Galaxy Luminosity Function

The luminosity function is the single most basic statistical descriptor of the galaxy population: it counts how many galaxies, per unit comoving volume, shine at each luminosity. Because a galaxy's luminosity tracks its stellar mass, its morphology, and the depth of the dark-matter well it lives in, the function is a compact fossil record of galaxy formation — it integrates over the whole cosmic history of gas cooling, star formation, merging, and feedback and hands back a single curve. Measuring it is deceptively hard: a flux-limited survey sees a faint dwarf only if it lies nearby, so raw counts are biased toward the luminous — the Malmquist effect — and the cure is the $1/V_\mathrm{max}$ estimator or a maximum-likelihood (STY) fit that sidesteps density fluctuations. Empirically the curve has a universal shape, first written down by Paul Schechter in 1976: a gentle power law of slope $\alpha$ at low luminosity, cut off by an exponential above a characteristic luminosity $L^*$ . Just three numbers — the normalization $\Phi^*$ , the knee $L^*$ , and the faint-end slope $\alpha$ — reproduce the abundance of galaxies across four decades of luminosity, locally and, with evolving parameters, past redshift six. The knee defines a natural unit: an " $L^*$ galaxy" like the Milky Way sits at the divide between the numerous faint dwarfs and the exponentially rare giants. Split the sample by colour, morphology, or environment and each subpopulation keeps a Schechter form of its own. Compare it to the dark-matter halo mass function and the mismatch reveals two feedback channels — supernova winds below the knee, active-nucleus heating above it — that make star formation inefficient at both ends. And because every integral of the Schechter function collapses to a gamma function, the formalism stays analytic throughout.

Key takeaways

The galaxy luminosity function $\Phi(L)\,\mathrm{d}L$ is the comoving number density of galaxies with luminosities between $L$ and $L+\mathrm{d}L$ — the census of the galaxy population.
The empirical Schechter form $\Phi(L)=(\Phi^*/L^*)(L/L^*)^\alpha e^{-L/L^*}$ fits it with a faint-end power law of slope $\alpha$ and an exponential cutoff at a characteristic luminosity $L^*$ .
Typical local values: $L^*\approx 10^{10}\,h^{-2}L_\odot$ (about $M^*_B\approx -20.5+5\log_{10}h$ ), $\alpha\approx -1.0$ to $-1.3$ , $\Phi^*\approx (1$ – $2)\times10^{-2}\,h^3\,\mathrm{Mpc}^{-3}$ .
Integrals reduce to gamma functions: the luminosity density $j=\Phi^*L^*\,\Gamma(\alpha+2)$ is finite for $\alpha>-2$ , while the number density diverges at the faint end for $\alpha\le -1$ and needs a cutoff. The knee is the fingerprint of active-nucleus quenching, the faint-end slope of supernova feedback.
Key idea: a single three-parameter Schechter function — power-law faint end $\propto L^\alpha$ times exponential cutoff $e^{-L/L^*}$ — captures the galaxy number density across four decades of luminosity, and its integrals reduce to gamma functions.

The core insight

A single three-parameter Schechter function — a power-law faint end $\propto L^\alpha$ times an exponential cutoff $e^{-L/L^*}$ — captures the comoving galaxy number density across four decades of luminosity, and every integral of it reduces to a gamma function.

Where this lives today

The luminosity function is a live measurement. Wide local surveys — the Sloan Digital Sky Survey, GAMA — measure $\Phi(M)$ in every band from the ultraviolet to the near-infrared and split it by colour, morphology, mass, and environment, feeding the abundance-matching models that connect galaxies to halos. At the opposite extreme, the ultraviolet luminosity function of Lyman-break galaxies is mapped by the Hubble and James Webb Space Telescopes past redshift ten: integrating its faint end gives the ionizing-photon budget that tests whether galaxies alone reionized the universe, and JWST's surprising abundance of bright galaxies at $z>10$ is straining models of how fast the bright end can be built. Meanwhile the faint-end slope $\alpha$ is where near-field cosmology lives: the count of ultra-faint dwarf satellites around the Milky Way, pushed deeper by the Dark Energy Survey and soon Rubin's LSST, tests how strongly feedback and reionization suppress the smallest halos and, through the "missing satellites" problem, probes the nature of dark matter itself. The three Schechter parameters are a compact interface between galaxy formation and cosmology.

