Shedding Light on the Galaxy Luminosity Function
aa r X i v : . [ a s t r o - ph . C O ] O c t Astron Astrophys Rev (2011)DOI 10.1007/s00159-011-0041-9
Shedding Light on the Galaxy Luminosity Function
Russell Johnston
Accepted 26 August 2011
Abstract
From as early as the 1930s, astronomers have tried to quantify the statistical na-ture of the evolution and large-scale structure of galaxies by studying their luminosity distri-bution as a function of redshift - known as the galaxy luminosity function (LF). Accuratelyconstructing the LF remains a popular and yet tricky pursuit in modern observational cos-mology where the presence of observational selection effects due to e.g. detection thresholdsin apparent magnitude, colour, surface brightness or some combination thereof can renderany given galaxy survey incomplete and thus introduce bias into the LF.Over the last 70 years there have been numerous sophisticated statistical approachesdevised to tackle these issues; all have advantages – but not one is perfect. This reviewtakes a broad historical look at the key statistical tools that have been developed over thisperiod, discussing their relative merits and highlighting any significant extensions and mod-ifications. In addition, the more generalised methods that have emerged within the last fewyears are examined. These methods propose a more rigorous statistical framework withinwhich to determine the LF compared to some of the more traditional methods. I also lookat how photometric redshift estimations are being incorporated into the LF methodology aswell as considering the construction of bivariate LFs. Finally, I review the ongoing devel-opment of completeness estimators which test some of the fundamental assumptions goinginto LF estimators and can be powerful probes of any residual systematic effects inherentmagnitude-redshift data.
Keywords
Galaxies: luminosity function, mass function – methods: statistical – cosmol-ogy: large-scale structure of the Universe
Understanding the origins and growth of structure that form the galaxies we observe todayis one of the many driving forces behind current cosmological research. The luminosityfunction (LF), denoted by Φ ( L ) (in units of ergs s − Mpc − ), provides one of the most University of the Western Cape, Modderdam Road, Bellville 7535, Cape Town, South AfricaSouth African Astronomical Observatory (SAAO), Observatory Road, Cape Town 7925, South AfricaUniversity of Glasgow, Kelvin Building, University Avenue, Glasgow, Scotland, UK G12 8QQE-mail: [email protected] Russell Johnston fundamental tools to probe the distribution of galaxies over cosmological time. It describesthe relative number of galaxies of different luminosities by counting them in a representativevolume of the Universe which then measures the comoving number density of galaxies perunit luminosity, L , such that dN = Φ ( L ) dLdV. (1)where dN is the observed number of galaxies within a luminosity range [ L, L + dL ]. Whenworking in luminosities it is common practice to apply log intervals of L . The quantity Φ ( L ) can be normalised such that ∞ Z Φ ( L ) dL = ρ, (2)where ρ is the number of objects per unit volume V , and thus Φ ( L ) dL gives the numberdensity of objects within a given luminosity range. In general, the density function can bedefined by ρ ( x ) , where x represents the 3D spatial cartesian co-ordinates such that the totalnumber N of objects per unit volume (Mpc − ) is N = Z V ρ ( x ) d x (3)However, it is common place to compute ρ from the measured redshifts z and angular co-ordinates. Thus, for a given sample within a respective minimum and maximum redshiftrange z min and z max and solid angle Ω at a distance r it is possible to compute, N = z max Z z min ρ ( z ) dV ′ dz dz, (4)where ρ ( z ) is now the density as a function of redshift and dV ′ ≡ Ωr dr is a solid angle-integrated differential volume element (see e.g. Chołoniewski 1987).The LF provides us with a robust handle to compare the difference between different setsof galaxies i.e. at different redshifts, galaxy types, environment etc... It allows us to assessthe statistical nature of galaxy formation and evolution and indeed, it seems that as soon as anew survey is carried out one of the first actions is to compute the LF, see e.g. Blanton et al.(2001); Fried et al. (2001); Norberg et al. (2002); Im et al. (2002); Blanton et al. (2003b);Liske et al. (2003); Wolf et al. (2003); Croom et al. (2004); Richards et al. (2006); Ilbert et al.(2006); Faber et al. (2007); Bouwens et al. (2008a); Siana et al. (2008); Crawford et al. (2009);Zucca et al. (2009); Haberzettl et al. (2009); Montero-Dorta & Prada (2009); Croom et al.(2009b); Willott et al. (2010); Rodighiero et al. (2010); Eales et al. (2010); Tilvi et al. (2010);Hill et al. (2010) to highlight just a small fraction of studies over the last ten years. This isperhaps indicative of the continuing popularity and relevance of this area of research.The study of galaxy LFs is now a vast subject area spanning a broad range of wave-lengths and probing back to the earliest galaxies. Therefore, it should be noted that thisreview is led with a strong emphasis on areas of work that have driven the development ofthe statistical methodology of LF estimation. Consequently, there will be areas of researchthat have not been cited, and so apologies are given in advance. Instead, the most relevantextensions and variations of the traditional approaches are examined in detail as well as con-sidering the most recent generalised statistical advances for LF estimation. Furthermore, Iexplore the branch of astro-statistics concerning completeness estimators that, at their core,represent a test of the validity of the assumption of separability between the LF, φ ( M ) and hedding Light on the Galaxy Luminosity Function 3 the density function ρ ( z ) , that is inherent in the LF methodology. I also discuss how suchestimators have been used to constrain evolutionary models.The format of this article will be as follows. The remainder of this section discussessome of the parameterisations of the LF that have led to the maximum likelihood estima-tors. There is then a brief introduction to the traditional non-parametric methods beforemoving to § § § § §
6, as well as incorporating photometric redshift estimation into currentLF methodology in §
7. In § § §
10 explores the tests of independence and completenessestimators that probe the separability assumption which underpins most LF estimators. Thisthen leads to a discussion in §
11 on some astrophysical applications for which accurate LFestimation has been crucially important. §
12 closes the review with final a summary anddiscussion.1.1 Parameterising the luminosity functionThe processes by which one can estimate the LF vary greatly. A popular parametric methoddeveloped (Sandage, Tammann, & Yahil 1979) is based on the Maximum Likelihood Esti-mator (MLE) where a parametric form of the LF is assumed. The most common of thesemodels is that of the Press-Schechter function named after William Press and Paul Schechter(Press & Schechter 1974; Schechter 1976) who originally derived it in the form of the massfunction during their studies of structure formation and evolution. It is typically written inthe form given by, Φ ( L ) dL = φ ∗ (cid:18) LL ∗ (cid:19) α exp (cid:18) − LL ∗ (cid:19) dLL ∗ , (5)where, φ ∗ is a normalisation factor defining the overall density of galaxies, usually quotedin units of h Mpc − , and L ∗ is the characteristic luminosity. The quantity α defines thefaint-end slope of the LF and is typically negative, implying large numbers of galaxies withfaint luminosities. LF studies of the local Universe ( z . . ) have estimated close to α ∼− . which could imply that there may be an infinite number of faint galaxies. However,integrating Equation 5 over luminosity provides us with the total luminosity density, j L (insolar luminosity units, h L ⊙ Mpc − ) and ensures the total luminosity density remains finite, j L = Z ∞ L lim L Φ ( L ) dL = φ ∗ L ∗ Γ (2 + α, L/L ∗ ) , (6)where Γ ( a, b ) is the incomplete gamma function defined as Γ ( a, b ) = ∞ Z b t ( a − e − b dt. (7) Russell Johnston
