Instantons
Hawking and I made progress with the HartleHawking proposal when we
realised there is a simple class of classical solutions to the field equations
for gravity coupled to an inflationary scalar field, which yield a finite
probability according to the DiracFeynman formula in imaginary time. The solutions we studied yield a classical
spacetime which is a real four dimensional manifold in either real time t or imaginary time τ provided that the transition t = iτ is performed on the sero time surface Σ shown in Figure (3).
The form of the solutions is generic for inflationary models with gently
sloping potentials V(Φ), which are the simplest models.
Figure 3:
An infinite open Universe emerges from a ‘pea’ instanton.
Region I is an infinite
inflating open ‘island Universe’.
Region II an approximately de Sitter region Φ is nearly constant) bounded by a
timelike singularity.
There are actually a one parameter family of finite probability
solutions, in which we have a onetoone relation


(1) 
where Ω_{0} is the current density parameter, and Φ_{0 }is the value of the scalar
field at the regular pole of the instanton. For the solutions described above, Ω_{o} < 1, but if one allows instantons which are singular at both
poles, and symmetric about the maximum of b,
these analytically continue to closed inflating Universes, and one can obtain
any value of Ω_{o} > 1.
The classical instanton solutions could then be used to compute the
quantum fluctuations about inflating spacetimes in a more precise manner than
had hitherto been possible. In
particular, previous calculations had to make more or less ad hoc choices for
the initial quantum state. But in these new solutions the initial quantum state
was initially and precisely defined from the no boundary proposal.I should also mention that doubts were raised about the HawkingTurok solutions
because they possess a singularity, albeit one mild enough that the quantum
fluctuations about it are well defined. These doubts have since been dealt with by
showing that by a suitable change of variables the singularity can actually be
removed. These developments are reviewed in N. Turok,
Before Inflation, preprint
(2000). For the purposes of the present
debate the precise predictions of the HawkingTurok calculation are not
relevant, though I must mention that work subsequent to this meeting has shown
that one can actually calculate the volume factors appropriate to inflation in
a slicingindependent way.When one includes these volume factors, the most probable instantons are those
in which the initial field Φ_{o }starts out large, and the
scalar field potential energy density is at the Planck density, at which
quantum gravitational effects become large. Therefore these calculations do
seem to resolve the question of the most probable starting point for inflation,
which is at the Planck density.
Contributed by: Dr. Neil Turok
