.. _literature_lf: 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 :ref:`literature_smf`. 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. .. contents:: On this page :local: :depth: 1 ---- 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): .. math:: \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: .. math:: \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: .. math:: 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: .. math:: m_\mathrm{obs}(z) = M_\mathrm{rest} + \mu(z) + K(z) where :math:`\mu(z) = 5\log_{10}(D_L/10\,\mathrm{pc})` is the distance modulus and K(z) is the K-correction: .. math:: 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 :math:`\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 :sup:`0.1`\ r, 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\ :sub:`lim`: .. math:: \hat{\Phi}(M) = \frac{1}{\Delta M} \sum_{i \in \mathrm{bin}} \frac{w_i}{V_{\max,i}} where :math:`V_{\max,i}` is the comoving volume within which galaxy :math:`i` (with absolute magnitude M\ :sub:`i`) would remain above the flux limit: :math:`m(z_{\max,i}) = m_\mathrm{lim}`. Weights :math:`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): .. math:: \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\ :sub:`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 :sup:`0.1`\ r-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\ :sub:`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 ----------------------- .. list-table:: :header-rows: 1 :widths: 22 12 10 9 9 11 13 * - Survey - Band - z range - Area (deg²) - N\ :sub:`gal` - V\ :sub:`eff` (h⁻³ Gpc³) - Reference * - SDSS_Blanton2003 - r (:sup:`0.1`\ r) - 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 ------- .. figure:: ../figures/literature/lit_lf_schechter_evolution.png :width: 90% :alt: 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. .. figure:: ../figures/literature/lit_lf_survey_coverage.png :width: 80% :alt: 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 ---------------------------------------- .. list-table:: :header-rows: 1 :widths: 22 10 12 12 10 22 * - Survey - Band - M★ − 5log(h) - ϕ★ (10⁻² h³ Mpc⁻³) - α - Note * - `Blanton+2003 `_ - :sup:`0.1`\ r (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\ :sub:`s` (Vega) - −23.44 - 1.08 - −0.96 - 2dFGRS+2MASS; low α typical of K-band * - `Kochanek+2001 `_ - K\ :sub:`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). .. list-table:: :header-rows: 1 :widths: 28 18 9 9 8 28 * - Directory / ID - Survey - z range - N\ :sub:`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. .. list-table:: :header-rows: 1 :widths: 28 18 9 9 8 28 * - Directory / ID - Survey - z range - N\ :sub:`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\ :sub:`s`-band (~2.2 μm). .. list-table:: :header-rows: 1 :widths: 28 18 9 9 8 28 * - Directory / ID - Survey - z range - N\ :sub:`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). .. list-table:: :header-rows: 1 :widths: 28 18 9 9 8 28 * - Directory / ID - Survey - z range - N\ :sub:`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). .. list-table:: :header-rows: 1 :widths: 28 20 9 9 8 26 * - Directory / ID - Survey - z range - N\ :sub:`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 :ref:`literature_smf` catalogue. The LF entry covers panchromatic luminosity functions; the SMF entry contains the digitised stellar mass function data files.