See e.g. Norberg et al. (2002) and Blanton et al. (2001) for estimates of j L from the respec-tive Two Degree Field Galaxy Redshift Survey (2dFGRS) and the Sloan Digital Sky Sur-vey (SDSS) redshift surveys respectively. A recent paper by Hill et al. (2010) combines thesurvey data from the Millennium Galaxy Catalogue (MGC Liske et al. 2003; Driver et al.2005), SDSS (York et al. 2000; Adelman-McCarthy et al. 2007) and the UKIRT InfraredDeep Sky Survey Large Area Survey (UKIDSS LAS Lawrence et al. 2007; Warren et al.2007) to determine LFs and luminosity densities over a broad range of wavelengths (UV toNIR) and thus probe the cosmic spectral energy distribution at z < . .By assuming the mass-to-light ratio relation, it is also possible to use j L to compute thetotal contribution from stars and galaxies to the mean mass density of the Universe ¯ ρ (e.g.Loveday et al. 1992).At this point it should be noted that it is common practice to convert Equation 5 from abso-lute luminosities L to absolute magnitudes M via the simple relation LL ∗ = 10 − . M − M ∗ ) . (8)This leads to the equivalent expression of the Schechter LF, Φ ( M ) = 0 . φ ∗ (cid:16) . M ∗ − M ) (cid:17) ( α +1) e . M ∗− M ) . (9)In a similar way for Φ ( L ) the LF in terms of magnitudes M is usually plotted in log space i.e log[ Φ ( M )] vs. M . For a galaxy with an observed apparent magnitude m , it is then straight-forward to determine its absolute magnitude M by M = m − ( d L ) − − A g ( l, b ) − K ( z ) − E ( z ) , (10)provided that the corrections for galactic extinction, A g ( l, b ) , K -correction (see § K ( z ) ,and evolution (see § E ( z ) , are well understood. The quantity d L is the luminositydistance, which, by invoking the standard Λ CDM cosmology, is defined as, d L = (1 + z ) (cid:18) cH (cid:19) Z z dz p (1 + z ) Ω m + Ω Λ , (11)where Ω m and Ω Λ represent the present-day dimensionless matter density and cosmologicalenergy density constant respectively. z is the redshift of the object, c is the speed of light and H is the Hubble constant.Although the Schechter form of the LF has been very successful as a generic fit to awide variety of survey data, it has also been shown that galaxy surveys sampled in the infra-red have yielded LFs that do not seem to fit the standard Press-Schechter formalism. Forexample, in Lawrence et al. (1986) the following power law analytical form was fitted todata obtained from the Infrared Astronomical Satellite (IRAS), φ ( L ) = dΦdL = φ ∗ L − β (cid:18) LL ∗ β (cid:19) − β , (12)where β and β define the slopes of the two power laws. In Saunders et al. (1990) a log-Gaussian form was adopted for a survey also using the IRAS given by φ ( L ) = dΦdL = φ ∗ (cid:18) LL ∗ (cid:19) − γ exp (cid:20) − σ log (cid:18) LL ∗ (cid:19)(cid:21) , (13) hedding Light on the Galaxy Luminosity Function 5 where γ is analogous to α in the Schechter formalism. Sanders et al. (2003) instead fitted abroken power-law of the form φ ( L ) ∝ L α , (14)to IRAS 60 µ m bright galaxy sample. This form of the LF has more recently been adoptedby Magnelli et al. (2009, 2011) when probing the infrared LF out to z = 2 . using Spitzer observations.A study of Early-Type galaxies within the Sloan Digital Sky Survey (SDSS) by Bernardi et al.(2003a) found that a Gaussian function of the following form best described the data: φ ( M ) = φ ∗ q πσ M exp (cid:18) − [ M i − M ∗ ] σ M (cid:19) . (15)For the study of quasi-stellar objects (QSO) Boyle et al. (1988a,b) demonstrated that a two-power law parameterisation is a reasonable fit to the data, Φ ( M ) = Φ ∗ . M − M ∗ ]( α +1) + 10 . M − M ∗ ]( β +1) , (16)with a break at M ∗ and a bright-end slope α steeper than the faint-end slope β .1.2 Toward more robust estimates of the luminosity functionComplementary to the parametric approach are the non-parametric methods which do notrequire any underlying assumption of the parametric form of the LF. The traditional ap-proaches for this class of estimator are the classical number count test (e.g Hubble 1936b;Christensen 1975), the Schmidt (1968) /V max estimator, the φ/Φ method (e.g. Turner1979), and the Lynden-Bell (1971) C − method. There is also the non-parametric coun-terpart of the MLE developed by Efstathiou, Ellis, & Peterson (1988) and often referred toas the Step-wise Maximum Likelihood method (SWML). A summary of all the traditionalmethods and their extensions is show in Table 1 at the end of § φ ( M ) , and the density function, ρ ( z ) (see § copula to construct the far-ultraviolet (FUV) - far-infrared (FIR) BLF.A non-parametric method by Le Borgne et al. (2009) has been developed and applied tomulti-wavelength high redshift IR and sub-mm data. The method differs from other non-parametric estimators by applying an inversion technique to galaxy counts that does notrequire the use of redshift information. With no assumption on the parametric form of theLF, the method uses an assumed set of spectral energy distribution (SED) templates to ex-plore the range of possible LFs for the observed number counts.All these diverse approaches perhaps underline the difficulty in its nature where intrin-sic bias can often lead to conflicting results. As a result it is quite often the case that several The copula is a function used to join multivariate distribution functions to their one-dimensional marginaldistribution function and is particularly useful for variables with co-dependence. Russell Johnston methods are applied to a given sample to help achieve consistency and quantify any biasfound. For example, Ilbert et al. (2005) developed their “Algorithm for Luminosity Func-tion” (ALF) tool that implements the /V max , C + (a modified version of Lynden-Bell’s C − method), SWML and the STY estimators applied to the VIMOS-VLT DEEP survey data(see also Zucca et al. 2006; Ilbert et al. 2006). More recently, these estimators were appliedtogether to the zCOSMOS survey (Zucca et al. 2009).There have been several reviews of the LF methodology as it has developed over theyears. The earliest of these was by Felten (1977) who performed nine determinations ofthe LF using variations of the classical method. BST88 gave a comprehensive review ofall non-parametric and parametric methods that had been developed up to 1988. However,key papers by Heyl et al. (1997), Willmer (1997), Takeuchi, Yoshikawa, & Ishii (2000) andIlbert et al. (2004) have made direct comparisons of the traditional methods with the use ofreal and simulated survey data. For the first time a rigorous exploration into their relativemerits was made by applying the major LF estimators to different scenarios from e.g. homo-geneous samples to ones with strong clustering properties, density evolution, varying LFs,observational bias from K -corrections etc. (see § We do not yet have a complete theory that fully describes how galaxies form and evolve intothe structures we observe today, and any model of formation and evolution must match theobserved data. Since the fledgling redshifts surveys during the 1970s, the mapping of the skyhas turned into a thriving industry where the largest survey to date, the Sloan Digital SkySurvey, has imaged in excess of 270 million galaxies with approximately a million of thesehaving spectroscopic redshifts. As as a result, luminosity function studies have been a vitaltool with which to analyse these surveys, since they allow us to probe the evolutionary pro-cesses of extragalactic sources as a function of redshift and thus place powerful constraintson current galaxy formation and evolutionary models.At around the same time redshift surveys came to fruition, the theoretical frameworkwas being laid for how we currently model galaxy formation and evolution. Moreover, ascomputational technology came of age, the field of numerical cosmology saw rapid devel-opment providing state-of-the-art N-body simulations based on our current knowledge ofthese physical processes and observations.2.1 Constructing a model of galaxy evolutionThe current standard Concordance model of cosmology represents the most concise modelto date, combining astronomical observations with theoretical predictions to explain the ori-gins, evolution, structure and dynamics of the Universe. The origin of the Concordancemodel is rooted in the Copernican principle - a fundamental assumption proposed by Nico-laus Copernicus in the 16th century that states we do not occupy a privileged position in