caribou-hi 3.0.0

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
  • HIModel
    • AbsorptionModel
    • EmissionModel
    • EmissionAbsorptionModel
  • HIPhysicalModel
    • AbsorptionPhysicalModel
    • EmissionPhysicalModel
    • EmissionAbsorptionPhysicalModel
  • 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 size-linewidth relationship. The clouds are assumed to have a log-normal column density distribution, such that the a cloud seen in absorption tends to have a lower column density than what is seen in emission.

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.

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.

HIModel

HIModel is the base "observational" caribou_hi model. EmissionModel, AbsorptionModel, and EmissionAbsorptionModel extend HIModel. This model is parameterized in terms of observational quantities such that it mimics the interpretation of traditional inverse modeling techniques. Note that this is still a forward model, but the observational quantities (i.e., brightness temperature) are treated as the "fundamental" parameters. The following table describes HIModel shared parameters in more detail.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
fwhm2	Square FWHM line width	km2 s-2	$\Delta V^2 \sim p\times{\rm ChiSquared}(\nu=1)$	500.0
log10_nHI	log10 HI volume density	cm-3	$\log_{10}n_{\rm HI} \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.0, 1.5]
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_n_alpha	log10 Lyα photon density	cm-3	$\log_{10}n_\alpha \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[-6.0, 2.0]
fwhm_L	Lorentzian FWHM line width	km s-1	$\Delta V_{L} \sim {\rm HalfNormal}(\sigma=p)$	None

AbsorptionModel

AbsorptionModel is a model that predicts 21-cm absorption (1-exp(-tau)) spectra. AbsorptionModel extends HIModel. 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 table describes the additional AbsorptionModel parameters in more detail.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
tau_total	Integrated optical depth	km s-1	$\int \tau_V dV \sim {\rm HalfNormal}(\sigma=p)$	10.0
tkin_factor	Kinetic temperature / max. kinetic temperature	``	$T_K/T_{K, \rm max} \sim {\rm Beta}(\alpha=p_0, \beta=p_1)$	[2.0, 2.0]

The tkin_factor parameter sets the kinetic temperature in the range from zero to the maximum kinetic temperature allowed by the FWHM.

EmissionModel

EmissionModel is similar to AbsorptionModel, except it predicts 21-cm emission brightness temperature spectra. EmissionModel extends HIModel. 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, and the subsequent table describes the additional EmissionModel parameters.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
filling_factor	Filling factor	``	$f \sim {\rm Uniform}(T_B/T_S, 1.0)$	``
TB_fwhm	Brightness temperature x FWHM	K km s-1	$T_B \Delta V \sim {\rm HalfNormal}(\sigma=p)$	1000.0
tkin_factor	Kinetic temperature / max. kinetic temperature	``	$T_K/T_{K, \rm max} \sim {\rm Beta}(\alpha=p_0, \beta=p_1)$	[2.0, 2.0]

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 $f_{\rm min} = T_B/T_S$ and one.

The tkin_factor parameter sets the kinetic temperature in the range from $T_B$ to the maximum kinetic temperature allowed by the FWHM.

EmissionAbsorptionModel

EmissionAbsorptionModel predicts both 21-cm emission (brightness temperature) and absorption (1-exp(-tau)) spectra. EmissionAbsorptionModel extends EmissionModel. The SpecData keys must be emission and absorption. The following diagram demonstrates the model, and the subsequent table describes the additional EmissionAbsorptionModel parameters.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
log10_wt_ff_tkin	Absorption weight / (filling factor x spin temperature)	K-1	$\log_{10} f/(w_\tau T_s) \sim {\rm Normal}(\mu, \sigma_{\log_{10} N_{\rm HI}})$	$\sigma_{\log_{10} N_{\rm HI}}$ = None

The absorption_weight parameter accounts for the difference between the column density probed in absorption and that probed in emission. Specifically, $N_{\rm HI, abs}/N_{\rm HI, em} = f/w_\tau$. By default, absorption_weight is assumed to be one; the column density seen in absorption is identical to that seen in emission. This is the default behavior, and when prior_sigma_log10_NHI = None. Otherwise, the absorption column density is drawn from a log-normal distribution with a width given by prior_sigma_log10_NHI. Note that the mode of this distribution is less than the mean, so the assumption of a log-normal column density will tend to decrease the column density probed by absorption.

