Galaxy Luminosity Function
The galaxy luminosity function ($\Phi$) specifies the number density of galaxies per unit comoving volume as a function of their intrinsic luminosity ($L$) or absolute magnitude ($M$). It is a fundamental tool for understanding the large-scale distribution of matter and the evolution of stellar populations across cosmic time.
The most widely used mathematical model for the global galaxy distribution is the Schechter function (1976). It describes the number of galaxies $\Phi(L)dL$ in the luminosity interval $[L, L + dL]$ as a combination of a power law at the faint end and an exponential cutoff at the bright end:
$$\Phi(L) \ dL = \left( \frac{\Phi^*}{L^*} \right) \left( \frac{L}{L^*} \right)^\alpha \exp(-L/L^*) \ dL$$
The function is defined by three primary parameters as illustrated in the figure above:
$L^*$ (Characteristic Luminosity): The “break” luminosity where the function transitions from power-law to exponential decay. In the blue band, $L_B^* \approx 1.2 \times 10^{10} h^{-2} L_\odot$, which is comparable to the luminosity of the Milky Way.
$\alpha$ (Faint-end Slope): This defines the abundance of faint galaxies. Typical values are $\alpha \approx -1.07$ for blue-band surveys. If $\alpha \leq -1$, the total number density of galaxies is formally infinite, though the physical function is cut off at a minimum luminosity.
$\Phi^*$ (Normalization): The number density of galaxies per Mpc$^3$, typically found to be $\Phi^* \approx 1.6 \times 10^{-2} h^3 \text{ Mpc}^{-3}$.
Astronomers often express the luminosity function in terms of absolute magnitude($M$). Given the logarithmic relationship between $L$ and $M$ ($M - M^* = -2.5 \log_{10}(L/L^*)$), the Schechter function is rewritten as:
$$\Phi(M) = (0.4 \ln 10) \Phi^* 10^{0.4(\alpha+1)(M^*-M)} \exp \left( -10^{0.4(M^*-M)} \right)$$
In the blue band, the characteristic absolute magnitude is $M^*_B \approx -19.7 + 5 \log_{10} h$. In the near-infrared K-band, which better traces the total stellar mass, $M^*_K \approx -23.1 + 5 \log_{10} h$.
Although faint galaxies are more numerous, the integrated luminosity density ($l_{tot}$) is dominated by galaxies near $L^*$. The total light emitted per unit volume is found by integrating the luminosity function:
$$l_{tot} = \int_{0}^{\infty} L \Phi(L) dL = \Phi^* L^* \Gamma(\alpha + 2)$$
where $\Gamma$ is the Gamma function. For a typical population where $\alpha \approx -1$, $l_{tot}$ is approximately $\Phi^* L^*$.
The “universal” Schechter function is often a simplification, as the luminosity distribution depends heavily on morphology and environment:
Color Bimodality: The galaxy population exhibits a bimodal distribution in color-magnitude space. Red sequence galaxies (older, early-type) dominate at high luminosities, while the blue cloud (star-forming, late-type) dominates at lower luminosities.
Environmental Dependence: In rich clusters, the luminosity function is dominated at the bright end by ellipticals and S0 galaxies, and at the faint end by dwarf ellipticals ($dE$). Clusters often contain a cD galaxy at the center that is significantly brighter than the Schechter $L^*$ predicts, representing a “light excess” at large radii.
Accurate determination of the luminosity function requires two critical corrections:
K-Correction: As galaxies are redshifted, a fixed observational filter samples bluer parts of their rest-frame spectra. The K-correction $K(z)$ is added to the observed magnitude to recover the rest-frame luminosity.
Malmquist Bias: In flux-limited surveys, intrinsically luminous galaxies are visible at much greater distances than dim ones. They are consequently overrepresented in samples, necessitating a volume-weighting correction.