Lagrangian Bias as a Gaussian Random Field

Abstract

Halo bias is typically treated as a set of coefficients in a perturbative expansion. We show instead that every point in a Gaussian density field has a well-defined scale-independent Lagrangian bias, thereby defining a bias field. This property can be extended to any linear operator acting on the Lagrangian density field, generating secondary bias fields. Halo bias then arises from geometric selection of Lagrangian patches within this pre-existing field, rather than being generated by collapse. We demonstrate that this framework predicts the measured $b(M)$ relation for halos. The multivariate Gaussian structure of the fields naturally explains the Gaussian distribution of halo bias at fixed mass and assembly bias. The results presented here motivate combining this framework with a forward model of halo collapse, yielding an ab initio model for halo clustering.

Figures (6)

• Figure 1: Demonstration of scale-independent pointwise bias using the initial conditions of a Quijote simulation at $z=127$, binned by grid-scale density percentile. Top left: The shell-averaged density $\langle\delta_k\rangle_{\mathbf q}$ at $5$ representative $k$-shells varies with $k$. Violins show the width of the underlying distribution. Bottom left: After normalizing by $\sigma_k^2$, the conditional mean $\langle\delta_k\rangle_{\mathbf q}/\sigma_k^2$ is the same across all five $k$-shells (symbols), collapsing to the theoretical prediction $\langle \delta(\mathbf{q})/\sigma^2\rangle$ (dotted lines). Right: Cross-power spectrum of each density bin with the full matter field. The inset shows the bias $P_{qm}/P_{mm}$; the measured values (symbols) agree with $\langle\delta(\mathbf{q})/\sigma^2\rangle$ (dotted lines) --- exactly the same prediction as in the bottom left panel, confirming the equivalence of the two perspectives.
• Figure 2: Measured halo bias $b(k) = P_{\rm cross}/P_{\rm matter}$ for two mass bins at $z=0$ (solid lines). The dashed horizontal lines show the prediction $b_E = \langle b_i\rangle/D + 1$ from the bias field framework, where $\langle b_i \rangle$ is the mean Lagrangian bias of all particles in the Lagrangian patches of the halos and $D$ is the linear growth factor.
• Figure 3: Assembly bias from the secondary bias field $\nabla^2\delta_s$, for halos in $[10^{14},\,2\times10^{14}]\,M_\odot/h$ at $z=0$. Left: Splitting at fixed $\delta_s$ evaluated over Lagrangian patches of halos by high/low $\nabla^2\delta_s$ (solid lines) produces a large bias split. The dashed lines represent the bias-field prediction $\langle b_\delta \rangle$ evaluated over the halo patch particles using only the uncontaminated low-$k$ ($k \ll 1/R_L$) modes. Mass-matched samples (dotted lines) show no bias split. Right: The same split produces differences in the stacked density profiles, with the more biased subsample being less concentrated. Both effects are a direct consequence of $\nabla^2\delta_s$ being an active secondary bias field of the GRF.
• Figure 4: Scatter plot of individual halo bias values, estimated from halo particles using Eqs. \ref{['eq:optimal_estimator']} and \ref{['eq:individual_halo_bias']}, divided by the growth rate $D(a)$. Points are colored by the $\delta_s - \nabla^2\delta_s$ splitting mentioned in the main text. The dotted line is the covariance direction of $\langle b_\delta b_{\nabla^2\delta}\rangle$ over all patches of radius $R_L$. The dashed line is the covariance direction defined by the halo population. The top and side panels show the marginal distributions. First, the distribution of $b_\delta$ over the full population is a Gaussian, as shown by the solid line in the top panel. Second the mean value of $b_{\nabla^2\delta}$ is shifted relative to the underlying Gaussian covariance direction, as shown by the difference between the solid line and the dotted line in the right panel. Third, splitting on $\nabla^2\delta_s$ at fixed $\delta_s$ produces a $b_\delta$ split, as shown by the two offset Gaussians in the top panel for the marginals of the red and green points in the scatter plot. This is the assembly bias effect.
• Figure S1: Eulerian bias (solid lines) of Lagrangian patches of fixed radius $R_L = 4.5\,{\rm Mpc}/h$, binned by smoothed density ($\delta_s$, $s = R_L$) percentile in the initial conditions. The prediction $b_E = \langle b_i\rangle /D(a) + 1$, where $\langle b_i\rangle$ is the mean bias of the particles in each bin, is shown as dashed lines.