Integral constraints in spectroscopic surveys

TL;DR

This work addresses biases introduced when the survey selection function is partially inferred from the data by deriving a complete formalism for integral constraints (IC) in spectroscopic surveys. It develops global, radial, and angular IC corrections, detailing how window functions and shot noise alter observed density fluctuations and 2-point statistics, and shows how to implement these corrections in RSD analyses using realistic mocks. The radial IC can significantly affect large-scale clustering, and the authors demonstrate that proper modelling removes biases in cosmological fits; they also propose an angular IC to mitigate unknown angular systematics and discuss combining ICs. The results provide practical prescriptions for unbiased large-scale analyses in current and upcoming surveys, with implications for primordial non-Gaussianity studies and beyond.

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.

Figures (15)

• Figure 1: Left: power spectrum multipoles (blue: monopole, red: quadrupole, green: hexadecapole) including the different contributions to the global integral constraint (see text). The complete result (dot-dashed curve) cannot be distinguished from the partial correction (dashed). Right: same, with the radial integral constraint. See section \ref{['sec:analysis_radial']} for more details about the survey configuration assumed in this figure.
• Figure 2: Normalised shot noise contributions from the global and radial integral constraints (blue: monopole, red: quadrupole, green: hexadecapole). As expected, the Poisson shot noise in $P^{\mathrm{c}}_{0}(k)$ is cancelled by the integral constraint contribution at large scales. The same survey case as in figure \ref{['fig:integral_constraint_terms_global_radial']} is assumed.
• Figure 3: Left: the window function multipoles $\mathcal{W}_{\ell}^{\delta,\delta}(s)$. Right: the window function multipoles $\mathcal{W}_{\ell p}^{\mathrm{rad},\mathrm{rad}}(s,\Delta)$.
• Figure 4: Left: the CMASS footprint, divided in $6$ different chunks. Right: different estimates of the redshift density $n(z)$, using bins $\Delta z = 0.005$: the redshift density of the true survey selection function (continuous black curve), and, for one mock data realisation, $n(z)$ measured in the whole CMASS footprint (dashed black curve) and in the $6$ chunks individually. The scatter between the $n(z)$ estimates is due to noise and clustering.
• Figure 5: Top panels: power spectrum multipoles (upper left: monopole, upper right: quadrupole, bottom: hexadecapole) measured from the 84 N-series mocks (see section \ref{['sec:mocks']}), using three different ways to model the redshift distribution in the random catalogues. The true selection function is used in the baseline (blue). The random redshift distribution of the binned (orange) and shuffled (green) schemes is inferred in $6$ separate chunks of each mock data realisation, as described in the text. For comparison, the red curve shows the effect of the binned scheme applied to the full CMASS footprint. The blue shaded area represents the standard deviation of the mocks. Bottom panels: difference of the shuffled and binned schemes to the baseline, with the standard deviation of the difference given by the error bars, normalised by the standard deviation of the mocks.