Statistical geometry and the microwave background

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Abstract
We investigate some aspects of the statistical geometry of 2D random fields relevant to studies of temperature anisotropics in the microwave background. We extend the work of Coles & Barrow to calculate the expectation values of the Euler–Poincaré characteristic of the excursion sets of Gaussian and non-Gaussian random fields above various threshold levels. Monte Carlo simulations of Gaussian and non-Gaussian fields possessing the covariance function expected in CDM models are used to assess the usefulness of this, and other, pattern statistics as discriminators between Gaussian and non-Gaussian random fields and hence between Gaussian and non-Gaussian primordial fluctuations. We find that such statistics should provide useful alternatives to ‘standard’ tests based on higher order correlations. We discuss the expected fractal behaviour of temperature contours. We discuss the implications of these results for the study of random fields in three dimensions, taking the view that the analytically simpler 2D case may provide a useful testing ground for ideas involved in the study of the 3D problem.

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