Zheng et al. (2007) Composite Model

This section of the documentation describes the basic behavior of the zheng07 composite HOD model. To see how this composite model is built by the PrebuiltHodModelFactory class, see zheng07_model_dictionary.

Overview of the Zheng et al. (2007) Model Features

This HOD-style model is based on Zheng et al. (2007), arXiv:0703457. 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 Zheng07Cens. 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 Zheng07Sats; satellites in this model follow an (unbiased) NFW profile, as governed by the NFWPhaseSpace class.

Building the Zheng et al. (2007) Model

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

>>> from halotools.empirical_models import PrebuiltHodModelFactory
>>> model = PrebuiltHodModelFactory('zheng07')

Customizing the Zheng et al. (2007) Model

There are three keyword arguments you can use to customize the instance returned by the factory, threshold, redshift and modulate_with_cenocc. In this section, we cover each of these keywords in turn.

Role of the threshold keyword

The threshold keyword argument pertains to the r-band absolute magnitude of the luminosity of the galaxy sample:

>>> model = PrebuiltHodModelFactory('zheng07', threshold = -20)

The only purpose of this keyword is to allow you to instantiate your model according to the best-fit values of the parameters taken from Table 1 of Zheng et al. (2007). After instantiation, the threshold attribute has no impact whatsoever on the behavior of the model. If you choose a different value for threshold than one of the values in Table 1 of Zheng et al. (2007), the model behavior will be set to the best-fit parameters of the default_luminosity_threshold variable set in the model_defaults module, and you can proceed to alter the param_dict however you like (see below).

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['logMmin'] = 12.5

The above line of code changes the minimum mass for a halo to host a central galaxy to \(10^{12.5}M_{\odot}\). See Parameters of the Zheng et al. (2007) model for a description of all parameters of this model.

Role of the redshift keyword

The redshift keyword argument must be set to the redshift of the halo catalog you might populate with this model.

>>> model = PrebuiltHodModelFactory('zheng07', threshold = -20, redshift = 2)

For the zheng07 model, the redshift attribute has no impact whatsoever on the behavior of the model; the purpose of this keyword for factory standardization purposes only, and also to remind users that this model has only been calibrated against z=0 observational data.

Role of the modulate_with_cenocc keyword

Note

As described in the docstring, if you are trying to strictly reproduce the results in Zheng et al 2007. you should set the modulate_with_cenocc keyword to True. Note that the default value is False, which for typical regions of parameter space gives very similar behavior to the published results.

The modulate_with_cenocc keyword argument controls how the Zheng07Sats.mean_occupation function is calculated. The default value is False, in which case the Zheng07Sats.mean_occupation function is calculated according to the following equation:

\[\langle N_{\mathrm{sat}} \rangle_{M} = \left( \frac{M - M_{0}}{M_{1}} \right)^{\alpha}\]

When modulate_with_cenocc is True, then the value of \(\langle N_{\rm sat}\rangle(M_{\rm vir})\) as calculated above will be multiplied by an overall factor of \(\langle N_{\rm cen}\rangle(M_{\rm vir})\), where the factor of \(\langle N_{\rm cen}\rangle\) is defined according to Zheng07Cens.mean_occupation:

\[\langle N_{\mathrm{cen}} \rangle(M_{\rm vir}) = \frac{1}{2}\left( 1 + \mathrm{erf}\left( \frac{\log_{10}M - \log_{10}M_{min}}{\sigma_{\log_{10}M}} \right) \right)\]

If you wish to modulate your satellite occupation statistics using alternative models for \(\langle N_{\rm cen}\rangle(M_{\rm vir})\), see Advanced usage of the zheng07 model.

Populating Mocks and Generating Zheng et al. (2007) 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('zheng07')
>>> model.populate_mock(halocat = 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 Zheng et al. (2007) Model Features

In addition to populating mocks, the zheng07 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)

Parameters of the Zheng et al. (2007) 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 publication, arXiv:0703457, for further details.

To see how the following parameters are implemented, see Zheng07Cens.mean_occupation.

  • param_dict[‘logMmin’] - Minimum mass required for a halo to host a central galaxy.
  • param_dict[‘sigma_logM’] - Rate of transition from \(\langle N_{\rm cen} \rangle = 0 \Rightarrow \langle N_{\rm cen} = 1 \rangle\).

To see how the following parameters are implemented, see Zheng07Sats.mean_occupation.

  • param_dict[‘alpha’] - Power law slope of the relation between halo mass and \(\langle N_{\rm sat} \rangle\).
  • param_dict[‘logM0’] - Low-mass cutoff in \(\langle N_{\rm sat} \rangle\).
  • param_dict[‘logM1’] - Characteristic halo mass where \(\langle N_{\rm sat} \rangle\) begins to assume a power law form.