frb.dm - Dispersion Measure Module

The frb.dm module provides comprehensive tools for calculating and modeling dispersion measure contributions from various sources along the line of sight to Fast Radio Bursts.

Module Overview

The dispersion measure (DM) is a key observable for FRBs, representing the integrated electron column density along the line of sight:

\[\text{DM} = \int_0^{d} n_e(l) \, dl\]

This module decomposes the total DM into contributions from:

  • Milky Way: Galactic disk and halo electrons

  • Intergalactic Medium (IGM): Diffuse cosmic electrons

  • Host Galaxy: Electrons in the FRB host galaxy

  • Intervening Systems: Galaxy halos and other structures

Submodules

The DM package contains the following submodules:

  • frb.dm.igm - Intergalactic Medium calculations

  • frb.dm.cosmic - Cosmic dispersion measure modeling

  • frb.dm.host - Host galaxy DM contributions

  • frb.dm.mcmc - MCMC analysis methods

  • frb.dm.prob_dmz - DM-redshift probability calculations

  • frb.dm.dm_ism_healpix_map.py - DM_ISM Healpix map from NE2001 or the like

frb.dm.igm - Intergalactic Medium

Module for IGM calculations

frb.dm.igm.fukugita04_dict()[source]

Data from Fukugita 2004, Table 1

Returns:

data dict.

Return type:

f04_dict (dict)

frb.dm.igm.average_fHI(z, z_reion=7.0)[source]

Average HI fraction 1 = neutral 0 = fully ionized

Parameters:

  • z (float or ndarray) – redshift

  • z_reion (float, optional) – redshift of reionization.

Returns:

float or ndarray

Return type:

fHI (float or ndarray)

frb.dm.igm.average_He_nume(z, z_HIreion=7.0)[source]

Average number of electrons contributed by He as a function of redshift per He nucleus

Following Kulkarni, Worseck & Hennawi 2018

https://arxiv.org/abs/1807.09774

Parameters:

  • z (float or ndarray) – Redshift

  • z_HIreion (float, optional) – Helium reionization redshift.

Returns:

Number of free electrons

per Helium nucelus.

Return type:

neHe (float or ndarray)

frb.dm.igm.z_from_DM(DM, cosmo=FlatLambdaCDM(name='Planck18', H0=<Quantity 67.66 km / (Mpc s)>, Om0=0.30966, Tcmb0=<Quantity 2.7255 K>, Neff=3.046, m_nu=<Quantity [0., 0., 0.06] eV>, Ob0=0.04897), coord=None, corr_nuisance=True)[source]

Report back an estimated redshift from an input IGM DM Any contributions from the Galaxy and/or host need to have been ‘removed’

Parameters:

  • DM (Quantity) – Dispersion measure.

  • cosmo (Cosmology, optional) – Cosmology of the universe. LambdaCDM with the Repo cosmology used by default.

  • coord (SkyCoord, optional) – If provided, use it to remove the ISM

  • corr_nuisance (bool, optional) – If True, correct for the MW Halo and the host with 100 DM units.

Returns:

Redshift

Return type:

z (float)

frb.dm.igm.f_diffuse(z, cosmo=FlatLambdaCDM(name='Planck18', H0=<Quantity 67.66 km / (Mpc s)>, Om0=0.30966, Tcmb0=<Quantity 2.7255 K>, Neff=3.046, m_nu=<Quantity [0., 0., 0.06] eV>, Ob0=0.04897), return_rho: bool = False, perturb_Mstar: float = None)[source]

Calculate the cosmic fraction of baryons in diffuse gas phase based on our empirical knowledge of baryon distributions and their ionization state.

Note that the default values use the standard values from Madau & Dickinson (2014) and Fukugita (2004). The former use a Salpeter IMF for rho_* which is no longer in fashion.

Parameters:

  • z (float or ndarray) – Redshift

  • cosmo (Cosmology, optional) – Cosmology of

  • universe. (the)

  • return_rho (bool, optional) – If true, the diffuse gas density is returned too.

  • perturb_Mstar (float, optional) – If provided, scale rho_Mstar by this value. Useful for exploring the uncertainty in f_diffuse

Returns:

Diffuse gas baryon fraction. rho_diffuse (Quantity, optional): Physical diffuse gas density.

Returned if return_rho is set to true.

Return type:

f_diffuse (float, ndarray)

frb.dm.igm.sigma_fd(z, rel_err_Mstar)[source]

Calculate the uncertainty in the diffuse fraction at a given redshift.

Parameters:

  • z (float or np.ndarray) – The redshift value.

  • rel_err_Mstar (float) – The relative error in the stellar mass.

