Modeling scale-dependent bias on the baryonic acoustic scale with the statistics of peaks of Gaussian random fields

TL;DR

The paper advances galaxy and halo clustering modeling by formulating a peak-based bias for Gaussian initial density fields, deriving second-order peak correlations, and applying peak-background split to obtain both k-independent and k-dependent bias at all orders. It then evolves these peak correlations within the Zel'dovich approximation, revealing velocity bias and mode-coupling effects that shape the BAO feature. The authors demonstrate that a residual, few-percent scale dependence of bias persists near the BAO at collapse, particularly for moderate peak heights, and show that massive halos in the MICE simulation exhibit a similar scale dependence around the BAO, well captured by the peak-based predictions. This work provides a more physically grounded and predictive framework than local bias alone, with direct relevance for interpreting BAO measurements and cosmological parameter estimation, while highlighting avenues for refinement with higher-order dynamics and non-Gaussian initial conditions.

Abstract

Models of galaxy and halo clustering commonly assume that the tracers can be treated as a continuous field locally biased with respect to the underlying mass distribution. In the peak model pioneered by BBKS, one considers instead density maxima of the initial, Gaussian mass density field as an approximation to the formation site of virialized objects. In this paper, the peak model is extended in two ways to improve its predictive accuracy. Firstly, we derive the two-point correlation function of initial density peaks up to second order and demonstrate that a peak-background split approach can be applied to obtain the k-independent and k-dependent peak bias factors at all orders. Secondly, we explore the gravitational evolution of the peak correlation function within the Zel'dovich approximation. We show that the local (Lagrangian) bias approach emerges as a special case of the peak model, in which all bias parameters are scale-independent and there is no statistical velocity bias. We apply our formulae to study how the Lagrangian peak biasing, the diffusion due to large scale flows and the mode-coupling due to nonlocal interactions affect the scale dependence of bias from small separations up to the baryon acoustic oscillation (BAO) scale. For 2-sigma density peaks collapsing at z=0.3, our model predicts a ~ 5% residual scale-dependent bias around the acoustic scale that arises mostly from first-order Lagrangian peak biasing (as opposed to second-order gravity mode-coupling). We also search for a scale dependence of bias in the large scale auto-correlation of massive halos extracted from a very large N-body simulation provided by the MICE collaboration. For halos with mass M>10^{14}Msun/h, our measurements demonstrate a scale-dependent bias across the BAO feature which is very well reproduced by a prediction based on the peak model.

Figures (9)