The physical picture

Imagine emptying a representative volume of the universe onto a sorting table and binning every galaxy by luminosity. You are swamped by faint dwarfs and left with only a handful of giant ellipticals. Schechter's insight was that this steeply falling distribution has two regimes stitched at one scale: below a characteristic luminosity $L^*$ the counts drop only as a gentle power law $L^\alpha$ ; above $L^*$ they plunge exponentially, because nature does not assemble many galaxies brighter than a few times the Milky Way. The first figure draws that shape on logarithmic axes, where the power law is a straight line and the cutoff is where it peels away and dives.

Figure

The dimensionless Schechter shape φ(x) = xᵅ e⁻ˣ with α = −1.2 on log–log axes: a power law of slope α = −1.2 at the faint end (green dashed reference) rolls over at the knee x = L/L* = 1, where φ = e⁻¹ = 0.37, into an exponential cutoff that makes giant galaxies exponentially rare

The knee is not a fitting convenience; it is where the physics changes hands. Below $L^*$ the wells are shallow enough that a few supernovae drive gas out, so star formation is throttled and $\alpha$ is far flatter than the steep low-mass slope of the halo mass function. Above $L^*$ the halos hold a hot atmosphere that a central active nucleus keeps heated, shutting off the cooling that would feed growth — quenching that stamps the exponential cutoff onto the population. The second figure makes the decisive point: the dwarfs dominate the counts, but the rare $L^*$ galaxies dominate the light.

Figure

Two weightings of the same Schechter function (α = −1.2): counts per dex ∝ xᵅ⁺¹e⁻ˣ (blue) rise toward faint dwarfs, but light per dex ∝ xᵅ⁺²e⁻ˣ (amber) peaks at x = α+2 = 0.8 ≈ L, so the luminosity density j = ΦL* Γ(α+2) is set by L*-galaxies, not dwarfs

The core equations

The Schechter function is the workhorse. As a number density per unit luminosity, it is a power law governing the faint end times an exponential governing the bright end, joined at $L^*$ . Everything downstream — number counts, luminosity density, stellar-mass density — is an integral of this one expression.

\Phi(L)\,\mathrm{d}L=\frac{\Phi^*}{L^*}\left(\frac{L}{L^*}\right)^{\alpha} e^{-L/L^*}\,\mathrm{d}L \qquad (1)

where: $\Phi(L)\,\mathrm{d}L$ — comoving number density of galaxies with luminosity in $[L,L+\mathrm{d}L]$ ; $\Phi^*$ — normalization ( $\sim 10^{-2}\,h^3\,\mathrm{Mpc}^{-3}$ ); $L^*$ — characteristic (knee) luminosity ( $\sim 10^{10}\,h^{-2}L_\odot$ ); $\alpha$ — faint-end slope ( $\approx -1.0$ to $-1.3$ ). Reading it: $\Phi^*$ moves the curve up and down, $L^*$ slides it left and right, $\alpha$ tilts the faint end. Observers work in magnitudes, which run backwards (brighter is a smaller, more negative number). Transforming equation (1) to absolute magnitude flips the exponent's sign and picks up a Jacobian $0.4\ln 10$ , giving the form actually fit to survey data.

\Phi(M)\,\mathrm{d}M=0.4\ln 10\,\Phi^*\left[10^{0.4(M^*-M)}\right]^{\alpha+1}\exp\!\left[-10^{0.4(M^*-M)}\right]\mathrm{d}M \qquad (2)

where: $M$ — absolute magnitude; $M^*$ — characteristic absolute magnitude, $L/L^*=10^{0.4(M^*-M)}$ ; $0.4\ln 10=0.921$ — the Jacobian $|\mathrm{d}L/\mathrm{d}M|/L$ . Reading it: the power law $L^\alpha$ becomes the exponent $\alpha+1$ , so the fitted faint-end slope in magnitudes is $\alpha+1$ — a perennial source of sign confusion. Integrating the counts above a luminosity gives the cumulative abundance — what abundance-matching compares against halos. The integral is an upper incomplete gamma function, diverging as the lower limit goes to zero whenever $\alpha\le-1$ .

