Primordial non-Gaussianity: local curvature method and statistical significance of constraints on f$_{NL$ from WMAP data}

We test the consistency of estimates of the non-linear coupling constant f$_{NL$ using non-Gaussian cosmic microwave background (CMB) maps generated by the method described in the work of Liguori, Matarrese \amp Moscardini. This procedure to obtain non-Gaussian maps differs significantly from the method used in previous works on the estimation of f$_NL$. Nevertheless, using spherical wavelets, we find results in very good agreement with Mukherjee \amp Wang, showing that the two ways of generating primordial non-Gaussian maps give equivalent results. Moreover, we introduce a new method for estimating the non-linear coupling constant from CMB observations by using the local curvature of the temperature fluctuation field. We present both Bayesian credible regions (assuming a flat prior) and proper (frequentist) confidence intervals on f$_NL$, and discuss the relation between the two approaches. The Bayesian approach tends to yield lower error bars than the frequentist approach, suggesting that a careful analysis of the different interpretations is needed. Using this method, we estimate f$_NL$=-10$^+270$$_-260$ at the 2$\sigma$ level (Bayesian) and f$_NL$=-10$^+310$$_-270$ (frequentist). Moreover, we find that the wavelet and the local curvature approaches, which provide similar error bars, yield approximately uncorrelated estimates of f$_NL$ and therefore, as advocated in the work of Cabella et al., the estimates may be combined to reduce the error bars. In this way, we obtain f$_NL$=-5 +/- 85 and f$_NL$=-5 +/- 175 at the 1$\sigma$ and 2$\sigma$ level respectively using the frequentist approach. }