Zu & Mandelbaum et al. (2015) Composite Model

This section of the documentation describes the basic behavior of the zu_mandelbaum15 composite HOD model. To see how this composite model is built by the PrebuiltHodModelFactory class, see zu_mandelbaum15_model_dictionary.

Overview of the Zu & Mandelbaum et al. (2015) Model Features

This HOD-style model is based on Zu & Mandelbaum et al (2015). The behavior of this model is governed by the Behroozi et al. (2010), but with parameters that have been refit to z=0 data, and with scatter that is allowed to vary with halo mass. The occupation statistics have the same functional form as the Leauthaud et al. (2011) Composite Model introduced in Leauthaud et al (2011).

In this model, there are two populations, centrals and satellites. Central occupation statistics are given by a nearest integer distribution with first moment given by an erf function; the class governing this behavior is ZuMandelbaum15Cens. Central galaxies are assumed to reside at the exact center of the host halo; the class governing this behavior is TrivialPhaseSpace.

Satellite occupation statistics are given by a Poisson distribution with first moment given by a power law that has been truncated at the low-mass end; the class governing this behavior is ZuMandelbaum15Sats; satellites in this model follow an (unbiased) NFW profile, as governed by the NFWPhaseSpace class.

Building the Zu & Mandelbaum et al. (2015) Model

You can build an instance of this model using the PrebuiltHodModelFactory class as follows:

>>> from halotools.empirical_models import PrebuiltHodModelFactory
>>> model = PrebuiltHodModelFactory('zu_mandelbaum15')

Customizing the Zu & Mandelbaum et al. (2015) Model

There are two keyword arguments you can use to customize the instance returned by the factory:

First, the threshold keyword argument pertains to the minimum stellar mass of the galaxy sample, in solar mass units with h=1:

>>> model = PrebuiltHodModelFactory('zu_mandelbaum15', threshold=10.75)

Second, the prim_haloprop_key keyword argument determines which halo mass definition will be used to populate a mock with this model. You are free to choose any halo mass definition you like, but you should be aware that the best-fit parameters of the Zu & Mandelbaum model are based on halo_m200m:

>>> model = PrebuiltHodModelFactory('zu_mandelbaum15', threshold=11, haloprop_key='halo_mvir')

The Colossus python package written by Benedikt Diemer can be used to convert between different halo mass definitions. This may be useful if you wish to use an existing halo catalog for which the halo mass definition you need is unavailable.

As described in Changing Composite Model Parameters, you can always change the model parameters after instantiation by changing the values in the param_dict dictionary. For example,

>>> model.param_dict['alphasat'] = 1.1

The above line of code changes the power law slope between halo mass and satellite occupation number, \(\langle N_{\rm sat} \rangle \propto M_{\rm halo}^{\alpha}\). See Parameters of the Zu & Mandelbaum et al. (2015) model for a description of all parameters of this model.

Populating Mocks and Generating Zu & Mandelbaum et al. (2015) Model Predictions

As with any Halotools composite model, the model instance can populate N-body simulations with mock galaxy catalogs. In the following, we’ll show how to do this with fake simulation data via the halocat argument.

>>> from halotools.sim_manager import FakeSim
>>> halocat = FakeSim()
>>> model = PrebuiltHodModelFactory('zu_mandelbaum15')
>>> model.populate_mock(halocat)  

See ModelFactory.populate_mock for information about how to populate your model into different simulations. See Tutorials on analyzing galaxy catalogs for a sequence of worked examples on how to use the mock_observables sub-package to study a wide range of astronomical statistics predicted by your model.

Studying the Zu & Mandelbaum et al. (2015) Model Features

In addition to populating mocks, the zu_mandelbaum15 model also gives you access to its underlying analytical relations. Here are a few examples:

>>> import numpy as np
>>> halo_mass = np.logspace(11, 15, 100)

To compute the mean number of each galaxy type as a function of halo mass:

>>> mean_ncen = model.mean_occupation_centrals(prim_haloprop=halo_mass)
>>> mean_nsat = model.mean_occupation_satellites(prim_haloprop=halo_mass)

To compute the mean stellar mass of central galaxies as a function of halo mass:

>>> mean_sm_cens = model.mean_stellar_mass_centrals(prim_haloprop=halo_mass)

Now suppose you wish to know the mean halo mass of a central galaxy with known stellar mass:

>>> stellar_mass = np.logspace(9, 12, 100)
>>> inferred_halo_mass = model.mean_halo_mass_centrals(stellar_mass)

Parameters of the Zu & Mandelbaum et al. (2015) model

The best way to learn what the parameters of a model do is to just play with the code: change parameter values, make plots of how the underying analytical relations vary, and also of how the mock observables vary. Here we just give a simple description of the meaning of each parameter. You can also refer to the original publications Leauthaud et al (2011), Behroozi et al. (2010) and Zu & Mandelbaum et al (2015) for more detailed descriptions of the meaning of each parameter.

To see how the following parameters are implemented, see ZuMandelbaum15Cens.mean_occupation and Behroozi10SmHm.mean_stellar_mass.

  • param_dict[‘smhm_m0’] - Characteristic stellar mass at redshift-zero in \(\langle M_{\ast} \rangle(M_{\rm halo})\).
  • param_dict[‘smhm_m1’] - Characteristic halo mass at redshift-zero in \(\langle M_{\ast} \rangle(M_{\rm halo})\).
  • param_dict[‘smhm_beta’] - Low-mass slope at redshift-zero of \(\langle M_{\ast} \rangle(M_{\rm halo})\).
  • param_dict[‘smhm_delta’] - High-mass slope at redshift-zero of \(\langle M_{\ast} \rangle(M_{\rm halo})\).
  • param_dict[‘smhm_gamma’] - Transition between low- and high-mass behavior at redshift-zero of \(\langle M_{\ast} \rangle(M_{\rm halo})\).
  • param_dict[‘u’smhm_sigma’] - Normalization of the log-normal scatter about the mean relation \(\langle M_{\ast} \rangle(M_{\rm halo})\).
  • param_dict[‘u’smhm_sigma_slope’] - Parameter controlling halo mass-dependence of the log-normal scatter.

To see how the following parameters are implemented, see ZuMandelbaum15Sats.mean_occupation and Behroozi10SmHm.mean_stellar_mass.

  • param_dict[‘alphasat’] - Power law slope of the relation between halo mass and \(\langle N_{\rm sat} \rangle\).
  • param_dict[‘betasat’] - Controls the amplitude of the power law slope \(\langle N_{\rm sat} \rangle\).
  • param_dict[‘bsat’] - Also controls the amplitude of the power law slope \(\langle N_{\rm sat} \rangle\).
  • param_dict[‘betacut’] - Controls the low-mass cutoff of \(\langle N_{\rm sat} \rangle\).
  • param_dict[‘bcut’] - Also controls the low-mass cutoff of \(\langle N_{\rm sat} \rangle\).