Cacciato et al. (2009) Composite Model

This section of the documentation describes the basic behavior of the cacciato09 composite CLF model. To see how this composite model is built by the PrebuiltHodModelFactory class, see cacciato09_model_dictionary.

For brevity, we describe this model as a “Conditional Luminosity Function” model. However, the associated classes work equally well as a model for the “Conditional Stellar Mass Function”. The only difference will be a change in the prim_galprop_key from luminosity to stellar_mass, and an accompanying change to the parameter values in the model dictionary.

Overview of the Cacciato et al. (2009) Model Features

This CLF-style model is based on Cacciato et al. (2009), arXiv:0807.4932. The behavior of this model is governed by an assumed mass-to-light relation for the centrals and a modified Schechter function for the satellites.

There are two populations, centrals and satellites. Central occupation statistics are given by a log-normal distribution; the class governing this behavior is Cacciato09Cens. 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 luminosities given by a modified Schechter function. the class governing this behavior is Cacciato09Sats; satellites in this model follow an (unbiased) NFW profile, as governed by the NFWPhaseSpace class.

Building the Cacciato et al. (2009) Model

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

>>> from halotools.empirical_models import PrebuiltHodModelFactory
>>> model = PrebuiltHodModelFactory('cacciato09')

Customizing the Cacciato et al. (2009) Model

The threshold keyword argument allows you to customize the luminosity threshold of the galaxy sample in units of Lsun with h=1 units:

>>> model = PrebuiltHodModelFactory('cacciato09', threshold = 11)

It is not permissible to dynamically change the threshold of the model instance. If you want to explore the effects of different thresholds, you should instantiate multiple models. Alternatively, you can always impose a higher threshold on an already existing galaxy catalog produced with a given model instance. The resulting reduced catalog will have the same statistical properties as if you ran the model with the higher threshold and same parameters.

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['sigma'] = 0.2

The above line of code changes the scatter in the mass-to-light ratio. See Parameters of the Cacciato et al. (2009) model for a description of all parameters of this model.

Populating Mocks and Generating Cacciato et al. (2009) 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('cacciato09')
>>> 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 Cacciato et al. (2009) Model Features

In addition to populating mocks, the cacciato09 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 median luminosity of central galaxies as a function of halo mass:

>>> median_lum = model.median_prim_galprop_centrals(prim_haloprop = halo_mass)

To compute the average number of satellites per halo as a function of halo mass:

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

By modifying the parameters stored in the param_dict, the underlying analytical relations such as those above allow you to study how the model behaves without the need to create Monte Carlo realizations of the Universe.

Parameters of the Cacciato et al. (2009) 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 Cacciato et al. (2009) publication, arXiv:0807.4932. The fiducial values of the cacciato09 model instance implemented in Halotools are drawn from the WMAP3 analysis of that publication.

The model also generalizes the CLF model of Cacciato et al. (2009) by allowing modifications of the high-luminosity cut-off of the satellite population. Briefly, changing the delta parameters should only affect the abundance of satellites that have luminosities similar to the central luminosity. On the other hand, faint satellites should be unaffected. The details of the 2 delta parameters are described in Lange et al. (2017), arXiv:1705.05043. Setting both to 0, as done by default, is equivalent to the model of Cacciato et al. (2009).

  • param_dict[‘log_L_0’] - Normalization of central mass-to-light ratio.
  • param_dict[‘log_M_1’] - Characteristic mass of central mass-to-light ratio.
  • param_dict[‘gamma_1’] - Low-mass slope of central mass-to-light ratio.
  • param_dict[‘gamma_2’] - High-mass slope of central mass-to-light ratio.
  • param_dict[‘sigma’] - Scatter in the log-normal mass-to-light distribution of centrals.
  • param_dict[‘a_1’] - Sets the faint-end slope of the satellite luminosity function.
  • param_dict[‘a_2’] - Determines the mass-dependence of the faint-end slope of the satellite luminosity function.
  • param_dict[‘log_M_2’] - Determines the mass-dependence of the faint-end slope of the satellite luminosity function.
  • param_dict[‘b_0’] - Modifies normalization of the satellite luminosity function.
  • param_dict[‘b_1’] - Modifies normalization of the satellite luminosity function.
  • param_dict[‘b_2’] - Modifies normalization of the satellite luminosity function.
  • param_dict[‘delta_1’] - Modifies the high-luminosity exponential cut-off of the satellite luminosity function.
  • param_dict[‘delta_2’] - Modifies the high-luminosity exponential cut-off of the satellite luminosity function.