Conditional luminosity functions#

This page shows how to evaluate conditional luminosity functions with LFKit’s public API. A conditional luminosity function describes the abundance of galaxies as a function of absolute magnitude while allowing the model to depend on another variable.

A conditional luminosity function has the form \(\Phi(M \mid x)\), where \(M\) is absolute magnitude and \(x\) is the conditioning variable. The quantity

\[\Phi(M \mid x)\,dM\]

is the number density of galaxies with absolute magnitudes between \(M\) and \(M+dM\) at fixed \(x\). The conditioning variable is generic: it can represent redshift, halo mass, environment, galaxy type, richness, stellar mass, or another quantity.

LFKit exposes conditional luminosity functions through lfkit.ConditionalLuminosityFunction. Each constructor returns a lfkit.LuminosityFunction object, so the resulting model can be evaluated with lf.phi and integrated with the usual lf.integrals namespace.

The examples below use redshift as the conditioning variable because this is the most common use case for luminosity function evolution. In that case, \(\Phi(M \mid z)\) describes how the magnitude distribution of galaxies changes with redshift. The same API can be used with any other conditioning variable by replacing z with the desired quantity.

The examples include:

a conditional Schechter luminosity function,
a conditional Schechter model using LFKit parameter models,
a lognormal component,
a modified Schechter-like component,
a two-component lognormal plus modified-Schechter model,
integrated number densities and component fractions,
a halo-mass conditional example,
selection-limited number densities.

The number-density units follow the normalization supplied to the luminosity function. For example, if the amplitudes are supplied in \({\rm Mpc}^{-3}\,{\rm mag}^{-1}\), then \(\Phi(M \mid z)\) has units of \({\rm Mpc}^{-3}\,{\rm mag}^{-1}\).

Conditional Schechter luminosity function#

A conditional Schechter luminosity function is a Schechter model whose parameters depend on an external variable. Instead of using one fixed normalization, characteristic magnitude, and faint-end slope, the model can let one or more of these quantities vary with the condition.

This example uses redshift as the condition. The normalization \(\phi_\star(z)\) and characteristic magnitude \(M_\star(z)\) evolve with redshift, while the faint-end slope \(\alpha\) is kept fixed. Each curve is therefore a Schechter luminosity function at one redshift. Comparing the curves shows how the abundance and bright-end turnover shift as the population evolves.

import numpy as np
import matplotlib.pyplot as plt
import cmasher as cmr

from lfkit import ConditionalLuminosityFunction

LABEL_SIZE = 15
TICK_SIZE = 13
TITLE_SIZE = 17
LEGEND_SIZE = 15

absolute_mag = np.linspace(-24.0, -14.0, 500)
redshifts = [0.1, 0.6, 1.1]

colors = cmr.take_cmap_colors(
    "cmr.guppy",
    len(redshifts),
    cmap_range=(0.0, 0.2),
)

lf = ConditionalLuminosityFunction.schechter(
    phi_star=lambda z: 1.0e-3 * (1.0 + z) ** 0.8,
    m_star=lambda z: -20.5 - 0.7 * (z - 0.1),
    alpha=-1.1,
)

fig, ax = plt.subplots(figsize=(7.0, 5.0))

for z_value, color in zip(redshifts, colors):
    z = np.full_like(absolute_mag, z_value)
    phi = lf.phi(absolute_mag, z)

    ax.plot(
        absolute_mag,
        phi,
        lw=3,
        color=color,
        label=rf"$z={z_value}$",
    )

ax.set_yscale("log")
ax.set_ylim(1.0e-8, 1.0e-1)
ax.invert_xaxis()
ax.set_xlabel("Absolute magnitude $M$", fontsize=LABEL_SIZE)
ax.set_ylabel(
    r"$\Phi(M \mid z)$ [$\mathrm{Mpc}^{-3}\,\mathrm{mag}^{-1}$]",
    fontsize=LABEL_SIZE,
)
ax.set_title("Conditional Schechter luminosity function", fontsize=TITLE_SIZE)
ax.tick_params(axis="both", labelsize=TICK_SIZE)
ax.legend(frameon=True, fontsize=LEGEND_SIZE, loc="best")

plt.tight_layout()

Conditional luminosity functions#

Conditional Schechter luminosity function#

Conditional Schechter surface#

Conditional Schechter model with LFKit parameter models#

Lognormal conditional component#

Modified Schechter conditional component#

Standard, modified, and lognormal component shapes#

Two-component conditional LF#

Two-component evolution#

Integrated conditional number density#

Component fractions#

Two-component LF surface#

Halo-mass conditional luminosity function#

Mean luminosity ratio from a conditional luminosity function#

Selection-limited conditional number density#

Selection fraction#