Galaxy Luminosity Functions

Twenty-four published galaxy luminosity function (GLF) measurements catalogued in literature_measurements/luminosity_function/, organised by photometric band: r-band (6), B/bJ-band (9), K-band (4), UV-band (2), and multiband (3). Each entry has a paper.json with full provenance (survey, method, cosmology, Schechter parameters per redshift bin and galaxy type).

Conventions: absolute magnitudes quoted as M − 5 log(h) with H₀ = 100 h km s⁻¹ Mpc⁻¹ unless noted; ϕ★ in h³ Mpc⁻³ mag⁻¹; Vega system for K-band, AB for r/B/UV unless noted.

Note

The stellar mass function (SMF) is the closely related observable in mass units rather than luminosity. See Stellar Mass Functions. The multiband GAMA_Driver2022 entry appears in both the LF and SMF catalogues; the LF entry covers panchromatic luminosity functions while the SMF entry contains the digitised stellar mass function.

Measurement methods 

Schechter function in magnitude units

The galaxy luminosity function is well described at all redshifts and bands by a Schechter function (Schechter 1976):

\[\Phi(L)\,\mathrm{d}L = \phi^\star \left(\frac{L}{L^\star}\right)^\alpha e^{-L/L^\star} \frac{\mathrm{d}L}{L^\star}\]

In the more convenient absolute magnitude form:

\[\Phi(M)\,\mathrm{d}M = \frac{\ln 10}{2.5}\,\phi^\star\, 10^{0.4(\alpha+1)(M^\star - M)}\, e^{-10^{0.4(M^\star - M)}}\,\mathrm{d}M\]

where

M★ — characteristic absolute magnitude (the “knee”);
ϕ★ — normalisation amplitude [h³ Mpc⁻³ mag⁻¹];
α — faint-end slope (α < −1 gives a rising faint end).

The luminosity density is:

\[j = \int_0^\infty L\,\Phi(L)\,\mathrm{d}L = \phi^\star L^\star \,\Gamma(\alpha + 2)\]

References: Schechter 1976, ApJ 203, 297; Press & Schechter 1974.

K-corrections

Unlike the stellar mass function, the luminosity function requires a K-correction to relate observed broadband photometry to rest-frame absolute magnitudes. The observed apparent magnitude is:

\[m_\mathrm{obs}(z) = M_\mathrm{rest} + \mu(z) + K(z)\]

where \(\mu(z) = 5\log_{10}(D_L/10\,\mathrm{pc})\) is the distance modulus and K(z) is the K-correction:

\[K(z) = -2.5\log_{10}\!\left[(1+z) \frac{\int F_\nu(\nu_\mathrm{e})\,R(\nu_\mathrm{o})\,\mathrm{d}\nu_\mathrm{o}} {\int F_\nu(\nu_\mathrm{o})\,R(\nu_\mathrm{o})\,\mathrm{d}\nu_\mathrm{o}} \right]\]

where \(\nu_\mathrm{e} = (1+z)\nu_\mathrm{o}\) and R(ν) is the bandpass response function. K(z) depends on both the galaxy SED type and the band, and must be computed for each galaxy individually.

Standard implementations: kcorrect (Blanton & Roweis 2007, arXiv:astro-ph/0606170) fits a non-negative combination of SED templates to the observed multi-band photometry. Blanton+2003 introduced the “z = 0.1 reference frame” (denoting bands as ^0.1r, etc.) to minimise K-correction uncertainties for SDSS samples centred at z ~ 0.1.

Assumptions:

Passband response R(ν) is known accurately (calibration + atmosphere).
Galaxy SED can be approximated by a linear combination of templates.
K-correction uncertainty propagates to Φ(M) at the 0.02–0.05 mag level for z < 0.3, growing to ~0.1 mag at z ~ 1.

1/Vmax estimator

For a flux-limited survey with apparent magnitude limit m_lim:

\[\hat{\Phi}(M) = \frac{1}{\Delta M} \sum_{i \in \mathrm{bin}} \frac{w_i}{V_{\max,i}}\]

where \(V_{\max,i}\) is the comoving volume within which galaxy \(i\) (with absolute magnitude M_i) would remain above the flux limit: \(m(z_{\max,i}) = m_\mathrm{lim}\). Weights \(w_i\) account for spectroscopic completeness and k-correction uncertainty.

The key assumption is spatial homogeneity; large-scale structure biases the estimate when survey volumes are small (VVDS, DEEP2, GOODS).

