A new framework for analyzing the effects of small scale inhomogeneities in cosmology

Abstract

We develop a new, mathematically precise framework for treating the effects of nonlinear phenomena occurring on small scales in general relativity. Our approach is an adaptation of Burnett's formulation of the "shortwave approximation", which we generalize to analyze the effects of matter inhomogeneities as well as gravitational radiation. Our framework requires the metric to be close to a "background metric", but allows arbitrarily large stress-energy fluctuations on small scales. We prove that, within our framework, if the matter stress-energy tensor satisfies the weak energy condition (i.e., positivity of energy density in all frames), then the only effect that small scale inhomogeneities can have on the dynamics of the background metric is to provide an "effective stress-energy tensor" that is traceless and has positive energy density---corresponding to the presence of gravitational radiation. In particular, nonlinear effects produced by small scale inhomogeneities cannot mimic the effects of dark energy. We also develop "perturbation theory" off of the background metric. We derive an equation for the "long-wavelength part" of the leading order deviation of the metric from the background metric, which contains the usual terms occurring in linearized perturbation theory plus additional contributions from the small-scale inhomogeneities. Under various assumptions concerning the absence of gravitational radiation and the non-relativistic behavior of the matter, we argue that the "short wavelength" deviations of the metric from the background metric near a point $x$ should be accurately described by Newtonian gravity, taking into account only the matter lying within a "homogeneity lengthscale" of $x$. Finally, we argue that our framework should provide an accurate description of the actual universe.

56 pages, minor corrections

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