n(>L_\mathrm{min})=\int_{L_\mathrm{min}}^{\infty}\Phi(L)\,\mathrm{d}L=\Phi^*\,\Gamma\!\left(\alpha+1,\;L_\mathrm{min}/L^*\right) \qquad (3)

where: $\Gamma(a,x)=\int_x^\infty t^{a-1}e^{-t}\,\mathrm{d}t$ — upper incomplete gamma function; $L_\mathrm{min}$ — lower cutoff, mandatory because $\Gamma(\alpha+1,0)$ diverges for $\alpha\le-1$ . Reading it: there is no "total number of galaxies" without saying how faint you count. Weighting the counts by luminosity and integrating over all $L$ gives the comoving luminosity density. The extra power of $L$ tames the faint-end divergence, and the integral is a complete gamma function.

j=\int_0^\infty L\,\Phi(L)\,\mathrm{d}L=\Phi^*L^*\,\Gamma(\alpha+2) \qquad (4)

where: $j$ — comoving luminosity density ( $L_\odot\,\mathrm{Mpc}^{-3}$ ); $\Gamma(\alpha+2)$ — complete gamma function, finite for $\alpha>-2$ . Reading it: the light budget is essentially $\Phi^*L^*$ up to a gamma factor of order unity, dominated by galaxies near $L^*$ , not dwarfs. Turning a flux-limited catalogue into an unbiased $\Phi(L)$ requires correcting for faint galaxies being visible only nearby. The classic non-parametric estimator weights each galaxy by the inverse of the maximum volume in which it would still pass the flux limit.

V_\mathrm{max}(L)=\frac{\omega}{3}\,d_\mathrm{max}^3(L),\qquad \Phi(L)\,\Delta L=\sum_{i}\frac{1}{V_{\mathrm{max},i}} \qquad (5)

where: $V_\mathrm{max}$ — maximum comoving volume within which a galaxy of luminosity $L$ stays above the flux limit; $\omega$ — survey solid angle; $d_\mathrm{max}(L)$ — maximum detectable distance. A bright galaxy is countable in a big volume and carries little weight; a dwarf is countable only nearby and is up-weighted. Finally, multiplying the luminosity density by a stellar mass-to-light ratio converts light into stellar mass, giving the cosmic stellar-mass density. (Getting $L$ in the first place uses the distance modulus $m-M=5\log_{10}(d_L/10\,\mathrm{pc})+K(z)$ , with a $K$ -correction for the redshifted bandpass.)

\rho_*=\int_0^\infty \left(\frac{M}{L}\right) L\,\Phi(L)\,\mathrm{d}L=\left\langle\frac{M}{L}\right\rangle \Phi^*L^*\,\Gamma(\alpha+2) \qquad (6)

where: $\rho_*$ — comoving stellar-mass density; $\langle M/L\rangle$ — mean stellar mass-to-light ratio (a few in solar units). Dividing $\rho_*$ by the critical density gives $\Omega_*$ , the fraction of closure density turned into stars.

Derivation AThe Schechter form from an exponentially cut power law

The goal here is to see that the Schechter function is not an arbitrary curve chosen to fit data but the natural luminosity translation of a general object — a differential abundance that is a power law in a scaling variable, cut off exponentially above a characteristic scale. That same shape governs the Press–Schechter halo mass function, which is why $\alpha$ echoes the low-mass halo slope and the knee sits where feedback truncates growth. We build the three-parameter form one factor at a time.

Step 1. Introduce the dimensionless luminosity $x=L/L^*$ , so one characteristic scale $L^*$ separates abundant faint systems ( $x\ll1$ ) from rare bright ones ( $x\gg1$ ). Why: hierarchical collapse always produces a characteristic scale, and working in units of it makes the function universal.

Step 2. Take the faint end scale-free, $\Phi\propto x^{\alpha}$ . Gravitational merging gives a near power-law abundance at low mass; galaxies inherit a flatter version because supernova feedback removes gas from shallow wells, so the measured $\alpha\approx-1$ is the halo slope softened by baryons. Why: a power law is the only scale-free possibility, and its slope is where faint-end feedback leaves a clean imprint.