Returns:

The uncertainty in the diffuse fraction.

Return type:

sigma_fd (float or np.ndarray)

frb.dm.igm.ne_cosmic(z, cosmo=FlatLambdaCDM(name='Planck18', H0=<Quantity 67.66 km / (Mpc s)>, Om0=0.30966, Tcmb0=<Quantity 2.7255 K>, Neff=3.046, m_nu=<Quantity [0., 0., 0.06] eV>, Ob0=0.04897), mu=1.3333333333333333)[source]

Calculate the average cosmic electron number density as a function of redshift. :param z: Redshift :type z: float or ndarray :param cosmo: Cosmology in :type cosmo: Cosmology, optional :param which the calculations are to be performed.: :param mu: Reduced mass :type mu: float

Returns:

Average physical number density of electrons in the unverse in cm^-3.

Return type:

ne_cosmic (Quantity)

frb.dm.igm.average_DM(z, cosmo=FlatLambdaCDM(name='Planck18', H0=<Quantity 67.66 km / (Mpc s)>, Om0=0.30966, Tcmb0=<Quantity 2.7255 K>, Neff=3.046, m_nu=<Quantity [0., 0., 0.06] eV>, Ob0=0.04897), cumul=False, neval=10000, mu=1.3333333333333333)[source]

Macquart Relation

Calculate the average cosmic DM ‘expected’ based on our empirical knowledge of baryon distributions and their ionization state.

This includes both the IGM and galactic halos, i.e. any and all diffuse gas

Parameters:

  • z (float) – Redshift

  • mu (float) – Reduced mass correction for He when calculating n_H

  • cumul (bool, optional) – Return the DM as a function of z

Returns:

DM values evaluated at

the required redshifts. An array is returned only if cumul is True.

zeval (ndarray): evaluation redshifts. Only returned if

cumul is True.

Return type:

DM (Quantity or Quantity array)

frb.dm.igm.average_DMhalos(z, cosmo=FlatLambdaCDM(name='Planck18', H0=<Quantity 67.66 km / (Mpc s)>, Om0=0.30966, Tcmb0=<Quantity 2.7255 K>, Neff=3.046, m_nu=<Quantity [0., 0., 0.06] eV>, Ob0=0.04897), f_hot=0.75, rmax=1.0, logMmin=10.3, logMmax=16.0, neval=10000, cumul=False)[source]

Average DM_halos term from halos along the sightline to an FRB

Parameters:

  • z (float) – Redshift of the FRB

  • cosmo (Cosmology) – Cosmology in which the calculations are to be performed.

  • f_hot (float, optional) – Fraction of the halo baryons in diffuse phase.

  • rmax (float, optional) – Size of a halo in units of r200

  • logMmin (float, optional) – Lowest mass halos to consider Cannot be much below 10.3 or the Halo code barfs The code deals with h^-1 factors, i.e. do not impose it yourself

  • logMmax (float, optional) – Highest halo mass. Default to 10^16 Msun

  • neval (int, optional) – Number of redshift values between 0 and z the function is evaluated at.

  • cumul (bool, optional) – Return a cumulative evaluation?

Returns:

One value if cumul=False

else evaluated at a series of z

zeval (ndarray): Evaluation redshifts if cumul=True

Return type:

DM_halos (Quantity or Quantity array)

frb.dm.igm.average_DMIGM(z, cosmo=FlatLambdaCDM(name='Planck18', H0=<Quantity 67.66 km / (Mpc s)>, Om0=0.30966, Tcmb0=<Quantity 2.7255 K>, Neff=3.046, m_nu=<Quantity [0., 0., 0.06] eV>, Ob0=0.04897), f_hot=0.75, rmax=1.0, logMmin=10.3, neval=10000, cumul=False, return_DMhalos=False)[source]

Estimate DM_IGM in a cumulative fashion

Parameters:

  • z (float) – Redshift of the FRB

  • cosmo (Cosmology, optional) – Cosmology in which the calculations are to be performed. LambdaCDM with the Repo cosmology assumed by default.

  • f_hot (float, optional) – Fraction of the halo baryons in diffuse phase.

  • rmax (float, optional) – Size of a halo in units of r200

  • logMmin (float, optional) –

    Lowest mass halos to consider. Cannot be much below 10.3 or the Halo code barfs. The code deals with

    h^-1 factors, i.e. do not impose it yourself

  • neval (int, optional) – Number of redshift values between 0 and z the function is evaluated at.

  • cumul (bool, optional) – Return a cumulative evaluation?

  • return_DMHalos (bool, optional) – Also return avgDM_halos?

