Schechter Luminosity Function

The Schechter luminosity function is a three-parameter analytical expression introduced by American astronomer Paul L. Schechter in 1976 that characterizes the comoving number density of galaxies per unit luminosity interval across field and cluster environments. Motivated theoretically by the self-similar gravitational condensation model of the Press–Schechter formalism (1974), the Schechter distribution combines a power-law slope α at low luminosities with an exp(−L/L*) exponential decay above a characteristic "knee" luminosity L*. Because its integral properties are expressed analytically through Euler gamma and incomplete gamma functions, the Schechter function serves as the universal benchmark in extragalactic astronomy for quantifying galaxy survey completeness, cosmic luminosity density, stellar mass functions, and the baryonic feedback processes that shape the galaxy–halo connection across cosmic time.

Let φ(L) dL denote the comoving number density of galaxies (typically measured in units of h³ Mpc⁻³) with bolometric or bandpass-filtered luminosity between L and L + dL. The Schechter luminosity function is defined by three structural parameters—the normalization density φ*, the characteristic knee luminosity L*, and the dimensionless faint-end slope α:

φ(L) dL = φ* (L / L*)^α exp(−L / L*) d(L / L*)

Each parameter governs a distinct regime of the galaxy population:

1. Characteristic Luminosity (L*): Marks the transition ("knee") between the power-law regime (L ≪ L*) and the exp(−L/L*) exponential cutoff (L ≫ L*). In the local universe (z < 0.1), L* in the optical r-band corresponds to an M_r* ≈ −20.44 + 5 log₁₀ h mag galaxy comparable in luminosity to the Milky Way or the Andromeda Galaxy.
2. Faint-End Slope (α): Controls the logarithmic slope d ln φ / d ln L ≈ α at low luminosities (L ≪ L*). Negative values (typically α ≈ −1.0 to −1.3 in the local optical universe, steepening to α ≈ −1.8 to −2.1 in high-redshift ultraviolet surveys) indicate that dwarf galaxies far outnumber bright L* spirals and ellipticals by number.
3. Normalization Density (φ*): Sets the overall comoving space density scale (in Mpc⁻³); at the knee luminosity L = L*, the differential density per unit L/L* evaluates to φ(L*) = φ* / e ≈ 0.3679 φ*.

Because observational galaxy catalogs record logarithmic fluxes as broadband absolute magnitudes M rather than linear luminosities via the Pogson relation L / L* = 10^{−0.4(M − M*)}, where M* is the characteristic absolute magnitude corresponding to L*, substituting d(L/L*) = −0.4 ln(10) 10^{−0.4(M − M*)} dM yields the magnitude-space Schechter function:

Φ(M) dM = (0.4 ln 10) φ* [10^{−0.4(M − M*)}]^{α + 1} exp[−10^{−0.4(M − M*)}] dM

where the prefactor 0.4 ln(10) ≈ 0.921034 converts differential intervals from natural logarithms to magnitudes. Note that in magnitude space, the faint-end slope scales as 10^{−0.4(α + 1)M}, meaning that a "flat" faint end in number per magnitude bin corresponds to α = −1.0 rather than α = 0.

Integrating the Schechter distribution above a minimum threshold luminosity L_min = x_min L* gives the cumulative comoving number density of galaxies n(>L_min) in terms of the upper incomplete Euler gamma function Γ(s, x) = ∫_x^∞ t^s−1 exp(−t) dt:

n(>L_min) = ∫_Lmin^∞ φ(L) dL = φ* Γ(α + 1, L_min / L*)

For α ≤ −1, the complete integral as L_min → 0 diverges because the power-law abundance of faint dwarf galaxies increases faster than L⁻¹, requiring a physical low-luminosity cutoff set by photoionization suppression of gas cooling during reionization. Conversely, weighting each galaxy by its luminosity L and integrating over all luminosities yields the total comoving luminosity density j_tot (in L_☉ Mpc⁻³):

j_tot = ∫₀^∞ L φ(L) dL = φ* L* Γ(α + 2)

Because the complete gamma function Γ(α + 2) converges for all α > −2, the total cosmic light budget remains finite even when the total galaxy count diverges (for −2 < α ≤ −1). Furthermore, the integrand L φ(L) peaks at L_peak = (α + 1)L* for α > −1, demonstrating that galaxies near the knee L* dominate the integrated starlight and stellar mass density of the universe.

A direct comparison between the theoretical dark-matter halo mass function predicted by the Press–Schechter formalism (or Sheth–Tormen model) and the observed galaxy Schechter luminosity function reveals a striking divergence at both the low-mass and high-mass extremes. Whereas the ΛCDM halo mass function scales steeply at low masses as dn/dM ∝ M^−1.9, the local galaxy luminosity and stellar mass functions exhibit a much shallower faint-end slope (α ≈ −1.1 to −1.3). Similarly, at the bright end, the galaxy luminosity function drops off much more sharply above L* than the dark-matter halo mass function above M*.

