Scaling solution for cosmological fluctuations and large-scale structure

We discuss the nonlinear multiplicative stochastic behavior of chaotic inflation with an emphasis on the relevance of the ‘‘scaling’’ effect for large-scale structure formation. We derive a simple criterion for the appearance of scaling behavior in the λφ4 inflaton model in an asymptotic limit which remains valid throughout the inflationary epoch. We show explicitly that in this limit the onset of the scaling regime does not require any special initial conditions and that it is independent of the self-coupling constant λ. A decrease in the dispersion of field fluctuations at late times is seen in the course of stochastic evolution, and shown to follow from the ‘‘squeezing’’ of probability distributions due to the nonlinear stochastic effect. Non-Gaussian statistics in adiabatic fluctuations are important only for superhorizon scales and the scaling regime does not lead to any significant statistical properties on currently observable scales. However, the scaling effect gives some cosmological consequences very different from what we expect in the naive diffusion approximation for quantum fluctuations. We show that the classical treatment (deterministic) of the inflaton field (essentially a quantum-mechanical object) becomes valid towards the end of inflation because of nonlinear effects.