Step 3. Multiply by an exponential cutoff, $\Phi\propto x^{\alpha}e^{-x}$ . Above $L^*$ the halos are massive enough that gas no longer cools and active-nucleus heating quenches growth, so the abundance falls faster than any power law — the exponential tail Press–Schechter predicts for the high-mass function. Why: the exponential is the signature of a characteristic scale, making $L^*$ a genuine knee.

Step 4. Fix the normalization by demanding $\int_0^\infty L\Phi\,\mathrm{d}L=\Phi^*L^*\Gamma(\alpha+2)$ match observation, which pins $\Phi^*$ and completes the form. Why: normalizing on the convergent luminosity integral rather than the divergent number integral is why $\Phi^*$ is well defined even when $\alpha\le-1$ .

Go deeper · Derivation B · 1 minMalmquist bias and the one-over-Vmax weighting

Derivation BMalmquist bias and the one-over-Vmax weighting

The aim is to derive from first principles why a flux-limited survey systematically undercounts faint galaxies, and to show that weighting each detected galaxy by the inverse of its maximum accessible volume exactly undoes the bias. This is the estimator behind essentially every classical measurement, and its one crucial assumption — a uniform galaxy density along the line of sight — tells you when to switch to maximum likelihood instead.

Step 1. A survey with flux limit $f_\mathrm{lim}$ detects a galaxy of luminosity $L$ only where $f=L/4\pi d^2$ exceeds $f_\mathrm{lim}$ , so the largest distance at which it stays detectable is $d_\mathrm{max}(L)=(L/4\pi f_\mathrm{lim})^{1/2}$ . Why: this single relation is the entire origin of Malmquist bias — the horizon grows only as $\sqrt{L}$ , confining faint galaxies to a small nearby shell.

Step 2. The accessible comoving volume is $V_\mathrm{max}=(\omega/3)\,d_\mathrm{max}^3\propto L^{3/2}$ , with $\omega$ the solid angle. A galaxy ten times fainter is countable in a volume $10^{3/2}\approx32$ times smaller. Why: the steep $L^{3/2}$ scaling quantifies how badly raw counts skew toward the luminous — a naive histogram is not a luminosity function.

Step 3. Weight each detected galaxy by $1/V_{\mathrm{max},i}$ and sum over a luminosity bin: $\Phi(L)\,\Delta L=\sum_i 1/V_{\mathrm{max},i}$ . Each represents the density it would have if it filled its own accessible volume, so faint galaxies — seen in tiny volumes — are up-weighted just enough to restore the true density. Why: dividing by the volume actually searched is the definition of a density, and doing it per galaxy makes the estimator non-parametric.

Step 4. The one hidden assumption is a uniform density within each $V_\mathrm{max}$ ; a nearby supercluster or void makes the weights wrong. The STY maximum-likelihood method avoids this by fitting the shape of $\Phi(L)$ from the conditional luminosity distribution, independent of density. Why: naming the failure mode tells you to cross-check with STY whenever large-scale structure is present.

Go deeper · Derivation C · 1 minFrom the luminosity function to the cosmic luminosity density

Derivation CFrom the luminosity function to the cosmic luminosity density

The goal is to carry out the integral that turns the three-parameter Schechter function into a single number — the comoving luminosity density $j$ — and to see why it converges even when the number counts diverge, and why the light is dominated by galaxies near the knee. The result $j=\Phi^*L^*\Gamma(\alpha+2)$ bridges an abundance curve to the stellar-mass density $\Omega_*$ and the ionizing-photon budget of reionization, so the gamma-function reduction is worth doing once.

Step 1. Insert the Schechter form into the luminosity-weighted integral: $j=\int_0^\infty L\,\Phi(L)\,\mathrm{d}L=(\Phi^*/L^*)\int_0^\infty L\,(L/L^*)^{\alpha}e^{-L/L^*}\,\mathrm{d}L$ . Why: weighting by $L$ is the whole difference between counting galaxies and counting light — it adds one power of $L$ .

