caribou-hi 2.0.1

A Bayesian Model of the Diffuse Neutral Interstellar Medium

caribou_hi

A Bayesian Model of the Diffuse Neutral Interstellar Medium

caribou_hi is a Bayesian model of the diffuse neutral interstellar medium written in the bayes_spec spectral line modeling framework, which enables inference from observations of neutral hydrogen (HI) 21-cm emission and absorption spectra.

Read below to get started, and check out the tutorials and guides here: https://caribou-hi.readthedocs.io.

• Installation
  • Basic Installation
  • Development Installation
• Notes on Physics & Radiative Transfer
• Models
  • AbsorptionModel
  • EmissionModel
  • EmissionAbsorptionModel
  • ordered
  • fwhm_L
• Syntax & Examples
• Issues and Contributing
• License and Copyright

Installation

Basic Installation

Install with pip in a conda virtual environment:

conda create --name caribou_hi -c conda-forge pymc cxx-compiler pip
conda activate caribou_hi
# Due to a bug in arviz, this fork is temporarily necessary
# See: https://github.com/arviz-devs/arviz/issues/2437
pip install git+https://github.com/tvwenger/arviz.git@plot_pair_reference_labels
pip install caribou_hi

Development Installation

Alternatively, download and unpack the latest release, or fork the repository and contribute to the development of caribou_hi!

Install in a conda virtual environment:

cd /path/to/caribou_hi
conda env create -f environment.yml
conda activate caribou_hi-dev
pip install -e .

Notes on Physics & Radiative Transfer

All models in caribou_hi apply the same physics and equations of radiative transfer.

The 21-cm excitation temperature (also called the spin temperature) is derived from the gas kinetic temperature, gas density, and Lyα photon density following Kim et al. (2014) equation 4.

Clouds are assumed to be homogenous and isothermal. The ratio of the column density to the volume density, both free parameters, thus determines the path length through the cloud. The non-thermal line broadening assumes a Larson law relationship.

The optical depth and radiative transfer prescriptions follow that of Marchal et al. (2019). By default, the clouds are ordered from nearest to farthest, so optical depth effects (i.e., self-absorption) may be present.

Notably, since these are forward models, we do not make assumptions regarding the optical depth. These effects are predicted by the model. There is one exception: the ordered argument, described below.

Models

The models provided by caribou_hi are implemented in the bayes_spec framework. bayes_spec assumes that the source of spectral line emission can be decomposed into a series of "clouds", each of which is defined by a set of model parameters. Here we define the models available in caribou_hi.

AbsorptionModel

Note that this model struggles to fit absorption-only data. We recommend sticking to EmissionModel and EmissionAbsorptionModel, so hopefully you have some emission data too!

AbsorptionModel is a model that predicts 21-cm absorption spectra. The SpecData key for this model must be absorption. The following diagram demonstrates the relationship between the free parameters (empty ellipses), deterministic quantities (rectangles), model predictions (filled ellipses), and observations (filled, round rectangles). Many of the parameters are internally normalized (and thus have names like _norm). The subsequent tables describe the model parameters in more detail.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
log10_NHI	log10 HI column density	cm-2	$\log_{10}N_{\rm HI} \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[20.0, 1.0]
log10_depth	log10 line-of-sight depth	pc	$\log_{10}d \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[1.0, 1.0]
log10_pressure	log10 pressure	K cm-3	$\log_{10}P_{\rm th} \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[3.0, 1.0]
velocity	Velocity (same reference frame as data)	km s-1	$V \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.0, 10.0]
log10_nth_fwhm_1pc	Non-thermal FWHM at 1 pc	km s-1	$\log_{10}\Delta V_{\rm 1 pc} \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.2, 0.1]
depth_nth_fwhm_power	Nonthermal size-linewidth power law index	unitless	$\alpha \sim {\rm Gamma}(\mu=p_0, \sigma=p_1)$	[0.3, 0.1]

Hyper Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
log10_n_alpha	log10 Lyα photon density	cm-3	$\log_{10}n_\alpha \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[-6.0, 1.0]
fwhm_L	Lorentzian FWHM line width	km s-1	$\Delta V_{L} \sim {\rm HalfNormal}(\sigma=p)$	None

The parameters describing the non-thermal line broadening, log10_nth_fwhm_1pc and depth_nth_fwhm_power, are treated as cloud parameters by default and with hyper_depth_linewidth = False. With hyper_depth_linewidth = True, however, these parameters are treated as hyper parameters (and thus shared between all clouds). The later imposes a bias by assuming that all clouds share the same size-linewidth relationship.

