Eternal Inflation Isn't

The detailed predictions of the Hawking-Turok 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 = m²φ²the duration of inflation turns out to be only of order M_plm^-2, or about 10¹² 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