What galaxy surveys really measure

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 source velocities but they can be used for nonlinear matter power spectra. We shall see that the complications due to the fact that we only observe on our background light cone, and that we do not truly know the distance of the observed galaxy but only its redshift, not only cause an additional difficulty, but provide even more a new opportunity for future galaxy surveys.

Figure 1

The (linear) matter power spectrum on the uniform curvature hypersurface (top curve, green), in longitudinal gauge (middle curve, red), and in comoving 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 ($\mathrm{\text{lensing}}+\mathrm{\text{potential}}$) 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 line), redshift-space distortion (green line), the correlation of density with redshift-space distortion (blue line), lensing (magenta line), and Doppler (cyan line); see Table . 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 a fixed value of $\ell =20$: density (red line), redshift-space distortion (green line), the correlation of density with redshift-space distortion (blue line), lensing (magenta line), Doppler (cyan line), and potential (black line); see Table . Solid lines denote positive contributions whereas dashed lines denote negative contributions. Bottom panel: The ratio between the new contributions ($\mathrm{\text{lensing}}+\mathrm{\text{potential}}$) and the total angular power spectrum as a function of $z_S$ for a fixed value of $\ell =20$.

Figure 5

Different terms of $C_{\ell }(z_S,z_{S^{\prime }})$ at $\ell =20$ for redshifts (from top to bottom) $z_S=0.1$, 0.5, 1, and 3, plotted as a function of $z_{S^{\prime }}$: standard term, i.e. $C_{\ell }^{DD}+C_{\ell }^{zz}+2C_{\ell }^{Dz}$ (blue line), lensing (magenta line), Doppler (cyan line), and potential (black line); see Table . Solid lines denote positive contributions whereas dashed lines denote negative contributions.

Figure 6

The effect of a window function on the density contribution (top panel), redshift-space distortion (middle panel), and lensing contribution (bottom panel). We have chosen $z_S=0.1$ and $\Delta z_S=0$ (no window, top curve, red), $\Delta z_S=0.002$ (middle curve, green), and $\Delta z_S=0.01$ (bottom curve, blue).

Figure 7

The effect of a window function with width $\Delta z_S=0.1z_S$ on the power spectrum $C_{\ell }(z_S)$ for redshifts (from top to bottom) $z_S=0.1$, 0.5, 1, and 3. The different curves are as follows: density (red line), redshift-space distortion (green line), the correlation of density with redshift-space distortion (blue line), lensing (magenta line), Doppler (cyan line), and gravitational potential (black line); see Table . Solid lines denote positive contributions whereas dashed lines denote negative contributions.

Figure 8

Top panel: The total power spectrum at redshifts (from top to bottom) $z_S=0.1$, $z_S=0.5$, $z_S=1$, and $z_S=3$ smeared by a window function with width $\Delta z_S=0.1z_S$. Bottom panel: The ratio between the new contributions ($\mathrm{\text{lensing}}+\mathrm{\text{potential}}$) 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 9

The various terms as a function of $z_S$ for a fixed value of $\ell =20$ and smeared by a window function with width $\Delta z_S=0.1z_S$: density (red line), redshift-space distortion (green line), the correlation of density with redshift-space distortion (blue line), lensing (magenta line), Doppler (cyan line), and potential (black line); see Table . Here the correlations between the lensing and Doppler and the lensing and potential are not negligible at large $z_S$, and they are included in the Doppler (cyan line), respectively, potential (black line) curves. Solid lines denote positive contributions; dashed lines denote negative contributions. Bottom panel: The ratio between the new contributions ($\mathrm{\text{lensing}}+\mathrm{\text{potential}}$) and the total angular power spectrum, smeared by a window function with width $\Delta z_S=0.1z_S$, plotted as a function of $z_S$ for a fixed value of $\ell =20$.

Figure 10

Cross correlations between different redshift bins $C_{\ell }(z_S,z_{S^{\prime }})$ at $\ell =20$ with a 10% window function and plotted as a function of $z_{S^{\prime }}$. From top to bottom $z_S=0.1$, 0.5, 1, and 3. The standard term, i.e. $C_{\ell }^{DD}+C_{\ell }^{zz}+2C_{\ell }^{Dz}$ (blue line), lensing (magenta line), Doppler (cyan line), and potential (black line) are shown; see Table . Solid lines denote positive contributions whereas dashed lines denote negative contributions.

Figure 11

Integrated power spectrum with a flat distribution between $z_{\min }=0.1$ and $z_{\max }=2$ with Gaussian tails at both ends. The density term is plotted in red and the lensing term in magenta. Note that the lensing term is completely dominated by its anticorrelation with the density and hence is negative.