theUniverse. This concept was generalised into what is termed, the Cosmological Principle, inwhich we assume that the Universe is both homogeneous and isotropic. In 1922 AlexanderFriedman provided a formal description in terms of a metric that was later independentlyimproved upon by Howard Robertson, Arthur Walker and Georges Lemaˆıtre into what is hedding Light on the Galaxy Luminosity Function 7 now referred to as the Friedmann-Lemaˆıtre-Robertson-Walker metric, or more simply - theFLRW metric.In its current, simplest form, the standard model is known as Λ Cold Dark Matter( Λ CDM). Whilst dark matter particles have not yet been directly detected, there are a numberof independent observations derived from high velocity dispersions of galaxies observed inclusters (Zwicky 1933) and of flat galaxy rotation curves (Rubin et al. 1980; Sofue & Rubin2001), which support its inferred existence. The Λ term refers to the non-zero cosmologicalconstant in general relativity theory which implies the Universe is currently undergoing aperiod of cosmic acceleration (see e.g. Peebles & Ratra 1988; Steinhardt et al. 1999). Therecent observational techniques derived from distance measurements of Type Ia supernovaeby Riess et al. (1998) and Perlmutter et al. (1999) have provided evidence of this accelera-tion; later studies from baryonic acoustic oscillations (Eisenstein et al. 2005), the integratedSachs-Wolf (ISW) effect (Giannantonio et al. 2008) and weak lensing (Schrabback et al.2010) have helped strengthen support of the Λ CDM model.By the 1960s a theory of galaxy formation was proposed by Eggen et al. (1962) knownas monolithic collapse (or the ‘top-down’ scenario) in which galaxies originate from largeregions of primordial baryonic gas. This baryonic mass then collapses to form stars withinthe central region allowing the most massive galaxies to form first. Seminal work by JamesPeebles (see e.g. Peebles 1970; Gunn & Gott 1972; Peebles 1973, 1974, 1980) establishedthe theoretical underpinnings for structure formation known as the gravitational instabil-ity paradigm. In essence, this theory states that small scale density fluctuations in the earlyUniverse grew by gravitational instability into the structures we see today. This led to the es-tablishment of the
Hierarchical clustering model (or ‘bottom-up’ scenario) in which largerobjects evolve from mergers with smaller objects (see e.g. Searle & Zinn 1978). The bottom-up scenario is now the more favoured of the two models and is fundamental to galaxy forma-tion models, ultimately leading to the theoretical and numerical modelling applied in currentN-body simulations.The work by Press & Schechter (1974) demonstrated how the structure of halos in theearly Universe could form through gravitational condensation in a density field using aGaussian random field of gas-like particles. Thus, the first analytical treatment of the massfunction was derived. White & Rees (1978) developed these ideas formulating a more so-phisticated model of galaxy formation in which dark matter halos formed hierarchically andconsequently baryons cooled and condensed at their centres to form galaxies. Thus, smallproto-galaxies form early in the history of the Universe and through merging processes buildup into larger galaxies we see today.The production of early N-body codes of the 1960s and 1970s (e.g. Aarseth 1963;Gingold & Monaghan 1977; Lucy 1977) began to see rapid development during the 1980s.Aarseth et al. (1979) was one of the first to develop simulations of galaxy clustering. How-ever, as the Cold Dark Matter model (Blumenthal et al. 1984) garnered momentum, Davis et al.(1985) provided the first simulations within the hierarchical clustering CDM framework.Over the next two decades hydrodynamics were integrated into simulations that would in-clude processes such as star formation, feedback and gas cooling (see also e.g. Fall & Efstathiou1980; White et al. 1987; Schaeffer & Silk 1988; White & Frenk 1991; Katz & Gunn 1991;Navarro & Benz 1991; Cen & Ostriker 1992; Katz & White 1993; Cole et al. 1994; Lacey & Cole1994; Navarro et al. 1994; Springel et al. 2001; Springel 2005; Kereˇs et al. 2005; Bower et al.2006; De Lucia & Blaizot 2007; Bertone et al. 2007; Weinberg et al. 2008; Dav´e et al. 2008;Kereˇs et al. 2009, highlighting just a few).However, despite the tremendous achievements made in the development of cosmologi-cal simulations, there remains a number of discrepancies between astronomical observations
Russell Johnston and what has been simulated within a Λ CDM framework. This has led some to question thevalidity of the current concordance model. Such discrepancies include the lack of directdetection of dark matter; the missing satellite problem, where observations of the local Uni-verse have found orders of magnitude less dwarf galaxies than predicted by simulations (e.g.Klypin et al. 1999; Moore et al. 1999; Willman et al. 2004; Kravtsov 2010); the core-cuspproblem, in which observations indicate a constant dark matter density in the inner partsof galaxies (de Blok 2010), whilst simulations prefer a power-law cusp as originally devel-oped by Navarro et al. (2004); and finally, the angular momentum problem, in which Λ CDMsimulations have had difficulties reproducing disk-dominated and bulgeless galaxies.Whilst the existence of dark matter remains an illusive quantity, possible solutions toaddressing the remaining issues may lie with not only in improving simulations with higherresolutions and more rigorous treatment of physical mechanism within galaxies (e.g. su-pernovae feedback), but also obtaining higher resolution observations from next genera-tion telescopes (see e.g. Gnedin & Zhao 2002; Governato et al. 2004; Robertson et al. 2004;Simon & Geha 2007; Governato et al. 2007; Simon & Geha 2007; McConnachie et al. 2009,for recent developments). Of course there exists the possibility that maybe as these improve-ments are made we may require an entirely different model beyond Λ CDM.2.2 The rise and rise of redshift surveys
The difference between redshifts obtained spectroscopically or photometrically essentiallyreduces to the precision of the methods. Spectroscopy is the most precise and, consequently,the most popular way to measure redshifts. The technique requires identification of spectrallines (typically emission lines) and record their wavelength λ . The relative shift of a theselines compared to their known position measured in the laboratory ( λ o ) allows us to measurethe redshift.The acquisition of redshift data in this way can be described as a two-stage process.One firstly images galaxies in a region of space using low resolution photometry and thentargets these galaxies to perform high resolution spectroscopy requiring long integrationtimes to achieve sufficiently high signal-to-noise. Redshift surveys such as the Two DegreeField Galaxy Redshift Survey (2dFGRS) and the Sloan Digital Sky Survey (SDSS) usedmulti-fibre spectrographs which could, respectively, record up to 400 and 600 redshifts si-multaneously.Alternatively, the use of photometry, offers a much quicker way to estimate redshifts to amuch greater depth and has thus gained in popularity in recent times. However, the precisionto which they are currently measured remains poor, with a typical uncertainty in the range . . δz . . . Moreover, effects from e.g. absorption due to galactic extinction and theLyman-alpha forest can contribute to systematics (see e.g. Massarotti et al. 2001). The tech-niques of photometric estimation of redshifts can be traced back to Baum (1962) who firstdeveloped a method using multi-band photometry in 9 filters for elliptical galaxies. Furtherwork by Loh & Spillar (1986) and Connolly et al. (1995) saw this area develop into tworespective techniques: the template fitting methods and empirical fitting methods . Templatefitting requires a library of theoretical or empirical SEDs to be generated coupled with spec-troscopic redshifts (for calibration purposes) which are then fitted to the observed colours ofthe galaxies, where redshift is a fitted parameter (see e.g. Bolzonella et al. 2000). Empirical hedding Light on the Galaxy Luminosity Function 9 fitting methods, whilst similar, instead use empirical relations with a training set of galax-ies with