Halo bias in Lagrangian space: Estimators and theoretical predictions

We present several methods to accurately estimate Lagrangian bias parameters and substantiate them using simulations. In particular, we focus on the quadratic terms, both the local and the non-local ones, and show the first clear evidence for the latter in the simulations. Using Fourier space correlations, we also show for the first time, the scale dependence of the quadratic and non-local bias coefficients. For the linear bias, we fit for the scale dependence and demonstrate the validity of a consistency relation between linear bias parameters. Furthermore, we employ real-space estimators, using both cross-correlations and the peak-background split argument. This is the first time the latter is used to measure anisotropic bias coefficients. We find good agreement for all the parameters among these different methods, and also good agreement for local bias with Excursion set constraints τ theory predictions. We also try to exploit possible relations among the different bias parameters. Finally, we show how including higher order bias reduces the magnitude and scale dependence of stochasticity of the halo field.