The Local Bias Model in the Large Scale Halo Distribution

TL;DR

The paper investigates halo bias in large-scale structure by testing the local deterministic bias model against a large N-body simulation and by comparing predictions from the peak-background split mass function. It directly fits the local relation $\delta_h = F[\delta_m]$ to extract $b_1$ and $b_2$ (via $c_2=b_2/b_1$) and studies their dependence on smoothing scale $R_s$, finding convergence around $R_s \sim 30$–$60\,\mathrm{Mpc}/h$. It demonstrates that two-point halo clustering is well described by the local model at $r \gtrsim 20$–$15$\,Mpc/$h$ with percent-level accuracy when using large $R_s$, while the three-point function shows systematic deviations for lower-mass halos; smoothed moments require non-linear and discreteness corrections at the 10–20% level. The study also shows that mass-function-based PBS predictions generally underpredict the linear bias by about 5–10%, implying systematic errors in mass calibration from clustering, and it outlines the regimes where the local bias model suffices versus where additional physics is needed for precision cosmology.

Abstract

We explore the biasing in the clustering statistics of halos as compared to dark matter (DM) in simulations. We look at the second and third order statistics at large scales of the (intermediate) MICEL1536 simulation and also measure directly the local bias relation h = f(δ) between DM fluctuations, δ, smoothed over a top-hat radius Rs at a point in the simulation and its corresponding tracer h (i.e. halos) at the same point. This local relation can be Taylor expanded to define a linear (b1) and non-linear (b2) bias parameters. The values of b1 and b2 in the simulation vary with Rs approaching a constant value around Rs > 30 - 60 Mpc/h. We use the local relation to predict the clustering of the tracer in terms of the one of DM. This prediction works very well (about percent level) for the halo 2-point correlation ξ(r_12) for r_12 > 15 Mpc/h, but only when we use the biasing values that we found at very large smoothing radius Rs > 30 - 60 Mpc/h. We find no effect from stochastic or next to leading order terms in the f(δ) expansion. But we do find some discrepancies in the 3-point function that needs further understanding. We also look at the clustering of the smoothed moments, the variance and skewness which are volume average correlations and therefore include clustering from smaller scales. In this case, we find that both next to leading order and discreetness corrections (to the local model) are needed at the 10 - 20% level. Shot-noise can be corrected with a term σe^2/n where σe^2 < 1, i.e., always smaller than the Poisson correction. We also compare these results with the peak-background split predictions from the measured halo mass function. We find 5-10% systematic (and similar statistical) errors in the mass estimation when we use the halo model biasing predictions to calibrate the mass.

Figures (17)

• Figure 1: Scatter plots showing halo density contrast $\delta_h$, smoothed over top-hat cells, for halos of 50 or more particles versus dark matter density fluctuations $\delta_m$ smoothed over the same cells. Results are shown for a different cell sizes with equivalent radius $R_s$ as labeled in the figure. Results are for simulation data at redshift z=0 (left panels) and z=0.5 (right panels). In a continuous line we show the least square fit to the local bias parabola.
• Figure 2: Variation of $b_1$ (left panels) and $c_2$ (right panels) as a function of the smoothing radius $R$. Top panels corresponds to $z=0.5$ and bottom panels to $z=0$. Results are shown for different minimum number of particles per halo, $n$. In each panel from bottom to top n=25 (black), n=50 (red), n=100 (green), n=200 (blue) and n=400 (yellow). Being the particle mass $23.42 \times 10^{10} M_{\sun}$ it yields, after correcting for resolution effects, minimum masses of 0.5 (black), 1.06 (red), 2.19 (green), 4.49 (blue), and 9.11 (yellow) $10^{12} M_{\sun}$
• Figure 3: Symbols with JK errorbar show the 2-point correlation function $\xi(r)$ from simulations for different minimum number of particles ($N>25, 100$ or $400$) per halo as labeled in the figure. The bottom errorbars corresponds to the measurements in the DM distribution. The bottom continuous (dashed) lines in each panel shows the RPT (linear) theory prediction. The upper continuous lines show the best fit amplitude for the RPT prediction shape, whose amplitudes $b^2$ are shown in the bottom labels. Top (bottom) panel correspond to z=0 (z=0.5).
• Figure 4: Bias from the ratio of 2-point correlation function $\xi(r)$ for different minimum number of particles per halo N=25, 100, 400 (from bottom to top). Top panel shows results for $z=0$ and bottom panel for $z=0.5$. The dashed lines show the values of the linear bias fit in the range $25<r<40$Mpc/h.
• Figure 5: Comparison of different estimates for the linear bias as a function of the minimal halo mass. Continuous line correspond to the local model fit to the scatter relation $\delta_m-\delta_h$ in Fig.\ref{['figb1Rs']} at R=60Mpc/h. Triangles correspond to bias from the 2-point function on large 30-80 Mpc/h scales (open triangles) and intermediate 20-40 Mpc/h scales (filled triangles).