Optimal linear reconstruction of dark matter from halo catalogues

We derive the weight function w(M) to apply to dark-matter haloes that minimizes the stochasticity between the weighted halo distribution and its underlying mass density field. The optimal w(M) depends on the range of masses being used in the estimator. While the standard biased-Poisson model of the halo distribution predicts that bias weighting is optimal, the simple fact that the mass is comprised of haloes implies that the optimal w(M) will be a mixture of mass-weighting and bias-weighting. In N-body simulations, the Poisson estimator is up to 15x noisier than the optimal. Implementation of the optimal weight yields significantly lower stochasticity than weighting haloes by their mass, bias or equal weighting in most circumstances. Optimal weighting could make cosmological tests based on the matter power spectrum or cross-correlations much more powerful and/or cost effective. A volume-limited measurement of the mass power spectrum at k = 0.2 h Mpc-1 over the entire z < 1 universe could ideally be done using only 6 million redshifts of haloes with mass M > 6 x 1013 h-1 M-circle dot (1 x 1013) at z = 0 (z = 1); this is 5x fewer than what the Poisson model predicts. Using halo occupancy distributions (HODs) we find that uniformly weighted catalogues of luminous red galaxies require >= 3x more redshifts than an optimally weighted halo catalogue to reconstruct the mass to the same accuracy. While the mean HODs of galaxies chosen to lie above a threshold luminosity are fortuitously very similar to the optimal w(M), the stochasticity of the halo occupation degrades the mass estimator. Blue or emission-line galaxies are approximate to 100x less efficient at reconstructing mass than an optimal weighting scheme. This suggests an efficient observational approach of identifying and weighting haloes with a deep photo-z survey before conducting a spectroscopic survey. The optimal w(M) and mass-estimator stochasticity predicted by the standard halo model for M > 1012 h-1 M-circle dot are in reasonable agreement with our measurements, with the important exceptions that the haloes must be assumed to be linearly biased samples of a 'halo field' that is distinct from the mass field. Halo catalogues extending below 1012 h-1 M-circle dot are more stochastic than the halo model predicts, suggesting that halo exclusion or other effects violate the assumption that haloes sample this halo field via a Poisson process.