Slow-down of expanding bubbles in the early Universe

We study slow-down effects for bubbles generated during a cosmological first-order phasetransition focusing on deflagrations and hybrids, where the bubble wall is preceded by a shockwave of heated plasma. These effects have been observed in multi-bubble simulations togetherwith a suppression of the gravitational wave emission, most dominantly for slow walls and largeshocks. In order to study the impact of the shock waves on the wall velocity around the time ofpercolation, we consider steady-state single-bubble solutions and incorporate the possible heat-ing effects by two different mechanisms. First, we investigate the slow-down as experiencedby a bubble expanding in the medium of an impeding shockwave, where the temperature ishigher than at nucleation, and the fluid is no longer at rest. Taking into account such heatingand kinematic effects, we find that the most significant slow-down occurs for the fastest walls,and thus cannot explain the suppression of the gravitational waves observed in the simulations.However, these effects turn out to be stronger for phase transitions with a Standard-Model-likechange in degrees of freedom compared to what usually implemented in the simulations, sug-gesting that the degrees of freedom can be an important additional parameter for characterizingthe gravitational wave spectrum. For the second slow-down mechanism, we study steady-statesolutions corresponding to heated droplets of false vacuum that shrink towards the end of thephase transition. By implementing a suitable boundary condition motivated by energy con-servation, we show how the droplet velocity, interpreted here as the late-time velocity of thebubble walls around percolation, can actually be calculated and predicted from the propertiesof the initial deflagration/hybrid, finding remarkable agreement with the numerical simulations.Droplets are found to generally shrink more slowly for stronger phase transitions and slowerdeflagrations, with a mild dependence on the change of degrees of freedom. Such slow dropletsnaturally correlate with a suppressed gravitational wave emission, while geometrical propertiessuch as the shock width play an important role as well.