we can use the collisionless/ballistic approximation.Then12 δf i = f i − f eq i (z; T (z)) ≈ f in i (T + ) − f eq i (z; T (z)) = 1 e γ + (|p|+v + p z )/T + ± 1 This shows that ∆δp adds more friction to the wall.Outside of the ultrarelativistic regime, one might expect f i (z) ∈ [f in i (T + ), f eq i (z; T (z))] and f in i ≈ f in;eq i .Assuming T (z) ∼ T + , v(z) ∼ v + , one again has δf i (z) ≥ 0 due to the suppression from the masses in f eq i (z; T (z)) compared with f in;eq i (T + ).Thus, both ∆δp and ∆s increase the resistance from the plasma on the wall expansion.The limit ∆δp → 0 and ∆s → 0 is indeed a lower bound on the pressure.
This confirms that, for γ w ≫ 1, P friction = P LTE + P dissipative → P BM .Note that one in general does not have P LTE + P dissipative = P BM outside of the ultrarelativistic regime so that one cannot conclude that P BM is bigger than P LTE based on P dissipative > 0. To have P friction = P BM , we have assumed (i) f i take the form of Eq. ( 92); (ii) the particles in front of the wall are in thermal equilibrium; and (iii) T + = T n .When any of these assumptions are not satisfied, one cannot conclude P friction = P BM .In particular, in the presence of the heating effects, T + > T n , and thus when the other two assumptions are satisfied, we have P LTE + P dissipative > P BM .

Figure 1 :
Figure 1: Rest frame of the bubble wall used for the matching conditions.

Figure 2 :
Figure2: Left: Example of P LTE − P driving (in arbitrary units) as a function of the wall velocity ξ w for a given phase transition strength α n , based on results of Ref.[63].P max LTE occurs slightly before the Jouguet velocity ξ J .Plot taken from Ref.[79].Right: Example of the total frictional pressure due to transmission and reflection in the ballistic picture that are assumed to become more and more exact for a larger and larger γ w .It is a monotonically increasing function and saturates at the BM value.

Figure 3 :
Figure 3: α det n,max as a function of Ψ n for the bag model (µ = ν = 4, blue line) and for µ = 4.1, ν = 4.2, and their approximations given in Eq. (56).The lines for the approximations almost overlap for the chosen value of ν.

Figure 4 :
Figure 4: Transition from a hybrid solution to a detonation solution at the Jouguet point ξ w → ξ J .As ξ w approaches ξ J , the shock wave (light red region) becomes thinner and thinner until it disappears when ξ w = ξ J .The light green band represents the rarefaction wave behind the wall.

Figure 6 :
Figure6: Left: Plot of α + as a function of α n in the limit ξ w → ξ J from the hybrid branch.We observe a saturation at α + (α n → ∞) → 0.0668.This saturation indicates that from the point of view of the phase boundary, the effective phase transition strength saturates as we increase α n .Right: Comparison between the driving and maximal LTE pressure on the bubble wall: When the maximal LTE friction (blue line) is always higher than the driving force (Red line), the heating prevents the runaway, for any α n .
BM only (red region)

Figure 8 :
Figure 8: Left: Comparison between α LTE n,crit and α BM n,crit for µ = ν = 4 and M/T n = 2, 5, 10.Right: Modified runaway curve obtained from max{α LTEn,crit , α BM n,crit }, above which the bubble wall's acceleration can be stopped by neither the hydrodynamic obstruction nor the leadingorder BM thermal friction.All the shaded regions (red and green) give non-runaway walls by the modified criterion, while the red region would give runaway walls if hydrodynamic obstruction is not taken into account.Recall our comment below Eq. (3) on the term "runaway" used in this paper.

√ p 2
+m 2 (z)+v(z)p z /T (z) ± 1 > 0(92) when assuming T (z) ≈ T + and v(z) ≈ v + ≈ 1.The signs ± correspond to the case of fermions/bosons entering the wall.Above f in i are the distribution functions for the incoming particles.We thus obtain ∆δp = P dissipative = −

Further, in the 3
ultrarelativistic regime we have T + = T n and f in i = f in;eq i (incoming particles are in thermal equilibrium at the nucleation temperature).Then one can recognize ∆δp = − 2E i f eq i (p, z; T (z)) = P BM − P LTE .