Fitting the Global 21-cm Signal

It’s relatively straightforward to call the Markov-Chain Monte Carlo code emcee (Foreman-Mackey et al. (2013)), and perform a fits to:

• The turning points in the global 21-cm signal.
• The galaxy luminosity function
• Something really cool I haven’t even thought of yet!

Here, we’ll focus on the global signal application.

Fitting Global 21-cm Extrema

The fastest model to fit is one treating the Lyman-\(\alpha\), ionization, and thermal histories as a tanh function, so that’s what we’ll use in this example.

import numpy as np

# These go to every calculation
base_pars = \
{
 'problem_type': 101,
 'tanh_model': True,
 'blob_names': [['tau_e'], ['cgm_h_2', 'igm_Tk', 'igm_dTb']],
 'blob_ivars': [None, np.arange(6, 21)],
 'blob_funcs': None,
}

Note

These blob_* parameters were covered in Simple Parameter Study: 2-D Model Grid, so if you have yet to go through that example, now might be a good time!

Now, initialize a fitter:

import ares

# Initialize fitter
fitter = ares.inference.FitGlobal21cm(**base_pars)

and the turning points to be fit:

fitter.turning_points = True

# Assume default parameters
fitter.data = {'tanh_model': True}

# Set errors
fitter.error = {tp:[1.0, 5.] for tp in list('BCD')}

Now, we set the parameters to be varied in the fit and whether or not to explore their values in log10,

# Set axes of parameter space
fitter.parameters = ['tanh_xz0', 'tanh_xdz', 'tanh_Tz0', 'tanh_Tdz']
fitter.is_log = [False]*4

as well as the priors on the parameters, which in this case we’ll take to be uninformative over a relatively broad range:

from ares.inference import PriorSet
from ares.inference.Priors import UniformPrior

ps = PriorSet()
ps.add_prior(UniformPrior(5, 20), 'tanh_xz0')
ps.add_prior(UniformPrior(0.1, 20), 'tanh_xdz')
ps.add_prior(UniformPrior(5, 20), 'tanh_Tz0')
ps.add_prior(UniformPrior(0.1, 20), 'tanh_Tdz')
fitter.prior_set = ps

Finally, we set the number of Goodman-Weare walkers

fitter.nwalkers = 128

and run the fit:

fitter.run(prefix='test_tanh', burn=10, steps=50, save_freq=10)

This will result in a series of files named test_tanh*.pkl. See the example on Analyzing MCMC Calculations to proceed with inspecting the above dataset.

Fitting Global 21-cm Signal

To fit the entire spectrum, rather than just the turning points, the above example requires only minor modification.

Whereas previously we set

fitter.turning_points = True

# Assume default parameters
fitter.data = {'tanh_model': True}

# Set errors
fitter.error = {tp:[1.0, 5.] for tp in list('BCD')}

now, we must provide errors at a specified set of frequencies:

fitter.turning_points = False
fitter.frequencies = np.arange(50, 200) # assumed to be in MHz

# Assume default parameters
fitter.data = {'tanh_model': True}

# Set errors to be a constant 10 mK across the band
fitter.error = 10. * np.ones_like(fitter.frequencies)

That’s it!