The effect of the linear term on the wavelet estimator of primordial non-Gaussianity

In this work, we present constraints on different shapes of primordial non-Gaussianity using the Wilkinson Microwave Anisotropy Probe (WMAP) seven-year data and the spherical Mexican hat wavelet f_nl estimator including the linear term correction. In particular, we focus on the local, equilateral and orthogonal shapes. We first analyse the main statistical properties of the wavelet estimator and show the conditions to reach optimality. We include the linear term correction in our estimators and compare the estimates with the values already published using only the cubic term. The estimators are tested with realistic WMAP simulations with anisotropic noise and the WMAP KQ 75 sky cut. The inclusion of the linear term correction shows a negligible improvement (≤1 per cent) in the error bar for any of the shapes considered. The results of this analysis show that, in the particular case of the wavelet estimator, the optimality for WMAP anisotropy levels is basically achieved with the mean subtraction, and in practical terms there is no need of including a linear term once the mean has been subtracted. Our best estimates are now f̂ nl(loc)=39.0±21.4, f̂ nl(eq)=-62.8±154.0 and f̂ nl(ort)=-159.8±115.1. We have also computed the expected linear term correction for simulated Planck maps with anisotropic noise at 143 GHz following the Planck Sky Model and including a mask. The improvement achieved in this case for the local f_nl error bar is also negligible (0.4 per cent).