Allowed parameter ranges#

The version of 21cmFAST that comes with 21CMMC uses numerous interpolation tables. As such, several parameter combinations/choices may have restricted ranges. This, in addition to limits set by observations restricts the ranges for several of the astrophysical parameters that are available within 21CMMC.

Cosmology#

At the present only flat cosmologies are allowed as some in-built cosmological functions do not allow for non-flat cosmologies. This will likely be remedied in the near future. Thus, only the dark matter energy density (\(\Omega_m\)) is available to be varied as \(\Omega_\Lambda = 1 - \Omega_m\) is enforced. In all cases, sensible ranges are restricted by Planck and other cosmological probes. For example, see Kern et al., 2017 for varying cosmology with 21CMMC.

  • \(\Omega_m\) - should be allowed to vary across (0, 1]

  • \(\Omega_b\) - should be allowed to take any non-zero value

  • \(h\) - should be allowed to take any non-zero value

  • \(\sigma_8\) - should be allowed to take any non-zero value. Extreme values may cause issues in combination with the astrophysical parameters governing halo mass (either \(M_{min}\) or \(T_{min,vir}\))

  • \(n_s\) - should be allowed to take any non-zero value

Ionisation astrophysical parameters#

Allowed astrophysical parameter ranges are determined by which parameter set is being used. Dictated by the choice of USE_MASS_DEPENDENT_ZETA in the FlagOptions struct.

If USE_MASS_DEPENDENT_ZETA = False then the user is selecting the older astrophysical parameterisation for the ionising sources. These include any of: (i) \(\zeta\) - the ionising efficiency (ii) \(R_{mfp}\) - minimum photon horizon for ionising photons or (iii) \(T_{min,vir}\) - minimum halo mass for the ionising sources, \(T_{min,vir}\).

  • \(\zeta\) - Largest range used thus far was [5,200] in Greig & Mesinger, 2015. In principle it can be less than 200 or even extended beyond 200. Going below 5 is also plausible, but starts to cause numerical issues.

  • \(R_{mfp}\) - Only if INHOMO_RECO=False. Typically [5,20] is adopted. Again, going below 5 Mpc cause numerical problems, but any upper limit is allowed. However, beyond 20 Mpc or so, \(R_{mfp}\) has very little impact on the 21cm power spectrum.

  • \(T_{min,vir}\) - Typically [4,6] is selected, corresponding to \(10^4\) - \(10^6\) K corresponding to atomically cooled haloes. It becomes numerically unstable to go beyond \(10^6\) K owing to interpolation tables with respect to halo mass. It is possible to consider \(<10^4\) K, however, in doing so it will generate a discontinuity. Internally, \(T_{min,vir}\) is converted to \(M_{min}\) using Equation 26 of Barkana and Loeb, 2001 whereby the mean molecular weight, \(\mu\) differs according to the IGM state (\(\mu = 1.22\) for \(<10^4\) K and \(\mu = 0.59\) for \(>10^4\) K)

If USE_MASS_DEPENDENT_ZETA = True then the user is selecting the newest astrophysical parameterisation allowing for mass dependent ionisation efficiency as well as constructing luminosity functions (e.g. Park et al., 2019). This allows an expanded set of astrophysical parameter including: (i) \(f_{\ast,10}\) - star formation efficiency normalised at \(10^{10} M_{\odot}\) (ii) \(\alpha_{\ast}\) - power-law scaling of star formation efficiency with halo mass (iii) \(f_{esc,10}\) - escape fraction normalised at \(10^{10} M_{\odot}\) (iv) \(\alpha_{esc}\) - power-law scaling of escape fraction with halo mass (v) \(M_{turn}\) - Turn-over scale for the minimum halo mass (vi) \(t_{star}\) - Star-formation time scale and (vii) \(R_{mfp}\) - minimum photon horizon for ionising photons.

  • \(f_{\ast,10}\) - Typically [-3., 0.] corresponding to a range of \(10^{-3} - 1\). In principle \(f_{\ast,10}\) can exceed unity, as it is the normalisation at \(10^{10} M_{\odot}\) and would depend on the power-law index, \(\alpha_{\ast}\) as to whether or not the star-formation efficiency exceeds unity. However, probably no need to consider this scenario.

  • \(\alpha_{\ast}\) - Typically [-0.5,1]. Could be modified for stronger scalings with halo mass.

  • \(f_{esc,10}\) - Typically [-3,0.] corresponding to a range of \(10^{-3} - 1\). In principle \(f_{esc,10}\) can exceed unity, as it is the normalisation at \(10^{10} M_{\odot}\) and would depend on the power-law index, \(\alpha_{esc}\) as to whether or not the escape fraction exceeds unity. However, probably no need to consider this scenario.

  • \(\alpha_{esc}\) - Typically [-1.0,0.5]. Could be modified for stronger scalings with halo mass.

  • \(M_{turn}\) - Typically [8,10] corresponding to a range of \(10^8\) - \(10^{10} M_{\odot}\) . In principle it could be extended, though less physical. To have \(M_{turn} > 10^{10} M_{\odot}\) would begin to be inconsistent with existing observed luminosity functions. Could go lower than \(M_{turn} < 10^{8} M_{\odot}\) though it could begin to clash with internal limits in the code for interpolation tables (which are set to \(M_{min} = 10^6 M_{\odot}\) and \(M_{min} = M_{turn}/50\).

  • \(t_{star}\) - Typically (0,1). This is represented as a fraction of the Hubble time. Thus, cannot go beyond this range

  • \(R_{mfp}\) - same as above

Heating astrophysical parameters#

For the epoch of heating, there are three additional parameters that can be set. These, can only be used if USE_TS_FLUCT=True which performs the heating. These include: (i) \(L_{X<2keV}/SFR\) - the soft-band X-ray luminosity of the heating sources (ii) \(E_{0}\) - the minimum threshold energy for X-rays escaping into the IGM from their host galaxies and (iii) \(\alpha_{X}\) - the power-law spectral index of the X-ray spectral energy distribution function.

  • \(L_{X<2keV}/SFR\) - Typically [38, 42], corresponding to a range of \(10^{38} - 10^{42} erg\,s^{-1} M^{-1}_{\odot} yr\). This range could easily be extended depending on the assumed sources. This range corresponds to high mass X-ray binaries.

  • \(E_{0}\) - [100, 1500]. Range is in \(eV\) corresponding to 0.1 - 1.5 \(keV\). Luminosity is determined in the soft-band (i.e. < 2 keV), thus wouldn’t want to expand this upper limit too much. Observations limit the lower range to ~0.1 keV.

  • \(\alpha_{X}\) - Typically [-2.,2] but depends on the population of sources being considered (i.e. what is producing the X-ray’s). Note, the X-ray SED is defined as \(\propto \nu^{-\alpha_X}\)