Article,
Estimators for local non-Gaussianities
(2006)cite arxiv:astro-ph/0606001 Comment: 26 pages. v2: added comments about the approximations used, published JCAP version.
Abstract
We study the Likelihood function of data given f_NL for the so-called local
type of non-Gaussianity. In this case the curvature perturbation is a
non-linear function, local in real space, of a Gaussian random field. We
compute the Cramer-Rao bound for f_NL and show that for small values of f_NL
the 3-point function estimator saturates the bound and is equivalent to
calculating the full Likelihood of the data. However, for sufficiently large
f_NL, the naive 3-point function estimator has a much larger variance than
previously thought. In the limit in which the departure from Gaussianity is
detected with high confidence, error bars on f_NL only decrease as 1/ln Npix
rather than Npix^-1/2 as the size of the data set increases. We identify the
physical origin of this behavior and explain why it only affects the local type
of non-Gaussianity, where the contribution of the first multipoles is always
relevant. We find a simple improvement to the 3-point function estimator that
makes the square root of its variance decrease as Npix^-1/2 even for large
f_NL, asymptotically approaching the Cramer-Rao bound. We show that using the
modified estimator is practically equivalent to computing the full Likelihood
of f_NL given the data. Thus other statistics of the data, such as the 4-point
function and Minkowski functionals, contain no additional information on f_NL.
In particular, we explicitly show that the recent claims about the relevance of
the 4-point function are not correct. By direct inspection of the Likelihood,
we show that the data do not contain enough information for any statistic to be
able to constrain higher order terms in the relation between the Gaussian field
and the curvature perturbation, unless these are orders of magnitude larger
than the size suggested by the current limits on f_NL.
