CosmoCoffee

This paper examines various approximations to CMB likelihood functions, and attempts to generalise to polarisation. The conclustion seems to be that accurately getting the shape of the polarisation function from simple fittings is difficult - I agree. In Eq 43 the authors appear to be taking the log of things like C^{TE} that can be negative - so does this not fail to return sensible likelihoods for some models?

There is a nice family of approximations the authors have not discussed: C^{1/3} and C^{-1/3} parameterizations are both much better than Gaussian + log normal - see the appendix of astro-ph/0511703. I did spend a little time myself trying to generalise this to polarisation, but came to the conclusion that it was non-trivial.

Note that WMAP are currently using small scale polarized likelihood functions that are quite inadequate once the data becomes much better (e.g. Planck).
Given the complications, for near-full sky observations perhaps it would be more accurate just to use the full 'Wishart' distribution with some effective degrees of freedom, neglecting (small) off-l correlations? (obviously with exact likelihood on large scales)

Hi Antony,

Thanks - we have now read astro-ph/0511703 and their likelihood fit is very interesting. We'll include this in the next version of the paper.

With the offset lognormal function, it is true that there is the possiblity that you end up trying to take the logarithm of a negative number for certain models. This was true even for an auto-power spectrum with a negative "a" parameter (the free parameter in Eq 43) multiplied by the measured power. However, letting [tex]a \, \to \infty[/tex] (for +ve measured [tex]C_l[/tex]) or [tex]a \, \to -\infty[/tex] (for -ve measure [tex]C_l[/tex]) will make the distribution tend to a Gaussian, and the fit is OK for all models in this limit. In general, we're only concerned with the slightly non-Gaussian regime where this taking the log of a negative number isn't a problem (except for extremely wacky models). An alternative way of thinking about this is to say that with this fitting function you can only compare certain models that end up taking a log of a positive number. All other models are assumed to be highly unlikely. This doesn't stop this function being used as a fitting function -- in fact, as we show in the paper, it can match the all-sky likelihood (which we showed is given by a Wishart distribution) well in the directions of the cross- and auto-power spectra.

cheers,
Michael.