Integral constraints in spectroscopic surveys
  Clustering analyses of spectroscopic surveys are based upon density
fluctuations, which are estimated by comparing the observed tracer density
field to a selection function accounting for the survey density and geometry.
However, this survey selection function is commonly partly inferred from the
observed data itself, leading to so-called integral constraints, for which we
propose a complete derivation. We discuss the normalisation of the introduced
window functions, the shot noise contribution to the integral constraint
corrections and wide-angle effects. Using this formalism, we review the
well-known global integral constraint, arising when the expected mean galaxy
density is taken to be the measured one. Another, stronger, constraint is
imposed when the radial selection function itself is estimated from the data
redshift distribution, as is often the case in the literature. We find that the
impact of such a radial integral constraint can be as significant as the window
function effect at large scales, depending on the survey geometry. Equations
for this radial integral constraint are derived within our general formalism.
We assess the validity of our approach by performing a Redshift Space
Distortions (RSD) analysis on mock catalogues and emphasise that our results
may be even more useful for analyses focusing on larger scales. Finally, as a
further application, we show that unknown angular systematics can be mitigated
by nulling the density fluctuations on a chosen angular scale. The induced loss
of clustering is modelled by an angular integral constraint which can be
combined with the radial one.