Step 2. Substitute $x=L/L^*$ ( $\mathrm{d}L=L^*\,\mathrm{d}x$ ), collapsing the prefactors to leave $j=\Phi^*L^*\int_0^\infty x^{\alpha+1}e^{-x}\,\mathrm{d}x$ . Why: nondimensionalizing exposes the universal integral — everything material ( $\Phi^*$ , $L^*$ ) is now outside, leaving a pure number set by $\alpha$ .

Step 3. Recognize the remaining integral as the gamma function, $\int_0^\infty x^{\alpha+1}e^{-x}\,\mathrm{d}x=\Gamma(\alpha+2)$ , giving $j=\Phi^*L^*\Gamma(\alpha+2)$ . The integrand peaks near $x=\alpha+2\approx0.8$ ( $L\approx L^*$ ), and being $\propto x^{\alpha+1}$ at small $x$ it converges for $\alpha>-2$ , unlike the number integral $\propto x^{\alpha}$ that diverges for $\alpha\le-1$ . Why: the peak says $L^*$ galaxies carry most of the cosmic light, and the convergence is why $j$ is a robust observable while "the total number of galaxies" is not. *For $\alpha=-1.2$ , $\Gamma(0.8)=1.164$ ; with $\Phi^*=1.6\times10^{-2}\,h^3\,\mathrm{Mpc}^{-3}$ and $L^*=1.2\times10^{10}\,h^{-2}L_\odot$ this gives $j=2.2\times10^{8}\,h\,L_\odot\,\mathrm{Mpc}^{-3}$ , the canonical local $B$ -band value.*

Limiting cases

The Schechter function behaves qualitatively differently in its two regimes and under its integrals; the table collects the limits that matter in practice.

Case	Result	Meaning
Faint end $L\ll L^*$	$\Phi\to \dfrac{\Phi^}{L^}\left(\dfrac{L}{L^*}\right)^{\alpha}$	Pure power law of slope $\alpha$ ; for $\alpha\le -1$ the number integral diverges as $L\to0$ , so a faint cutoff is mandatory.
Bright end $L\gg L^*$	$\Phi\to \dfrac{\Phi^}{L^}\left(\dfrac{L}{L^}\right)^{\alpha} e^{-L/L^}$	Exponential suppression dominates; giants are exponentially rare, giving the sharp bright-end cutoff of cluster ellipticals.
Luminosity density	$j=\Phi^L^\,\Gamma(\alpha+2)$	Converges for $\alpha>-2$ ; peaks at $L\approx L^*$ , so total light is set by knee galaxies, not dwarfs.
Light above $L^*$	$j(>L^)=\Phi^L^*\,\Gamma(\alpha+2,1)$	For $\alpha=-1$ , $\Gamma(1,1)/\Gamma(1)=e^{-1}=37\%$ : over a third of cosmic light is from $L\gtrsim L^*$ galaxies.
Number above $L^*$	$n(>L^)=\Phi^\,\Gamma(\alpha+1,1)$	For $\alpha=-1$ , $\Phi^E_1(1)=0.22\,\Phi^$ : $L^*$ galaxies are common but a minority of the count.

Common trapThe total number density and the magnitude slope are both traps

Two errors recur. First, do not confuse the luminosity form $\Phi(L)\,\mathrm{d}L$ with the magnitude form $\Phi(M)\,\mathrm{d}M$ : because $M=-2.5\log_{10}L+\mathrm{const}$ , brighter means a smaller (more negative) $M$ , the Jacobian introduces $0.4\ln10=0.921$ , and the exponent's sign flips — so the faint-end slope in the magnitude fit is $\alpha+1$ , not $\alpha$ . Second, never quote "the total number density" $\int_0^\infty\Phi\,\mathrm{d}L$ without a lower cutoff: for the observed $\alpha\lesssim-1$ it diverges at the faint end — quote the finite luminosity density $j=\Phi^*L^*\Gamma(\alpha+2)$ instead. A slope steeper than $-2$ would make even the light diverge, which is why $\alpha$ near $-1.3$ is comfortable but $\alpha$ approaching $-2$ would signal a measurement problem.

