Separate Universe Consistency Relation and Calibration of Halo Bias

Linear halo bias is the response of dark matter halo number density to a long wavelength fluctuation in the dark matter density. Using abundance matching between separate universe simulations which absorb the latter into a change in the background, we test the consistency relation between the change in a one point function, the halo mass function, and a two point function, the halo-matter cross correlation in the long wavelength limit. We find excellent agreement between the two at the $1-2\%$ level for average halo biases between $1 \lesssim \bar b_1 \lesssim 4$ and no statistically significant deviations at the $4-5\%$ level out to $\bar b_1 \approx 8$. Halo bias inferred assuming instead a universal mass function is significantly different and inaccurate at the 10\% level or more. The separate universe technique provides a way of calibrating linear halo bias efficiently for even highly biased rare halos in the $\Lambda$CDM model. Observational violation of the consistency relation would indicate new physics, e.g.~in the dark matter, dark energy or primordial non-Gaussianity sectors.