Models with Multiple Source Populations

ARES can handle an arbitrary number of source populations. To access this functionality, create a dictionary representing each source population of interest. Below, we’ll create a population representative of PopII stars and another representative of PopIII stars.

Before we start, it is important to note that in ARES, source populations are identified by their spectra over some contiguous interval in photon energy. This can be somewhat counterintuitive. For example, though UV emission from stars and X-ray emission from their compact remnants, e.g., X-ray binary systems, are both natural byproducts of star formation, we treat them as separate source populations in ARES even though the emission from each type of source is related to the same rate of star formation. However, because stars and XRBs have very different spectra, whose normalizations are parameterized differently, it is more convenient in the code to keep them separate. Because of this, what you might think of as a single source population (stars and their remnants) actually constitutes two source populations in ARES.

Let’s start with a PopII source population:

pars = \
{
 'problem_type': 100,              # Blank slate global 21-cm signal calculation

 # Setup star formation
 'pop_Tmin{0}': 1e4,               # atomic cooling halos
 'pop_fstar{0}': 1e-1,             # 10% star formation efficiency

 # Setup UV emission
 'pop_sed_model{0}': True,
 'pop_sed{0}': 'bb',               # PopII stars -> 10^4 K blackbodies
 'pop_temperature{0}': 1e4,
 'pop_rad_yield{0}': 1e42,
 'pop_fesc{0}': 0.2,
 'pop_Emin{0}': 10.19,
 'pop_Emax{0}': 24.6,
 'pop_EminNorm{0}': 13.6,
 'pop_EmaxNorm{0}': 24.6,
 'pop_lya_src{0}': True,
 'pop_ion_src_cgm{0}': True,
 'pop_heat_src_igm{0}': False,

 # Setup X-ray emission
 'pop_sed{1}': 'pl',
 'pop_alpha{1}': -1.5,
 'pop_rad_yield{1}': 2.6e38,
 'pop_Emin{1}': 2e2,
 'pop_Emax{1}': 3e4,
 'pop_EminNorm{1}': 5e2,
 'pop_EmaxNorm{1}': 8e3,

 'pop_lya_src{1}': False,
 'pop_ion_src_cgm{1}': False,
 'pop_heat_src_igm{1}': True,

 'pop_sfr_model{1}': 'link:sfrd:0',
}

Note

See Problem Types for more information about why we chose problem_type=100 here.

We might as well go ahead and run this to establish a baseline:

import ares

sim = ares.simulations.Global21cm(**pars)
sim.run()

ax, zax = sim.GlobalSignature(color='k')

Now, let’s add a PopIII-like source population. We’ll assume that PopIII sources are brighter on average (in both the UV and X-ray) but live in lower mass halos. We could just copy-pase the dictionary above, change the population ID numbers and, for example, the UV and X-ray pop_rad_yield parameters. Or, we could use some built-in tricks to speed this up.

First, let’s take the PopII parameter set and make a ParameterBundle object:

popII = ares.util.ParameterBundle(**pars)

This let’s us easily extract parameters according to their ID number, i.e.,

popIII_uv = popII.pars_by_pop(0, True)
popIII_uv.num = 2
popIII_xr = popII.pars_by_pop(1, True)
popIII_xr.num = 3

The second argument tells ARES to remove the parameter ID numbers.

Now, we can simply reset the ID numbers and update a few important parameters:

popIII_uv['pop_Tmin{2}'] = 300
popIII_uv['pop_Tmax{2}'] = 1e4
popIII_uv['pop_rad_yield{2}'] = 1e43
popIII_uv['pop_temperature{2}'] = 1e5
popIII_uv['pop_fstar{2}'] = 1e-3

popIII_xr['pop_sfr_model{3}'] = 'link:sfrd:2'
popIII_xr['pop_rad_yield{3}'] = 2.6e39

Now, let’s make the final parameter dictionary and run it:

pars.update(popIII_uv)
pars.update(popIII_xr)

sim = ares.simulations.Global21cm(**pars)
sim.run()

ax, zax = sim.GlobalSignature(color='b', ax=ax)

import matplotlib.pyplot as pl
pl.savefig('ares_gs_multipop.png')

Example calculations with a single population (black) and multiple source populations (blue).

Note that the parameter file hangs onto the parameters of each population separately. To verify a few key changes, you could do:

for key in ['pop_Tmin', 'pop_fstar', 'pop_rad_yield']:
    print key, sim.pf.pfs[0][key], sim.pf.pfs[2][key]

Note

These are very simple models for PopII and PopIII stars. For more sophisticated approaches, see More Realistic Galaxy Populations and Including Models for Population III Stars.

Note in the final plot command, we supplied the previous ax object to overplot the results of the single population calculation on the same axes as before.

Alternative Population ID Tagging Syntax

Using the curly braces to denote population ID numbers will lead to problems if you don’t want to create a dictionary of parameters, but instead want to supply the parameters as keyword arguments directly to a simulation class. For this reason, it is also acceptable to bracket population ID numbers with underscores in parameter names. For example, instead of

pars['pop_Tmin{0}'] = 1e4

you could do

pars['pop_Tmin_0_'] = 1e4

Linking Populations

If you are fitting a realization of the 21-cm signal with a multi-population model, you may want to have parameters common to both models that are allowed to vary. To link two parameters together, you can simply replace a parameter value of one population (usually a number) to the name of a parameter for another population. For example, to make the PopII and PopIII star formation efficiencies the same (using the parameter dictionary above), you could do

pars['pop_fstar{2}'] = 'pop_fstar{0}'

and any change to pop_fstar{0} will automatically propagate to pop_fstar{2}.