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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 $C_{ij}\equiv<(\delta_i -b_i\delta_m)(\delta_j-b_j\delta_m)>$, where $\delta_m$ is the dark matter overdensity in Fourier space, $\delta_i$ 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.Comment: 17 pages, 14 figures, matched the published version in Phys. Rev. D including one new figur

We present a simple empirical function for the average density profile of cosmic voids, identified via the watershed technique in ΛCDM N -body simulations. This function is universal across void size and redshift, accurately describing a large radial range of scales around void centers with only two free parameters. In analogy to halo density profiles, these parameters describe the scale radius and the central density of voids. While we initially start with a more general four-parameter model, we find two of its parameters to be redundant, as they follow linear trends with the scale radius in two distinct regimes of the void sample, separated by its compensation scale. Assuming linear theory, we derive an analytic formula for the velocity profile of voids and find an excellent agreement with the numerical data as well. In our companion paper [Sutter et al., Mon. Not. R. Astron. Soc. 442, 462 (2014)] the presented density profile is shown to be universal even across tracer type, properly describing voids defined in halo and galaxy distributions of varying sparsity, allowing us to relate various void populations by simple rescalings. This provides a powerful framework to match theory and simulations with observational data, opening up promising perspectives to constrain competing models of cosmology and gravity. Introduction.-While tremendous effort has been conducted studying the properties of dark matter halos, cosmic voids have largely been unappreciated by the broad scientific community. However, as voids occupy the most underdense regions in the Universe, and constitute the dominant volume fraction of it, they are promising independent probes to test our theories of structure formation and cosmology. For example, voids are the ideal laboratories for studies of dark energy (e.g., Refs. [1-5]) and modified gravity (e.g., Refs. [6-9]), as the importance of ordinary gravitating matter is mitigated in their interior. Unlike dark matter halos, voids are in addition more closely related to the initial conditions of the Universe, thanks to the limited number of phase-space foldings occurring inside of them [10][11][12][13][14][15].A fundamental quantity to describe the structure of voids in a statistical sense is their spherically averaged density profile. In contrast to the well-known formulas parametrizing density profiles of simulated dark matter halos (e.g., Refs. [16][17][18][19]), rather few models for void density profiles have been developed, mainly focusing on the central regions [3,[20][21][22][23], rarely taking into account the compensation walls outside the void [24]. In this Letter we present a simple formula that is able to accurately describe the density profile around voids of any size and redshift, out to large distances from their center. Although we focus our attention on dark matter simulations here, our companion paper [25] extends the analysis to voids defined in other tracer types, such as dark matter halos and mock galaxies of various number densities, yielding consistent results. Thus, give...