The Covariance of Photometric and Spectroscopic Two-Point Statistics: Implications for Cosmological Parameter Inference

TL;DR

This paper derives an analytic expression for the covariance between photometric and spectroscopic two-point statistics, showing it is negligible for cosmological parameter inference in large-scale surveys.

Contribution

It provides the first analytic derivation of the covariance between photometric and spectroscopic power spectra under Gaussian assumptions and plane-parallel approximation.

Findings

Covariance exists on large radial scales but is negligible for parameter inference.

Neglecting this covariance is justified due to low information content of these modes.

Analytic expression aids future combined analyses of photometric and spectroscopic data.

Abstract

To combine information from measurements of the redshift-space power spectrum from spectroscopic data with angular weak lensing, galaxy clustering and galaxy-galaxy lensing power spectra from photometric surveys (i.e. the $3 \times 2$ point statistics), we must account for the covariance between the two probes. Currently any covariance between the two types of measurements is neglected as existing photometric and spectroscopic surveys largely probe different cosmological volumes. This will cease to be the case as data arrives from Stage-IV surveys. In this paper we derive an analytic expression for the covariance between photometric 2D angular power spectra and the 3D redshift-space power spectrum for Gaussian fields under the plane-parallel approximation. We find that the two probes are covariant on large radial scales, but because the information content of these modes is extremely low due to sample variance, we forecast that it is safe to neglect this covariance when performing cosmological parameter inference.