STY maximum-likelihood estimator

The STY estimator (Sandage, Tammann & Yahil 1979) maximises the likelihood that each galaxy is observed at its luminosity given the selection function. It is unbiased by density fluctuations but assumes a parametric form (Schechter function):

\[\ln\mathcal{L} = \sum_i \ln\phi(L_i) - \ln\!\int_{L_{\min}(z_i)}^{\infty} \phi(L)\,\mathrm{d}L\]

SWML / non-parametric extension

The Stepwise Maximum Likelihood (SWML) estimator (Efstathiou, Ellis & Peterson 1988) is the non-parametric extension of STY. It estimates Φ in discrete magnitude bins without assuming a Schechter shape, enabling detection of the turnover at M★ or departures from a power law at the faint end. All three estimators are unbiased by density fluctuations; 1/Vmax is sensitive to clustering but is the most commonly reported form.

Open questions and current status 

Faint-end slope and completeness

The faint-end slope α is robustly measured at α ~ −1.0 to −1.3 in r-band and B-band locally. The principal uncertainty is surface brightness incompleteness: low-surface-brightness dwarf galaxies (M > −14 in r-band) are missed by automated photometry pipelines, causing the observed LF to flatten below the completeness limit. Correcting for incompleteness at M_r > −14 requires injection- recovery tests with 20–50% uncertainties.

Luminosity evolution and M★(z)

M★ brightens with look-back time in all bands (passive + star-formation evolution). In rest-frame B/r-band, M★ brightens by ~0.5–1.0 mag from z = 0 to z = 1. The slope of this evolution depends on the sample selection (early-type vs late-type) and the IMF. At z > 1 the brightening slows as star formation rates peak at z ~ 2–3.

In rest-frame K-band, M★ is approximately constant with redshift out to z ~ 2 (stellar mass dominated), making K-band LF a better proxy for the stellar mass function. The UV LF traces the instantaneous star formation rate and evolves strongly by ~2 mag in M★ from z = 0 to z = 4.

Colour-type decomposition

The total LF is the sum of red (early-type, passive) and blue (late-type, star-forming) populations. These have systematically different α and M★: red galaxies have a steeper bright end; blue galaxies contribute the faint-end upturn. The evolution of each sub-population is distinct. Accurately separating populations at high redshift requires rest-frame colour information that becomes degenerate with photo-z uncertainties.

Cosmic variance

VVDS, DEEP2, and FORS Deep Field cover only 1–10 deg², subtending 50–100 Mpc at z ~ 0.5–1. The LF inferred from such fields is biased by 5–20% relative to the fair-sample mean. GAMA and SDSS resolve this at z < 0.3; VIPERS resolves it at z ~ 0.5–1.2 over 24 deg².

High-redshift UV LF (z > 3)

The rest-frame UV LF at z > 3 is now measured to M_UV ~ −13 by HST deep fields and JWST. The faint-end slope steepens to α ~ −2 at z > 5. This steep UV LF is a key input to reionisation models (ionising photon budget). JWST is extending measurements to even fainter luminosities and higher redshifts, with significant sample-to-sample variance remaining between different analyses.

Progress over two decades 

2001–2002: Cole+2001 (2dFGRS+2MASS K-band) and Norberg+2002 (2dFGRS bJ-band) establish the local LF from the largest redshift surveys of the era (~10–100k galaxies). The Schechter shape is confirmed; M★(K) ~ −23.4 at z ~ 0.05.
2003: Blanton+2003 (SDSS DR1, ~148k galaxies) introduces the K-corrected ^0.1r-band convention and produces the first precise SDSS LF with red/blue sub-samples. Becomes the canonical local r-band benchmark.
2005–2006: VVDS (Ilbert+2005) and DEEP2 (Willmer+2006) push the B-band LF to z ~ 1.5 and z ~ 1.35 respectively with spectroscopic redshifts. The first clear evidence for ~0.5–1 mag brightening of M★ with look-back time.
2008–2009: SDSS DR6 (MonteroDorta+2009, 321k galaxies) refines the r-band LF; zCOSMOS (Zucca+2009, 10k galaxies, z < 1) and VVDS (Cucciati+2012, UV LF) extend to higher redshift.
2012–2015: GAMA I (Loveday+2012, 16k galaxies) and GAMA II (Loveday+2015, 180k galaxies) produce multi-band LFs in ugrizYJHK with well-controlled completeness. PRIMUS (Cool+2012, 40k galaxies, 9 deg²) bridges z = 0.2–1 in r-band.
2013–2016: VIPERS PDR-1/PDR-2 (Davidzon+2013/2016, 54–72k galaxies, z = 0.45–1.3) delivers the B-band LF at z ~ 1 from a large spectroscopic survey. FDF (Gabasch+2004) and GOODS (Dahlen+2005) extend to z ~ 5 via photometric redshifts.
2017: GAMA DR3 (Wright+2017, 221k galaxies) produces the final panchromatic LF (FUV–MIR, 21 bands) establishing the energy output of the z < 0.65 Universe.
2022: GAMA DR4 (Driver+2022, 300k galaxies) refines the local LF to z ~ 0.65 across all bands, including the deepest spectroscopic completeness correction to date.