In modern galaxy formation theory, this U-shaped stellar-to-halo mass ratio (M_* / M_halo, which peaks at ~20% conversion efficiency near halo mass M_halo ≈ 10¹² M_☉) is explained by two complementary baryonic feedback mechanisms:

• Faint-End Suppression (M_halo < 10¹¹ M_☉): In shallow gravitational potentials with low circular velocities (v_c < 100 km s⁻¹ in an Navarro–Frenk–White profile), ultraviolet background photoionization during cosmic reionization and core-collapse supernova winds expel cold gas from the disk, flattening the faint-end slope α.
• Bright-End Quenching (M_halo > 10^12.5 M_☉): In high-mass elliptical and group halos, virial shock heating of infalling gas combined with mechanical and radiative feedback from central supermassive black holes (governed by the M–sigma relation) shuts down cooling flows, preventing galaxies from growing indefinitely through star formation and enforcing the sharp exp(−L/L*) Schechter cutoff.

To capture the distinct evolutionary pathways of star-forming disks and quenched spheroids in deep surveys, contemporary analyses frequently employ a double Schechter function, summing two components that share a common L* but have separate normalizations (φ₁*, φ₂*) and slopes (α₁ ≈ −0.5 for quiescent red-sequence galaxies and α₂ ≈ −1.5 for blue star-forming systems).

Why does the Schechter luminosity function diverge at zero luminosity when α ≤ −1?

For α ≤ −1, the differential number density φ(L) ∝ L^α rises at least as steeply as 1/L as L → 0, causing the cumulative number integral ∫₀^∞ φ(L) dL to diverge logarithmically (for α = −1) or as a power law (for α < −1). However, the integrated luminosity density ∫₀^∞ L φ(L) dL = φ* L* Γ(α + 2) remains completely finite for all α > −2 because each progressively fainter dwarf galaxy contributes vanishingly little light.

Do Brightest Cluster Galaxies (BCGs) follow the Schechter luminosity function?

No. While satellite galaxies within rich galaxy clusters follow a Schechter distribution, the central Brightest Cluster Galaxy (BCG)—typically a giant cD elliptical residing at the bottom of the cluster's gravitational potential well—forms an ultra-luminous outlier bump above the exponential cutoff L*. BCGs grow predominantly via dissipationless dry mergers and dynamical friction rather than in-situ gas cooling.

How does the Schechter function evolve with cosmological redshift?

Observations from the Hubble Space Telescope and James Webb Space Telescope (JWST) show that the rest-frame ultraviolet faint-end slope α steepens systematically with redshift, from α ≈ −1.2 at z = 0 to α ≈ −2.1 at z = 8–10, while the normalization φ* declines by more than an order of magnitude, reflecting the bottom-up hierarchical assembly of early dwarf galaxies.

1. Schechter, P. (1976). "An analytic expression for the luminosity function for galaxies." The Astrophysical Journal, 203, 297–306. https://doi.org/10.1086/154079
2. Blanton, M. R., Hogg, D. W., Bahcall, N. A., Brinkmann, J., Britton, M., Connolly, A. J., Csabai, I., Fukugita, M., Loveday, J., Meiksin, A., Munn, J. A., Nicholl, R. C., Okamura, S., Quinn, T., Schneider, D. P., Shimasaku, K., Strauss, M. A., Tegmark, M., Vogeley, M. S., & Weinberg, D. H. (2003). "The galaxy luminosity function and luminosity density at redshift z = 0.1." The Astrophysical Journal, 592(2), 819–838. https://doi.org/10.1086/375776
3. Baldry, I. K., Driver, S. P., Brough, S., Hopkins, A. M., Liske, J., Loveday, J., & Robotham, A. S. G. (2012). "Galaxy And Mass Assembly (GAMA): the galaxy stellar mass function at z < 0.06." Monthly Notices of the Royal Astronomical Society, 421(1), 621–634. https://doi.org/10.1111/j.1365-2966.2012.20340.x
4. Bouwens, R. J., Illingworth, G. D., Oesch, P. A., Trenti, M., Labbé, I., Bradley, L., Carollo, M., van Dokkum, P. G., Gonzalez, V., Holwerda, B., Franx, M., Spitler, L., Smit, R., & Magee, D. (2015). "UV luminosity functions at redshifts z ∼ 4 to z ∼ 10: 10,000 galaxies from HST legacy fields." The Astrophysical Journal, 803(1), 34. https://doi.org/10.1088/0004-637X/803/1/34
5. Benson, A. J., Bower, R. G., Frenk, C. S., Lacey, C. G., Baugh, C. M., & Cole, S. (2003). "What shapes the luminosity function of galaxies?" The Astrophysical Journal, 599(1), 38–49. https://doi.org/10.1086/379160