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 power spectrum (along with the mean y-parameter) does not provide a complete description of the SZ effect. Therefore, SZ-based constraints on cosmological parameters and on cluster gastrophysics which assume Gaussianity will be biased. We derive an analytic expression for the n-point joint PDF of the SZ power spectrum. Our derivation, which is based on the halo model, has several advantages: it is expressed in an integral form which allows quick computation; it is applicable to any given survey and any given angular scale; and it is straightforward to incorporate many of the complexities which arise when modeling the SZ signal. To illustrate, we use our expression to estimate p(C-l), the one-point PDF of the SZ power spectrum. For small sky coverage (applicable to BIMA/CBI and the Sunyaev Zel'dovich Array experiments), our analysis shows that p(C-l) on the several arcminute scale is expected to be strongly skewed, peaking at a value well below the mean and with a long tail which extends to high C-l values. In the limit of large sky coverage (applicable to the South Pole Telescope and Planck), p(C-l) approaches a Gaussian form. However, even in this limit, the variance of the power spectrum is very different from the naive Gaussian-based estimate, because different l models are strongly correlated, making the cosmic variance of the SZ bandpower much larger than the naive estimate. Our analysis should also be useful for modeling the PDF of the power spectrum induced by gravitational lensing at large l.