Quick Lyman-alpha (QLA) model

This section of the documentation gives a brief description of the different components of the quick Lyman-alpha sub-grid model. We mostly focus on the parameters and values output in the snapshots.

Given the nature of the model, no feedback or black holes are used. The star formation model is minimalist and the chemistry/cooling models are limited to primordial abundances.

Gas entropy floor

The gas particles in the QLA model are prevented from cooling below a certain temperature. The temperature limit depends on the density of the particles. The floor is implemented as a polytropic “equation of state”\(P = P_c \left(\rho/\rho_c\right)^\gamma\) (all done in physical coordinates), with the constants derived from the user input given in terms of temperature and Hydrogen number density. We use \(gamma=1\) in this model. The code computing the entropy floor is located in the directory src/entropy_floor/QLA/ and the floor is applied in the drift and kick operations of the hydro scheme.

An additional over-density criterion above the mean baryonic density is applied to prevent gas not collapsed into structures from being affected. To be precise, this criterion demands that the floor is applied only if \(\rho_{\rm com} > \Delta_{\rm floor}\bar{\rho_b} = \Delta_{\rm floor} \Omega_b \rho_{\rm crit,0}\), with \(\Delta_{\rm floor}\) specified by the user, \(\rho_{\rm crit,0} = 3H_0/8\pi G\) the critical density at redshift zero [1], and \(\rho_{\rm com}\) the gas co-moving density. Typical values for \(\Delta_{\rm floor}\) are of order 10.

The model is governed by 3 parameters for each of the two limits. These are given in the QLAEntropyFloor section of the YAML file. The parameters are the Hydrogen number density (in \(cm^{-3}\)) and temperature (in \(K\)) of the anchor point of the floor as well as the minimal over-density required to apply the limit. To simplify things, all constants are converted to the internal system of units upon reading the parameter file.

For a normal quick Lyman-alpha run, that section of the parameter file reads:

QLAEntropyFloor:
  density_threshold_H_p_cm3: 0.1       # Physical density above which the entropy floor kicks in expressed in Hydrogen atoms per cm^3.
  over_density_threshold:    10.       # Over-density above which the entropy floor can kick in.
  temperature_norm_K:        8000      # Temperature of the entropy floor at the density threshold expressed in Kelvin.

SWIFT will convert the temperature normalisations and Hydrogen number density thresholds into internal energies and densities respectively assuming a neutral gas with primordial abundance pattern. This implies that the floor may not be exactly at the position given in the YAML file if the gas has different properties. This is especially the case for the temperature limit which will often be lower than the imposed floor by a factor \(\frac{\mu_{\rm neutral}}{\mu_{ionised}} \approx \frac{1.22}{0.59} \approx 2\) due to the different ionisation states of the gas.

Recall that we additionally impose an absolute minimum temperature at all densities with a value provided in the SPH section of the parameter file. This minimal temperature is typically set to 100 Kelvin.

Gas cooling: Wiersma+2009a with fixed primordial metallicity

The gas cooling is based on the redshift-dependent tables of Wiersma et al. (2009)a that include element-by-element cooling rates for the 11 elements (H, He, C, N, O, Ne, Mg, Si, S, Ca and Fe) that dominate the total rates. The tables assume that the gas is in ionization equilibrium with the cosmic microwave background (CMB) as well as with the evolving X-ray and UV background from galaxies and quasars described by the model of Haardt & Madau (2001). Note that this model ignores local sources of ionization, self-shielding and non-equilibrium cooling/heating. The tables can be obtained from this link which is a re-packaged version of the original tables. The code reading and interpolating the table is located in the directory src/cooling/QLA/.

The Wiersma tables containing the cooling rates as a function of redshift, Hydrogen number density, Helium fraction (\(X_{He} / (X_{He} + X_{H})\)) and element abundance relative to the solar abundance pattern assumed by the tables (see equation 4 in the original paper). Since the quick Lyman-alpha model is only of interest for gas outside of haloes, we can make use of primordial gas only. This means that the particles do not need to carry a metallicity array or any individual element arrays. Another optimization is to ignore the cooling rates of the metals in the tables.

Above the redshift of Hydrogen re-ionization we use the extra table containing net cooling rates for gas exposed to the CMB and a UV + X-ray background at redshift nine truncated above 1 Rydberg. At the redshift or re-ionization, we additionally inject a fixed user-defined amount of energy per unit mass to all the gas particles.

In addition to the tables we inject extra energy from Helium II re-ionization using a Gaussian model with a user-defined redshift for the centre, width and total amount of energy injected per unit mass. Additional energy is also injected instantaneously for Hydrogen re-ionisation to all particles (active and inactive) to make sure the whole Universe reaches the expected temperature quickly (i.e not just via the interaction with the now much stronger UV background).

The cooling itself is performed using an implicit scheme (see the theory documents) which for small values of the cooling rates is solved explicitly. For larger values we use a bisection scheme. The cooling rate is added to the calculated change in energy over time from the other dynamical equations. This is different from other commonly used codes in the literature where the cooling is done instantaneously.

We note that the QLA cooling model does not impose any restriction on the particles’ individual time-steps. The cooling takes place over the time span given by the other conditions (e.g the Courant condition).

Finally, the cooling module also provides a function to compute the temperature of a given gas particle based on its density, internal energy, abundances and the current redshift. This temperature is the one used to compute the cooling rate from the tables and similarly to the cooling rates, they assume that the gas is in collisional equilibrium with the background radiation. The temperatures are, in particular, computed every time a snapshot is written and they are listed for every gas particle:

Name	Description	Units	Comments
Temperatures	Temperature of the gas as computed from the tables.	[U_T]	The calculation is performed using quantities at the last time-step the particle was active

Note that if one is running without cooling switched on at runtime, the temperatures can be computed by passing the --temperature runtime flag (see Command line options). Note that the tables then have to be available as in the case with cooling switched on.

The cooling model is driven by a small number of parameter files in the QLACooling section of the YAML file. These are the re-ionization parameters and the path to the tables. A valid section of the YAML file looks like:

QLACooling:
  dir_name:     /path/to/the/Wiersma/tables/directory # Absolute or relative path
  H_reion_z:            11.5      # Redshift of Hydrogen re-ionization
  H_reion_ev_p_H:        2.0      # Energy injected in eV per Hydrogen atom for Hydrogen re-ionization.
  He_reion_z_centre:     3.5      # Centre of the Gaussian used for Helium re-ionization
  He_reion_z_sigma:      0.5      # Width of the Gaussian used for Helium re-ionization
  He_reion_ev_p_H:       2.0      # Energy injected in eV per Hydrogen atom for Helium II re-ionization.

Star formation

The star formation in the Quick Lyman-alpha model is very simple. Any gas particle with a density larger than a multiple of the critical density for closure is directly turned into a star. The idea is to rapidly eliminate any gas that is found within bound structures since we are only interested in what happens in the inter-galactic medium. The over-density multiple is the only parameter of this model.

The code applying this star formation law is located in the directory src/star_formation/QLA/.

For a normal Quick Lyman-alpha run, that section of the parameter file reads:

# Quick Lyman-alpha star formation parameters
QLAStarFormation:
  over_density:              1000      # The over-density above which gas particles turn into stars.

[1]

Recall that in a non-cosmological run the critical density is set to 0, effectively removing the over-density constraint of the floors.