Beware of commonly used approximations II: estimating systematic biases
  in the best-fit parameters
  Cosmological parameter estimation from forthcoming experiments promise to
reach much greater precision than current constraints. As statistical errors
shrink, the required control over systematic errors increases. Therefore,
models or approximations that were sufficiently accurate so far, may introduce
significant systematic biases in the parameter best-fit values and jeopardize
the robustness of cosmological analyses. We present a general expression to
estimate a priori the systematic error introduced in parameter inference due to
the use of insufficiently good approximations in the computation of the
observable of interest or the assumption of an incorrect underlying model.
Although this methodology can be applied to measurements of any scientific
field, we illustrate its power by studying the effect of modeling the angular
galaxy power spectrum incorrectly. We also introduce Multi_CLASS, a new, public
modification of the Boltzmann code CLASS, which includes the possibility to
compute angular cross-power spectra for two different tracers. We find that
significant biases in most of the cosmological parameters are introduced if one
assumes the Limber approximation or neglects lensing magnification in modern
galaxy survey analyses, and the effect is in general larger for the
multi-tracer case, especially for the parameter controlling primordial
non-Gaussianity of the local type, $f_{\rm NL}$.