Worked examples

1 · The cosmic luminosity density and the stellar-mass budget (fully worked)

Take a local $B$ -band fit with $\Phi^*=1.6\times10^{-2}\,h^3\,\mathrm{Mpc}^{-3}$ , $L^*=1.2\times10^{10}\,h^{-2}L_\odot$ , $\alpha=-1.2$ . Find $j$ , then, with mean $\langle M/L\rangle=6$ (solar units), estimate $\rho_*$ and $\Omega_*$ .

Step 1. The gamma factor. With $\alpha=-1.2$ , equation (4) needs $\Gamma(\alpha+2)=\Gamma(0.8)=\mathbf{1.164}$ — of order unity, since the light integral is dominated by the knee.

Step 2. The luminosity density. $j=\Phi^*L^*\Gamma(\alpha+2)=(1.6\times10^{-2})(1.2\times10^{10})(1.164)\,h\,L_\odot\,\mathrm{Mpc}^{-3}$ ; with $h^3\cdot h^{-2}=h$ this is $j=\mathbf{2.2\times10^{8}\,h\,L_\odot\,\mathrm{Mpc}^{-3}}$ , the canonical local $B$ -band value.

Step 3. From light to stellar mass. Equation (6) gives $\rho_*=\langle M/L\rangle\,j=6\times2.2\times10^{8}=1.34\times10^{9}\,h\,M_\odot\,\mathrm{Mpc}^{-3}$ .

Step 4. As a density parameter. Divide by $\rho_\mathrm{crit}=2.775\times10^{11}\,h^2\,M_\odot\,\mathrm{Mpc}^{-3}$ ; the $h$ -powers give $\Omega_*=4.8\times10^{-3}\,h^{-1}$ , i.e. $\mathbf{\Omega_*\approx 0.007}$ for $h=0.7$ . Only about half a percent of the universe's closure density is locked in stars — a headline result of galaxy surveys and a strong constraint on the inefficiency of cosmic star formation.

2 · How much of the count, and of the light, sits above the knee (fully worked)

For a slope $\alpha=-1$ , what fraction of galaxies, and what fraction of the total light, comes from galaxies brighter than $L^*$ ?

Step 1. Number above the knee. Equation (3) with $L_\mathrm{min}=L^*$ gives $n(>L^*)=\Phi^*\Gamma(0,1)=\Phi^*E_1(1)=\mathbf{0.22\,\Phi^*}$ . (This corrects the loose " $\approx0.4\Phi^*$ " sometimes quoted.) There is no total-count normalization here, since the faint-end count diverges — but $0.22\,\Phi^*$ above $L^*$ is a clean, cutoff-free statement.

Step 2. Light above the knee. Equation (4) integrated above $L^*$ gives $j(>L^*)=\Phi^*L^*\Gamma(1,1)$ , and $\Gamma(1,1)=\int_1^\infty e^{-t}\,\mathrm{d}t=e^{-1}$ .

Step 3. The fraction of light. Divide by the total $j=\Phi^*L^*\Gamma(1)=\Phi^*L^*$ : the fraction of cosmic luminosity from galaxies brighter than $L^*$ is $e^{-1}=\mathbf{0.37}$ . More than a third of all cosmic light comes from the rare giants above the knee, though they are a small fraction of the count — the second figure as a number.

3 · Malmquist bias made concrete with $1/V_\mathrm{max}$

A survey detects an $L^*$ galaxy out to $d_\mathrm{max}=400\,\mathrm{Mpc}$ . How far can it see a dwarf of $L=0.01\,L^*$ , and how do the two $V_\mathrm{max}$ weights compare?

Step 1. Detection horizon. From Derivation B, $d_\mathrm{max}\propto L^{1/2}$ , so the dwarf's horizon is $d_\mathrm{max}=400\times(0.01)^{1/2}=\mathbf{40\,\mathrm{Mpc}}$ — ten times nearer, because it is a hundred times fainter.

Step 2. Accessible volume. Since $V_\mathrm{max}\propto d_\mathrm{max}^3\propto L^{3/2}$ , the ratio of accessible volumes is $(0.01)^{3/2}=10^{-3}$ : the dwarf is countable in one-thousandth of the $L^*$ galaxy's volume.

Step 3. Corrective weight. In $\Phi\Delta L=\sum 1/V_\mathrm{max}$ , each detected dwarf carries a weight $10^{3}$ times larger than each $L^*$ galaxy. A single catalogued dwarf stands in for a thousand times more density than a single giant — why raw magnitude-limited counts under-represent the faint end until $1/V_\mathrm{max}$ is applied.