spectroscopically obtained redshifts (see also e.g. Brunner et al. 1997; Wang et al.1998). In the case of Connolly et al. (1995), for example, they used linear regression wherethe redshift was assumed to be a linear or quadratic function of the magnitudes.Both methods have been improved upon by the incorporation of, for example, Bayesianinference (Kodama et al. 1999; Ben´ıtez 2000; Stabenau et al. 2008; Wolf 2009), nearestneighbour weighting schemes (e.g. Ball et al. 2008; Lima et al. 2008), boosted decision trees(Gerdes et al. 2010), and artificial neural networks (Firth et al. 2003; Vanzella et al. 2004;Collister et al. 2007; Oyaizu et al. 2008; Y`eche et al. 2010). Alternatively, others have at-tempted more generalised approaches that incorporate aspects of both the traditional meth-ods (e.g. Budav´ari 2009).Improvements on photometry have also assisted the case for photo-z. For example, theCOMBO-17 photometric redshift survey produced multi-colour data in a total of 17 opticalfilters - five broad-band filters ( UBVRI ) and 12 medium-band covering a wavelength rangeof 400 to 930 nm (see e.g. Wolf et al. 2004). Having this many filters allowed significantincreases in resolution improving galaxy redshift accuracy to δz ∼ . (Wolf et al. 2003;Bonfield et al. 2010). Galaxy redshift surveys have played, and will continue to play a vital role in our under-standing of the formation, evolution and distribution of galaxies in the Universe. Prior tothe 1970’s, models of the structure of the Universe were based on the observed distributionof galaxies projected onto the plane of the sky. Whilst early pioneers had already identifiedthe clustering nature of galaxies from 2-D samples (e.g. Hubble 1936b; Charlier, C. V. L.1922), it would take the move to three-dimensional data-sets before the wider astronomycommunity would accept these claims. As a consequence, this required the measurementof redshifts on a much grander scale. To make a large enough survey where redshifts ofthousands of galaxies could be measured would require a lot of dedicated telescope timeand funding. Nevertheless, it was in 1977 that these investments were made and dedicatedredshift surveys began.The first major breakthroughs in mapping large-scale structure began with the CfA sur-vey which ran from 1977 to 1982 (Huchra et al. 1983) and measured spectroscopic redshiftsfor a total of 2401 galaxies out to a limiting apparent magnitude of m lim ≤ . mag. Thissurvey represented the first large area maps of large-scale structure in the nearby universeand confirmed the 3-D clustering properties of galaxies already proposed a little over 50years previously. CfA2 (Geller & Huchra 1989) continued the survey measuring a total of18,000 redshifts out to 15,000 kms − and m lim ≤ . mag (see Figure 1). With survey dataprovided by the Southern Sky Redshift Survey (SSRS/SSRS2) da Costa et al. (1998, 1994),CfA was extended to include the southern hemisphere. Despite this tremendous achieve-ment, spectrographic technological constraints allowed only one galaxy at a time to be ob-served, making the whole process extremely time consuming. However, the technology de-velopments during the 1980s provided the first multi-object fibre spectrographs that allowedbetween 20 and 200 galaxies to be observed simultaneously during one exposure.A new era of space-based telescopes began with the Infrared Astronomical Satellite(IRAS), launched 1983. The IRAS Point Source Catalogue (PSCz) redshift survey ran from1992 to 1995 and mapping 15,411 galaxies over 84% of the sky out to 0.6 Jy in the far-IR (60 µ m) (Saunders et al. 2000) marking the first all-sky survey (see Figure 2 left). The HubbleSpace Telescope (HST) was launched in 1990 and the Hubble Deep Field-North (HDF-N) Fig. 1
Extract strips from the CfA redshift surveys. The strip on the sky was 6 degrees wide and 130degrees long with our origin begin at the apex of the wedge. Image courtesy of the Smithsonian AstrophysicalObservatory Emilio Falco et al. (1999). survey in 1995 (Williams et al. 1996) was the next landmark, allowing unprecedented detailof faint galaxy populations to a magnitude of m V = 30 mag out to high redshift. In 1998 thefollow-up survey, HDF-South, sampled a random field in the southern hemisphere sky withequal success (see e.g. Fern´andez-Soto et al. 1999). The more recent Hubble Ultra DeepField (HUDF Beckwith et al. 2006) ran from 2003 to 2004 and has allowed evolutionaryLF constraints on the very faintest galaxies to redshifts reaching the end of the epoch ofre-ionisation, z photo ∼ (e.g. Ferguson et al. 2000; Kneib et al. 2004; Bouwens et al. 2009;Zheng et al. 2009; Finkelstein et al. 2010; McLure et al. 2010; Oesch et al. 2010). hedding Light on the Galaxy Luminosity Function 11 Fig. 2
Throughout the 1990s there were a number of surveys that paved the way to con-strain the local LF out to z ∼ . such as the Stromlo-APM Redshift Survey (S-APM)(Loveday et al. 1992), Las Campanas (LCRS) (Lin et al. 1996) and the ESO Slice Project(ESP) (Zucca et al. 1997). However, the next milestone from ground-based telescopes wasin the form of the Two Degree Field Galaxy Redshift Survey (2dFGRS) (Colless 1998).This survey ran from 1998 to 2003 and used the multi-fibre spectrograph on the Anglo-Australian Telescope (AAT), which could measure up to 400 galaxy redshifts simultane-ously. The photometry was taken from the Automatic Plate Measuring (APM) scans of theUK Schmidt Telescope (UKST) plates with measured magnitudes out to m lim b J = 19 . mag.The 2dFGRS team recovered a total of 245,591 redshifts, 220,000 of which were galaxies(see Figure 2 middle). With this increase in instrumentation precision and the vast number ofobjects catalogued, the scientific goals became equally ambitious. Some of 2dFGRS goalsincluded measuring the power spectrum of the galaxy distribution on scales up to few hun-dred M pc − , determining the galaxy LF, clustering amplitude and the mean star-formationrate out to a redshift z ∼ . .At around the same time as the 2dFGRS another team was carrying out a survey calledthe Two Micron All Sky Survey (2MASS) (Skrutskie et al. 2006). This saw the return of anear-infrared full sky survey and was the first all-sky photometric survey of galaxies brighterthan m K = 13 . mag cataloguing approximately 100,000 galaxies.However, it is the Sloan Digital Sky Survey (SDSS) that takes the prize as the most ambi-tious survey to date by mapping a quarter of the entire sky to a median spectroscopic redshift z spec ∼ . using multi-band photometry with unprecedented accuracy (e.g. Abazajian et al.2003; Adelman-McCarthy et al. 2006; Abazajian et al. 2009). Using the dedicated, 2.5-metertelescope on Apache Point, New Mexico, USA and multi-fibre spectrographs, SDSS has im-aged over 200 million galaxies and obtained just over 1 million spectroscopic redshifts. Thecombination of 2dFGRS and SDSS has provided the most accurate maps of the nearby Uni-verse placing strong constraints on the LF out to z ∼ . (Norberg et al. 2002; Blanton et al.2003b; Montero-Dorta & Prada 2009).Exploring galaxy evolution out to z ∼ . and beyond has been and continues to be ex-plored with surveys such as the Canada-France Redshift Survey (CFRS) (Lilly et al. 1995),Autofib I & II (Ellis et al. 1996; Heyl et al. 1997), the Canadian Network for Observa-tional Cosmology survey (CNOC1 & 2) (Lin et al. 1997, 1999), COMBO-17 (Wolf et al.2003),VIMOS-VLT Deep Survey (VVDS) (e.g. Ilbert et al. 2005), the Deep Extragalactic Evolutionary Probe 2 (DEEP2) (e.g. Willmer et al. 2006) and the more recent zCOSMOS(Lilly 2007; Zucca et al. 2009). All have been instrumental in constraining the statistical na-ture of the evolutionary processes of early-type to late-type galaxies at intermediate redshiftsthrough the study of LF as a function of color (e.g. Bell et al. 2004).There appears to be no sign of a slowing down of redshift surveys. In fact, future projectssuch as the Dark Energy Survey and those involving the Large Synoptic Survey Telescope(LSST), the Square Kilometre Array demonstrator telescopes, ASKAP and MeerKAT, andthe James Web Space Telescope (JWST) will provide the next generation of surveys withdata-sets orders of magnitude larger, with greater wavelength coverage, and probing bothfainter and more distant galaxies.
