Fitting EDGES-like Signals

In March of 2018, the EDGES collaboration reported an anamolously-strong absorption signal in the sky-averaged spectrum at 78 MHz (Bowman et al. (2018)). This is roughly where one might expect the global 21-cm signal, though its amplitude is 2-3x larger than the most extreme cases in a \(\Lambda \text{CDM}\) framework. This has led to a variety of exotic explanations, such as milli-charged dark (see, e.g., Barkana (2018), Fialkov et al. (2018), Berlin et al. (2018), Kovetz et al. (2018)) matter and excess power in the Rayleigh-Jeans tail of the microwave background (see, e.g., Feng & Holder (2018), Ewall-Wice et al. (2018), Fraser et al. (2018), Pospelov et al. (2018)). These lists are horribly incomplete, so apologies for that!

In ARES, we have not included any specific models of dark matter. However, in Mirocha & Furlanetto (2019) we explored two general possibilities:

• A parametric “excess cooling” model, in which the thermal history at very high redshift is allowed to depart from the predictions of standard recombination codes in \(\Lambda \text{CDM}\) cosmologies.

• An astrophysically-generated radio background, whose strength scales with the star formation of galaxies as \(L_R \propto f_R \text{SFR}\). The parameter \(f_R\) was left as a free parameter.

In this section, we will show how to run these models within ARES.

Excess Cooling Models

At high-z the temperature of a mean-density gas parcel evolves between \(T(z) \propto (1+z)\) and \(T(z) \propto (1+z)^2\) – the \((1+z)\) dependence a signature that Compton scattering tightly couples the CMB, spin, and kinetic temperatures, and the \((1+z)^2\) dependence indicating that Compton scattering has become inefficient, allowing the gas to cool adiabatically. This thermal history can be modeled accurately by noting that the log-cooling rate, \(d\log T/ d\log t\), of a mean-density gas parcel transitions smoothly between -2/3 at very high-z and -4/3 in a matter-dominated cosmology. So, rather than modeling the thermal history directly, we take

\[\frac{d\log T}{d\log t} = \frac{\alpha}{3} - \frac{(2+\alpha)}{3} \bigg\{1 + \exp \left[-\left(\frac{z}{z_0}\right)^{\beta} \right] \bigg \} \label{eq:Thist}\]

and integrate to obtain the thermal history. We have constructed this relation such that \(\alpha=-4\) reproduces the typical thermal history, and while varying $alpha$ can change the late-time cooling rate, the cooling rate as \(z \rightarrow \infty\) tends to \(d\log T/ d\log t = -2/3\), as it must to preserve the thermal history during the recombination epoch. See Section 2.3.1 of our paper for more discussion of this model.

To switch to this parameteric cooling model, you can add the following updates to any dictionary of parameters you would usually supply to ARES:

# New base_kwargs
cold = \
{
 'load_ics': 'parametric',
 'approx_thermal_history': 'exp',  # other models are available
 'inits_Tk_p0': 189.5850442,     # z0
 'inits_Tk_p1': 1.26795248,      # Beta
 'inits_Tk_p2': -4,              # alpha
}

The parameter values listed above are adopted to reproduce the standard \(\Lambda \text{CDM}\) result, which ARES draws from CosmoRec. To convince yourself of this, go ahead and compare to the standard scenario:

import ares
import matplotlib.pyplot as pl

# Default simulation, turn off astrophysical sources.
sim1 = ares.simulations.Global21cm(radiative_transfer=False)
sim1.run()

ax, zax = sim1.GlobalSignature(label='CosmoRec')

# Use the parameteric cooling
sim2 = ares.simulations.Global21cm(radiative_transfer=False, **cold)
sim2.run()

sim2.GlobalSignature(ax=ax, color='b', ls='--', label=r'$\alpha=-4$')

Now, if you make cooling faster than adiabatic:

colder = cold.copy()
colder['inits_Tk_p2'] = -6

sim3 = ares.simulations.Global21cm(radiative_transfer=False, **colder)
sim3.run()

sim3.GlobalSignature(ax=ax, color='r', label=r'$\alpha=-6$')
ax.legend()
pl.savefig('ares_edges_cold.png')

Voila!

Comparison of parametric excess cooling models with CosmoRec solution for \(\Lambda \text{CDM}\) cosmology.

By the way, if you would like to add the EDGES models you can do so via

b18 = ares.util.read_lit('bowman2018')
b18.plot_recovered(ax=ax, color='k', alpha=0.2)

If you want to use the exact models presented in Mirocha & Furlanetto (2019), you can summon the requisite parameters using the ParameterBundle framework.

pars = ares.util.ParameterBundle('mirocha2019:base') \
     + ares.util.ParameterBundle('mirocha2019:cold')

sim4 = ares.simulations.Global21cm(**pars)
sim4.run()

sim4.GlobalSignature(ax=ax, color='g', lw=3, ls='--', label='MF18 cooling', ymin=-600)
ax.legend()

pl.savefig('ares_edges_mf18_cooling.png')

Various “excess cooling” models for the global 21-cm signal compared to the EDGES 78 MHz signal(s).

