A Bayesian Estimate of the Skewness of the Cosmic Microwave Background

We propose a formalism for estimating the skewness and angular power spectrumof a general Cosmic Microwave Background data set. We use the EdgeworthExpansion to define a non-Gaussian likelihood function that takes into accountthe anisotropic nature of the noise and the incompleteness of the sky coverage.The formalism is then applied to estimate the skewness of the publiclyavailable 4 year Cosmic Background Explorer (COBE) Differential MicrowaveRadiometer data. We find that the data is consistent with a Gaussian skewness,and with isotropy. Inclusion of non Gaussian degrees of freedom has essentiallyno effect on estimates of the power spectrum, if each $C_\ell$ is regarded as aseparate parameter or if the angular power spectrum is parametrized in terms ofan amplitude (Q) and spectral index (n). Fixing the value of the angular powerspectrum at its maxiumum likelihood estimate, the best fit skewness is$S=6.5\pm6.0\times10^4(\muK)^3$; marginalizing over Q the estimate of theskewness is $S=6.5\pm8.4\times10^4(\muK)^3$ and marginalizing over n one has$S=6.5\pm8.5\times10^4(\muK)^3$.Comment: submitted to Astrophysical Journal Letter