Bias Modelling

We can compute higher order biases and scale-dependent biases from PNG from a given Halo Occupation Distribution (HOD) and a Halo Mass Function (HMF). This is implemented in the HOD subpackage, which contains four modules:

  • hods — HOD models. All HOD models inherit from the abstract base class BaseHOD, which defines the interface (get_hod_params, HOD). Currently provides the YP (Yankelevich & Porciani 2018) and Smith_BGS (Smith et al. 2024) models.

  • hmf — Halo mass function multiplicity functions, Lagrangian biases, and halo number density.

  • peak_background_bias — Peak-background split galaxy bias computation using the HOD.

Cosmological quantities needed by the HOD/HMF pipeline (sigma_R, rho_m, M(z,R), etc.) are precomputed and cached on ClassWAP via setup_hod_hmf so they are only computed once regardless of the number of tracers.

Barring equilateral and orthogonal PNG — the biases can also be given from polynomial fits — see SetSurveyFunctions.

HOD Models

All HOD models inherit from BaseHOD, which defines the interface:

class cosmo_wap.HOD.hods.BaseHOD

Abstract base class for Halo Occupation Distribution models.

abstractmethod get_hod_params(zz, m_c=None)

Return HOD parameters as a tuple for the given redshift(s).

abstractmethod HOD(zz, *args)

Return the mean number of galaxies per halo N(M) at redshift zz.

YP — Yankelevich & Porciani (2018) [arXiv:1807.07076]. Two free parameters (M0, NO) fitted to the survey’s linear bias and number density. Used by default for spectroscopic surveys (Euclid, Roman, SKA, etc.).

Smith_BGS — Smith et al. (2024). Five-parameter HOD (Mmin, sigma, M0, M1, alpha) with best-fit parameters from AbacusSummit. Used for the DESI BGS survey, where the HOD parameters are functions of a threshold apparent magnitude m_c.

To add a new HOD model, subclass BaseHOD and implement get_hod_params and HOD (and optionally fit_params if the model has free parameters that need fitting).

HMF

The HMF class provides halo mass function multiplicity functions, Lagrangian bias models, and the halo number density n_h.

class cosmo_wap.HOD.hmf.HMF(cosmo_funcs, hmf=’Tinker10’)

Parameters:

  • cosmo_funcs: An instance of ClassWAP providing cached cosmological quantities (sig_R, delta_c, rho_m, M).

  • hmf: Choice of HMF. ’Tinker10’ (default) uses the Tinker 2010 multiplicity function with analytic Lagrangian biases. ’ST’ uses the Sheth-Tormen multiplicity function. Any other value falls back to Tinker 2010 multiplicity with numeric Lagrangian biases.

Key attributes:

  • nu_func — peak height \(\nu(z,R) = \delta_c / \sigma(z,R)\)

  • multiplicity — the selected multiplicity function \(f(\nu)\)

  • lagbias — Lagrangian bias model (LagBias_Tinker10, LagBias_ST, or numeric LagBias)

Methods:

n_h(zz)

Halo number density per unit mass dn/dM at redshift zz. Returns an array over the radius grid R.

PBBias

The PBBias class computes non-Gaussian biases using the Peak Background Split (PBS) approach. It uses an HMF instance for the halo mass function and Lagrangian biases, and an HOD model (selected via the hod parameter) to relate halo masses to galaxy occupation numbers.

For the default YP HOD, the free parameters are fit to the survey’s linear bias and number density, so the only inputs needed are b_1(z), n_g(z), and cosmology. For Smith_BGS, the HOD parameters are derived from the apparent magnitude cut m_c.

class cosmo_wap.HOD.peak_background_bias.PBBias(cosmo_funcs, survey_params, hmf=’Tinker10’, hod=’YP’, m_c=None)

This class computes second-order bias and non-Gaussian biases from the HMF and HOD for a given survey and cosmology. These are then transferred onto the SetSurveyFunctions object via the add_bias_attr method, making them available for use in power spectrum and bispectrum calculations.

Parameters:

  • cosmo_funcs: An instance of ClassWAP that contains cosmological information and cached HOD/HMF quantities (sig_R, R, M, rho_m, delta_c).

  • survey_params: An instance of SurveyParams containing survey parameters, where the relevant parameters are the linear bias (b_1) and the number density (n_g).

  • hmf: Choice of HMF passed to HMF. ’Tinker10’ (default) or ’ST’ (Sheth-Tormen).

  • hod: Choice of HOD model. ’YP’ (default) or ’Smith_BGS’.

  • m_c: Apparent magnitude cut, only used by Smith_BGS.

Computed biases (passed to SetSurveyFunctions via add_bias_attr):

  • b_2 — second-order Eulerian bias

  • g_2 — tidal bias (from local Lagrangian approximation)

Non-Gaussian bias parameters for each PNG type:

  • loc (local-type PNG, \(A=1, \alpha=0\))

  • eq (equilateral-type PNG, \(A=3, \alpha=2\))

  • orth (orthogonal-type PNG, \(A=-3, \alpha=1\))

Each of these contains:

  • b_01 (\(b_{\psi}\))

  • b_11 (\(b_{\psi \delta}\))

Usage Example

Here’s how you can use the bias modelling with compute_bias=True:

import matplotlib.pyplot as plt
import cosmo_wap as cw
from cosmo_wap.lib import utils

cosmo = utils.get_cosmo()
survey = cw.SurveyParams.Euclid(cosmo)

# compute_bias=True triggers HOD/HMF pipeline
cosmo_funcs = cw.ClassWAP(cosmo, survey, compute_bias=True)

# Access biases via the survey tracer
zz = cosmo_funcs.z_survey
tracer = cosmo_funcs.survey[0]

# Compare non-Gaussian biases for local, equilateral, orthogonal PNG
plt.plot(zz, tracer.loc.b_11(zz), label=’local’)
plt.plot(zz, tracer.eq.b_11(zz), label=’equilateral’)
plt.plot(zz, tracer.orth.b_11(zz), label=’orthogonal’)
plt.legend()
plt.show()

For the BGS survey with the Smith HOD:

survey_bgs = cw.SurveyParams.BGS(cosmo, cut=20)
cosmo_funcs = cw.ClassWAP(cosmo, survey_bgs, compute_bias=True)

Extending

Adding a new HOD: Subclass BaseHOD from cosmo_wap.HOD.hods and implement get_hod_params and HOD. Override fit_params if the model has free parameters to calibrate against survey data.

The HMF class can be easily extended to support additional halo mass functions using the hmf libraries.