This section begins with a brief overview of the K -correction and the methods adoptedto constrain source evolution for a given population of galaxies. The final part discussesthe two important fundamental assumptions common to all the traditional LF estimators,namely, completeness of observed data and separability between the luminosity and densityfunctions.3.1 The K -correctionThe modelling of galaxy evolution and K -correction is a vital part of most galaxy surveyanalysis, which, if not accounted for properly, can adversely affect accurate determinationof the LF. The use of K -correction can be traced backed to early 20th century pioneeringobservers such as Hubble (1936a) and Humason et al. (1956). The observed wavelengthfrom a galaxy is different from the one that was emitted due to cosmological redshift, z .The K -correction allows us to transform from the observed wavelength, λ o when measuredthrough a particular filter (or bandpass) at z , into the emitted wavelength, λ e in the rest frameat z = 0 . Following the derivation from Hogg et al. (2002) (see also e.g. Oke & Sandage1968) we consider an object with an apparent magnitude m R that has been observed in the R bandpass. However, it is desirable to obtain the absolute magnitude M Q in the rest-framebandpass Q . Therefore, we firstly consider the relation between the corresponding emittedfrequency, ν e and the observed frequency, ν o , given by ν e = (1 + z ) ν o , (17)where z is redshift at which the source object was observed. Hogg et al. then go onto showthat the corresponding K -correction can be determined by K = − . " [1 + z ] R dν o ν o f ν ( ν o ) S R ( ν o ) R dν e ν e g Qν ( ν e ) S Q ( ν e ) R dν o ν o g Rν ( ν o ) S R ( ν o ) R dν e ν e f ν (cid:0) ν e z (cid:1) S Q ( ν e ) , (18)where S is often referred to as the transmission (or sensitivity) function of the detector forwhich at each frequency ν is the mean contribution of a photon of frequency ν to the outputsignal from the detector in the respective bandpass. The quantity g is the spectral density offlux for a zero-magnitude or standard source. Thus determining the K -corrected absolutemagnitude in the rest frame of the object is simply given by M Q = m R − d L ) − K, (19)where d L is the cosmology dependent luminosity distance already defined in Equation 11. hedding Light on the Galaxy Luminosity Function 13 Fig. 3
The characteristic shape of the Schechter luminosity function. The LF is typically presented in termsof absolute magnitudes M with the number density on the y-axis as the log ( Φ ) . For this example the LFhas been modelled using constrained LF parameter values based on the 2dFGRS results. In panel (a) thisimplies: α = − . , M ∗ = − . and φ ∗ = 1 . × − h Mpc − . The steepness of the faint-endslope is determined by the α parameter and the characteristic magnitude, M ∗ , indicates the ‘knee’ of the LF.Panel (b) shows the effect of introducing a pure luminosity evolution (PLE) model. In terms of magnitudesthis model is shown as E ( z ) and is dependent on the evolutionary parameter β . The red line shows the LFfrom panel (a) and the dotted lines (from left to right on the plot) represents the same LF for a range of β =1.5, 1.0, 0.5, -0.5, -1.0 and -1.5, at a redshift of z = 0 . . In panel (c) a pure density evolution (PDE) modelis alternatively introduced in to the LF. As with panel (b), the dotted lines (from top to bottom on the plot)represent LFs for the range of γ = 6.0, 4.0, 2.0 and -2.0, -4.0 and -6.0 at the same redshift. L ∗ ( z ) = L ∗ (0)(1 + z ) β (20)where β is the evolution parameter and although it is galaxy-type dependent, a global cor-rection is often adopted. Applying this correction to magnitudes, the evolution correction E ( z ) is the written as M ∗ = M ∗ (0) − E ( z ) (21) E ( z ) = − β . (1 + z ) (22)For the study of quasi-stellar objects (QSOs) in the 2df QSO redshift survey, Croom et al.(2004) instead, characterised luminosity evolution as a second-order polynomial luminositygiven by L ∗ ( z ) = L ∗ (0)10 k z + k z (23) M ∗ ( z ) = M ∗ (0) − . k z + k z ) (24)where the model requires a two parameter fit given by k and k .Pure Density (Number) Evolution (PDE) assumes that galaxies were more numerous inthe past but have since merged. The treatment of this type of evolution can be modelled in asimilar way as PLE, φ ∗ ( z ) = φ ∗ ( z )(1 + z ) γ , (25)where the parameter, γ , is the number evolution parameter. By assuming either of the abovemodels (or a combination of both) one can apply this technique to observables and simul-taneously constrain both the LF and E ( z ) parameters via a maximum likelihood technique.Whilst this has proved to be a popular approach it should be noted that the wrong evolution-ary model may be adopted in the analysis yielding questionable results.The second alternative approach is termed as a free-form technique that, as the name sug-gests, requires no strict parametric assumption for the evolutionary model. An early methodby Robertson (1978, 1980) developed an iterative procedure for exploring evolution of theradio luminosity function (RLF) whereby E ( z ) was sampled over multiple log( z ) slices un-til convergence. Whilst this approach allowed a free-form analysis in redshift, assumptionsregarding luminosity evolution were still required and there was no built-in measure of therange of evolution allowed by the data (see e.g. Zawislak-Raczka & Kumor-Obryk 1986,1987, 1990, for applications and extensions). Peacock & Gull (1981) and Peacock (1985)developed a fully free-form methodology that instead assumes a series expansion for theevolutionary model and employs a χ minimisation technique to constrain its coefficients.A more recent development in this area by Dye & Eales (2010) extends the ideas of Peacockand Gull by introducing a reconstruction technique that discretises the functions of redshiftand luminosity and employs an adaptive gridding in ( L − z ) space to constrain evolution.Recent observational techniques (e.g. Blanton et al. 2003a; Bell et al. 2004) have iden-tified a bimodal distribution in color across a broad range of redshifts allowing clear segre-gation of early- (red) and late- (blue star forming) type galaxies. This has allowed deeperexploration into the evolutionary properties of these populations beyond the simple e.g PLEapproach (see § Φ ( M ) is plotted in log space. The inputparameters are, α = − . , M ∗ = − . and φ ∗ = 1 . × − h Mpc − . The panelindicates the various components that make up the shape of the LF i.e. the power-law slope atthe faint end the steepness of which is governed by α ; M ∗ , which characterises the turnoverfrom the bright-end to the faint end; and the exponential cut-off which constrains the brightend of the LF. Panel (b) then shows how pure luminosity evolution (PLE) affects the shapeof the LF by incorporating the PLE model from Equation 22 into the Schechter function fora range of β given by [1.5, 1.0, 0.5, -0.5, -1.0 , -1.5] at a fixed redshift of z = 0 . . These hedding Light on the Galaxy Luminosity Function 15 are indicated, respectively, from left to right on the plot by the dotted lines. The red solidline shows the Universal LF from panel (a). As one would expect, the LF is affected only inits position in magnitude and is independent of any shift in number density. Panel (c) nowshows the case where only pure density evolution (PDE) is present by adopting Equation 25for a range of γ given by [6.0, 4.0, 2.0 and -2.0, -4.0, -6.0] as, respectively, indicated bythe dotted lines from top to bottom in the plot. It is clear that the shape of the