[Paper Review] The PDF perspective on the tracer-matter connection: Lagrangian bias and non-Poissonian shot noise

Oliver Friedrich, Anik Halder+6 more|arXiv (Cornell University)|Jul 5, 2021

Galaxies: Formation, Evolution, Phenomena71 references20 citations

TL;DR

This paper introduces a Lagrangian bias model within the PDF framework to better describe the tracer-matter connection, showing it outperforms Eulerian models in capturing ⟨δtracer|δm⟩ at second order. It also validates a two-parameter non-Poissonian shot-noise model, significantly improving joint PDF modeling for galaxy and matter density fluctuations in N-body simulations and mock catalogs.

ABSTRACT

We study the connection of matter density and its tracers from the PDF perspective. One aspect of this connection is the conditional expectation value $\langle \delta_{\mathrm{tracer}}|\delta_m angle$ when averaging both tracer and matter density over some scale. We present a new way to incorporate a Lagrangian bias expansion of this expectation value into standard frameworks for modelling the PDF of density fluctuations and counts-in-cells statistics. Using N-body simulations and mock galaxy catalogs we confirm the accuracy of this expansion and compare it to the more commonly used Eulerian parametrization. For halos hosting typical luminous red galaxies, the Lagrangian model provides a significantly better description of $\langle \delta_{\mathrm{tracer}}|\delta_m angle$ at second order in perturbations. A second aspect of the matter-tracer connection is shot-noise, \ie the scatter of tracer density around $\langle \delta_{\mathrm{tracer}}|\delta_m angle$. It is well known that this noise can be significantly non-Poissonian and we validate the performance of a more general, two-parameter shot-noise model for different tracers and simulations. Both parts of our analysis are meant to pave the way for forthcoming applications to survey data.

Motivation & Objective

• To improve the modeling of the conditional expectation ⟨δtracer|δm⟩ in the context of the joint probability density function (PDF) of matter and tracer density fluctuations.
• To address the limitations of Eulerian bias parametrizations by introducing a Lagrangian bias expansion that aligns better with standard PDF modeling frameworks.
• To validate a generalized two-parameter shot-noise model to account for non-Poissonian scatter in tracer density around the mean conditional expectation.
• To enable more accurate cosmological inference from future survey data by providing a robust, physically motivated model for the tracer-matter connection in PDF statistics.
• To lay the groundwork for joint analyses of PDF and 2-point function statistics by establishing consistency between bias parameters and stochasticity models.

Proposed method

• Formulates a Lagrangian bias expansion for the conditional expectation ⟨δtracer|δm⟩ using a cumulant generating function (CGF) approach, derived from the joint PDF of matter and tracer density fluctuations.
• Integrates the Lagrangian bias model into standard PDF modeling frameworks based on symmetric collapse and saddle-point approximations, enabling consistent treatment of non-linear density field evolution.
• Employs a two-parameter shot-noise model (amplitude and scale dependence) to describe the scatter of tracer density around ⟨δtracer|δm⟩, generalizing beyond Poissonian assumptions.
• Calibrates and tests the model using high-resolution N-body simulations (Quijote and T17) and mock galaxy catalogs (Molino), comparing predictions to measured joint PDFs.
• Performs parameter fitting for both Lagrangian and Eulerian bias models across multiple redshifts, smoothing scales, and halo mass bins to assess accuracy and consistency.
• Validates consistency relations between Lagrangian and Eulerian parameters and compares results with 2-point function measurements and analytical bias predictions.

Experimental results

Research questions

• RQ1 Does a Lagrangian bias expansion provide a more accurate description of ⟨δtracer|δm⟩ than the standard Eulerian parametrization in the context of PDF modeling?
• RQ2 How well does the two-parameter non-Poissonian shot-noise model capture the scatter in tracer density around the conditional mean across different tracers and simulation setups?
• RQ3 Are the best-fitting Lagrangian and Eulerian bias parameters consistent with each other and with analytical predictions based on halo mass and redshift?
• RQ4 To what extent does the joint PDF p(δtracer, δm) contain cosmological information beyond what is accessible through 2-point statistics alone?
• RQ5 Can the proposed model be effectively generalized to projected, line-of-sight statistics relevant to real survey data, such as cosmic shear and galaxy overdensity maps?

Key findings

• For halos hosting typical luminous red galaxies, the Lagrangian bias model provides a significantly better description of ⟨δtracer|δm⟩ at second order in perturbations compared to the Eulerian parametrization.
• The two-parameter non-Poissonian shot-noise model effectively captures deviations from Poissonian noise across diverse tracers and simulation setups, as confirmed by comparison with measured joint PDFs.
• Best-fitting Lagrangian and Eulerian bias parameters are consistent with each other and with analytical bias predictions based on halo mass, validating the theoretical consistency of the model.
• The joint PDF p(δtracer, δm) contains cosmological information that complements 2-point function statistics, particularly when combined with lensing or multi-scale analyses.
• The model is transferable to projected, line-of-sight statistics (e.g., lensing convergence and 2D galaxy density), making it directly applicable to real survey data such as DES and LSST.
• The framework enables a more efficient and physically motivated modeling of the tracer-matter connection, with potential to constrain nuisance parameters in 2-point function analyses through PDF-based priors.