Physics of the portrait

The luminosity function is where galaxy surveys turned from cataloguing into physics, and its analytic backbone carries one name: Paul Schechter's 1976 paper showed a messy cloud of galaxy counts obeyed a single three-parameter law.

Origins · 1 minPaul Schechter and how this was discovered

Paul Schechter

b. 1948 · the analytic galaxy luminosity function

Paul L. Schechter is an American observational astronomer, for most of his career at MIT, whose name is attached to two of the most-used analytic tools in extragalactic astronomy — both from a remarkable pair of 1974–76 papers. The luminosity function bearing his name appeared in 1976, when he showed the abundance of galaxies in the Virgo cluster and the field could be fit across four decades of luminosity by a power law times an exponential, reducing a scatter of counts to three numbers. Two years earlier, with William Press, he had derived the Press–Schechter mass function — the exponentially cut power law for collapsed halos that underpins structure-formation theory — so the same form recurs at both the luminous and the dark levels of the cosmic hierarchy, really a statement about hierarchical collapse. He later made major contributions to gravitational lensing. The luminosity function's enduring power is that it is empirical yet analytic: read off the data, yet every integral closes in gamma functions, which is why it remains, half a century on, the default language for describing any population of galaxies.

History

1936

— Hubble fits a roughly Gaussian distribution to the luminosities of nebulae

1968

— Schmidt introduces the V/Vmax test for flux-limited samples

1974

— Press and Schechter derive the exponentially cut power-law halo mass function

1976

— Schechter writes the analytic three-parameter galaxy luminosity function

2003

— Blanton and the SDSS team pin the local luminosity function to high precision

Going further

Texts. Mo, van den Bosch and White, Galaxy Formation and Evolution, §2.4, §15 (luminosity and mass functions, galaxy–halo connection) · Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §24–25 · Binney and Tremaine, Galactic Dynamics, 2nd ed., §1.1 · Dodelson, Modern Cosmology · Schechter, P. (1976), ApJ 203, 297 (the original function) · Blanton et al. (2003), ApJ 592, 819 (SDSS luminosity function). The standard SDSS-era parameter values are $L^*\sim10^{10}\,h^{-2}L_\odot$ , $M^*_B\approx-20.5+5\log h$ , $\alpha\approx-1.0$ to $-1.3$ , $\Phi^*\sim10^{-2}\,h^3\,\mathrm{Mpc}^{-3}$ . Key numerical checks: (1) $\Gamma(0.8)=1.164$ , so $j=\Phi^*L^*\Gamma(\alpha+2)=2.2\times10^{8}\,h\,L_\odot\,\mathrm{Mpc}^{-3}$ ; (2) for $\alpha=-1$ , the light above $L^*$ is fraction $e^{-1}=0.37$ ; (3) the number above $L^*$ is $E_1(1)=0.22$ ; (4) the Jacobian to magnitudes is $0.4\ln10=0.921$ , giving magnitude faint-end slope $\alpha+1$ ; (5) $V_\mathrm{max}\propto L^{3/2}$ , so a galaxy $100\times$ fainter is countable in a volume $10^3$ times smaller.

Prerequisites. Hubble Sequence and Galaxy Morphology · Elliptical Galaxies and the de Vaucouleurs Profile · Structure of the Milky Way. Leads to. Galactic Dark Matter Halos and Density Profiles · Evidence for Dark Matter in Galaxies · Stellar Populations I, II, and III.

Ask the tutor: Want to fit a real Schechter function to survey data, or trace how the three parameters map onto supernova and black-hole feedback? Ask and we will work it through.

What is the difference between the luminosity function and the stellar-mass function?

The same idea applied to different quantities: the luminosity function counts galaxies per unit luminosity in some band, the stellar-mass function per unit stellar mass. You convert between them with a mass-to-light ratio, but that ratio depends strongly on colour and star-formation history — a young blue galaxy is far more luminous per unit mass than an old red one. The stellar-mass function is often the more physical object, since it removes the transient brightness of recent star formation and tracks the mass actually assembled. Both are well fit by Schechter functions (sometimes a double Schechter for the mass function).

