Beware of commonly used approximations I: errors in forecasts
  In the era of precision cosmology, establishing the correct magnitude of
statistical errors in cosmological parameters is of crucial importance.
However, widely used approximations in galaxy surveys analyses can lead to
parameter uncertainties that are grossly mis-estimated, even in a regime where
the theory is well understood (e.g., linear scales). These approximations can
be introduced at three different levels: in the form of the likelihood, in the
theoretical modelling of the observable and in the numerical computation of the
observable. Their consequences are important both in data analysis through
e.g., Markov Chain Monte Carlo parameter inference, and when survey instrument
and strategy are designed and their constraining power on cosmological
parameters is forecasted, for instance using Fisher matrix analyses. In this
work, considering the galaxy angular power spectrum as the target observable,
we report one example of approximation for each of such three categories:
neglecting off-diagonal terms in the covariance matrix, neglecting cosmic
magnification and using the Limber approximation on large scales. We show that
these commonly used approximations affect the robustness of the analysis and
lead, perhaps counter-intuitively, to unacceptably large mis-estimates of
parameters errors (from few~$10\%$ up to few~$100\%$) and correlations.
Furthermore, these approximations might even spoil the benefits of the nascent
multi-tracer and multi-messenger cosmology. Hence we recommend that the type of
analysis presented here should be repeated for every approximation adopted in
survey design or data analysis, to quantify how it may affect the results. To
this aim, we have developed \texttt{Multi\_CLASS}, a new extension of
\texttt{CLASS} that includes the angular power spectrum for multiple (galaxy
and other tracers such as gravitational waves) populations.
