Phenomenological Models for the Global 21-cm Signal

Two common phenomenological parameterizations for the global 21-cm signal are included in ARES and get their own set of pre-defined parameters: the tanh and Gaussian models. To generate them (without default parameters) one need only do:

import ares
import numpy as np
import matplotlib.pyplot as pl

sim_1 = ares.simulations.Global21cm(tanh_model=True)
sim_2 = ares.simulations.Global21cm(gaussian_model=True)

# Have a look
ax, zax = sim_1.GlobalSignature(color='k', fig=1)
ax, zax = sim_2.GlobalSignature(color='b', ax=ax)

Now, you might say “I could have done that myself extremely easily.” You’d be right! However, sometimes there’s an advantage in working through ARES even when using simply parametric forms for the global 21-cm signal. For example, you can tap into ARES’ inference module and fit data, perform forecasting, or run large sets of models. In each of these applications, ARES can take care of some annoying things for you, like tracking the quantities you care about and saving them to disk in a format that can be easily analyzed later on. For more concrete examples, check out the following pages:

• Inline Analysis

• Fitting the Global 21-cm Signal

• Monte-Carlo Sampling Higher Dimensional Spaces

• Analyzing MCMC Calculations

In the remaining sections we’ll cover different ways to parameterize the signal.

Parameterizing the IGM

Whereas the Gaussian absorption model makes no link between the brightness temperature and the underlying quantities of interest (ionization history, etc.), the tanh model first models \(J_{\alpha}(z)\), \(T_K(z)\), and \(x_i(z)\), and from those histories produces \(\delta T_b(z)\).

Now, let’s assemble a set of parameters that will generate a global 21-cm signal using ParameterizedQuantity objects for each main piece: the thermal, ionization, and Ly-\(\alpha\) histories. We’ll assume that the thermal and ionization histories are tanh functions, but take the Ly-\(\alpha\) background evolution to be a power-law in redshift:

pars = \
{
    'problem_type': 100,           # blank slate global 21-cm signal problem
    'parametric_model': True,      # in lieu of, e.g., tanh_model=True

    # Lyman alpha history first: ParameterizedQuantity #0
    'pop_Ja': 'pq[0]',
    'pq_func[0]': 'pl',         # Ja(z) = p0 * ((1 + z) / p1)**p2
    'pq_func_var[0]': '1+z',
    'pq_func_par0[0]': 1e-9,
    'pq_func_par1[0]': 20.,
    'pq_func_par2[0]': -7.,

    # Thermal history: ParameterizedQuantity #1
    'pop_Tk': 'pq[1]',         # Tk(z) = p1 + (p0 - p1) * 0.5 * (1 + tanh((p2 - z) / p3))
    'pq_func[1]': 'tanh_abs',
    'pq_func_var[1]': 'z',
    'pq_func_par0[1]': 1e3,
    'pq_func_par1[1]': 0.,
    'pq_func_par2[1]': 8.,
    'pq_func_par3[1]': 6.,

    # Ionization history: ParameterizedQuantity #2
    'pop_xi': 'pq[2]',        # xi(z) = p1 + (p0 - p1) * 0.5 * (1 + tanh((p2 - z) / p3))
    'pq_func[2]': 'tanh_abs',
    'pq_func_var[2]': 'z',
    'pq_func_par0[2]': 1,
    'pq_func_par1[2]': 0.,
    'pq_func_par2[2]': 8.,
    'pq_func_par3[2]': 2.,
}

Note

The thermal history automatically includes the adiabatic cooling term, so users need not add account for that explicitly.

To run it, as always:

sim_3 = ares.simulations.Global21cm(**pars)
sim_3.GlobalSignature(color='r', ax=ax)
pl.savefig('ares_gs_phenom.png')

Comparing three phenomenological models for the global 21-cm signal.

Now, because the parameters of these models are hard to intuit ahead of time, it can be useful to run a set of them. As per usual, we can use some built-in machinery.

blob_pars = ares.util.BlobBundle('gs:basics') \
          + ares.util.BlobBundle('gs:history')

base_pars = pars.copy()
base_pars.update(blob_pars)

mg = ares.inference.ModelGrid(**base_pars)

Let’s focus on the \(J_{\alpha}(z)\) parameters:

mg.axes = {'pq_func_par1[0]': np.arange(15, 26, 1),
           'pq_func_par2[0]': np.arange(-9, -2.5, 0.5)}

mg.run('test_Ja_pl', clobber=True)

Just to do a quick check, let’s look at where the absorption minimum occurs in this model grid:

anl = ares.analysis.ModelSet('test_Ja_pl')

anl.Scatter(anl.parameters, c='z_C', fig=4, edgecolors='none')

pl.savefig('ares_gs_Ja_grid.png')

Basic exploration of a 2-D parameter grid.