Abstract
Random noise arises in many physical problems in which the observer is not tracking the full system. A case in point is inflationary cosmology, the current paradigm for describing the very early universe, where one is often interested only in the time-dependence of a subsystem. In inflationary cosmology it is assumed that a slowly rolling scalar field leads to an exponential increase in the size of space. At the end of this phase, the scalar field begins to oscillate and transfers its energy to regular matter. This transfer typically involves a parametric resonance instability. This article reviews work which the author has done in collaboration with Walter Craig studying the role which random noise can play in the parametric resonance instability of matter fields in the presence of the oscillatory inflation field. We find that the particular idealized form of the noise studied here renders the instability more effective. As a corollary, we obtain a new proof of finiteness of the localization length in the theory of Anderson localization.
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Notes
- 1.
There is, in fact, a small hole in our proof of Theorem 1: in the case of values of k in the resonance band of the noiseless system, the Ψ j are not necessarily identically distributed on SL(2) because of the exponential factor which enters. We still obtain the rigorous result μ(q) > 0 for all values of k, and numerical evidence confirms the validity of the statement μ(q) > μ(0) even for values of k which are in the resonance band. The application of our result to Anderson localization involves values of k which are in the stability bands of the noiseless system and is hence robust - I thank Walter Craig for pointing out this issue.
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Acknowledgements
I wish to thank P. Guyenne, D. Nicholls and C. Sulem for organizing this conference in honor of Walter Craig, and for inviting me to contribute. Walter Craig deserves special thanks for collaborating with me on the topics discussed here, for his friendship over many years, and for comments on this paper. The author is supported in part by an NSERC Discovery Grant and by funds from the Canada Research Chair program.
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Brandenberger, R.H. (2015). Partial Differential Equations with Random Noise in Inflationary Cosmology. In: Guyenne, P., Nicholls, D., Sulem, C. (eds) Hamiltonian Partial Differential Equations and Applications. Fields Institute Communications, vol 75. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2950-4_12
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