Quantum State During and after $O(4)$-Symmetric Bubble Nucleation with   Gravitaional Effects
  We extend our previous analysis of the quantum state during and after $O(4)$-symmetric bubble nucleation to the case including gravitational effects. We find that there exists a simple relationship between the case with and without gravitational effects. In a special case of a conformally coupled scalar field which is massless except on the bubble wall, the state is found to be conformally equivalent to the case without gravity. 
Takahiro TANAKA and Misao SASAKI