The detailed predictions of the HawkingTurok instantons are not
crucial to the present discusssion. What is important is the claim of Guth and
others that if such an instanton started the Universe, and started inflation,
that there would be an infinite period of inflation between the instanton and
ourselves. This is an assertion one can check by performing an analysis of the
fluctuations about the instanton solution and the inflating spacetime which
emerges from it. My conclusion from these calculations, at least within the
context of the causal formulation given above, is that our past light cone
contained only a brief episode of inflation, and if we follow it back we are
led after a very short time right back to the Planck density. For example in a theory with V = m^{2}φ^{2}the duration of
inflation turns out to be only of order M_{pl}m^{2}, or about 10^{12} Planck times, or about 10^{31}
seconds.
The reason my finding is so very different to that of Guth et al. is that I am asking a different
question. Whereas Guth makes a
comparison between the classical rolling of the field downhill and the quantum
jumps uphill in a single Hubble time,
I am asking for the integrated effect of the quantum jumps over the entire spacetime
being considered. I view the classical solution as a first approximation to the
entire classical spacetime, and the issue is then whether the effects of the
quantum jumps, when integrated up over the past, significantly modify the
classical solution. The main point is
that the classical solution is coherent,
so that the scalar field Φ
changes only in one direction as one tracks back in time: for a solution which starts at the Planck
density (which is the case where the quantum effects are greatest) the scalar
field monotonically increases as one goes back in time towards the Planck
density.
However, the quantum jumps are diffusive
in character. Because they are either positive or negative, their effect does
not add up coherently. If the potential
V(Φ) were nearly constant, then the quantum
diffusion would grow as the square root of the time. This lack of coherence
means that the quantum jumps do not severely correct the classical solution until one reaches the Planck density. At
the Planck density, all calculations break down and there is no point in
arguing whether inflation is eternal or not at the Planck density because that
really is the realm of the unknown.
What I would say, however, is that I think the study of inflating
spacetimes is a challenging problem which forces us to carefully think through
exactly what we should calculate. I think this is a very important and
interesting exercise even if all it teaches us is about the limitations of
existing theoretical frameworks.
Acknowledgements
I acknowledge my debt to many physicists for discussions of these
issues, in particular Jim Hartle, A. Guth, A. Linde, V. Rubakov and A.
Vilenkin. I also acknowledge my
collaborators S. Gratton, S. Hawking, T. Hertog, K. Kirklin and T. Wiseman.
Contributed by: Dr. Neil Turok