LF is nowdominated by shifts in the φ direction.3.3 Completeness, separability and the selection functionIn general, when compiling a magnitude-redshift catalogue, we would like to be able toquantify in some way, how close we are to having a representative sample of the underlyingdistribution of galaxies. However, there are a number of constraints preventing us from ob-serving all objects in the sky. This is termed a measure of completeness for a given survey.The ability to interpret and measure it accurately is not trivial. There are many diverse con-tributing sources of in -completeness that have to be corrected for and understood to constructaccurately the LF. Nevertheless, an assumption of LF estimation is one of completeness ofa volume- or flux-limited survey. For clarity, I now discuss two general interpretations ofcompleteness that commonly appear in the observational cosmology literature: This can be described by the process of combining photometric and spectroscopic galaxysurvey catalogues. In the most general sense, an observer will measure apparent magnitudes, m , of galaxies in a portion of the sky out to a faint limiting apparent magnitude, m flim imposed by the physical limitations of the telescope. CCD instrumentation can be affectedby pixel saturation due to very bright objects. This imposes a bright limit to the survey, m blim .At this stage the catalogue consists of measured magnitudes and sky positions only, withouttheir 3D spatial redshift distribution. Therefore, each galaxy is then targeted to obtain ameasure of its redshift. The most accurate approach is by multi-fibre spectroscopy, whereoptical fibres are positioned on a plate which has holes drilled at the positions of the sourcesmeasured from the photometry. However, a drawback of measuring redshifts in this wayarises from spatial limitations. For example, in a region that has a high density of objects, alot of galaxies may be missed since there is a physical spatial limit on how close the fibrescan be placed i.e. fibre collisions . Therefore, redshift completeness can be presented as thepercentage of successfully measured redshifts over a list of targeted galaxies within a survey.Whilst this is in itself an important contributing factor to the overall completeness picture,we can describe the overarching completeness in terms of magnitude completeness . At this stage we can now ask the question, how do we know our percentage of successfullymatched targets is complete up to (or within) the apparent magnitude limit(s), m flim (and m blim )? Some of the contributing factors that affect this specific type of completeness canbe summarised as: galaxies that are missed because they are located close to bright stars orlie close to the edge or defected part of the CCD image; the wrong K -corrections applied;the wrong evolutionary model adopted; galaxies with the same surface brightness that mayor may not be detected depending on their shape and overall extent i.e. a compact object is more likely to have enough pixels above the detection limit than a very diffuse galaxy of thesame brightness; and adverse effects from a varying magnitude limit over a photographicplate or CCD image.Cosmological surface brightness dimming of objects due to the expansion of the uni-verse has been examined by e.g. Lanzetta et al. (2002). By studying the distribution of star-formation rates as a function of redshift in the Hubble Deep Field they demonstrated thatsuch an effect would bias derived quantities such as the luminosity density.An effect explored by Ilbert et al. (2004) that can potentially bias the shape of the globalLF arises from a wide range of K -corrections being applied across different galaxy types.Such an effect results in a varying galaxy-type dependent absolute magnitude limit wherecertain galaxy populations will not be detectable out to the full extent of the magnitude limitof the survey. This form of incompleteness is particularly crucial toward the faint end of theLF and discussed in greater detail in § Malmquist Bias (see e.g. Hendry & Simmons 1990).As one images out to higher redshifts only intrinsically bright sources will be observed.Since bright objects at large distances are rare, one observes a decrease in the number densityof imaged objects as a function of redshift. A way to quantify this effect is to compute the selection function , the probability that a galaxy at a redshift z will be included in a givenmagnitude- (or flux) limited survey. Thus, in the simplest scenario, the selection function S ( z ) may be expressed as S ( z ) = R M lim ( z ) −∞ Φ ( M ) dM R ∞−∞ Φ ( M ) dM (26)where M lim is the absolute magnitude limit of the survey. In practical terms this obser-vational effect implies we are sampling less of the underlying distribution of galaxies atincreasing redshifts. To correct for this effect it is common to apply a weighting scheme byincorporating the inverse of the selection function in the LF estimation such that w ∝ S ( z ) . (27)More generally, both magnitude and redshift completeness definitions can now be groupedin terms of, the probability that a galaxy of apparent magnitude, m , is observable. Weighting galaxies via the selection function introduces a very large assumption that iscentral to all the traditional methods for constructing the LF - separability between theprobability densities φ ( M ) and ρ ( z ) . This directly implies that the absolute magnitudes M (or luminosities L ) are statistically independent from their spatial distribution and thus theLF has a Universal form. As such, the joint probability density of M and z , P ( M, z ) , can beexpressed in a separable form as the product of the two univariate distributions such that P ( M, z ) ∝ φ ( M ) ρ ( z ) (28)The assumption of separability between the probability densities, can thus be thought ofas an extension to the idea of magnitude completeness. In this scenario it is assumed thatthe absolute magnitudes, M , have also been corrected, where deemed necessary, for any hedding Light on the Galaxy Luminosity Function 17 luminosity and/or density evolution (Equations 20 and 25, respectively), galactic extinctionand K -correction. In terms of the LF, these corrections can also be thought of as additionalcontributing factors to completeness. Therefore, only when such corrections have been ac-curately made can the separability assumption be valid. This review begins with the maximum likelihood estimator (MLE). As a statistical tool, theMLE is by no means a recent development. Its origins can be traced as far back as e.g.Bernoulli (1769, 1778) through to R. A. Fisher who provided a more formal derivation ofthe MLE in his seminal papers Fisher (1912, 1922). For a comprehensive historical reviewsee e.g. Aldrich (1997) and Stigler (2008). In terms of its application within the context ofobservational cosmology it was Sandage, Tammann, & Yahil (1979), hereafter STY79, whowere the first to see it as powerful approach to galaxy LF estimation. This is a parametrictechnique which therefore assumes an analytical form for the LF and thus eliminates theneed for the binning of data (as usually required by most non-parametric methods). Theequally popular non-parametric counterpart of the MLE, the step-wise maximum likelihood(SWML), is discussed in § x , a continuous random variable, thatis described by a probability distribution function (PDF) given by f ( x ; θ ) , (29)where θ represent the parameter we wish to estimate. As we shall see, in practice θ representsmore than one parameter of the LF. If x represents our observed data then