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====== 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:

$$\Phi(L) \ dL = \left( \frac{\Phi^*}{L^*} \right) \left( \frac{L}{L^*} \right)^\alpha \exp(-L/L^*) \ dL$$

[{{ :courses:ast403:phi_l.jpg?600 |Fig 1: A schematic plot of the Schechter function.}}]

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.

[{{ :courses:ast403:galaxy_color.jpg?600 |Fig2: The density of galaxies in color–magnitude space.The color of ∼ 70 000 galaxies with redshifts $0.01 \le z \le 0.08$ from the Sloan Digital Sky Survey is measured by the rest-frame $u−r$, i.e., after a (small) correction for their redshift was applied.}}]

**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.

[{{ :courses:ast403:phil_env.jpg?400 | Fig 2: The luminosity function for different Hubble types
of field galaxies (top) and galaxies in the Virgo Cluster of
galaxies (bottom). }}]

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.