# Hearin et al. (2015) Composite Model¶

This section of the documentation describes the basic behavior of the hearin15 composite HOD model. To see how this composite model is built by the PrebuiltHodModelFactory class, see hearin15_model_dictionary.

## Overview of the Hearin et al. (2015) Model Features¶

This HOD-style model is based on Hearin et al. (2015), arXiv:1512.03050. The behavior of this model is identical to Leauthaud et al. (2011), except this model implements assemby bias using decorated HOD methods in the HeavisideAssembias class.

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 Leauthaud11Cens. 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 Leauthaud11Sats; satellites in this model follow an (unbiased) NFW profile, as governed by the NFWPhaseSpace class.

For both centrals and satellites, the occupation statistics are decorated with assembly bias.

## Building the Hearin 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('hearin15')


## Customizing the Hearin et al. (2015) Model¶

There are numerous keyword arguments you can use to customize the instance returned by the factory.

The threshold keyword argument and the redshift keyword argument behave in the exact same way as they do in the leauthaud11 model. See Leauthaud et al. (2011) Composite Model for further details.

The sec_haloprop_key keyword argument determines the secondary halo property used to modulate the assembly bias. So, if you want halos at fixed mass with above- or below-average concentration to have above- or below-average mean occupations, you would set sec_haloprop_key to halo_nfw_conc.

The central_assembias_strength keyword argument determines how strong the assembly bias is in the occupation statistics of central galaxies. For constant assembly bias strength at all masses, set this variable to be a float between -1 to 1. For assembly bias strength that has mass-dependence, you should provide a list of control values, which will be interpreted as the strength at each of the input central_assembias_strength_abscissa control points.

As described below in Parameters of the Hearin et al. (2015) Model, the strength of assembly bias can always be modulated after instantiation by changing the appropriate values in param_dict. However, the central_assembias_strength_abscissa cannot be determined dynamically after instantiation. If you want to change the abscissa, you must instantiate a new model.

Exactly analogous comments apply to the satellite_assembias_strength and satellite_assembias_strength_abscissa keyword arguments.

The split keyword argument determines how your halo population is divided into two sub-populations at each mass. So, if you want halos in the upper 75th percentile of concentration at each mass to have different occupation statistics than halos in the lower 25th percentile, you would set split to 0.75. You can study the impact of mass-dependence of subpopulation division by passing in a list of control values for the split keyword, which will be interpreted as the splitting fraction at each of the input split_abscissa control points.

Once you make a choice for split, you cannot change this after instantiating a model. If you want to study the effects of different choices for split, you must instantiate a new model.

## Populating Mocks and Generating Hearin 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('hearin15')
>>> 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 Hearin et al. (2015) Model Features¶

In addition to populating mocks, the hearin15 model also gives you access to its underlying analytical relations. Firstly, the hearin15 model naturally inherits all of the methods of the leauthaud11 model, which you can read about in Leauthaud et al. (2011) Composite Model. Here we give examples of the additional methods that are unique to this model.

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


The mean_occupation methods of the hearin15 model operate slightly differently than they do in the leauthaud11 model, because here two halo properties govern the behavior of the model, not just one. Suppose you wish to compute the mean occupation of centrals for halos in the upper-percentile split. In this case you can force the mean_occupation_centrals method to interpret the halos as being in the upper-percentile division via the sec_haloprop_percentile keyword:

>>> mean_ncen_upper = model.mean_occupation_centrals(prim_haloprop = halo_mass, sec_haloprop_percentile=1)


For halos in the lower-percentile split:

>>> mean_ncen_lower = model.mean_occupation_centrals(prim_haloprop = halo_mass, sec_haloprop_percentile=0)


If you have a mixed population, just pass in a second array storing the actual value values of the secondary halo property via the sec_haloprop keyword:

>>> fake_sec_prop = np.random.random(len(halo_mass))
>>> mean_ncen = model.mean_occupation_centrals(prim_haloprop=halo_mass, sec_haloprop=fake_sec_prop)


To compute the strength of assembly bias as a function of halo mass:

>>> assembias_cens = model.assembias_strength_centrals(prim_haloprop=halo_mass)
>>> assembias_sats = model.assembias_strength_satellites(prim_haloprop=halo_mass)


## Parameters of the Hearin et al. (2015) Model¶

The hearin15 model naturally inherits all of the parameters of the leauthaud11 model, which you can read about in Parameters of the Leauthaud et al. (2011) model. Here we only describe the parameters that are unique to the hearin15 model.

• param_dict[‘mean_occupation_centrals_assembias_param1’] - controls the strength of assembly bias in the centrals population as specified at the first control point. If you passed in a float to the centrals_assembias_strength keyword argument, there will only be one such parameter. If you passsed in a list, there will be one parameter per list element. Changing the values of this parameter modulates the strength of assembly bias. The only permissible values are between -1 to 1; values outside this range will be interpreted as endpoint values.