# Models for Star Formation in Galaxies¶

The are a number of different ways to model star formation in ARES. The method employed is determined by the value of the parameter pop_sfr_model, which can take on any of the following values:

• 'fcoll'
Relate the global star formation rate density (SFRD) to the rate at which matter collapses into halos above some threshold.
• 'sfrd-func'
Model the SFRD with a user-supplied function of redshift.
• 'sfe-func'
Model the star formation efficiency (SFE) as a function of halo mass and (optionally) redshift.
• 'link:sfrd:ID'
Link the SFRD to that of the population with given ID number.

Each of these is discussed in more detail below.

Note

In what follows, we show isolated examples for illustrative purposes, i.e., initialization of a single source population and verification of its properties. To implement these star formation models in global 21-cm or meta-galactic background calculations with multiple source populations, you’ll need to add population ID numbers to each star formation parameter. For example, pop_sfr_model{0} instead of pop_sfr_model, and so on.

## fcoll models¶

In this case the SFRD is modeled as:

$\mathrm{SFRD} = f_{\ast} \bar{\rho}_b^0 \frac{d f_{\mathrm{coll}}}{dt}$

where $$f_{\ast}$$ is the efficiency of star formation, $$\bar{\rho}_b^0$$ is the mean baryon density today, and $$f_{\mathrm{coll}}$$ is the fraction of mass in collapsed halos above some threshold.

A basic set of 'fcoll' parameters can be summoned via:

import ares

pars = ares.util.ParameterBundle('pop:fcoll')


To initialize a population, just do:

pop = ares.populations.GalaxyPopulation(**pars)

# Print SFRD at redshift 20.
print pop.SFRD(20.)


This will be a very small number because ARES uses cgs units internally, which means the SFRD is in units of $$\mathrm{g} \ \mathrm{s}^{-1} \ \mathrm{cm}^{-3}$$, with the volume assumed to be co-moving. To convert to the more familiar units of $$M_{\odot} \ \mathrm{year}^{-1} \ \mathrm{cMpc}^{-3}$$,

from ares.physics.Constants import rhodot_cgs

print pop.SFRD(20.) * rhodot_cgs


Note

You can also provide pop_Tmax (or pop_Mmax) to relate the SFRD to the rate of collapse onto halos above pop_Tmin and below pop_Tmax (or pop_Mmax).

## sfrd-func models¶

If pop_sfr_model=='sfrd_func' you’ll need to provide your SFRD function via the pop_sfrd parameter. You can use a ParameterBundle if you’d like, though in this case it is particularly short:

pars = ares.util.ParameterBundle('pop:sfrd-func')


A really simple example would be just to make this population have a constant star formation history:

pars['pop_sfrd'] = lambda z: 1e-2


However, you could also use a ParameterizedHaloProperty here (see param_populations for more details). This might be advantageous if, for example, you want to vary the parameters of the SFRD in a model grid or Monte Carlo simulation.

Let’s make a power-law SFRD. For example, the following:

pars['pop_sfr_model'] = 'sfrd-func'
pars['pop_sfrd'] = 'pq'
pars['pq_func'] = 'pl'
pars['pq_func_var'] = '1+z'
pars['pq_func_par0'] = 1e-2
pars['pq_func_par1'] = 7.
pars['pq_func_par2'] = -6


sets the SFRD to be

$\mathrm{SFRD} = 10^{-2} \left(\frac{1 + z}{7} \right)^{-6} M_{\odot} \ \mathrm{year}^{-1} \ \mathrm{cMpc}^{-3}$

## sfrd-tab models¶

Alternatively, you can supply a lookup table for the SFRD. To do this, modify your parameters as follows:

pars['pop_sfr_model'] = 'sfrd-tab'
pars['pop_sfrd'] = (z, sfrd)


where z and sfrd are arrays you’ve generated yourself. ARES will construct an interpolant from these arrays using scipy.interpolate.interp1d, using the method supplied in pop_sfrd_interp. By default, this will be a 'cubic' spline, but you can also supply, e.g., pop_sfrd_interp='linear'.

By default, ARES assumes your SFRD is in units of $$\mathrm{g} \ \mathrm{s}^{-1} \ \mathrm{cm}^{-3}$$ (co-moving) (corresponding to pop_sfrd_units='internal'), but if you can change this to ‘msun/yr/cmpc^3’ if you’d prefer the more sensible units of $$M_{\odot} \ \mathrm{yr}^{-1} \ \mathrm{cMpc}^{-3}$$! In fact, these are the only two options, so as long as pop_sfrd_units != 'internal', ARES assumes the $$M_{\odot} \ \mathrm{yr}^{-1} \ \mathrm{cMpc}^{-3}$$ units.

## sfe-func models¶

Rather than parameterizing the SFRD directly, it is possible to parameterize the star formation efficiency as a function of halo mass and redshift, and integrate over the halo mass function in order to obtain the global SFRD.

Grab a few parameters to begin:

pars = ares.util.ParameterBundle('pop:sfe-func')


This set of parameters assumes a double power-law for the SFE as a function of halo mass with sensible values for the parameters. To create a population instance, as per usual,

pop = ares.populations.GalaxyPopulation(**pars)


To test the SFE model,

import numpy as np
import matplotlib.pyplot as pl

Mh = np.logspace(7, 13, 100)
pl.loglog(Mh, pop.SFE(z=10, M=Mh))


and the SFRD:

pop.SFRD(10.)