Returns:

DM_IGM (Quantity or Quantity array): One value if cumul=False

else evaluated at a series of z

zeval (ndarray, optional): Evaluation redshifts if cumul=True DM_halos (ndarray, optinal):

Return type:

float or list

frb.dm.igm.avg_rhoISM(z, cosmo=FlatLambdaCDM(name='Planck18', H0=<Quantity 67.66 km / (Mpc s)>, Om0=0.30966, Tcmb0=<Quantity 2.7255 K>, Neff=3.046, m_nu=<Quantity [0., 0., 0.06] eV>, Ob0=0.04897), perturb_Mstar: float = None)[source]

Co-moving Mass density of the ISM

Interpolates from z=0 values to z=1 where we assume M_ISM = M* and also for z>1

Parameters:

  • z (float or ndarray) – Redshift

  • cosmo (Cosmology, optional) – Cosmology in which the calculations are to be performed. LambdaCDM with defs.frb_cosmo parameters assumed by default.

  • perturb_Mstar (float, optional) – If provided, scale rho_Mstar by this value. Useful for exploring the uncertainty in f_diffuse

Returns:

Units of Msun/Mpc^3

Return type:

rhoISM (Quantity)

frb.dm.igm.avg_rhoMstar(z, remnants=True)[source]

Return mass density in stars as calculated by Madau & Dickinson (2014).

Note: these authors assumed the Salpeter IMF which is no longer in fashion.

Parameters:

  • z (float or ndarray) – Redshift

  • remnants (bool, optional) – Include remnants in the calculation?

Returns:

Units of Msun/Mpc^3

Return type:

rho_Mstar (Quantity)

frb.dm.igm.avg_rhoSFR(z)[source]

Average SFR density

Based on Madau & Dickinson (2014)

Parameters:

z (float or ndarray) – Redshift

Returns:

SFR – Units of Msun/yr/Mpc^3

Return type:

Quantity

frb.dm.igm.z_to_array(z)[source]

Convert input scalar or array to an array

Parameters:

z (float or ndarray) – Redshift

Returns:

  • z (ndarray)

  • flg_z (int) – 0 – Input was a scalar 1 – Input was an array

Key Functions

frb.dm.igm.ne_cosmic(z, cosmo=FlatLambdaCDM(name='Planck18', H0=<Quantity 67.66 km / (Mpc s)>, Om0=0.30966, Tcmb0=<Quantity 2.7255 K>, Neff=3.046, m_nu=<Quantity [0., 0., 0.06] eV>, Ob0=0.04897), mu=1.3333333333333333)[source]

Calculate the average cosmic electron number density as a function of redshift. :param z: Redshift :type z: float or ndarray :param cosmo: Cosmology in :type cosmo: Cosmology, optional :param which the calculations are to be performed.: :param mu: Reduced mass :type mu: float

Returns:

Average physical number density of electrons in the unverse in cm^-3.

Return type:

ne_cosmic (Quantity)

frb.dm.igm.z_from_DM(DM, cosmo=FlatLambdaCDM(name='Planck18', H0=<Quantity 67.66 km / (Mpc s)>, Om0=0.30966, Tcmb0=<Quantity 2.7255 K>, Neff=3.046, m_nu=<Quantity [0., 0., 0.06] eV>, Ob0=0.04897), coord=None, corr_nuisance=True)[source]

Report back an estimated redshift from an input IGM DM Any contributions from the Galaxy and/or host need to have been ‘removed’

Parameters:

  • DM (Quantity) – Dispersion measure.

  • cosmo (Cosmology, optional) – Cosmology of the universe. LambdaCDM with the Repo cosmology used by default.

  • coord (SkyCoord, optional) – If provided, use it to remove the ISM

  • corr_nuisance (bool, optional) – If True, correct for the MW Halo and the host with 100 DM units.

Returns:

Redshift

Return type:

z (float)

frb.dm.cosmic - Cosmic DM Modeling

Utility pieces for DM calculations

frb.dm.cosmic.DMcosmic_PDF(Delta, C0, sigma, A=1.0, alpha=3.0, beta=3.0)[source]

PDF(Delta) following the McQuinn formalism describing the DM_cosmic PDF

See Macquart+2020 for details

Parameters:

  • Delta (float or np.ndarray) – DM / averageDM values

  • C0 (float) – parameter

  • sigma (float)

  • A (float, optional)

  • alpha (float, optional)

  • beta (float, optional)

Return type:

float or np.ndarray

frb.dm.cosmic.deviate1(C0, sigma, beta, orig=False)[source]

Calculate deviate to solve fo C0

Parameters:

  • C0 (float)

  • sigma (float)

  • beta (float)

