Halo Abundance Matching

This technique relates measurements of the galaxy luminosity function (LF) to the dark matter halo mass function by assuming galaxies of luminosity \(L\) with number density \(\phi(L)\) live in halos of mass \(M_h\), whose number density is given by the HMF, \(n(M_h)\). By enforcing the condition:

\[\int_{L(M_h)} \phi(L) dL = \int_{M_h} n(M_h) dM_h\]

we can solve for the mass-to-light relationship, \(L(M_h)\). This allows us to model the galaxy population at lower luminosities and higher redshifts than have so far been observed, provided some model for extrapolating \(L(M_h)\) in mass and redshift.

The two most critical ingredients in this model are constraints on the luminosity function, and a model for the HMF. Let’s grab the constraints from Bouwens et al. (2015) to start:

b15 = ares.util.read_lit('bouwens2015')

The object b15 has four important attributes:

• info

• redshifts

• data

• fits

The first gives a full reference to the paper, while the data and fits entries tell us where in the paper the data is located.

For example, the best-fit Schecter parameters are:

print b15.fits['lf']['pars']

with corresponding redshifts:

print b15.redshifts

Note

Check out the primer on Using Data from the Literature if you haven’t already!

To initialize a population, we must importantly set pop_model='ham':

pars = \
{
 'pop_model': 'ham',
 'pop_Macc': 'mcbride2009',         # Halo MAR
 'pop_constraints': 'bouwens2015',  # Galaxy LF
 'pop_kappa_UV': 1.15e-28,          # Luminosity / SFR
}

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

This population instance has an attribute ham, which carries along all of the abundance matching info. For starters, let’s just plot up the constraints on the LF at each redshift, which have been converted from magnitudes to rest-frame \(1500 \AA\) luminosities:

L = np.logspace(27., 30.)
for i, z in enumerate(pop.ham.constraints['z']):
    Lh, phi = pop.ham.LuminosityFunction(z)
    pl.loglog(Lh, phi, ls='--', lw=3)

pl.xlabel(r'$L \ (\mathrm{erg} \ \mathrm{s}^{-1} \ \mathrm{Hz}^{-1})$')
pl.ylabel(r'$\phi(L)$')
pl.xlim(1e27, 1e30)
pl.ylim(1e-39, 1e-29)
pl.show()

Now, the key quantity yielded by the abundance matching procedure is \(L_h(M_h)\), which we convert to a star-formation efficiency (SFE), \(f_{\ast}\), assuming a model for the halo mass accretion rate (MAR; from McBride et al. (2009); see pop_Macc parameter above):

colors = ['r', 'b', 'g', 'k', 'm']

# First, plot the discrete set of points for the SFE
for i, z in enumerate(pop.ham.redshifts):
    j = pop.ham.redshifts.index(z)
    Mh = pop.ham.MofL_tab[j]
    pl.scatter(Mh, pop.ham.fstar_tab[j], label=r'$z=%g$' % z, color=colors[i], marker='o', facecolors='none')

pl.plot([1e8, 1e15], [0.2]*2, color='k', ls=':')
pl.plot([1e8, 1e15], [0.3]*2, color='k', ls=':')
pl.xlabel(r'$M_h / M_{\odot}$')
pl.ylabel(r'$f_{\ast}$')
pl.legend(ncol=1, frameon=False, fontsize=16, loc='lower right')

# Now, plot fits to the SFE over a range of masses
Marr = np.logspace(8, 14)
for i, z in enumerate(pop.ham.redshifts):
    j = pop.ham.redshifts.index(z)

    fast = pop.ham.SFE(z=z, M=Marr)
    pl.loglog(Marr, fast, color=colors[i])

You can also access the SFRD via pop.ham.SFRD, which just integrates the product of the SFE and MAR over the mass function.

Note

You can run simulations of the global 21-cm using the HAM model for source populations. Just be sure to pass in the appropriate parameters, as pop_model != 'ham' by default!