Minimizing the stochasticity of halos in large-scale structure surveys

TL;DR

The paper investigates stochasticity in halo tracers of the dark matter field, challenging the Poisson shot-noise assumption by defining a shot-noise matrix that includes cross-bin correlations. Through high-resolution N-body simulations and a halo-model framework, it identifies two nontrivial eigenmodes of the shot-noise matrix, with the lowest-eigenvalue mode providing an optimal weighting that suppresses stochasticity and enhances correlation with dark matter. The authors show that a modified mass weighting function w(M)=M+M_0 closely approximates the optimal eigenvector and that the suppression of stochasticity can be strong, especially with improved mass resolution and limited halo mass uncertainty. The halo-model analysis reproduces the key eigenvalues/eigenvectors and indicates further reduction in stochasticity if lower-mass halos are resolved, underscoring the potential for significantly improved cosmological parameter constraints in future surveys when using mass-weighted halo tracers.

Abstract

In recent work (Seljak, Hamaus and Desjacques 2009) it was found that weighting central halo galaxies by halo mass can significantly suppress their stochasticity relative to the dark matter, well below the Poisson model expectation. In this paper we extend this study with the goal of finding the optimal mass-dependent halo weighting and use $N$-body simulations to perform a general analysis of halo stochasticity and its dependence on halo mass. We investigate the stochasticity matrix, defined as $C_{ij}\equiv<(δ_i -b_iδ_m)(δ_j-b_jδ_m)>$, where $δ_m$ is the dark matter overdensity in Fourier space, $δ_i$ the halo overdensity of the $i$-th halo mass bin and $b_i$ the halo bias. In contrast to the Poisson model predictions we detect nonvanishing correlations between different mass bins. We also find the diagonal terms to be sub-Poissonian for the highest-mass halos. The diagonalization of this matrix results in one large and one low eigenvalue, with the remaining eigenvalues close to the Poisson prediction $1/\bar{n}$, where $\bar{n}$ is the mean halo number density. The eigenmode with the lowest eigenvalue contains most of the information and the corresponding eigenvector provides an optimal weighting function to minimize the stochasticity between halos and dark matter. We find this optimal weighting function to match linear mass weighting at high masses, while at the low-mass end the weights approach a constant whose value depends on the low-mass cut in the halo mass function. Finally, we employ the halo model to derive the stochasticity matrix and the scale-dependent bias from an analytical perspective. It is remarkably successful in reproducing our numerical results and predicts that the stochasticity between halos and the dark matter can be reduced further when going to halo masses lower than we can resolve in current simulations.

Figures (14)

• Figure 1: TOP: Autopower spectra for 10 consecutive halo mass bins (solid colored lines) and the dark matter (dotted black line). MIDDLE: Bias of the 10 halo bins determined from the cross power with the dark matter, the dotted lines show the scale-independent bias. BOTTOM: Cross-correlation coefficients of the 10 halo bins with the dark matter (solid colored lines) without shot noise subtraction. When the shot noise $C_{ii}$ is subtracted from $\langle\delta_i^2\rangle$, by definition the cross-correlation coefficient becomes unity (dashed lines). For reference, the value $r=1$ is plotted (dotted black line). The error bars on all three plots were computed from the ensemble of the 30 independent realizations of our simulations. They show the standard deviation on the mean of each quantity shown.
• Figure 2: Elements of the shot noise matrix as defined in Eq. (\ref{['sn']}) with 10 halo mass bins. Most of the diagonal components (solid lines with stars) agree with Poisson white noise, i.e., $C_{ii}=1/\bar{n}_i$ (dotted black line on top), except the highest-mass bin which is clearly suppressed (solid black line). There are both positive (dashed lines with circles, scaled in red) and negative (dotted lines with triangles, scaled in blue) off-diagonal components.
• Figure 3: The 10 eigenvalues (left) and eigenvectors (right) of the shot noise matrix from Fig. \ref{['snfig']} in corresponding colors. The black dotted line shows the value $1/\bar{n}_i$. The eigenvectors are averaged over the entire $k$-range.
• Figure 4: The normalized eigenvector $V_i^{-}$, corresponding to the lowest shot noise eigenvalue $\lambda^{-}$, computed for 10, 30, 100 uniformly weighted bins and 100 mass-weighted bins (from top to bottom). The latter was shifted downwards by a factor of 2 for visibility. The dotted (blue) and the dashed (red) lines represent linear and modified mass weighting, respectively. The best-fit values for $M_0$ are given in the bottom right for the respective cases.
• Figure 5: The normalized vector $\sum_{i}C_{ij}^{-1}b_i$ that provides an optimal estimator for the dark matter, computed for 100 uniformly weighted mass bins. Since this vector is very similar to $V_i^{-}$, modified mass weighting (dashed red line) still yields a reasonable fit with a slightly higher value of $M_0$ (bottom right).