  • orig (bool, optional) – use the original approach. Not recommended

Returns:

deviate

Return type:

float

frb.dm.cosmic.build_C0_spline(max_log10_sigma=0.0, npt=100, ret_all=False, beta=4.0)[source]

Generate a spline of C0 vs sigma values for the McQuinn formalism

Parameters:

  • max_log10_sigma (float, optional)

  • npt (int, optional)

  • ret_all (bool, optional) – if True, return more items

  • beta (float, optional)

Returns:

If ret_all, return f_C), sigmas, COs else return the spline

Return type:

float or tuple

frb.dm.cosmic.grab_sigma_spline()[source]

Load up the sigma spline

Parameters:

redo

Return type:

scipy.interpolate.InterpolatedUnivariateSpline

frb.dm.cosmic.grab_C0_spline(max_log10_sigma=0.0, npt=100, ret_all=False, redo=False, beta=3.0, ifile=None)[source]

Load up the C0 spline

Parameters:

  • max_log10_sigma

  • npt

  • ret_all

  • redo

  • beta

  • ifile

Return type:

scipy.interpolate.InterpolatedUnivariateSpline

frb.dm.host - Host Galaxy Contributions

Methods related to DM in FRB Host Galaxies

frb.dm.host.dm_host_from_Halpha(z: float, Halpha: Quantity, reff_ang: Quantity, AV: float = None)[source]

Estimate DM_Host from Halpha and angular size (and redshift)

Parameters:

  • z (float) – Redshift

  • Halpha (Quantity) – Total Halpha flux of the galaxy

  • reff_ang (Quantity) – Galaxy angular size

  • AV (float, optional) – Correct for dust if provided

Returns:

DM_host as observed (i.e. includes the 1/1+z term)

Return type:

Quantity

frb.dm.host.dm_host_from_ssfr(z: float, ssfr: Quantity)[source]

Estimate DM_host from the surface density of SFR

Parameters:

  • z (float) – Redshift

  • ssfr (Quantity) – Surface density of SFR (with units)

Returns:

DM_host as observed (i.e. includes the 1/1+z term)

Return type:

Quantity

frb.dm.host.dm_host_halo(R: Quantity, log10_Mstar: float, z: float, HMR: str = 'Moster', mNFW: ModifiedNFW = None)[source]

Calculate the DM contribution from the host galaxy’s halo

Parameters:

  • R (Quantity) – Projected radial distance from the center of the halo

  • log10_Mstar (float) – Logarithm (base 10) of the stellar mass of the host galaxy

  • z (float) – Redshift of the host galaxy

  • HMR (str, optional) – Halo mass relation to use (‘Moster’ or ‘Kravstov’). Default is ‘Moster’

  • mNFW (models.ModifiedNFW, optional) – Instance of the ModifiedNFW model. If None, a default model will be created. Default is None

Returns:

DM contribution from the host galaxy’s halo

Return type:

Quantity

Raises:

IOError – If the specified halo mass relation (HMR) is not supported

frb.dm.mcmc - MCMC Analysis

Methods for MCMC analysis of the Macquart relation

frb.dm.mcmc.grab_parmdict(tight_ObH=False)[source]

Generate the parameter dict for the MCMC run

Parameters:

tight_ObH (bool, optional) – [description]. Defaults to False.

Raises:

IOError – [description]

Returns:

[description]

Return type:

dict

frb.dm.mcmc.one_prob(Obh70, F, DM_FRBp, z_FRB, mu=150.0, lognorm_s=1.0, lognorm_floor=0.0, orig=False, beta=4.0)[source]

Calculate the probability for a single FRB

Parameters:

  • Obh70 (float) – Value of Omega_b * H_0

  • F (float) – Feedback parameter

  • DM_FRBp (np.ndarray) – Values of DM_FRBp for analysis

  • z_FRB (np.ndarray) – z values for evaluation

  • mu (float, optional) – Mean of log-normal PDF

  • lognorm_s (float, optional) – Sigma of log-normal PDF

  • lognorm_floor (float, optional) – Floor to the log-normal PDF

  • orig (bool, optional) – if True (not recommended!), use the original approach to calculating sigma

  • beta (float, optional) – Parameter for DM PDF

Returns:

Likelihood probability

Return type:

float

frb.dm.mcmc.mcquinn_DM_PDF_grid(Delta_values, C0, sigma, alpha=3.0, beta=3.0)[source]

PDF(Delta) for the McQuinn formalism describing the DM_cosmic PDF

Parameters:

  • Delta (2D ndarray) – DM / averageDM values

  • C0 (np.ndarray) – C0 values

  • sigma (np.ndarray) – sigma values

  • alpha (float, optional)

  • beta (float, optional)

Returns:

frb.dm.mcmc.all_prob(Obh70, F, in_DM_FRBp, z_FRB, mu=150.0, lognorm_s=1.0, lognorm_floor=0.0, beta=3.0)[source]

Calculate the probability for a set of FRBs

Parameters:

  • Obh70 (float) – Value of Omega_b * H_0

  • F (float) – Feedback parameter

  • in_DM_FRBp (np.ndarray) – Values of DM_FRBp for analysis Not used?!