What was solved: The overall shape and normalisation of the LF at 0 < z < 1 is well established across all major bands; the Schechter parameterisation is validated; the luminosity density ρ_L(z) evolution is traced to z ~ 4 in UV.

What remains open: The faint end at M > −14 (below SDSS completeness); the LF at z > 2 in rest-frame optical; the role of environment (cluster vs field) in LF shape; the connection between LF evolution and star formation quenching.

Survey parameter table 

Survey	Band	z range	Area (deg²)	N_gal	V_eff (h⁻³ Gpc³)	Reference
SDSS_Blanton2003	r (^0.1r)	0.02 – 0.22	2627	148 k	0.1	ApJ 592, 819
SDSS_MonteroDorta2009	r	0.005 – 0.25	7000	321 k	0.5	MNRAS 399, 1106
GAMA_Loveday2012	r	0.002 – 0.5	135	15.7 k	0.008	MNRAS 420, 1239
GAMA_Loveday2015	ugrizYJHK	0.002 – 0.5	180	180 k	0.01	MNRAS 451, 1540
PRIMUS_Cool2012	r	0.2 – 1.0	9	40 k	0.03	ApJ 748, 10
2dFGRS_Norberg2002	bJ (B)	0.002 – 0.30	1500	111 k	0.04	MNRAS 336, 907
VVDS_Ilbert2005	B	0.05 – 1.5	0.6	11.6 k	0.004	A&A 439, 863
DEEP2_Willmer2006	B	0.2 – 1.35	3	11 k	0.01	ApJ 647, 853
VIPERS_Davidzon2016	B	0.5 – 1.3	24	72 k	0.05	A&A 586, A23
FDF_Gabasch2004	B	0.5 – 5.0	0.04	5.5 k	0.001	A&A 421, 41
2dFGRS_Cole2001	K	0.0 – 0.2	1500	17 k	0.02	MNRAS 326, 255
2MASS_Kochanek2001	K	0.0 – 0.1	20000	4.2 k	0.002	ApJ 560, 566
VVDS_Cucciati2012	FUV (1500 Å)	0.05 – 4.5	1	11 k	0.005	A&A 539, A31
GAMA_Driver2022	FUV–MIR (21)	0.002 – 0.65	250	300 k	0.03	MNRAS 513, 439

Figures 

LF Schechter parameter evolution — Evolution of M★ (top) and faint-end slope α (bottom) with redshift for r-band (blue), B/bJ-band (orange), K-band (red), and UV (green). K-band M★ is approximately constant with redshift (stellar mass dominated); UV α steepens to ~ −1.6 at z > 1.

LF survey area vs redshift — Survey area vs median redshift for all catalogued LF surveys, coloured by band. Symbol size is proportional to log N_gal. Note the large spread in area-redshift coverage: 2MASS covers 20 000 deg² but only z < 0.1; FORS Deep Field covers 0.04 deg² but reaches z ~ 5.

Schechter parameters at z ~ 0 by band 

Survey	Band	M★ − 5log(h)	ϕ★ (10⁻² h³ Mpc⁻³)	α	Note
Blanton+2003	^0.1r (AB)	−20.44	1.49	−1.05	z ref = 0.1 frame; STY
MonteroDorta+2009	r (AB)	−20.71	1.08	−1.26	SDSS DR6; deeper sample
Norberg+2002	bJ (Vega)	−19.66	1.61	−1.21	2dFGRS; M* as M*−5log(h)
Cole+2001	K_s (Vega)	−23.44	1.08	−0.96	2dFGRS+2MASS; low α typical of K-band
Kochanek+2001	K_s (Vega)	−23.39	1.16	−1.09	2MASS extended source catalog
Arnouts+2005	FUV 1500 Å (AB)	−18.05	1.55	−1.21	GALEX+VVDS; z = 0–0.2

r-band 

Rest-frame r-band (~6200 Å).

