Halo Stochasticity from Exclusion and non-linear Clustering

TL;DR

This work shows that halo exclusion and non-linear clustering induce deviations from Poisson shot noise in the stochasticity of discrete tracers like haloes and galaxies, particularly a negative k^0 correction from exclusion and a positive correction from non-linear clustering outside the exclusion radius. The authors develop a Lagrangian-space bias and exclusion framework, derive how these effects modify P^(d)(k) through terms like P_lin, I_22(k), and W_R(k), and introduce a stochasticity matrix to handle multiple mass bins. They validate the approach with the zHORIZON simulations, revealing mass- and redshift-dependent corrections that can amount to a few percent on large scales, and show that galaxy samples with satellites can exhibit substantial departures from fiducial Poisson noise. The results highlight the necessity of including exclusion and clustering corrections for high-precision cosmology in upcoming surveys and suggest practical pathways for mitigating stochasticity via mass- and bias-informed weighting.

Abstract

The clustering of galaxies in ongoing and upcoming galaxy surveys contains a wealth of cosmological information, but extracting this information is a non-trivial task since galaxies and their host haloes are stochastic tracers of the matter density field. This stochasticity is usually modeled as Poisson shot noise, which is constant as a function of wavenumber with amplitude given by 1/n, where n is the number density of galaxies. Here we use dark matter haloes in N-body simulations to show evidence for deviations from this simple behaviour and develop models that explain the behaviour of the large scale stochasticity. First, haloes are extended, non-overlapping objects, i.e., their correlation function needs to go to -1 on small scales. This leads to a negative correction to the stochasticity relative to the Poisson value at low wavenumber k, decreasing to zero for wavenumbers large compared to the inverse exclusion scale. Second, haloes show a non-linear enhancement of clustering outside the exclusion scale, leading to a positive stochasticity correction. Both of these effects go to zero for high-k, making the stochasticity scale dependent even for k<0.1 h/Mpc. We show that the corrections in the low-k regime are the same in Eulerian and Lagrangian space, but that the transition scale is pushed to smaller scales for haloes observed at present time, relative to the initial conditions. These corrections vary with halo mass and redshift. We also discuss simple applications of these effects to the galaxy samples with non-vanishing satellite fraction, where the stochasticity can again deviate strongly from the fiducial Poisson expectation. Overall these effects affect the clustering of galaxies at a level of a few percent even on very large scales and need to be modelled properly if we want to extract high precision cosmological information from the upcoming galaxy surveys.

Figures (15)

• Figure 1: Cartoon version of the correlation function of discrete tracers. Continuous linear correlation function (black dashed) and non-linear correlation function (red dashed). The true correlation function of discrete tracers (green solid line) agrees with the non-linear continuous correlation function outside the exclusion scale and is -1 below, except for the delta function at the origin arising from discreteness. Thus, there are two corrections compared to the continuous linear bias model, a negative correction inside the exclusion radius (red shaded) and a positive one outside the exclusion radius due to non-linear clustering (blue shaded).
• Figure 2: Power spectrum of a randomly distributed halo sample obeying exclusion (red points) and corresponding model with (red solid line) and without (red dashed line) exclusion. In a second step we populate these haloes with $N_\text{gal}=2$ satellite galaxies, and calculate the auto power spectrum of the satellite galaxies (green points) and their cross power spectrum with the halo centers (blue points). The blue and green solid lines show our model predictions, whereas the dashed lines show the naive expectation of Poisson shot noise.
• Figure 3: Kaiser bias Kaiser:1984on in configuration and Fourier space. Left panel: Un-smoothed (black dashed) and $R=4\ h^{-1}\text{Mpc}$ smoothed (black solid) linearly biased matter correlation functions $b_{1,\text{tr}}^2\xi(r)$ and continuous correlation function of the thresholded regions $\xi_\text{tr}(r)$ (red dashed). The red solid line shows a simple implementation of exclusion imposed on the correlation function of the thresholded regions. Right panel: Power spectrum correction arising from the non-linear biasing (top line) and effect of increasing exclusion for $R=0,4,6,8 \ h^{-1}\text{Mpc}$ from top to bottom.
• Figure 4: Clustering of peaks in a one dimensional skewer through a density field smoothed with a Gaussian filter of scale $R=2\ h^{-1}\text{Mpc}$ ($M\approx 8.6\times 10^{12}\ h^{-1} M_\odot$). Left panel: For fixed peak height the correlation function flattens out on small scales (black), but with increasing bin width the exclusion becomes stronger. The width of the bin in peak height increases from dark to light red. The linear local bias expansion is the same for all of these models and is shown by the dashed line. For reference we overplot the Gaussian smoothing (dash-dotted) and the top-hat smoothing scale containing the same mass (dashed). Right panel: Corresponding stochasticity correction $\Delta P_\text{pk}(k)=\text{FT}[\xi_\text{pk}](k)-b_{1,\text{pk}}^2 P_\text{lin}(k)$ for the fiducial bin width.
• Figure 5: Example of the halo-halo correlation function of the traced back haloes for mass bin V. The vertical solid line is the fitted exclusion radius. The dot-dashed line shows the linear bias contribution, whereas the dashed line shows linear plus quadratic bias. Note that the second order bias parameter was fitted to the correlation function and does deviate quite strongly from the PBS prediction. The red solid line shows a simple model for halo exclusion Eq. \ref{['eq:smoothedstep']}. In the right panel we show the integrand of the Fourier transform $r^3 \xi_\text{hh}(r)$, which is of essential importance for the stochasticity modelling.