  • z_FRB (np.ndarray) – z values for evaluation

  • mu (float, optional) – Mean of log-normal PDF

  • lognorm_s (float, optional) – Sigma of log-normal PDF

  • lognorm_floor (float, optional) – Floor to the log-normal PDF

  • beta (float, optional) – Parameter for DM PDF

Returns:

Log like-lihood

Return type:

float

frb.dm.mcmc.pm_four_parameter_model(parm_dict: dict, tight_ObH=False, beta=3.0)[source]

Builds a pymc3 model for the 4-parameter MCMC

Parameters:

  • parm_dict (dict) – dict with the pymc3 parameters

  • tight_ObH (bool, optional) – If True, restrict the ObH0 value based on CMB. Defaults to False.

  • beta (float, optional) – PDF parameter. Defaults to 3..

Raises:

IOError – [description]

Returns:

pymc3 model

Return type:

pm.Model

frb.dm.mcmc.tt_spl_sigma(value)[source]
frb.dm.mcmc.tt_spl_C0_3(value)[source]

frb.dm.prob_dmz - DM-Redshift Probabilities

Code for calculations of P(DM|z) and P(z|DM)

class frb.dm.prob_dmz.P_DM_z[source]

Bases: object

frb.dm.prob_dmz.prob_DMcosmic_FRB(frb, DM_min=0.0, DM_max=5000.0, step=1.0, ISMfrac=0.1, DM_MWhalo=50.0)[source]

Generate P(DM_cosmic) for an input FRP

Parameters:

  • frb (frb.frb.FRB)

  • DM_min (float, optional) – Lowest DM for the calulation

  • DM_max (float, optional) – Highest DM for the calulation

  • step (float, optional) – Step size of DM array in units of pc/cm**3

  • ISMfrac (float, optional) – Fraction of DM_ISM to adopt as the 1-sigma error

  • DM_MWhalo (float, optional) – Fixed value to use for the MW halo

Returns:

numpy.ndarray, numpy.ndarray

DM_cosmic values (units of pc/cm**3), P(DM_cosmic) normalized to unity

Return type:

tuple

frb.dm.prob_dmz.grid_P_DMcosmic_z(beta=3.0, F=0.31, zvals=None, DM_cosmics=None, cosmo=FlatLambdaCDM(name='Planck18', H0=<Quantity 67.66 km / (Mpc s)>, Om0=0.30966, Tcmb0=<Quantity 2.7255 K>, Neff=3.046, m_nu=<Quantity [0., 0., 0.06] eV>, Ob0=0.04897))[source]

Generate a grid of P(DM_cosmic|z)

Parameters:

  • beta (float, optional) – sigma_DM_cosmic parameter

  • F (float, optional) – Feedback parameter (higher F means weaker feedback)

  • zvals (np.ndarray, optional) – Redshifts for the grid

  • DMcosmic (np.ndarray, optional) – DMs for the grid

  • cosmo (optional) – Cosmology

Returns:

z, DM_cosmic, P(DM_cosmic|z)

Return type:

tuple

frb.dm.prob_dmz.build_grid_for_repo(outfile: str)[source]

Build a P(DM,z) grid for the Repository

Parameters:

outfile (str) – Path+filename for output file

frb.dm.prob_dmz.grab_repo_grid(grid_name)[source]

Grab the grid from the Repository based on the given grid name

Parameters:

grid_name (str) – Name of the grid to grab

Returns:

