The Scale Dependence of Halo and Galaxy Bias: Effects in Real Space

TL;DR

This work demonstrates that scale-dependent bias from nonlinear dynamics and halo bias persists on very large scales, with a strong and mass-dependent imprint in both halo and galaxy clustering. It develops a physically motivated analytic framework, Halo-PT within the Halo Model, to predict 1-Loop corrections to halo-halo and halo-dark matter power spectra and to interpret shifts and damping of BAO features. Complemented by a suite of high-resolution N-body simulations, the study reveals a nontrivial, mass-dependent bias on large scales, demonstrates the importance of shot-noise corrections and halo exclusion for halo clustering, and shows that galaxy clustering inherits these scale-dependent biases via the HOD. The results have direct implications for interpreting BAO measurements and for deriving robust cosmological constraints from galaxy surveys, motivating future work to sharpen the model, extend to redshift space, and refine parameter inferences.

Abstract

We examine the scale dependence of dark matter, halo and galaxy clustering on very large scales (0.01<k[h/Mpc]<0.15), due to non-linear effects from dynamics and halo bias. We pursue a two line offensive: high resolution numerical simulations are used to establish several new results, and an analytic model is developed to understand their origins. Our simulations show: (i) that the z=0 dark matter power spectrum is suppressed relative to linear theory by ~5% on scales (0.05<k[h/Mpc]<0.075); (ii) that, indeed, halo bias is non-linear over the scales we probe and that the scale dependence is a strong function of halo mass. High mass haloes show no suppression of power on scales (k<0.07[h/Mpc]), and only show amplification on smaller scales, whereas low mass haloes show strong, ~5-10%, suppression over the range (0.05 <k[h/Mpc] <0.15). Our results have relevance for studies of the baryon acoustic oscillation features. Non-linear mode-mode coupling: (i) damps these features on progressively larger scales as halo mass increases; (ii) produces small shifts in the positions of the peaks and troughs which depend on halo mass. Our analytic model is described in the language of the `halo-model'. However, for the first time the halo-halo clustering term is propagated into the non-linear regime using `1-loop' perturbation theory and a non-linear halo bias model. We show that, with bias parameters derived from simulations, the model predictions are in agreement with the numerical results. We then use the model to explore the scale dependence of galaxies of different colour and find significant differences between the power spectra of the two populations. Thus understanding the scale dependent bias for a given galaxy sample will be crucial for deriving accurate cosmological constraints. (Abridged)

Figures (12)

• Figure 1: Halo power spectrum measurements and predictions, ratioed with a smooth 'No Baryon' dark matter power spectrum, for four bins in halo mass: results for massive haloes (the top two panels) are from the LR simulations, whereas lower masses (the two bottom panels) are from the HR simulations. Filled circles in the top, middle and bottom sections of each panel show the ensemble average non-linear $P^{\delta\delta}(k)$$P^{\rm h\rm h}(k)$, and $P^{\delta\rm h}$, respectively. The open circles in the middle sections show $P^{\rm h\rm h}(k)$ with the non-standard shot noise subtraction described in Appendix \ref{['app:HaloDiscrete']}. In all panels the linear theory dark matter, halo-halo and halo-cross power spectra are shown as dashed lines. The top panel also shows predictions from halofit (solid lines) Smithetal2003 and 1-Loop perturbation theory (dot-dash lines). Solid lines in the middle and bottom panels show our new analytic model, Halo-PT.
• Figure 2: Large scale halo bias derived directly from $N$-body simulations for four bins in halo mass. The top sections of each panel show the estimators $\hat{b}^{\delta\rm h}_{\rm NL}$ (solid blue points) and $\hat{b}^{\rm h\rm h}_{\rm NL}$ with and without shot noise correction (large and small red stars, respectively). The bottom panels show the same, but for $\hat{b}^{\delta\rm h}_{\rm Lin}$ and $\hat{b}^{\rm h\rm h}_{\rm Lin}$ (See equations \ref{['eq:biasestimator1']} and \ref{['eq:biasestimator2']} for definitions). The solid blue and red lines in each panel show the predictions for the bias from our Halo-PT model.
• Figure 3: Dark matter power spectrum measured as a function of wavenumber measured from the $z=0$ time slice of the Hubble Volume simulation Evrardetal2002. In the top panel the red and blue points show the estimates of the dark matter power spectrum measured from the simulation, with and without a Poisson shot noise correction. The dot-dash line shows the linear theory and the triple dot-dash line shows the Poisson correction. The blue dotted and dot-dash curves show the 1- and 2-Halo terms. The thick red dot-dash curve shows the 2-Halo term where $P_{\rm NL}$ has been used instead of $P_{\rm Lin}$. The bottom panel presents the ratio with respect to the halo model, but with equation (\ref{['eq:Lin2Halo']}) for the 2-Halo term. The solid red line shows the effect of the $P_{\rm NL}$ modification.
• Figure 4: First three halo bias parameters derived from the Sheth-Tormen ShethTormen1999 mass function as a function of halo mass Scoccimarroetal2001. The solid line shows $b_1$, the dashed line $b_2$, and the dot-dashed line $b_3$.
• Figure 5: '1-Loop' bias parameters $b^{\rm h\rm h}$ and $b^{\delta\rm h}$ in the ultra-large scale limit. The solid through to dotted curves show $b^{\rm h\rm h}$, measured on scales ($k_{\rm obs}=\left\{0.001,\ 0.005, 0.01,\ 0.05\right\} h{\rm Mpc}^{-1}$), derived from the 1-Loop halo-halo cross power spectrum as a function of halo mass. The triple-dot dash curve shows $b^{\delta\rm h}$ as derived from the 1-Loop halo-dark matter cross power spectrum.