• Figure 1: Left panel : Lagrangian bias coefficients characterizing the second order peak bias $\mathfrak{\tilde{b}}_{\rm{II}}(q_1,q_2)$, Eqs. (\ref{['eq:bvv']}) -- (\ref{['eq:bzz']}), as a function of peak height for a filtering radius $R_S=2.9\ {\rm {\it h}^{-1}Mpc}$ or, equivalently, a mass scale $M_S=3\times 10^{13}\ {\rm M_\odot/{\it h}}$. The shape parameter is $\gamma_1\approx 0.65$. For the $2\sigma$ peaks considered in subsequent illustrations, $\mathfrak{\tilde{b}}_{\rm{20}}$ is negative, $\mathfrak{\tilde{b}}_{\rm{20}}\approx -1.2$. Right panel : The second and fourth root $\mathfrak{\tilde{b}}_{\rm{11}}^{1/2}$ and $\mathfrak{\tilde{b}}_{\rm{02}}^{1/4}$ define a characteristic scale below which the scale dependence of $\mathfrak{\tilde{b}}_{\rm{II}}$ is large. In the limit $\nu\to\infty$, $\mathfrak{\tilde{b}}_{\rm{02}}$ becomes negative and converges towards $-\sigma_2^{-2}(1-\gamma_1^2)^{-1}$, whereas $\mathfrak{\tilde{b}}_{\rm{11}}$ asymptotes to the constant $(\gamma_1/\sigma_1)^2(1-\gamma_1^2)^{-1}$. Note that, in contrast to $\mathfrak{\tilde{b}}_{\rm{11}}^{1/2}$ and $\mathfrak{\tilde{b}}_{\rm{02}}^{1/4}$ that have units of length, $\mathfrak{\tilde{b}}_{\rm{20}}$ is dimensionless.
• Figure 2: The correlation of initial density peaks at the second order is shown as the long dashed-dotted (magenta) curve for 2$\sigma$ (left panel) and 3$\sigma$ (right panel) density peaks collapsing at redshift $z_0=0.3$ according to the spherical collapse prescription. For the Gaussian filter used in this paper, this corresponds to a mass scale $M_S=3\times 10^{13}$ and $2\times 10^{14}\ {\rm M_\odot/{\it h}}$, respectively. The individual contributions appearing in Eq. (\ref{['eq:xpkeasy']}) are shown separately. Namely, the solid (cyan) curve is the first order contribution $\mathfrak{\tilde{b}}_{\rm{I}}^2\xi_0^{(0)}$, whereas the second order term quadratic in $\mathfrak{\tilde{b}}_{\rm{II}}$, linear in $\mathfrak{\tilde{b}}_{\rm{II}}$ and independent of $\mathfrak{\tilde{b}}_{\rm{II}}$ are shown as the short dashed-dotted, short-dashed and long-dashed curve, respectively. A dotted line indicates negative values.
• Figure 3: A comparison between the initial unsmoothed density correlation $\xi(r,z_0)$ (black, dotted-dashed) and the initial peak correlation $\xi_{\rm pk}(\nu,R_S,r)$ (red, solid) around the BAO. To obtain the peak correlation, the density field was smoothed with a Gaussian filter on mass scale $M_S=3\times 10^{13}$ (left panel) and $2\times 10^{14}\ {\rm M_\odot/{\it h}}$ (right panel). The dotted-long dashed, short-dashed and long-dashed curves represent the individual contributions $\mathfrak{\tilde{b}}_{\rm{10}}^2\xi_0^{(0)}$, $2\mathfrak{\tilde{b}}_{\rm{10}}\mathfrak{\tilde{b}}_{\rm{01}}\xi_0^{(1)}$ and $\mathfrak{\tilde{b}}_{\rm{01}}^2\xi_0^{(2)}$ to the first order peak correlation (Eq.\ref{['eq:xpk']}). A nonzero $\mathfrak{\tilde{b}}_{\rm{01}}$ restores, and even amplifies the acoustic peak otherwise smeared out upon filtering the mass density field. The dotted curve indicates the second order correction to the peak correlation. Results are shown for the CDM transfer function considered in this paper.
• Figure 4: The evolved power spectrum $P_{\rm pk}(k,z)$ (dashed curve) for the 2$\sigma$ peaks as predicted by Eq.(\ref{['eq:xpkevol']}) at the redshift of collapse $z=z_0=0.3$ (left panel) and at $z=1$ (right panel). The first order term and the mode-coupling contribution are shown as solid curves together with the unsmoothed linear mass power spectrum $P_\delta(k,z_0)$. At low wavenumber, the scale-dependent contribution to the mode-coupling power, $P_{\rm MC}(k)-P_{\rm MC}(k=0)$, scales as $k^2$ since there is no mass-momentum conservation. The Poisson expectation $1/\bar{n}_{\rm pk}$ (not shown on this figure) is $\approx 10^4$.
• Figure 5: Matter and peak correlation functions at the redshift of collapse $z=z_0$ as predicted by Eqs.(\ref{['eq:ximza2']}) and (\ref{['eq:xpkevol']}), respectively. In configuration space, the mode-coupling contribution $\xi_{\rm MC}$ is always positive left to the BAO and negative right to the BAO, so it introduces a shift in the acoustic scale. The magnitude of $\xi_{\rm MC}$ depends strongly on the bias parameters.