Statistics and Topology of the COBE Differential Microwave Radiometer First-year Sky Maps

We use statistical and topological quantities to test the COBE DMR first-year sky maps against the hypothesis that the observed temperature fluctuations reflect Gaussian initial density perturbations with random phases. Recent papers discuss specific quantities as discriminators between Gaussian and non-Gaussian behavior, but the treatment of instrumental noise on the data is largely ignored. The presence of noise in the data biases many statistical quantities in a manner dependent on both the noise properties and the unknown cosmic microwave background temperature field. Appropriate weighting schemes can minimize this effect, but it cannot be completely eliminated. Analytic expressions are presented for these biases, and Monte Carlo simulations are used to assess the best strategy for determining cosmologically interesting information from noisy data. The genus is a robust discriminator that can be used to estimate the power-law quadrupole-normalized amplitude, Qrms - PS, independently of the two-point correlation function. The genus of the DMR data is consistent with Gaussian initial fluctuations with Qrms - PS = (15.7 ± 2.2) - (6.6 ± 0.3)(n - 1) μK, where n is the power-law index. Fitting the rms temperature variations at various smoothing angles gives Qrms - PS = 13.2 ± 2.5 μK and n = 1.7- 0.6 + 0.3. While consistent with Gaussian fluctuations, the first year data are only sufficient to rule out strongly non-Gaussian distributions of fluctuations.