The main free parameters in this model are:

• pop_Tmin{0}: The minimum virial temperature of star-forming halos [Kelvin].

• pop_Z{0}: Metallicity assumed for stellar models (BPASS v1.0 by default).

• pop_fesc{0}: Escape fraction of ionizing radiation.

• pop_fstar{0}: Efficiency of star formation. Actually many parameters, see The ParameterizedQuantity Framework to decipher the relevant parameters (e.g., pq_func_par0{0}[0], pq_func_par1{0}[0], etc.)

• pop_rad_yield{1}: The normalization of the X-ray luminosity SFR relation [\(\mathrm{erg} \ \mathrm{s}^{-1} \ (M_{\odot} \ \mathrm{yr})^{-1}\)]

• pop_alpha{1}: Spectral index of X-ray emission.

• pop_logN{1}: Typical column density of hydrogen in galaxies that hardens intrinsic X-ray spectrum [\(\log_{10} \mathrm{cm}^{-2}\)]

Astrophysical Radio Backgrounds

The simplest way to augment the radio background is to parameterize it. You can do so easily in ARES via the parameter Tbg, to which you can supply a Python function (assumed to be defined in terms of redshift), or pl, to indicate use of a power-law model. In the latter case, you must also supply the parameters Tbg_p0, Tbg_p1, and Tbg_p2 which define the power-law as

\[T_r(z) = p_0 \left(\frac{1+z}{1+p_1} \right)^{p_2}\]

Another way to implement a new radio background is to link emission to star formation, in analogy with how we generally scale the cosmic UV and X-ray backgrounds. In Mirocha & Furlanetto (2019), we adopted an empirical relation between the monochromatic 1.4 GHz luminosity and SFR (see, e.g., Gurkan et al. 2018),

\[L_R = 10^{22} f_R \left(\frac{\text{SFR}}{M_{\odot} \ \mathrm{yr}^{-1}} \right) \ \text{W} \ \text{s}^{-1} \ \text{Hz}^{-1}\]

and assumed a power-law spectrum with index \(\alpha=-0.7\).

Note

We have scaled the Gurkan et al. 150 MHz normalization to 1.4 GHz. See Section 2.3.2 in our paper for more details.

To create such a source population in ARES, we build off the standard approach calibrated to high-z UV luminosity functions. Because we’re assuming that the radio spectrum does not depend on host galaxy mass or time, we can simply link the SFRD of this new population to that of a pre-existing population, in this case with ID number 0:

from ares.physics.Constants import nu_0_mhz, h_p, erg_per_ev

# Need rest 21-cm frequency and spectral bounds in eV
E21 = nu_0_mhz * 1e6 * h_p / erg_per_ev # 1.4 GHz -> eV
Emin = 1e7 * (h_p / erg_per_ev)         # 10 MHz -> eV
Emax = 1e12 * (h_p / erg_per_ev)        # 100 GHz -> eV

# Setup new parameters
radio_pop = \
{
 'pop_sfr_model{2}': 'link:sfrd:0',    # Link to SFRD of population #0
 'pop_sed{2}': 'pl',
 'pop_alpha{2}': -0.7,
 'pop_Emin{2}': Emin,
 'pop_Emax{2}': Emax,
 'pop_EminNorm{2}': None,
 'pop_EmaxNorm{2}': None,
 'pop_Enorm{2}': E21, # 1.4 GHz
 'pop_rad_yield_units{2}': 'erg/s/sfr/hz', # Indicate this normalization is monochromatic

 'pop_solve_rte{2}': True,
 'pop_radio_src{2}': True,  # Only emit in the radio
 'pop_lw_src{2}': False,
 'pop_lya_src{2}': False,
 'pop_heat_src_igm{2}': False,
 'pop_ion_src_igm{2}': False,
 'pop_ion_src_cgm{2}': False,

 # Key parameters!
 'pop_rad_yield{2}': 1e22,
 'pop_zdead{2}': None,
}

Note

If you’re running with a different set of baseline parameters, you may need to use a different population ID number!

Again, these parameters are stored as a ParameterBundle, with the best-fitting values used in our paper, so we can simply execute:

pars = ares.util.ParameterBundle('mirocha2019:base') \
     + ares.util.ParameterBundle('mirocha2019:radio')

sim5 = ares.simulations.Global21cm(**pars)
sim5.run()

sim5.GlobalSignature(ax=ax, color='r', label='MF18 radio', ymin=-600)
ax.legend()

pl.savefig('ares_edges_mf18_radio.png')

Same figure as before, with the addition of a source-generated radio background.

By default, the parameter pop_zdead{2} is used, which instantaneously shuts down the radio emission at the supplied redshift. To see the impact of this, simply set pop_zdead{2} to None.