Numpy dict from the npz or npy save file

Return type:

dict

frb.dm.prob_dmz.get_DMcosmic_from_z(zarray, perc=0.5, redo_pdmz_grid=False, DMevals=array([1.00000000e+00, 3.00100100e+00, 5.00200200e+00, 7.00300300e+00, 9.00400400e+00, 1.10050050e+01, 1.30060060e+01, 1.50070070e+01, 1.70080080e+01, 1.90090090e+01, 2.10100100e+01, 2.30110110e+01, 2.50120120e+01, 2.70130130e+01, 2.90140140e+01, 3.10150150e+01, 3.30160160e+01, 3.50170170e+01, 3.70180180e+01, 3.90190190e+01, 4.10200200e+01, 4.30210210e+01, 4.50220220e+01, 4.70230230e+01, 4.90240240e+01, 5.10250250e+01, 5.30260260e+01, 5.50270270e+01, 5.70280280e+01, 5.90290290e+01, 6.10300300e+01, 6.30310310e+01, 6.50320320e+01, 6.70330330e+01, 6.90340340e+01, 7.10350350e+01, 7.30360360e+01, 7.50370370e+01, 7.70380380e+01, 7.90390390e+01, 8.10400400e+01, 8.30410410e+01, 8.50420420e+01, 8.70430430e+01, 8.90440440e+01, 9.10450450e+01, 9.30460460e+01, 9.50470470e+01, 9.70480480e+01, 9.90490490e+01, 1.01050050e+02, 1.03051051e+02, 1.05052052e+02, 1.07053053e+02, 1.09054054e+02, 1.11055055e+02, 1.13056056e+02, 1.15057057e+02, 1.17058058e+02, 1.19059059e+02, 1.21060060e+02, 1.23061061e+02, 1.25062062e+02, 1.27063063e+02, 1.29064064e+02, 1.31065065e+02, 1.33066066e+02, 1.35067067e+02, 1.37068068e+02, 1.39069069e+02, 1.41070070e+02, 1.43071071e+02, 1.45072072e+02, 1.47073073e+02, 1.49074074e+02, 1.51075075e+02, 1.53076076e+02, 1.55077077e+02, 1.57078078e+02, 1.59079079e+02, 1.61080080e+02, 1.63081081e+02, 1.65082082e+02, 1.67083083e+02, 1.69084084e+02, 1.71085085e+02, 1.73086086e+02, 1.75087087e+02, 1.77088088e+02, 1.79089089e+02, 1.81090090e+02, 1.83091091e+02, 1.85092092e+02, 1.87093093e+02, 1.89094094e+02, 1.91095095e+02, 1.93096096e+02, 1.95097097e+02, 1.97098098e+02, 1.99099099e+02, 2.01100100e+02, 2.03101101e+02, 2.05102102e+02, 2.07103103e+02, 2.09104104e+02, 2.11105105e+02, 2.13106106e+02, 2.15107107e+02, 2.17108108e+02, 2.19109109e+02, 2.21110110e+02, 2.23111111e+02, 2.25112112e+02, 2.27113113e+02, 2.29114114e+02, 2.31115115e+02, 2.33116116e+02, 2.35117117e+02, 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1.97398699e+03, 1.97598799e+03, 1.97798899e+03, 1.97998999e+03, 1.98199099e+03, 1.98399199e+03, 1.98599299e+03, 1.98799399e+03, 1.98999499e+03, 1.99199600e+03, 1.99399700e+03, 1.99599800e+03, 1.99799900e+03, 2.00000000e+03]), beta=3.0, F=0.31, cosmo=FlatLambdaCDM(name='Planck18', H0=<Quantity 67.66 km / (Mpc s)>, Om0=0.30966, Tcmb0=<Quantity 2.7255 K>, Neff=3.046, m_nu=<Quantity [0., 0., 0.06] eV>, Ob0=0.04897))[source]

Gives DMcosmic values of zarray, considering the percentile. Default is median (i.e. percentile=0.5)

Parameters:

  • zarray (float or np.array) – Redshift value or values If np.array, then it must start from 0

  • perc (float) – Percentile of the PDF(DM_cosmic) for each z, from 0 to zmax. Default = 0.5

  • redo_pdmz_grid (bool) – Whether to recompute the DMcosmic-z grid Default is False If True, it will be computational expensive

  • DMevals (np.array) – Array representing the DMcosmic evaluations of the PDF. Should start with 1. Default = np.linspace(1.,5000.,1000) Changing this only works when redo_pdmz_grid=True

  • beta (float) – Slope of a galactic halo parameter (See Macquart+2020) Default = 3.0 Changing this only works when redo_pdmz_grid=True

  • F (float) – Parameter that depends on galaxy feedback (See Macquart+2020) Default = 0.31 Changing this only works when redo_pdmz_grid=True

  • cosmo (astropy.cosmology object) – Cosmology

Returns:

  • zarray (np.array) – z values where the DMcosmic was computed

  • DMcosmic_array (np.array) – DMcosmic values corresponding to the z_array (in a given percentile)

Detailed Function Documentation

IGM Functions

frb.dm.prob_dmz.ne_cosmic(z, cosmo=defs.frb_cosmo, mu=4. / 3)

Calculate the average cosmic electron number density as a function of redshift.

