Bubble wall velocities in local equilibrium

Abstract

It is commonly expected that a friction force on the bubble wall in a first-order phase transition can only arise from a departure from thermal equilibrium in the plasma. Recently however, it was argued that an effective friction, scaling as γ2 w (with γ w being the Lorentz factor for the bubble wall velocity), persists in local equilibrium. This was derived assuming constant plasma temperature and velocity throughout the wall. On the other hand, it is known that, at the leading order in derivatives, the plasma in local equilibrium only contributes a correction to the zero-temperature potential in the equation of motion of the background scalar field. For a constant plasma temperature, the equation of motion is then completely analogous to the vacuum case, the only change being a modified potential, and thus no friction should appear. We resolve these apparent contradictions in the calculations and their interpretation and show that the recently proposed effective friction in local equilibrium originates from inhomogeneous temperature distributions, such that the γ2 w -scaling of the effective force is violated. Further, we propose a new matching condition for the hydrodynamic quantities in the plasma valid in local equilibrium and tied to local entropy conservation. With this added constraint, bubble velocities in local equilibrium can be determined once the parameters in the equation of state are fixed, where we use the bag equation in order to illustrate this point. We find that there is a critical value of the transition strength αcrit such that bubble walls run away for α>αcrit.

Electroweak bubble wall expansion: gravitational waves and baryogenesis in Standard Model-like thermal plasma

“…If for a given strength, the above formula yields a velocity above the Jouguet value then the transition is too strong and there cannot be thermodynamic equilibrium. Let us also note that the above expression can be derived from recent results on the wall velocity in local equilibrium [97]. In particular, equating the two pressures in formula (17) of that reference and assuming constant velocity with thermal equilibrium one arrives at the same result.…”

“…Furthermore we draw a comparison with unsophisticated estimates for the wall velocity in the thin-and thick-wall approximations and we found remarkable agreement with the thick-wall formula. We provide a simple derivation for this formula and make a connection with recent results of wall velocities in thermal equilibrium [97]. A similar derivation for the wall thickness is carried out that gives a remarkably good approximation for the Higgs wall-thickness but overestimates the singlet thickness by a factor of about ≈ 7/5 in all cases.…”

“…Investigations of the wall properties assuming case (2), local equilibrium, include [94][95][96][97] and in this case the only friction on the wall propagation comes from hydrodynamic effects of the plasma. On the other hand, if the wall is thin (but thicker than the particles thermal wavelength so that WKB approximation is valid), case (2), reflection and transmission of particles is appropriate to quantify the friction.…”

Detection of Early-Universe Gravitational Wave Signatures and Fundamental Physics

“…Recently, progress has been made in characterizing the importance of the equilibrium part of the distribution function, where variation of the plasma temperature, which is a function of the position relative to the wall and v w , plays a role. These variations are tied to hydrodynamic effects in the plasma, which can induce a backreaction force on the wall [93,[214][215][216][217]. For ultrarelativistic bubble walls, with a Lorentz factor γ(v w ) = 1/ 1 − v 2 w 10, equilibration cannot be maintained across the bubble wall.…”

Gravitational waves from patterns of electroweak symmetry breaking: an effective perspective

“…For other form of friction coefficient h(γ) in ∆p fr ≡ ∆p LO + h(γ)∆p N LO , one might have to numerically integrate (4.45) to obtain the efficiency factor κ φ for the bubble wall collision. Note that our general method also applies to the recent revisits on the friction forms of ∆p fr ∝ (γ 2 − 1) [275] or ∆p fr ∝ γ [276]. This is particularly important for the case of strong first-order phase transitions when the friction force is much smaller than the driving force so that the bubble walls would take longer time to approach a terminal velocity, before which they had already been colliding with each other if the initial separation is so small (namely a very large decay rate) that there is no enough time for the bubble wall to be ever close to a terminal velocity.…”