Sample Variance of non-Gaussian Sky Distributions

Non-Gaussian distributions of cosmic microwave background (CMB) anisotropies have been proposed to reconcile the discrepancies between different experiments at half-degree scales (Coulson et al. 1994). Each experiment probes a different part of the sky, furthermore, sky coverage is very small, hence the sample variance of each experiment can be large, especially when the sky signal is non-Gaussian. We model the degree-scale CMB sky as a χ2n field with n-degrees of freedom and show that the sample variance is enhanced over that of a Gaussian distribution by a factor of (n+ 6)/n. The sample variance for different experiments are calculated, both for Gaussian and non-Gaussian distributions. We also show that if the distribution is highly non-Gaussian (n <∼ 4) at half-degree scales, then the nonGaussian signature of the CMB could be detected in the FIRS map, though probably not in the COBE map. Subject headings: cosmic microwave background cosmology: theory Recently, several groups have reported results on the measurement of anisotropies of the cosmic microwave background (CMB) at degree scales (de Bernardis et al. 1994; Cheng et al. 1993; Gunderson et al. 1993; Meinhold et. al 1993; Schuster et al. 1993; Tucker et al. 1993; Wollack et al. 1993). The beam size, beam throw, most sensitive angular scale, sky coverage and quoted rms temperature anisotropies are summarized in Table 1. The results from these experiments do not agree with each other, in particular the results from the same experiment, MAX, for two different part of the sky, the Gamma Ursa Minor (GUM) region and the mu-Pegasi (MuP) region, contradict each other at 2σ level. A way to reconcile these measurements is to have a non-Gaussian distribution of temperature anisotropies at halfdegree scales (e.g. Coulson et al. 1994). At present, there are still large uncertainties in all experiments due to foreground subtractions, therefore the need to invoke non-Gaussian temperature distributions remains to be established. Since different experiments probe different part of the sky and the sky coverage of each experiment is small, the sample variance of each experiment can be large, especially when the sky signal is non-Gaussian. The goal of this paper is to quantify the difference in the expected sample variance between non-Gaussian and Gaussian fields in order to determine if this effect could be responsible for the discrepancy between experiments. We also estimate the minimum sample size for each experiment in order for the sample variance to be less than 20μK, both for Gaussian and non-Gaussian distributions. Regarding the statistical analysis, the most important quantity is the number of independent measurements in a single sample. However, the data points in CMB experiments are correlated and therefore contain less statistical information. Thus, it is useful to determine the effective number of data points, Ne, defined as the number of independent measurements of temperature anisotropies in each experiment (we will discuss how to estimate Ne for each experiment later). Expressed in terms of Ne, the sample-averaged rms temperature