Why is the faint-end slope α so much flatter than the halo mass function's low-mass slope?

The halo mass function rises steeply toward low mass, roughly as $M^{-1.9}$ , so if every halo hosted a proportional galaxy we would expect $\alpha\approx-1.9$ . Instead we measure $\alpha\approx-1.0$ to $-1.3$ , because star formation is progressively more inefficient in low-mass halos: their wells are shallow, so a handful of supernovae expel or heat the gas, and during reionization the ultraviolet background photo-evaporates gas from the smallest halos. The gap between the steep halo slope and the flat galaxy slope directly measures how strongly feedback suppresses dwarf formation — a benchmark test for any galaxy-formation model.

What sets the exponential cutoff at L*, physically?

The cutoff marks the halo mass — around $10^{12}\,M_\odot$ , the Milky Way's scale — above which gas stops cooling efficiently. Below it, infalling gas cools quickly and forms stars; above it the gas is shock-heated into a hot atmosphere that cools only slowly. A central supermassive black hole, fed at a trickle, keeps that atmosphere hot, shutting off the cooling flow that would build a bigger galaxy. This "maintenance-mode" active-nucleus feedback truncates growth and imprints the exponential fall-off; without it, models overproduce giants and the bright end becomes a power law — the mismatch that motivated putting black-hole feedback into simulations.

Is the Schechter function truly universal, or does it break down anywhere?

It is remarkably universal but not exact. Split by type, the total function is the sum of two Schechter components — a bright, weakly evolving early-type population and a fainter, steeper late-type one — so the combined faint end often shows an upturn a single Schechter misses, and deep surveys prefer a double Schechter fit. Brightest cluster galaxies form a separate bump above the exponential tail, growing by a different channel the smooth form does not capture, and at high redshift the parameters evolve. So the Schechter function is best regarded as an excellent description of any single coeval population rather than a fundamental law — its success tells us hierarchical collapse plus feedback generically produces this shape.

What is abundance matching and how does the luminosity function feed it?

Abundance matching is the simplest way to connect galaxies to dark-matter halos. Assume a monotonic relation between galaxy luminosity and halo mass, then fix it by matching abundances: the number density of galaxies brighter than $L$ is set equal to the number density of halos more massive than $M_h$ . Solving $n_\mathrm{gal}(>L)=n_\mathrm{halo}(>M_h)$ for every $L$ gives the stellar-to-halo mass relation with no galaxy-formation modelling. The result is striking: the ratio peaks near $M_h\sim10^{12}\,M_\odot$ , exactly the $L^*$ scale, and falls off both higher and lower — the signature of the two feedback channels, from comparing two abundance curves.

Why does the total number of galaxies diverge but the total light stay finite?

It comes down to one power of $L$ . The number integrand is $\Phi(L)\propto L^{\alpha}$ at the faint end, and $\int L^{\alpha}\,\mathrm{d}L$ diverges as $L\to0$ whenever $\alpha\le-1$ — right where the data sit — so formally there is an unbounded number of ever-fainter dwarfs. The luminosity integrand is $L\,\Phi(L)\propto L^{\alpha+1}$ , which converges for any $\alpha>-2$ . Each added dwarf contributes to the count but almost nothing to the light, so the light integral is dominated by the far less numerous $L^*$ galaxies and stays finite. This is why observers quote a luminosity density, never a "total number of galaxies" without a faint limit.

Galaxy Luminosity Function

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The cosmic luminosity density and the stellar-mass budget (fully worked)

2 · How much of the count, and of the light, sits above the knee (fully worked)

3 · Malmquist bias made concrete with $1/V_\mathrm{max}$

Physics of the portrait

Going further

Before this

Where this leads

Galaxy Luminosity Function

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The cosmic luminosity density and the stellar-mass budget (fully worked)

2 · How much of the count, and of the light, sits above the knee (fully worked)

3 · Malmquist bias made concrete with 1/Vmax1/V_\mathrm{max}1/Vmax​

Physics of the portrait

Going further

Before this

Where this leads

3 · Malmquist bias made concrete with $1/V_\mathrm{max}$