This function computes the physical number density of electrons in the universe, accounting for the cosmic baryon density and ionization fraction.

Parameters:

  • z (float) – Redshift

  • cosmo (astropy.cosmology.Cosmology) – Cosmology in which the calculations are performed

  • mu (float) – Reduced mass (default: 4/3 for fully ionized H and He)

Returns:

Average physical number density of electrons in the universe

Return type:

astropy.units.Quantity

Example:

from frb.dm.igm import ne_cosmic

# Calculate electron density at z=1
z = 1.0
n_e = ne_cosmic(z)
print(f"Electron density at z={z}: {n_e}")
frb.dm.prob_dmz.DM_cosmic(z, cosmo=defs.frb_cosmo, cumul=False)

Calculate the cosmic dispersion measure contribution.

Integrates the cosmic electron density from z=0 to the specified redshift, including corrections for cosmological expansion.

Parameters:

  • z (float) – Redshift of the FRB

  • cosmo (astropy.cosmology.Cosmology) – Cosmology for calculations

  • cumul (bool) – Return cumulative DM evolution with redshift

Returns:

Cosmic dispersion measure or array if cumul=True

Return type:

astropy.units.Quantity

Example:

from frb.dm.igm import DM_cosmic

# Calculate cosmic DM to z=0.5
z_frb = 0.5
dm_cosmic = DM_cosmic(z_frb)
print(f"Cosmic DM to z={z_frb}: {dm_cosmic}")
frb.dm.prob_dmz.z_from_DM(DM, cosmo=defs.frb_cosmo, coord=None, corr_nuisance=True)

Estimate redshift from observed dispersion measure.

Uses the cosmic DM-redshift relation to estimate the redshift of an FRB given its observed dispersion measure, after correcting for Galactic and other contributions.

Parameters:

  • DM (astropy.units.Quantity) – Observed dispersion measure

  • cosmo (astropy.cosmology.Cosmology) – Cosmology

  • coord (astropy.coordinates.SkyCoord) – Sky coordinates (optional)

  • corr_nuisance (bool) – Apply corrections for nuisance parameters

Returns:

Estimated redshift

Return type:

float

Example:

from frb.dm.igm import z_from_DM
from astropy import units as u

# Estimate redshift from DM
DM_obs = 500 * u.pc / u.cm**3
z_est = z_from_DM(DM_obs)
print(f"Estimated redshift: {z_est:.2f}")
frb.dm.prob_dmz.DM_halos(z, cosmo, f_hot=0.75, rmax=2., logMmin=10.3, logMmax=16., neval=500, cumul=False)

Calculate dispersion measure contribution from galaxy halos.

Models the DM contribution from hot gas in galaxy halos along the line of sight, using a halo mass function approach.

Parameters:

  • z (float) – Redshift of the FRB

  • cosmo (astropy.cosmology.Cosmology) – Cosmology for calculations

  • f_hot (float) – Fraction of halo baryons in diffuse phase

  • rmax (float) – Size of halo in units of r200

  • logMmin (float) – Log of minimum halo mass (cannot be much below 10.3)

  • logMmax (float) – Log of maximum halo mass (default: 10^16 Msun)

  • neval (int) – Number of redshift evaluation points

  • cumul (bool) – Return cumulative evaluation

Returns:

DM contribution from halos

Return type:

astropy.units.Quantity

Utility Functions

frb.dm.prob_dmz.f_diffuse(z, cosmo=defs.frb_cosmo, return_rho=False)

Calculate the fraction of baryons in the diffuse IGM.

Parameters:

  • z (float) – Redshift

  • cosmo (astropy.cosmology.Cosmology) – Cosmology

  • return_rho (bool) – Also return the diffuse gas density

Returns:

Diffuse fraction (and density if requested)

Return type:

float or tuple

frb.dm.prob_dmz.sigma_DM_cosmic(DM_cosmic, rel_err_Mstar=0.1)

Calculate uncertainty in cosmic DM due to stellar mass uncertainties.