the likelihoodfunction, L , can be written as f ( x , x , ..., x N | θ , θ , ..., θ k ) = L = N Y i =1 f ( x i ; θ , θ , ..., θ k ) (30)where x i are N independent observations. It is often the case that the likelihood function isexpressed in terms of the logarithmic likelihood such that ln {L} = N X i =1 ln f ( x i ; θ ,θ , ..., θ k ) (31)Constraining the θ , θ , ..., θ k parameters is, in principle, a straightforward matter of max-imising the likelihood function L ( θ ) or ln {L ( θ ) } such that ∂ ( L or ln {L} ) ∂θ j = 0 , j = 1 , , ..., k (32)In the context of estimating the parameters of the LF we consider a galaxy at redshift z for which we can define the cumulative luminosity function (CLF) and thus determine theprobability that the galaxy will have an absolute magnitude brighter than M as p ( M | z ) = M R −∞ φ ( M ′ ) ρ ( z ) f ( M ′ ) dM ′∞ R −∞ φ ( M ′ ) ρ ( z ) f ( M ′ ) dM ′ , (33) where ρ ( z ) is the density function for the redshift distribution, f ( M ′ ) is the completenessfunction which for a 100 % complete survey would be f ( M ′ ) = , M brightlim ≤ M ′ ≤ M faintlim , otherwise . (34)It follows that the probability density for detected galaxies is given by the partial derivativeof P ( M, z ) with respect to M , p ( M i , z i ) = ∂p ( M, z ) ∂M = φ ( M i ) M bright( zi ) R M faint( zi ) φ ( M ′ ) dM ′ (35)Note that the density functions have cancelled thus rendering the technique insensitiveto density inhomogeneities. Finally, the likelihood is maximised to give L = N Y i =1 p ( M i , z i ) . (36)Most commonly a Schechter function is assumed where the parameters that we wish toestimate are α and M ∗ as defined in Equation 9 on page 4.4.1 Normalisation, goodness of fit & error estimatesAlthough the MLE method has become more popular than other traditional non-parametricmethods there are aspects not to be overlooked. This approach does not determine the nor-malisation parameter φ ∗ of the LF and consequently has to be estimated by independentmeans. The approach originally described by Davis & Huchra (1982) and later adoptedby e.g. Loveday et al. (1992); Lin et al. (1996); Willmer (1997); Springel & White (1998);Blanton et al. (2003c); Montero-Dorta & Prada (2009) incorporates a minimum variancedensity estimator to determine the mean density of objects. The method can be summarisedas follows. The normalisation can be cast in terms of ¯ n = P N gal j =1 w ( z j ) R dV S ( z ) w ( z ) , (37)where N gal is the number of galaxies in the sample, w ( z ) is a weighting function for eachgalaxy defined by the inverse of the second moment of the two-point correlation functiongiven by w ( z ) = 11 + ¯ nJ S ( z ) , (38)and S ( z ) is the selection function for the survey defined within a maximum and minimumredshift range, S ( z ) = R L max ( z ) L min ( z ) dL Φ ( L, z ) R L max L min dL Φ ( L, z ) , (39) hedding Light on the Galaxy Luminosity Function 19 where the quantity J is the integral of the correlation function given by J = Z ∞ dr r ξ ( r ) . (40)The normalisation is then calculated iteratively and the error can be computed by D δ ¯ n E / = (cid:20) ¯ n R dV φ ( z ) w ( z ) (cid:21) / . (41)Davis & Huchra (1982) and Willmer (1997) point out that whilst this method is robust itcan return a biased estimate if the survey sample has significant inhomogeneities. In a morerecent paper by Hill et al. (2010) it was commented that further bias may be introduced dueto incompleteness at higher redshifts resulting in over weighting of φ ∗ . For more explorationinto this and other normalisation methods, see Willmer (1997).In terms of the goodness-of-fit of the adopted parametric form of the LF, this too, ashighlighted by Springel & White (1998), is not built into the MLE and, therefore, has to beassessed independently. For survey samples that may not so obviously be described by aSchechter function, caution should be taken as this implies that nearly any functional formcould be made to fit a given data-set. Furthermore, the nature of the method effectively de-termines the slope of the LF at any point. One can, however, apply a simple χ minimisationtest to probe the goodness-of-fit. Aside from this, if the survey sample is not complete nearthe apparent magnitude limit sources close to the limit will be underestimated thus makingthe slope of the LF underestimated (Saunders et al. 1990).A standard approach for estimating the relative error on the LF parameters M ∗ and α wasadopted by Efstathiou et al. (1988). This involves jointly varying these parameters aroundthe maximum likelihood value to find where the likelihood increases by the β -point of the χ distribution; we have ln L = ln L max − χ β ( N ) (42)where N is the number of degrees of freedom corresponding to ∆χ = 2 . for σ limit( . confidence interval) and ∆χ =6.17 for σ limit ( confidence interval).4.2 Further extensionsThe STY79 method remains one of the most widely applied LF estimators to date and asa result has been modified over the years. For example, Marshall et al. (1983) (hereafter,M83) extended its use for quasars by simultaneously fitting evolution parameters with theluminosity function parameters. For this they test both pure density and pure luminositymodels. In their analysis the probability distribution in the likelihood for the observablesis described instead by Poisson probabilities. The luminosity and redshift space ( L − z )distribution is gridded such that the likelihood is defined as the product of the probabilitiesof observing either 1 or 0 quasars in each cell such that L = N Y i λ ( z i , L i ) dzdL e − λ ( z i ,L i ) dzdL N Y j e − λ ( z j ,L j ) dzdL , (43)where the quantity λ ( z i , L i ) dzdL represents the expected number of objects in each cellin the L − z plane. The j index takes into account cells where no objects were observed. This form of the likelihood has proved popular and has been widely applied. Chołoniewski(1986) adopted the method and applied it to the CfA survey data. More recent examples byBoyle et al. (2000) studied the quasar LF in the 2df-QSO survey and Wall et al. (2008) forexploring a sub-millimetre galaxy sample from the GOODS-N survey. As I will discuss inmore detail in §
7, Christlein et al. (2009) also drew on this approach when incorporatingphotometric redshift estimates into the MLE.Saunders et al. (1990) used the STY79 approach not only to constrain the LF but in-stead integrate over the comoving volume to determine the radial density field. In this wayno knowledge of the LF is required. They demonstrated that by parameterising the radialdensity function ρ ( | r − | ) they can fit it as a step function and obtain the variation on the MLEas L = N Y i =1 ρ ( z i ) R ρ ( z i )( dV /dz ) dz . (44)Heyl et al. (1997) generalised STY79 by constructing a statistical framework to explorehow the LF evolves with redshift. This generalisation also allowed for the combination ofmultiple samples (see also Avni & Bahcall (1980) extension of V /V max in § signal (the cluster LF), the other accounting for background galaxycounts from the observations of many individual events (the galaxies luminosities), withoutknowledge of which event is the signal and which is background. Given j data-sets of e.g.cluster 1, cluster 2... etc.. each comprising of N j galaxies, the likelihood is maximised suchthat ln L = X clusters, j X galaxies, i ln p i − s j , (45)where p i = p ( m i ) is the probability of the i th galaxy of the j th cluster to have an apparentmagnitude m i . The quantity s is the expected number of galaxies given the model, evaluatedby s = Z m flim ,j m blim ,j φ ( m ) dm, (46)where m blim and m flim are the respective bright and faint limiting magnitudes of the j th clus-ter data in this example. In their analysis the LF φ is modelled as a convolved power-law andSchechter function. The goodness-of-fit was determined adopting the χ test. In Andreon(2006) and Andreon et al. (2006) they extend this approach within a bayesian frameworkand apply a Markov Chain Monte Carlo algorithm to constrain the LF parameters (MCMCMetropolis et al. 1953). See also e.g. Andreon et al. (2008); Andreon (2010) for further ap-plications of the method.In § One of the main difficulties in constructing an accurate LF from flux-limited survey samplesis the issue of completeness. I have discussed some of the major problems with galaxy detec-tion. However, in general terms we are hindered observationally by the notorious
