Gravitational radiation from colliding vacuum bubbles: Envelope approximation to many-bubble collisions

We introduce an approximation to calculate the gravitational radiation produced by the collision of true-vacuum bubbles that is simple enough to allow the simulation of a phase transition by the collision of hundreds of bubbles. This "envelope approximation" neglects the complicated "overlap" regions of colliding bubbles and follows only the evolution of the bubble walls. The approximation accurately reproduces previous results for the gravitational radiation from the collision of two scalar-field vacuum bubbles. Using a bubble nucleation rate given by $\Gamma ={\Gamma }_0{e}^{\beta t}$, we simulate a phase transition by colliding 20 to 200 bubbles; the fraction of vacuum energy released into gravity waves is $\frac{E_{\mathrm{GW}}}{E_{\mathrm{vac}}}=0.06\mathrm{}{\left(\frac{H}{\beta }\right)}^2$ and the peak of the spectrum occurs at ${\omega }_{max}=1.6\mathrm{}\beta$ ($H^2=\frac{8\pi G\rho }{3}$ is the Hubble constant associated with the false-vacuum phase). The spectrum is very similar to that in the two-bubble case, except that the efficiency of gravity-wave generation is about five times higher, presumably due to the fact that a given bubble collides with many others. Finally, we consider two further "statistical" approximations, where the gravitational radiation is computed as an incoherent sum over individual bubbles weighted by the distribution of bubble sizes. These approximations provide reasonable estimates of the gravitational-wave spectrum with far less computation.