Reducing sample variance: halo biasing, non-linearity and stochasticity

Comparing clustering of differently biased tracers of the dark matter distribution offers the opportunity to reduce the cosmic variance error in the measurement of certain cosmological parameters. We develop a formalism that includes bias non-linearities and stochasticity. Our formalism is general enough that can be used to optimise survey design and tracers selection and optimally split (or combine) tracers to minimise the error on the cosmologically interesting quantities. Our approach generalises the one presented by McDonald & Seljak (2009) of circumventing sample variance in the measurement of $f\equiv d \ln D/d\ln a$. We analyse how the bias, the noise, the non-linearity and stochasticity affect the measurements of $Df$ and explore in which signal-to-noise regime it is significantly advantageous to split a galaxy sample in two differently-biased tracers. We use N-body simulations to find realistic values for the parameters describing the bias properties of dark matter haloes of different masses and their number density. We find that, even if dark matter haloes could be used as tracers and selected in an idealised way, for realistic haloes, the sample variance limit can be reduced only by up to a factor $\sigma_{2tr}/\sigma_{1tr}\simeq 0.6$. This would still correspond to the gain from a three times larger survey volume if the two tracers were not to be split. Before any practical application one should bear in mind that these findings apply to dark matter haloes as tracers, while realistic surveys would select galaxies: the galaxy-host halo relation is likely to introduce extra stochasticity, which may reduce the gain further.