# Integral constraints in spectroscopic surveys

## TL;DR

This paper derives a comprehensive formalism for integral constraints in spectroscopic surveys, addressing their impact on clustering analyses, and proposes methods to mitigate systematic effects, especially on large scales.

## Contribution

The paper introduces a complete derivation of integral constraints, including normalisation, shot noise, and wide-angle effects, and applies it to improve large-scale clustering analyses.

## Key findings

- Radial integral constraints can significantly affect large-scale measurements.
- Angular systematics can be mitigated by nulling density fluctuations on specific scales.
- The formalism is validated with mock catalogues in RSD analysis.

## Abstract

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.