HIPhysicalModel

HIPhysicalModel is the base "physical" caribou_hi model. EmissionPhysicalModel, AbsorptionPhysicalModel, and EmissionAbsorptionPhysicalModel extend HIModel. This model is parameterized in terms of physical quantities, which enables us to include additional physical constraints such as a size-linewidth relationship. The assumed value for the size-linewidth power law index is set at initialization time by the parameter depth_nth_fwhm_power. The following table describes HIPhysicalModel shared parameters in more detail.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
fwhm2	Square FWHM line width	km2 s-2	$\Delta V^2 \sim p\times{\rm ChiSquared}(\nu=1)$	500.0
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_n_alpha	log10 Lyα photon density	cm-3	$\log_{10}n_\alpha \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[-6.0, 2.0]
nth_fwhm_1pc	Non-thermal FWHM line width at 1 pc depth	km s-1	$\Delta V_{\rm nth} \sim {\rm TruncatedNormal}(\mu=p_0, \sigma=p_1, {\rm lower}=0.0)$	[1.75, 0.25]
fwhm_L	Lorentzian FWHM line width	km s-1	$\Delta V_{L} \sim {\rm HalfNormal}(\sigma=p)$	None

AbsorptionPhysicalModel

AbsorptionPhysicalModel is a model that predicts 21-cm absorption (1-exp(-tau)) spectra. AbsorptionPhysicalModel extends HIPhysicalModel. 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 table describes the additional AbsorptionPhysicalModel parameters in more detail.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
NHI_fwhm2_thermal	Column density / thermal FWHM^2	cm-2 km-2 s2	$N_{\rm HI}/\Delta V_{\rm th}^2 \sim {\rm HalfNormal}(\sigma=p)$	1.0e20
fwhm2_thermal_fraction	Thermal FWHM^2 / total FWHM^2	``	$\Delta V_{\rm th}^2/\Delta V^2 \sim {\rm Beta}(\alpha=p_0, \beta=p_1)$	[2.0, 2.0]

EmissionPhysicalModel

EmissionPhysicalModel is similar to AbsorptionModel, except it predicts 21-cm emission brightness temperature spectra. EmissionPhysicalModel extends HIPhysicalModel. 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, and the subsequent table describes the additional EmissionPhysicalModel parameters.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
filling_factor	Filling factor	``	$f \sim {\rm Uniform}(T_B/T_S, 1.0)$	``
ff_NHI	Filling factor x column density	cm-2	$f N_{\rm HI} \sim {\rm HalfNormal}(\sigma=p)$	1.0e21
fwhm2_thermal_fraction	Thermal FWHM^2 / total FWHM^2	``	$\Delta V_{\rm th}^2/\Delta V^2 \sim {\rm Beta}(\alpha=p_0, \beta=p_1)$	[2.0, 2.0]

The fwhm2_thermal_fraction parameter sets the thermal line width in the range from the minimum, set by the optically thin brightness temperature, to the total line width.

EmissionAbsorptionPhysicalModel

EmissionAbsorptionPhysicalModel predicts both 21-cm emission (brightness temperature) and absorption (1-exp(-tau)) spectra. EmissionAbsorptionPhysicalModel extends EmissionPhysicalModel. The SpecData keys must be emission and absorption. The following diagram demonstrates the model, and the subsequent table describes the additional EmissionAbsorptionPhysicalModel parameters.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
log10_wt_ff_tkin	Absorption weight / (filling factor x spin temperature)	K-1	$\log_{10} f/(w_\tau T_s) \sim {\rm Normal}(\mu, \sigma_{\log_{10} N_{\rm HI}})$	$\sigma_{\log_{10} N_{\rm HI}}$ = 0.5

The absorption_weight parameter accounts for the difference between the column density probed in absorption and that probed in emission. Specifically, $N_{\rm HI, abs}/N_{\rm HI, em} = f/w_\tau$. The absorption column density is drawn from a log-normal distribution with a width given by prior_sigma_log10_NHI.

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) 2024-2025 Trey Wenger

Trey V. Wenger; [e-mail]

This code is licensed under MIT license (see LICENSE for details)