What galaxy surveys really measure

TL;DR

This work derives gauge-invariant, first-order expressions for the observable galaxy density on the light cone, Δ({\mathbf n},z)=δ_z({\mathbf n},z)+δV({\mathbf n},z)/V(z), accounting for redshift-space distortions, volume distortions, lensing, and ISW effects. It expresses δ_z in terms of the gauge-invariant density D_g, line-of-sight velocity V, and metric perturbations Ψ,Φ, including an ISW integral, and computes the corresponding volume perturbation δV/V to obtain the full observable perturbation. The angular power spectra are then derived, with C_ℓ(z_S) built from transfer-function combinations F_ℓ(k,z_S) that encode density, velocity, and lensing contributions, and with a parallel radial power spectrum C_ℓ(z_S,z_S'). The analysis shows that while density and redshift-space distortions dominate at low to moderate redshifts, lensing and potential terms become progressively important at higher redshifts or with wide redshift windows, offering a powerful, bias-free framework for cosmological parameter estimation and consistency tests of LCDM with future surveys such as Euclid.

Abstract

In this paper we compute the quantity which is truly measured in a large galaxy survey. We take into account the effects coming from the fact that we actually observe galaxy redshifts and sky positions and not true spatial positions. Our calculations are done within linear perturbation theory for both the metric and the observer velocities but they can be used for non-linear matter power spectra. We shall see that the complications due to the fact that we only observe on our background lightcone and that we do not truly know the distance of the observed galaxy, but only its redshift is not only an additional difficulty, but even more a new opportunity for future galaxy surveys.

Figures (11)

• Figure 1: The (linear) matter power spectrum on the uniform curvature hypersurface (top curve, green), in longitudinal gauge (middle curve, red) and in co-moving gauge (bottom curve, blue).
• Figure 2: Top panel: The transversal power spectrum at (from top to bottom) $z_S=0.1$, $z_S=0.5$, $z_S=1$ and $z_S=3$. Bottom panel: The ratio between the new contributions (lensing+potential) and the total angular power spectrum at (from top to bottom) $z_S=3$, $z_S=1$, $z_S=0.5$ and $z_S=0.1$. Solid lines denote positive contributions whereas dashed lines denote negative contributions.
• Figure 3: The dominant terms at redshifts (from top to bottom) $z_S~=~0.1,~0.5,~1$ and $3$: density (red), redshift space distortion (green), the correlation of density with redshift space distortion (blue), lensing (magenta), Doppler (cyan), see Table \ref{['t:color']}. The potential terms are too small to appear on our log-plot.
• Figure 4: Top panel: The various terms as a function of $z_S$ for fixed value of $\ell=20$: density (red), redshift space distortion (green), the correlation of density with redshift space distortion (blue), lensing (magenta), Doppler (cyan) and potential (black), see Table \ref{['t:color']}. Solid lines denote positive contributions whereas dashed lines denote negative contributions. Bottom panel: The ratio between the new contributions (lensing+potential) and the total angular power spectrum as a function of $z_S$ for fixed value of $\ell=20$.
• Figure 5: Different terms of $C_\ell(z_S,z_{S'})$ at $\ell=20$ for redshifts (from top to bottom) $z_S=0.1,~0.5,~1$ and $3$, plotted as a function of $z_{S'}$: standard term, i.e. $C^{DD}_\ell+C^{zz}_\ell+2C^{Dz}_\ell$ (blue), lensing (magenta), Doppler (cyan), potential (black), see Table \ref{['t:color']}. Solid lines denote positive contributions whereas dashed lines denote negative contributions.