Open inflation and the singular boundary

The singularity in Hawking and Turok’s model of open inflation has some appealing properties. We suggest that this singularity should be regularized with matter. The singular instanton can then be obtained as the limit of a family of “no-boundary” solutions where both the geometry and the scalar field are regular. Using this procedure, the contribution of the singularity to the Euclidean action is just 1/3 of the Gibbons-Hawking boundary term. Unrelated to this question, we also point out that gravitational backreaction improves the behaviour of scalar perturbations near the singularity. As a result, the problem of quantizing scalar perturbations and gravity waves seems to be very well posed. Recently, Hawking and Turok [1,2] have suggested that an open universe can be created from nothing. This is an attractive possibility because it would allow to construct open models of inflation with very simple inflationary potentials (see also [3–5]). The new ingredient that makes their construction possible is that they allow their instanton solution to be singular. There is some justification for this, since the Euclidean action Electronic address: [email protected] 1 is integrable near the singularity. Moreover, if we think of the singularity as the boundary of spacetime, the Gibbons-Hawking boundary term [6] is non-vanishing and finite. This is rather coincidental, since it requires the extrinsic curvature of the boundary to increase just at the same rate as the inverse of its volume as the singularity is approached. In this paper, we suggest that the singularity should be regularized with matter, so that the instanton can be obtained as the limit of a family of nonsingular geometries where the scalar field is also well behaved. The simplest way to do this is to introduce a membrane coupled to the scalar field. The Euclidean action is given by