High-Order Correlations of Peaks and Halos: a Step toward Understanding Galaxy Biasing

TL;DR

The authors present an analytic framework to predict high-order, hierarchical correlation amplitudes $S_j(R)$ for density peaks and dark halos in the quasi-linear regime, bridging peak theory and extended Press-Schechter with a spherical-collapse evolution. By deriving bias coefficients $b_k$ for halos and peaks, they compute $S_j$ for galaxies in the quasilinear limit and test the predictions against diverse N-body simulations, finding good agreement when halos are identified earlier than the measurement epoch. The results imply that observed high-order moments, such as those from the APM survey, can constrain the degree of galaxy bias relative to mass, with clusters naturally exhibiting higher moments due to their larger masses. The work highlights the potential of $S_j$ measurements to diagnose bias and informs interpretations of galaxy and cluster clustering, while acknowledging uncertainties in linking galaxies to halos and the role of nongravitational processes.

Abstract

We develop an analytic model for the hierarchical correlation amplitudes S_j(R)= \bxi_j(R)/\bxi_2^{j-1}(R) of density peaks and dark matter halos in the quasi-linear regime. The statistical distribution of density peaks and dark matter halos within the initial density field are determined by the peak formalism and by an extension of the Press-Schechter formalism, respectively. Modifications of these distributions caused by gravitationally induced motions are treated using a spherical collapse model. We test our model against results from a variety of N-body simulations. The model works well for peaks and for halos that are identified earlier than the time when the moments are calculated. Because halos are spatially exclusive at the time of their identification, our model is only qualitatively correct for halos identified at the same time as the moments are calculated. The S_j depend only weakly on the bias parameter b for large b but increase rapidly with decreasing b at b\sim 1. Thus if galaxies are associated with peaks in the initial density field, or with dark halos formed at high redshifts, a measurement of S_j in the quasilinear regime should determine whether galaxies are significantly biased relative to the mass. We use our model to interpret the observed high order correlation functions of galaxies and clusters. We find that if the values of S_j for galaxies are as high as those given by the APM survey, then APM galaxies should not be significantly biased.

Figures (17)

• Figure 1: The skewness of density peaks with different heights $\nu$ predicted by our model (solid curves) compared with that derived from N-body simulations (circles). The dashed curves show the skewness of the mass density distribution in the simulation. Results are shown for the standard cold dark matter model with $(\Omega, \Gamma, \sigma_8)=(1,0. 5, 0.62)$. The thick ticks on the horizontal axis show the values of $R$ where ${\overline {\xi}} _2 (R)=1$.
• Figure 2: The same as Fig. 1a for a model with $(\Omega, \Gamma, \sigma_8)=(1, 0.5, 1.24)$.
• Figure 3: The same as Fig. 1a for a model with $(\Omega, \Gamma, \sigma_8)=(1, 0.2, 0.5)$.
• Figure 4: The same as Fig. 1a for a model with $(\Omega, \Gamma, \sigma_8)=(1, 0.2, 1)$.
• Figure 5: The kurtosis of density peaks with different heights $\nu$ predicted by our model (solid curves) compared with that derived from N-body simulations (circles). The dashed curves show the kurtosis of the mass density distribution in the simulation. Results are shown for the standard cold dark matter model with $(\Omega, \Gamma, \sigma_8)=(1,0. 5, 0.62)$. The thick ticks on the horizontal axis show the values of $R$ where ${\overline {\xi}} _2 (R)=1$.