EmissionModel

EmissionModel is similar to AbsorptionModel, except it predicts 21-cm emission brightness temperature spectra. The SpecData key for this model must be emission. EmissionModel takes an additional initialization argument, bg_temp, which is the assumed background brightness temperature (by default it is bg_temp=3.77, an estimate for the cosmic microwave background and Galactic synchrotron emission at 21-cm). The following diagram demonstrates the model.

EmissionAbsorptionModel

EmissionAbsorptionModel predicts both 21-cm emission (brightness temperature) and absorption (1-exp(-tau)) spectra. The SpecData keys must be emission and absorption. EmissionAbsorptionModel takes an additional initialization argument, bg_temp, which is the assumed background brightness temperature (by default it is bg_temp=3.77, an estimate for the cosmic microwave background and Galactic synchrotron emission at 21-cm). The following diagram demonstrates the model, and the subsequent table describe the additional model parameters.

Note that the filling factor, $f$, is unconstrained, so the reported column densities are really $N_{\rm HI}/f$.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
filling_factor	Filling Factor	``	$f \sim {\rm Uniform}(0, 1)$	``
absorption_weight	Absorption weight	``	$w_\tau \sim {\rm Beta}(\alpha=0, \beta=1-f)$	``

The filling_factor parameter accounts for beam dilution in the emission spectrum. The expected brightness temperature contribution from a cloud is multiplied by filling_factor, which takes values between zero and one.

The absorption_weight parameter accounts for the probability that a cloud that appears in the emission beam is not traced by the absorption sightline. The absorption optical depth of a cloud is multiplied by absorption_weight, which takes values between zero and one. The prior is chosen to require absorption_weight=1 when filling_factor=1.

ordered

An additional parameter to set_priors for these models is ordered. By default, this parameter is False, in which case the order of the clouds is from nearest to farthest. Sampling from these models can be challenging due to the labeling degeneracy: if the order of clouds does not matter (i.e., the emission is optically thin), then each Markov chain could decide on a different, equally-valid order of clouds.

If we assume that the emission is optically thin, then we can set ordered=True, in which case the order of clouds is restricted to be increasing with velocity. This assumption can drastically improve sampling efficiency. When ordered=True, the velocity prior is defined differently:

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
velocity	Velocity	km s-1	$V_i \sim p_0 + \sum_0^{i-1} V_i + {\rm Gamma}(\alpha=2, \beta=1.0/p_1)$	[0.0, 1.0]

fwhm_L

The velocity of a cloud can be challenging to identify when spectral lines are narrow and widely separated. We overcome this limitation by modeling the line profiles as a "pseudo-Voight" profile, which is a linear combination of a Gaussian and Lorentzian profile. The parameter fwhm_L is a latent hyper-parameter (shared among all clouds) that characterizes the width of the Lorentzian part of the line profile. When fwhm_L is zero, the line is perfectly Gaussian. This parameter produces line profile wings that may not be physical but nonetheless enable the optimization algorithms (i.e, MCMC) to converge more reliably and efficiently. Model solutions with non-zero fwhm_L should be scrutinized carefully. This feature can be turned off by supplying None (default) to prior_fwhm_L, in which case the line profiles are assumed Gaussian.

It can also be useful to give Variational Inference a head-start by initializing it evenly spread over the velocity prior. Supply start = {"velocity_norm", np.linspace(-3.0, 3.0, model.n_clouds)} to model.fit() or in init_kwargs to model.sample(). Check out the bayes_spec tutorial for an example.

Syntax & Examples

See the various tutorial notebooks under docs/source/notebooks. Tutorials and the full API are available here: https://caribou-hi.readthedocs.io.

Issues and Contributing

Anyone is welcome to submit issues or contribute to the development of this software via Github.

License and Copyright

Copyright (c) 2024 Trey Wenger

GNU General Public License v3 (GNU GPLv3)

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/.