Gravitational radiation from colliding vacuum bubbles

In the linearized-gravity approximation we numerically compute the amount of gravitational radiation produced by the collision of two true-vacuum bubbles in Minkowski space. The bubbles are separated by distance d and we calculate the amount of gravitational radiation that is produced in a time τ∼d (in a cosmological phase transition τ corresponds to the duration of the transition, which is expected to be of the order of the mean bubble separation d). Our approximations are generally valid for τ≲${\mathit{H}}^{\mathrm{-}1}$. We find that the amount of gravitational radiation produced depends only upon the grossest features of the collision: the time τ and the energy density associated with the false-vacuum state, ${\mathrm{\rho }}_{\mathrm{vac}}$. In particular, the spectrum ${\mathit{dE}}_{\mathrm{GW}}$/dω∝${\mathrm{\rho }}_{\mathrm{vac}}^2$ ${\mathrm{\tau }}^6$ and peaks at a characteristic frequency ${\mathrm{\omega }}_{\max }$≃3.8/τ, and the fraction of the vacuum energy released into gravitational waves is about 1.3×${10}^{\mathrm{-}3}$(τ/${\mathit{H}}^{\mathrm{-}1}$ ${)}^2$, where ${\mathit{H}}^2$=8πG${\mathrm{\rho }}_{\mathrm{vac}}$/3 (τ/${\mathit{H}}^{\mathrm{-}1}$ is expected to be of the order of a few percent). We address in some detail the important symmetry issues in the problem, and how the familiar ‘‘quadrupole approximation’’ breaks down in a most unusual way: it overestimates the amount of gravitational radiation produced in this highly relativistic situation by more than a factor of 50. Most of our results are for collisions of bubbles of equal size, though we briefly consider the collision of vacuum bubbles of unequal size. Our work implies that the vacuum-bubble collisions associated with strongly first-order phase transition are a very potent cosmological source of gravitational radiation.