Can we really measure fNL from the galaxy power spectrum?

The scale-dependent galaxy bias generated by primordial non-Gaussianity (PNG) can be used to detect and constrain deviations from standard single-field inflation. The strongest signal is expected in the local model for PNG, where the amplitude of non-Gaussianity can be expressed by a set of parameters (fnl, gnl, ...). Current observational constraints from galaxy clustering on fnl and gnl assume that the others PNG parameters are vanishing. Using two sets of cosmological N-body simulations where both fnl and gnl are non-zero, we show that this strong assumption generally leads to biased estimates and spurious redshift dependencies of the parameters. Additionally, if the signs of fnl and gnl are opposite, the amplitude of the scale-dependent bias is reduced, possibly leading to a false null detection. Finally we show that model selection techniques like the Bayesian evidence can (and should) be used to determine if more than one PNG parameter is required by the data.