Optimal weighting in galaxy surveys: Application to redshift-space distortions

Using multiple tracers of large-scale structure allows to evade the limitations imposed by sampling variance for some parameters of interest in cosmology. We demonstrate the optimal way of carrying out a multitracer analysis in a galaxy redshift survey by considering the principal components of the shot noise matrix from two-point clustering statistics. We show how to construct two tracers that maximize the benefits of sampling variance and shot noise cancellation using optimal weights. On the basis of high-resolution $N$-body simulations of dark matter halos we apply this technique to the analysis of redshift-space distortions and demonstrate how constraints on the growth rate of structure formation can be substantially improved. The primary limitations are nonlinear effects, which cause significant biases in the method already at scales of $k<0.1h{\mathrm{Mpc}}^{-1}$, suggesting the need to develop nonlinear models of redshift-space distortions in order to extract the maximum information from future redshift surveys. Nonetheless we find gains of a factor of a few in constraints on the growth rate achievable when merely the linear regime of a galaxy survey like EUCLID is considered.

Figure 1

Three different eigenvectors of the shot noise matrix obtained from a halo catalog that has been split into 30 mass bins of equal number density. The two non-Poisson eigenvectors (upper red and lower blue stars) are overplotted with their best-fit weighting functions $w_{\pm }$ as solid lines with functional form given in the top left of the panel in corresponding colors. One representative Poisson eigenvector is shown as green circles; when used as a weighting function it yields a very low clustering signal-to-noise ratio $\Sigma$, due to its oscillatory behavior.

Figure 2

Real-space auto-power and cross-power spectra of the two fields ${\delta }_{+}$ and ${\delta }_{-}$ (as indicated in the upper right corner in corresponding order), obtained through weighting the halo catalog by the two non-Poisson eigenvectors of the shot noise matrix $w_{+}$ and $w_{-}$. The long-dashed black line shows the dark matter power spectrum from the simulation and the dotted line its linear theory prediction. The three different estimators for the relative bias $\alpha$ between ${\delta }_{+}$ and ${\delta }_{-}$, as defined in Eq. (11), are depicted in a dot-dashed line ($C_{+-}/C_{++}$), dot-dot-dot-dashed line ($C_{--}/C_{+-}$) and dashed line ($\sqrt{C_{--}/C_{++}}$). For visibility, they were shifted upwards by a factor of ${10}^3$.

Figure 3

Sampling variance statistic ${\sigma }_{\mathrm{SV}}$ for five different tracer selections at $z=0$ as described in the text, $20/80$ (dot-dot-dashed blue line), $50/50$ (dot-dashed green line), $80/20$ (dashed red line), $u/w_{-}$ (solid orange line) and $w_{+}/w_{-}$ (long-dashed yellow line). Dotted lines show the corresponding results for uncorrelated modes by swapping the real and imaginary parts of one of the tracer’s Fourier modes.

Figure 4

Fisher matrix element $F_{ff}$ as a function of $k$ at $\mu =1$ and $z=0$ for various tracer selections (indicated in the top right of each panel). Results are shown for a single-tracer analysis with the first (solid blue line) and the second tracer (dotted green line) used separately, both of the tracers combined in a multitracer analysis (dashed red line), and each of the tracers combined with idealistic dark matter observations (dot-dashed orange and long-dashed yellow lines, respectively).

Figure 5

Fit for the redshift-space distortion parameter $\beta$ (top) and the product $f{\sigma }_8$ (bottom) from the two-point clustering statistics of halos in an $N$-body simulation with effective volume $V_{\mathrm{eff}}\simeq 6.6h^{-3}{\mathrm{Gpc}}^3$ and halo-mass resolution $M_{\min }\simeq 9.4\times {10}^{11}{h}^{-1}{\mathrm{M}}_{\odot }$ at $z=0$. LEFT: Conventional single-tracer analysis utilizing all halos (not weighted) from the same catalog. RIGHT: Multitracer analysis with the two fields ${\delta }_{+}$ and ${\delta }_{-}$, obtained through weighting the halo catalog with its principal components $w_{+}$ and $w_{-}$. The fitting results are shown in logarithmic bins of $k$ (points with $1\mathrm{\text{−}}\sigma$ error bars), as well as cumulative as a function of $k=k_{\max }$ with fixed $k_{\min }=0.0048h{\mathrm{Mpc}}^{-1}$ (red solid lines with shaded region). Dotted lines show the linear theory prediction with $f={\Omega }_{\mathrm{m}}^{0.55}$ and ${\sigma }_8=0.81$, dashed lines the cumulative sampling variance limit.

Figure 6

Fit for the growth rate $f$ from the two-point clustering statistics of halos and the dark matter combined. LEFT: All uniform halos (not weighted) and the dark matter. RIGHT: All halos, weighted with the lowest stochasticity weight $w_{-}$, and the dark matter. The meaning of lines and symbols is the same as in Fig. 5.

Figure 7

Ratios of the binned one-sigma error bars (from Fig. 5) on $\beta$ (upper solid blue line) and $f{\sigma }_8$ (dashed blue line) when comparing the optimal multitracer to the single-tracer analysis at $z=0$ (left panel) and $z=1$ (right panel). Additionally, the improvement in constraints on $\beta$ from a combined clustering analysis of halos and dark matter as compared to the single-tracer case is shown in the middle solid green line (no weighting of halos) and the lower solid red line (optimally weighted halos). Adding a log-normal scatter of ${\sigma }_{\ln M}=0.1$ (dot-dashed line) and ${\sigma }_{\ln M}=0.5$ (dotted line) to the halo masses results in a degradation of the constraints from the weighted fields.

Figure 8

Halo model prediction for the improvement on ${\sigma }_{\beta }$ from the single-tracer analysis to the optimal multitracer analysis (upper blue lines) at $k\simeq 0.016h{\mathrm{Mpc}}^{-1}$ as a function of the lower halo-mass threshold $M_{\min }$ at $z=0$ (solid) and $z=1$ (dashed). The results for the combined analysis of uniform halos (not weighted) and dark matter (middle green lines), as well as the combination of optimally weighted halos with the dark matter (lower red lines) are also depicted.