The Probability Distribution Function of the Sunyaev‐Zel’dovich Power Spectrum: An Analytical Approach

The Sunyaev Zel'dovich (SZ) signal is highly non-Gaussian, so the SZ powerspectrum (along with the mean $y$ parameter) does not provide a completedescription of the SZ effect. Therefore, SZ-based constraints on cosmologicalparameters and on cluster gastrophysics which assume Gaussianity will bebiased. We derive an analytic expression for the $n$-point joint PDF of the SZ powerspectrum. Our derivation, which is based on the halo model, has severaladvantages: it is expressed in an integral form which allows quick computation;it is applicable to any given survey and any given angular scale; it isstraightforward to incorporate many of the complexities which arise whenmodeling the SZ signal. To illustrate, we use our expression to estimate$p(C_\ell)$, the one-point PDF of the SZ power spectrum. For small sky coverage(applicable to BIMA/CBI and the Sunyaev Zel'dovich Array experiments), ouranalysis shows that $p(C_\ell)$ on the several arc-minute scale is expected tobe strongly skewed, peaking at a value well below the mean and with a long tailwhich extends to tail high $C_\ell$ values. In the limit of large sky coverage(applicable to the South Pole Telescope and Planck), $p(C_\ell)$ approaches aGaussian form. However, even in this limit, the variance of the power spectrumis very different from the naive Gaussian-based estimate. This is becausedifferent $\ell$ models are strongly correlated, making the cosmic variance ofthe SZ band-power much larger than the naive estimate. Our analysis should alsobe useful for modeling the PDF of the power spectrum induced by gravitationallensing at large $\ell$.Comment: v2: added one figure and discussions. Will appear in ApJ December issue. 12 pages, 8 figure