Parameters:

  • DM_cosmic (astropy.units.Quantity) – Cosmic dispersion measure

  • rel_err_Mstar (float) – Relative error in stellar mass

Returns:

DM uncertainty

Return type:

astropy.units.Quantity

Physical Models

Ionization Models

The module includes models for cosmic reionization history:

Hydrogen Reionization: - Reionization redshift: z_HI ~ 6-8 - Affects IGM electron fraction at high redshift

Helium Reionization: - HeII reionization: z_HeII ~ 3-4 - Increases electron density at intermediate redshift

Implementation:

from frb.dm.igm import f_ionization_He

# Calculate helium ionization fraction
z = 3.0
f_He = f_ionization_He(z)
print(f"Helium ionization fraction at z={z}: {f_He}")

Halo Models

Galaxy halo contributions use the halo mass function and hot gas profiles:

Key Parameters: - Halo mass range: 10^10.3 to 10^16 M_sun - Hot gas fraction: ~75% of cosmic baryon fraction - Radial extent: typically 2 × r200

Scaling Relations: - Gas density profiles (e.g., beta models) - Mass-temperature relations - Feedback effects on gas distribution

Usage Examples

Basic DM Analysis

from frb.dm import igm
from astropy import units as u

# Calculate total cosmic DM budget
z_max = 2.0
dm_cosmic_total = igm.DM_cosmic(z_max)

# Calculate contributions at different redshifts
redshifts = [0.1, 0.5, 1.0, 1.5, 2.0]
for z in redshifts:
    dm_z = igm.DM_cosmic(z)
    print(f"DM(z={z:.1f}) = {dm_z:.0f}")

Redshift Estimation

from frb.dm.igm import z_from_DM
from astropy import units as u
from astropy.coordinates import SkyCoord

# Observed FRB parameters
DM_obs = 750 * u.pc / u.cm**3
coord = SkyCoord(ra=123.45*u.deg, dec=12.34*u.deg)

# Estimate redshift
z_est = z_from_DM(DM_obs, coord=coord)
print(f"Estimated redshift: {z_est:.2f}")

# Calculate expected cosmic DM at this redshift
dm_cosmic_expected = igm.DM_cosmic(z_est)
print(f"Expected cosmic DM: {dm_cosmic_expected:.0f}")

# Calculate excess DM (host + local contributions)
dm_excess = DM_obs - dm_cosmic_expected
print(f"Excess DM: {dm_excess:.0f}")

Halo Contribution Analysis

from frb.dm.igm import DM_halos
from astropy.cosmology import Planck18

# Calculate halo DM contribution
z_frb = 1.0
dm_halos = DM_halos(z_frb, Planck18)

print(f"Halo DM contribution to z={z_frb}: {dm_halos:.1f}")

# Compare different halo parameters
f_hot_values = [0.5, 0.75, 1.0]
for f_hot in f_hot_values:
    dm_h = DM_halos(z_frb, Planck18, f_hot=f_hot)
    print(f"Halo DM (f_hot={f_hot}): {dm_h:.1f}")

Advanced Usage

Custom Cosmologies

from astropy.cosmology import FlatLambdaCDM
from frb.dm.igm import DM_cosmic

# Define custom cosmology
custom_cosmo = FlatLambdaCDM(H0=70, Om0=0.3)

# Compare DM calculations
z = 1.0
dm_planck = DM_cosmic(z)  # Default Planck18
dm_custom = DM_cosmic(z, cosmo=custom_cosmo)

print(f"DM (Planck18): {dm_planck:.0f}")
print(f"DM (Custom):   {dm_custom:.0f}")

Monte Carlo Analysis

import numpy as np
from frb.dm.igm import z_from_DM
from astropy import units as u

# Simulate DM uncertainties
DM_mean = 500 * u.pc / u.cm**3
DM_err = 50 * u.pc / u.cm**3
n_trials = 1000

# Generate DM samples
DM_samples = np.random.normal(
    DM_mean.value, DM_err.value, n_trials
) * DM_mean.unit

# Calculate redshift distribution
z_samples = [z_from_DM(dm) for dm in DM_samples]

z_mean = np.mean(z_samples)
z_std = np.std(z_samples)
print(f"Redshift: {z_mean:.2f} ± {z_std:.2f}")

Error Handling

The DM module includes robust error handling:

from frb.dm.igm import z_from_DM
from astropy import units as u

try:
    # This might fail for very high DM values
    z = z_from_DM(10000 * u.pc / u.cm**3)
except ValueError as e:
    print(f"DM too high: {e}")

try:
    # This might fail for negative DM
    z = z_from_DM(-100 * u.pc / u.cm**3)
except ValueError as e:
    print(f"Invalid DM: {e}")

Performance Notes

Computational Efficiency: - DM calculations use vectorized numpy operations - Cosmological integrals are cached for repeated calls - Halo mass function integration is optimized for speed

Memory Usage: - Large redshift arrays may require significant memory - Use cumul=False for single-point calculations - Consider chunking for very large datasets

Accuracy: - Integration tolerances balance speed and precision - Default parameters provide ~1% accuracy for most applications - High-precision calculations may require custom tolerances

See Also

Note

The DM module assumes a standard cosmological model by default. For non-standard cosmologies, explicitly pass the cosmology parameter to relevant functions.