The integrated bispectrum as a test of cosmic microwave background non-Gaussianity: detection power and limits on f$_{NL$ with WMAP data}

We propose a fast and efficient bispectrum statistic for cosmic microwave background (CMB) temperature anisotropies to constrain the amplitude of the primordial non-Gaussian signal measured in terms of the non-linear coupling parameter f$_{NL$. We show how the method can achieve a remarkable computational advantage by focusing on subsets of the multipole configurations, where the non-Gaussian signal is more concentrated. The detection power of the test increases roughly linearly with the maximum multipole, as shown in the ideal case of an experiment without noise and gaps. The CPU-time scales as l$^3$$_max$ instead of l$^5$$_max$ for the full bispectrum, which for Planck resolution l$_max$ \~{} 3000 means an improvement in speed of a factor of 10$^7$ compared with the full bispectrum analysis with minor loss in precision. This approach is complementary to the fast method introduced by Komatsu, Spergel \amp Wandelt using a reconstruction of the primordial fluctuation field. We find that the introduction of a galactic cut partially destroys the optimality of the configuration, which will then need to be dealt with in the future. We find for an ideal experiment with l$_max$ = 2000 that upper limits of f$_NL$ \lt 8 can be obtained at 1$\sigma$. For the case of the WMAP experiment, we would be able to put limits of |f$_NL$| \lt 40 if no galactic cut were present. Using the real data with a galactic cut, we obtain an estimate of -80 \lt f$_NL$ \lt 80 and -160 \lt f$_NL$ \lt 160 at 1 and 2$\sigma$, respectively.