Composite models

Composite models combine multiple luminosity function components. They are useful when a population is better described as a mixture rather than by one single analytic shape.

Two-component luminosity function

The two-component model combines a lognormal component with a cutoff-modified Schechter component. The lognormal component describes a localized population, while the modified Schechter component contributes a broader luminosity function with a suppressed bright end.

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

from lfkit import LuminosityFunction

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

lognormal = LuminosityFunction.lognormal(
    mean_absolute_mag=-21.0,
    sigma_log_luminosity=0.10,
    amplitude=6.0e-4,
)

composite = LuminosityFunction.two_component(
    lognormal_mean_absolute_mag=-21.0,
    lognormal_sigma_log_luminosity=0.25,
    lognormal_amplitude=6.0e-4,
    modified_phi_star=1.0e-3,
    modified_alpha=-1.1,
    modified_luminosity_fraction=0.562,
)

absolute_mag = np.linspace(-24.0, -14.0, 500)
colors = cmr.take_cmap_colors("cmr.guppy", 2, cmap_range=(0.0, 0.2))

fig, ax = plt.subplots(figsize=(7.0, 5.0))
ax.plot(
    absolute_mag,
    lognormal.phi(absolute_mag),
    lw=3,
    color=colors[0],
    label="Lognormal component",
)
ax.plot(
    absolute_mag,
    composite.phi(absolute_mag),
    lw=3,
    color=colors[1],
    label="Two-component model",
)

ax.set_yscale("log")
ax.set_ylim(1e-8, 1e-1)
ax.invert_xaxis()
ax.set_xlabel("Absolute magnitude $M$", fontsize=LABEL_SIZE)
ax.set_ylabel(
    r"$\Phi(M)$ [$\mathrm{Mpc}^{-3}\,\mathrm{mag}^{-1}$]",
    fontsize=LABEL_SIZE,
)
ax.set_title("Composite 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()

(png)

Luminosity cutoff modifier

A luminosity cutoff can be applied to an existing luminosity function object. This keeps the base model unchanged and returns a new object whose lfkit.LuminosityFunction.phi() values are multiplied by a bright-end cutoff.

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

from lfkit import LuminosityFunction

LABEL_SIZE = 15
TICK_SIZE = 13
TITLE_SIZE = 17
LEGEND_SIZE = 14

absolute_mag = np.linspace(-24.0, -14.0, 500)

models = {
    "Schechter": LuminosityFunction.schechter(
        phi_star=1.0e-3,
        m_star=-20.5,
        alpha=-1.1,
    ),
    "Double Schechter": LuminosityFunction.double_schechter(
        phi_star=1.0e-3,
        m_star=-20.5,
        alpha=-1.1,
        beta=-0.6,
        m_transition=-18.5,
    ),
    "Double power law": LuminosityFunction.double_power_law(
        phi_star=1.0e-3,
        m_star=-20.5,
        alpha=-1.4,
        beta=-0.4,
    ),
}

colors = cmr.take_cmap_colors(
    "cmr.guppy",
    len(models),
    cmap_range=(0.0, 0.35),
)

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

for (label, base), color in zip(models.items(), colors):
    cutoff = base.with_luminosity_cutoff(
        m_star=-20.5,
        cutoff_power=1.0,
        cutoff_amplitude=0.08,
    )

    ax.plot(
        absolute_mag,
        base.phi(absolute_mag),
        lw=2.5,
        ls="--",
        color=color,
        label=f"{label}",
    )
    ax.plot(
        absolute_mag,
        cutoff.phi(absolute_mag),
        lw=3,
        color=color,
        label=f"{label} + cutoff",
    )

ax.set_yscale("log")
ax.set_ylim(1e-8, 1e-1)
ax.invert_xaxis()
ax.set_xlabel("Absolute magnitude $M$", fontsize=LABEL_SIZE)
ax.set_ylabel(
    r"$\Phi(M)$ [$\mathrm{Mpc}^{-3}\,\mathrm{mag}^{-1}$]",
    fontsize=LABEL_SIZE,
)
ax.set_title("Applying one luminosity cutoff to multiple LF models", fontsize=TITLE_SIZE)
ax.tick_params(axis="both", labelsize=TICK_SIZE)
ax.legend(frameon=True, fontsize=LEGEND_SIZE, loc="best")
plt.tight_layout()

(png)