Cosmological bubble friction in local equilibrium

Abstract

In first-order cosmological phase transitions, the asymptotic velocity of expanding bubbles is of crucial relevance for predicting observables like the spectrum of stochastic gravitational waves, or for establishing the viability of mechanisms explaining fundamental properties of the universe such as the observed baryon asymmetry. In these dynamic phase transitions, it is generally accepted that subluminal bubble expansion requires out-of-equilibrium interactions with the plasma which are captured by friction terms in the equations of motion for the scalar field. This has been disputed in works pointing out subluminal velocities in local equilibrium arising either from hydrodynamic effects in deflagrations or from the entropy change across the bubble wall in general situations. We argue that both effects are related and can be understood from the conservation of the entropy of the degrees of freedom in local equilibrium, leading to subluminal speeds for both deflagrations and detonations. The friction effect arises from the background field dependence of the entropy density in the plasma, and can be accounted for by simply imposing local conservation of stress-energy and including field dependent thermal contributions to the effective potential. We illustrate this with explicit calculations of dynamic and static bubbles for a first-order electroweak transition in a Standard Model extension with additional scalar fields.

Field-theoretic derivation of bubble-wall force

“…which is covariantly conserved, ∂ µ T Ψ µν = 0. 16 We wish to calculate (A.8) at the one-loop order, which corresponds to the Feynman diagram in figure 13, where the solid oriented line is the free thermal fermionic propagator and the cross indicates the fermionic energymomentum insertion, which is obtained by varying (A.8) with respect to Ψ(y) andΨ(y ).…”

“…e βEp + 1 , (A.10) 16 For simplicity we assume that the background spacetime is expanding adiabatically slowly, such that time derivatives of the scale factor a can be neglected. Then the way the expansion enters the problem is through temperature's dependence on the scale factor, whose precise form can be obtained from conservation of the entropy density s, (d/dt)s ≈ 0.…”

“…After submission of the first version of this paper, two related works appeared[15,16] 2. Even though an expanding bubble explicitly breaks Lorentz symmetry, one can speak of an approximate Lorentz symmetry maintained by the state, if it can be characterized by a Lorentz covariant distribution function.…”

“…Variation of temperature and velocity across the bubble wall has been studied in ref [16]. which appeared after the submission of the first version of this article 5.…”

Bubble wall velocity at strong coupling

“…This topic has been the subject of an intense analysis in the literature. Results based on fluctuation-dissipation theorems [6,7] are complemented by perturbative computations in specific models based on kinetic theory [8][9][10][11][12][13][14][15][16][17][18][19] as well as by the analysis of hydrodynamic effects [20,21]. Several regimes are envisaged depending on the bubble wall velocity and the friction force.…”