Directory / ID	Survey	z range	N_gal	Cites	Reference
SDSS_Blanton2003	SDSS spec-z	0.02 – 0.22	148 k	~985	ApJ 592, 819 (arXiv:astro-ph/0210215)
SDSS_MonteroDorta2009	SDSS DR6 spec-z	0.005 – 0.25	321 k	~350	MNRAS 399, 1106 (arXiv:0806.4930)
GAMA_Loveday2012	GAMA I spec-z	0.002 – 0.5	15.7 k	~340	MNRAS 420, 1239 (arXiv:1111.0166)
GAMA_Loveday2015	GAMA II spec-z	0.002 – 0.5	180 k	~150	MNRAS 451, 1540 (arXiv:1505.01003)
GAMA_McNaughtRoberts2014	GAMA II spec-z	0.04 – 0.26	50 k	~120	MNRAS 445, 2125 (arXiv:1404.3748)
PRIMUS_Cool2012	PRIMUS prism-z	0.2 – 1.0	40 k	~230	ApJ 748, 10 (arXiv:1108.4933)

B/bJ-band 

Rest-frame B-band (~4400 Å) / bJ-band.

Directory / ID	Survey	z range	N_gal	Cites	Reference
2dFGRS_Norberg2002	2dFGRS spec-z	0.002 – 0.30	111 k	~800	MNRAS 336, 907 (arXiv:astro-ph/0111011)
VVDS_Ilbert2005	VVDS spec-z	0.05 – 1.5	11.6 k	~500	A&A 439, 863 (arXiv:astro-ph/0409133)
VVDS_Zucca2006	VVDS spec-z	0.05 – 1.5	11.6 k	~280	A&A 455, 879 (arXiv:astro-ph/0606371)
DEEP2_Willmer2006	DEEP2 spec-z	0.2 – 1.35	11 k	~560	ApJ 647, 853 (arXiv:astro-ph/0506041)
FDF_Gabasch2004	FORS Deep Field photo-z	0.5 – 5.0	5.5 k	~240	A&A 421, 41 (arXiv:astro-ph/0312089)
GOODS_Dahlen2005	GOODS spec+photo-z	0.0 – 2.0	5 k	~330	ApJ 631, 126 (arXiv:astro-ph/0507005)
zCOSMOS_Zucca2009	zCOSMOS spec-z	0.1 – 1.0	10.6 k	~200	A&A 508, 1217 (arXiv:0904.3621)
VIPERS_Davidzon2013	VIPERS PDR-1 spec-z	0.45 – 1.1	54 k	~130	A&A 558, A23 (arXiv:1305.2745)
VIPERS_Davidzon2016	VIPERS PDR-2 spec-z	0.5 – 1.3	72 k	~110	A&A 586, A23 (arXiv:1511.05146)

K-band 

Near-infrared K_s-band (~2.2 μm).

Directory / ID	Survey	z range	N_gal	Cites	Reference
2dFGRS_Cole2001	2dFGRS+2MASS spec-z	0.0 – 0.20	17 k	~1100	MNRAS 326, 255 (arXiv:astro-ph/0012429)
2MASS_Kochanek2001	2MASS spec-z (extended)	0.0 – 0.10	4.2 k	~700	ApJ 560, 566 (arXiv:astro-ph/0011456)
6dFGS_Jones2006	6dFGS spec-z	0.0 – 0.15	11 k	~300	MNRAS 369, 25 (arXiv:astro-ph/0603015)
VVDS_Prescott2009	VVDS spec-z	0.2 – 1.0	8 k	~130	MNRAS 395, 1591 (arXiv:0903.0383)

UV-band 

Rest-frame FUV (~1500 Å).

Directory / ID	Survey	z range	N_gal	Cites	Reference
VVDS_Arnouts2005	GALEX + VVDS spec-z	0.0 – 1.2	1 k	~400	ApJ 619, L43 (arXiv:astro-ph/0412525)
VVDS_Cucciati2012	VVDS spec-z	0.05 – 4.5	11 k	~280	A&A 539, A31 (arXiv:1109.1828)

Multiband (panchromatic)

Panchromatic LF across many bands simultaneously (FUV to MIR).

Directory / ID	Survey	z range	N_gal	Cites	Reference
GAMA_Driver2012	GAMA spec-z (21 bands)	0.013 – 0.10	104 k	~350	MNRAS 422, 1527 (arXiv:1204.1508)
GAMA_Wright2017	GAMA DR3 spec-z (21 bands)	0.002 – 0.65	221 k	~160	MNRAS 470, 283 (arXiv:1702.04713)
GAMA_Driver2022	GAMA DR4 spec-z (21 bands)	0.002 – 0.65	300 k	~120	MNRAS 513, 439 (arXiv:2201.07439)

Note

GAMA_Driver2022 also appears in the Stellar Mass Functions catalogue. The LF entry covers panchromatic luminosity functions; the SMF entry contains the digitised stellar mass function data files.