Cosmology and Nongalactic Astrophysics · arXiv:1211.4128 v2 · submitted 17 Nov 2012
A local bias approach to the clustering of discrete density peaks
The paper says
“In this paper, we show that the 2-point correlation function of maxima of a homogeneous and isotropic Gaussian random field can be thought of, up to second order at least, as arising from a local bias expansion formulated in terms of rotationally invariant variables.”
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Abstract
Maxima of the linear density field form a point process that can be used to understand the spatial distribution of virialized halos that collapsed from initially overdense regions. However, owing to the peak constraint, clustering statistics of discrete density peaks are difficult to evaluate. For this reason, local bias schemes have received considerably more attention in the literature thus far. In this paper, we show that the 2-point correlation function of maxima of a homogeneous and isotropic Gaussian random field can be thought of, up to second order at least, as arising from a local bias expansion formulated in terms of rotationally invariant variables. This expansion relies on a unique smoothing scale, which is the Lagrangian radius of dark matter halos. The great advantage of this local bias approach is that it circumvents the difficult computation of joint probability distributions. We demonstrate that the bias factors associated with these rotational invariants can be computed using a peak-background split argument, in which the background perturbation shifts the corresponding probability distribution functions. Consequently, the bias factors are orthogonal polynomials averaged over those spatial locations that satisfy the peak constraint. In particular, asphericity in the peak profile contributes to the clustering at quadratic and higher order, with bias factors given by generalized Laguerre polynomials. We speculate that our approach remains valid at all orders, and that it can be extended to describe clustering statistics of any point process of a Gaussian random field. Our results will be very useful to model the clustering of discrete tracers with more realistic collapse prescriptions involving the tidal shear for instance.
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abstract compiled · 2026-09-23 · no language-model review
How the main claim is stated: asserted 0.97
What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.73 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.19 · compares with prior results 0.47 · mentions code or data availability 0.02
The abstract, sentence by sentence
This table labels the abstract only (arXiv listing abstract v2). The number is the probability the classifier gave the label.
| Sentence | Role | The authors' own work | States the main conclusion | Strength |
|---|---|---|---|---|
| Maxima of the linear density field form a point process that can be used to understand the spatial distribution of virialized halos that collapsed from initially overdense regions. | background 1.00 | 0.09 | 0.03 | asserted 0.88 |
| However, owing to the peak constraint, clustering statistics of discrete density peaks are difficult to evaluate. | gap 0.49 | 0.05 | 0.02 | asserted 1.00 |
| For this reason, local bias schemes have received considerably more attention in the literature thus far. | background 0.98 | 0.03 | 0.02 | asserted 1.00 |
| In this paper, we show that the 2-point correlation function of maxima of a homogeneous and isotropic Gaussian random field can be thought of, up to second order at least, as arising from a local bias expansion formulated in terms of rotationally invariant variables. | contribution 0.65 | 0.94 | 0.77 | asserted 0.97 |
| This expansion relies on a unique smoothing scale, which is the Lagrangian radius of dark matter halos. | method 0.68 | 0.63 | 0.05 | asserted 1.00 |
| The great advantage of this local bias approach is that it circumvents the difficult computation of joint probability distributions. | interpretation 0.61 | 0.26 | 0.11 | asserted 1.00 |
| We demonstrate that the bias factors associated with these rotational invariants can be computed using a peak-background split argument, in which the background perturbation shifts the corresponding probability distribution functions. | method 0.90 | 0.87 | 0.08 | asserted 1.00 |
| Consequently, the bias factors are orthogonal polynomials averaged over those spatial locations that satisfy the peak constraint. | result 0.88 | 0.83 | 0.12 | asserted 1.00 |
| In particular, asphericity in the peak profile contributes to the clustering at quadratic and higher order, with bias factors given by generalized Laguerre polynomials. | result 0.95 | 0.86 | 0.13 | asserted 0.99 |
| We speculate that our approach remains valid at all orders, and that it can be extended to describe clustering statistics of any point process of a Gaussian random field. | future work 0.75 | 0.22 | 0.19 | speculative 1.00 |
| Our results will be very useful to model the clustering of discrete tracers with more realistic collapse prescriptions involving the tidal shear for instance. | future work 0.77 | 0.17 | 0.06 | hedged possibility 0.51 |
What kind of paper the classifier reports this is: theoretical result 0.91 · evidence from proof or derivation 0.99
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 8,212 input tokens, $0.0003 reported.
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Cited by 6 Pith papers
Reviewed papers on Pith that cite this one. The full citation record
- Relativistic wide-angle galaxy bispectrum on the light-cone astro-ph.CO · May 2017
- Non-local halo bias with and without massive neutrinos astro-ph.CO · May 2014
- Peak-Background Split, Renormalization, and Galaxy Clustering astro-ph.CO · Dec 2012
- Large-Scale Galaxy Bias astro-ph.CO · Nov 2016
- Cosmological Parameter Estimation with Large Scale Structure Observations astro-ph.CO · Aug 2013
- DESI and other dark energy experiments in the era of neutrino mass measurements astro-ph.CO · Aug 2013
How the literature uses it
12 arXiv papers cite this in 37 sentences
Papers citing it · 24 sentences without a confident label
Builds on it 1 paper
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Third, we generalize these results to excursion set peaks [ref] and, in particular, provide a local bias expansion along the lines of [ref] that includes the first-crossing condition.
Non-Gaussian bias: insights from discrete density peaks · Introduction · 0.84
Uses its method or tool 3 papers
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Here, we will use the excursion set peaks (ESP) approach to halo clustering [ref] , which we briefly summarise in Section [ref] and Appendix [ref].
Squeezing the halo bispectrum: a test of bias models · Introduction · 0.98
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In this Section, we derive the non-Gaussian corrections to the 2-point correlation function $(r;ν_c)$ in two different ways: from a direct computation of the joint probability density $P_2(_i;ν_i,η_i,ζ_i)$ and from the effective local bias approach [ref] .
Non-Gaussian bias: insights from discrete density peaks · Peak 2-point correlation with non-Gaussian initial conditions · 0.82
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[equation] where the various bias factors $b_ij^L$ and $χ_ij^L$ can be obtained from a peak-background split applied to the halo mass function (see [ref] for details).
Non-local halo bias with and without massive neutrinos · Scale-dependent bias · 0.81
Compares with it 1 paper
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First, we demonstrate that the non-Gaussian bias obtained from a direct calculation of the non-Gaussian 2-point probability density agrees with that derived from the effective, local bias expansion advocated by [ref] .
Non-Gaussian bias: insights from discrete density peaks · Introduction · 0.89
Cites it as support 1 paper
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After all, all the ESP bias factors can be derived from a peak-background split, as shown in [ref] .
Squeezing the halo bispectrum: a test of bias models · Conclusion · 0.99
Cites it as background 2 papers
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The formalism of scale-dependent bias was developed in a series of papers: [ref] δ_h() = ∑^∞_n=1 1n!∫d^3_1 (2π)^3⋯∫d^3_n (2π)^3 c^L_n(_1,…,_n)δ_L(_1)⋯δ_L(_n)δ_D(-_1…n); _1…n ≡_1+⋯+_n.
Symmetries, Invariants and Generating Functions: Higher-order Statistics of Biased Tracers · Scale-Dependent Bias · 0.92
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In the remainder of the present work we shall therefore not address the interesting question of bias, see e.g. [ref] .
Cosmological Parameter Estimation with Large Scale Structure Observations · Number counts versus the real space power spectrum · 0.83
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