Halo stochasticity in global clustering analysis

Galaxy clustering and cosmic magnification can be used to estimate the dark matter power spectrum if the theoretical relation between the distribution of galaxies and the distribution of dark matter is precisely known. In the present work, we study the statistics of haloes, which in the halo model determines the distribution of galaxies. Haloes are known to be biased tracer of dark matter, and at large scales it is usually assumed there is no intrinsic stochasticity between the two field (i.e. r = 1). Following the work of Seljak & Warren, we explore how correct this assumption is and, moving a step further, we try to qualify the nature of stochasticity. We use principal component analysis applied to the outputs of a cosmological N-body simulation as a function of mass to: (i) explore the behaviour of stochasticity in the correlation between haloes of different masses; and (ii) explore the behaviour of stochasticity in the correlation between haloes and dark matter. We show results obtained using a catalogue with 2.1 million haloes, from a PMFAST simulation with box size of 1000 h(-1) Mpc and with about four billion particles. In the relation between different populations of haloes, we find that stochasticity is not negligible even at large scales. In agreement with the conclusions of Tegmark & Bromley, who studied the correlations of different galaxy populations, we found that the shot noise subtracted stochasticity is qualitatively different from 'enhanced' shot noise and, specifically, it is dominated by a single stochastic eigenvalue. We call this the 'minimally stochastic' scenario, as opposed to shot noise-like stochasticity which is 'maximally stochastic'. In the correlation between haloes and dark matter, we find that the stochasticity is minimized, as expected, near the dark matter peak (k similar to 0.02 h Mpc(-1) for a Lambda cold dark matter cosmology), and, even at large scales, it is of the order of 15 per cent above the shot noise. Moreover, we find that the reconstruction of the dark matter distribution is improved when we use the principal component eigenvectors as tracers of the bias, but still the reconstruction is not perfect, due to stochasticity.