The impact of the redshift-dependent selection effect of halos on the redshift-space power spectrum

TL;DR

This work analyzes how redshift-dependent selection of halos (and by extension galaxies) biases the redshift-space power spectrum. By constructing AM catalogs that reproduce observed $n(z)$ and comparing to single-mass-threshold catalogs, the authors quantify the bias across linear to quasi-nonlinear scales, supported by both N-body simulations and emulator-based theory. They find that for SDSS-like LOWZ/CMASS samples the fractional changes in monopole and quadrupole moments are typically below 1–2% at $k \,\lesssim\ 0.3~h\,\text{Mpc}^{-1}$, with larger deviations only for strongly redshift-dependent $n(z)$ cases. A Fisher-matrix analysis indicates such selection effects are unlikely to bias cosmological parameter estimates beyond the statistical uncertainties for present survey volumes, though they may matter for future all-sky, large-volume missions like DESI or Euclid. Overall, the results validate the robustness of current analyses to modest selection effects and provide a method to incorporate potential biases in planning next-generation surveys.

Abstract

In a wide-area spectroscopic survey of galaxies, it is nearly impossible to obtain a homogeneous sample of galaxies with respect to galaxy properties such as stellar mass and host halo mass across a range of redshifts. Despite the selection effect, theoretical templates in most analyses assume single tracers when compared with the measured clustering quantities. We demonstrate analytically that the selection effect inevitably introduces a bias in the redshift-space power spectrum on scales from linear to nonlinear scales. To quantitatively assess the impact of the selection effect, we construct mock galaxy catalogs from halos in N-body simulations by selecting halos above redshift-dependent mass thresholds such that the resulting redshift distribution of the halos, $n(z)$, matches that of SDSS-like galaxies. We find that the selection effect causes fractional changes of up to only 1% and 2% in the monopole and quadrupole moments of the redshift-space power spectrum at $k\lesssim 0.3~h{\rm{Mpc}}^{-1}$, respectively, compared to the moments for the single mass-threshold (therefore single tracer) sample, for $n_{\rm g}(z)$ of the SDSS-like galaxy samples. We also argue that the selection effect is unlikely to cause a significant bias in the estimation of cosmological parameters using the Fisher matrix method, provided that the redshift-dependent selection effect is modest.

Figures (8)

• Figure 1: The number density of LOWZ ( left panel) and CMASS ( middle panel) galaxies as a function of redshift, denoted as $n(z)$, taken from the SDSS DR11 catalog. Note that $n(z)$ is the number density per unit redshift interval. For illustrative purpose we also show the comoving $z$-coordinate on the bottom axis, computed using the inverse relation between redshift and the comoving radial distance at each redshift, $z=z^{-1}(\chi)$, based on the Planck 2018 cosmology. This $z$-coordinate can be compared to the size of AbacusSumitt simulation box ($2~h^{-1}$Gpc) used in this paper. The LOWZ galaxies are taken from the redshift range $z\simeq[0.09,0.5]$, while the CMASS galaxies are from the range $z\simeq [0.4,0.8]$. Note that we set the comoving $z$-coordinate at the lowest redshift to $\chi=0$ for each galaxy sample. We also consider a case of $n(z)$ following a power-law scaling given as $n(z)\propto z^{-\alpha}$ ( right panel) to study the impact of galaxy selection effect as a general case.
• Figure 2: Halo mass thresholds as a function of redshift, $M_{\rm th}(z)$, for the LOWZ ( left panel) and CMASS ( middle panel) samples, respectively. We select halos with mass above $M_{\rm th}(z)$ in each redshift bin, so that the resulting number density of halos reproduces the number density of each sample, $n(z)$, in Fig. \ref{['fig:SDSSdens']} (also see Section \ref{['ssub:mocks_w-selectioneffect']}). We determine the mass threshold function based on the average of 25 realizations of the halo catalog. The three solid lines in the right panel show the mass threshold for each of the three power-law density cases in Fig. \ref{['fig:SDSSdens']}.
• Figure 3: A figure illustrating the method for incorporating the window effect on the single-mass threshold sample, used to quantify the impact of selection effects in comparison to the results of the AM catalog. In each panel, the dashed line shows the number density which corresponds to the mass threshold in the legend. For the single mass-threshold sample, we randomly select halos above this single mass threshold in each redshift bin such that the redshift dependence of the number density is proportional to that for the LOWZ, CMASS or power-law sample in Fig. \ref{['fig:SDSSdens']} (here we show results only the for the LOWZ and CMASS samples) (see Section \ref{['ssub:mock_single_mass_threshold']} for details on the single mass-threshold sample). This single mass-threshold hold sample has the same window function as that for the AM sample.
• Figure 4: A comparison of the two power spectra, $P_{\rm AM}$ and $P_{M_{\rm th}}$, for the abundance-matching (AM) sample and the single mass-threshold ($M_{\rm th}$) sample both of which reproduce $n(z)$ of the LOWZ sample. We show the ratio for the real-space power spectrum ($\textit{left panel}$), and the monopole ($\textit{middle panel}$) and quadrupole ($\textit{right panel}$) moments of the redshift-space power spectrum. The data points in each panel are the mean of the $50$ realizations and the error bars are the $1\sigma$ error on the mean, estimated by dividing the standard deviation by $\sqrt{50}$. The black dashed vertical line in each panel indicates the half of the Nyquist wavenumber, and the solid line is the ratio computed using the Dark Emulator in Kobayashi et al. kobayashi2022fullshapecosmologyanalysissdssiii (see text for the details).
• Figure 5: Similar to the previous figure, but the results for the CMASS sample.