Halo bias in Lagrangian Space: Estimators and theoretical predictions

TL;DR

This work advances the measurement of halo bias in Lagrangian space by employing Fourier-space cross-spectra, real-space Hermite/Laguerre-based estimators, and Separate-Universe PBS to extract $b_{10}$, $b_{20}$, and $b_{s^2}$, including their scale dependence. It provides strong evidence for a non-zero tidal bias $b_{s^2}$ and demonstrates consistent results across methods, with linear and quadratic biases largely matching ESP$\tau$ theory for scale dependence but showing shortcomings in predicting tidal bias. The study reveals near-universal redshift behavior when biases are expressed against the peak height $\nu$ and derives relations among bias parameters, including a numerical Eulerian tidal-bias fit. By illustrating how higher-order bias reduces stochasticity, the results offer practical priors and a framework for more accurate LSS analyses in current and future surveys.

Abstract

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 ESP$τ$ 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.

Figures (12)

• Figure 1: Bias parameters as function of $k$: We show $b_1$, $b_2$ and $b_{s^2}$ as measured from Fourier space estimator. For clarity, we only show 2 mass bins from every box and different line-styles (dashed, solid and dotted) correspond to different boxes (of size $L = 690, 1380, 3000\ Mpc/h$ respectively), from which the corresponding mass bins (specified by color) are picked. The dependence on the wavenumber is the shared by all the parameters: constant piece on large scales followed by $k^2$-like piece on intermediate scales followed by a cutoff around the halo scale. This scale is shown with vertical lines in top-panel in corresponding colors for different masses.
• Figure 2: $b_{11}$ as a function of mass: The values are estimated from the best fit values for $b_{1}$ and using Eq. (\ref{['eq:b1k']}), assuming a tophat window $W(kR)$ at the halo scale. We will also follow the scheme of the colors blue, green and red (B, G, R) representing different box sizes throughout this paper unless explicitly mentioned otherwise, or when the box size is not important.
• Figure 3: Bias estimators $b_1$, $b_2$ and $b_{s^2}$ as function of mass: agreement between real space, Fourier space estimator and $\text{ESP}\tau$ theory. For real space, we mention the large scale used to calculate the bias parameters. We use Eq. (\ref{['eq:b1real']}) and Eq. (\ref{['eq:b2real']}) for $b_{10}$ and $b_{20}$ respectively, and the smoothing scales used to extract scale independent part are mentioned. Real space points have been shifted along x-direction for clarity.
• Figure 4: $b_2(M)$ without $b_{s^2}$: Fourier space estimator for $b_2$ with (red points) and without (blue) including the shear in Eq. (\ref{['eq:b2k-bs2k']}) for 3 Gpc/h box. For comparison, we show the real space estimates (green boxed, shifted along x-axis for clarity) which are independent of the shear by construction. The two estimators are in agreement only if a non-vanishing tidal bias in Fourier space is considered. Though not shown here, the fit for $b_2$ without shear is also much worse for other boxes as well as PBS estimator, especially for high mass halos. Overall consistency with real space estimates is presented in Figure \ref{['fig-b1M']}.
• Figure 5: PBS bias measurements from different boxlet sizes: As explained in the text, the size of error bars increase with increasing boxlet sizes (decreasing number of boxlets). Markers are shifted along x-direction with decreasing numbers of boxlets for clarity.