Malmquist hedding Light on the Galaxy Luminosity Function 21 bias effect. This effect means we are biased to observe intrinsically brighter objects at higherredshifts and observe only the fainter objects over smaller volumes nearby. In this section Iwill discuss all the various non-parametric weighting schemes that have been devised overthe years to correct for this bias.5.1 The classical approachThe classical method, as coined by Felten (1977), represents the first rudimentary binnednumber count approach to determining the LF and was initially developed and applied bye.g. Hubble (1936b), van den Bergh (1961), Kiang (1961), and Shapiro (1971). However,as pointed out in BST88 the method was not formally introduced until the publications ofChristensen (1975), Schechter (1976) and Felten (1977).The underlying assumption of the method is that the distribution of sources within thedata-set in question is spatially homogeneous i.e. with no strong large-scale clustering. Fromthis starting point we count the number of galaxies N within a volume V such that Φ ≡ NV (47)The volume, V ( M ) , is calculated for the maximum distance that each galaxy with an abso-lute magnitude, M i , could have and still remain in the sample. As an example, Felten (1977)applies the following expression (neglecting K -corrections) to calculate the volume, V ( M ) = 43 π exp[0 . m lim − M i − (48) × (cid:20) E (0 . α ln 10) − E (0 . α ln 10 csc b min )csc b min (cid:21) where m lim is the apparent magnitude limit of the survey, α and b min are related to thedirectional-dependent galactic absorption calculation, and E ( x ) is a second exponentialintegral (Abramowitz & Stegun 1964, Chap 5).The number of galaxies, N , within the absolute magnitude limits of the survey, M , isbinned into a histogram (see Felten 1977; BST88) with each bin divided by V ( M ) to convertthe histogram to units of mag − Mpc − and return a differential estimate of the LF, φ ( M ) .Whilst this method is relatively straightforward to apply, its basic assumption of homo-geneity is well understood to be a handicap. At the time when galaxy surveys were shallowit was common practice to exclude clustered regions such as the Virgo cluster and membersof the Local Group to try and avoid biasing in the shape and thus the parameters of the LF(Felten 1977). Also, Felten (1976) showed mathematically that the classical test gives a bi-ased estimate of the LF and provides expressions for the fractional bias and the variance of Φ .5.2 The /V max and V /V max estimatorsA natural development from the classical approach is now the famous
V /V max test first de-scribed by Kafka (1967); Rowan-Robinson (1968) but more formally detailed and applied inthe much celebrated paper by Schmidt (1968) to assess the uniformity and the cosmologicalevolution of quasars at high redshift (see also Schmidt 1972, 1976). As with the classical
Fig. 4
The construction of the traditional Schmidt (1968)
V/V max test. The basic construction of the methodconsiders the ratio between the volume V , in which a galaxy is observed to the volume V max , the maximumvolume the said galaxy could occupy and still be observed. method, V /V max assumes spatial homogeneity.
V /V max is essentially a completeness esti-mator and Fig. 4 illustrates its construction. The basic principle of the test is simple and isdefined by considering two volumes: – V , the volume of the sphere of radius R , where R is the distance at which a galaxy wasactually detected, compared to – V max , the maximum volume within which a galaxy could have been detected and stillremain in the catalogue in question. Thus, V max is the volume enclosed at maximumredshift, z max at which the galaxy in question could still have been observed.Assuming that the distribution of objects within the survey sample is homogeneous, then itfollows that the value V /V max is expected to be uniformly distributed in the interval [0,1].Thus, for a complete sample with no evolution
V /V max has expectation value (cid:28) VV max (cid:29) = 12 , (49) with an often quoted statistical uncertainty of / (12 N ) / , where N is the total number ofobjects in the sample (see e.g. Hudson & Lynden-Bell 1991). In reality, the value calculatedfrom V /V max for a survey will deviate from / . By how much the value deviates from / is usually considered to be either a signature of incompleteness (e.g. Malmquist bias) and/oran indication of evolution: a value that is greater than / would imply a density evolutionwhere galaxies were more numerous in the past, where as a value less than / would implythat galaxies were less numerous in the past.In the same paper, Schmidt also outlined a variation of this statistic that could be usedto estimate the LF under the condition where the maximum distance r max an object couldhave and still be included in the sample was independent of its direction, Φ = N X i =1 V max ,i . (50) hedding Light on the Galaxy Luminosity Function 23 Once again it was Felten (1976) who would dub this estimator as the ‘Schmidt’s estimator’.The comoving volume can be determined by evaluating, V max = cH Z Ω Z z max z min d L (1 + z ) A ( z ) dz dΩ, (51)where Ω is the solid angle of the survey, d L is the luminosity distance as given by Equa-tion 11, and z min and z max is redshift range of the sample. The quantity A ( z ) is related tothe transverse comoving distance and defined as A ( z ) ≡ p Ω m (1 + z ) + Ω k (1 + z ) + Ω Λ , (52)where Ω m , Ω k and Ω Λ are, respectively, the matter density, curvature and cosmologicalenergy density constants.If one assumes Poisson fluctuations and homogeneity within each bin then a standardapproach (see e.g. Condon 1989) to error estimation is simply computing the rms on eachbin, σ i = " N X i =1 V max ,i / . (53)As we shall see in the following section, if these assumptions breakdown Eales (1993) pro-vided an more rigorous modified approach to error estimation. V max estimator Although strictly speaking
V /V max is a completeness estimator I have included its develop-ment in this section as it is so intimately linked with the /V max estimator for constructingthe LF.Since its inception, /V max has remained a popular estimator for determining lumi-nosity functions and as a probe of evolution, most likely due to its simplicity and ease ofimplementation. However, as we will see in § V max estimators haveevolved, been improved and refined over the years to accommodate the many different typesof survey that have steadily grown in size and complexity. Selected below is a summary ofsome of the most significant developments of the V max method.Huchra & Sargent (1973) were the first to extend its use to galaxies from the Markarianlists I to IV (see Markarian 1967, 1969a,b; Markarian, Lipovetskij, & Lipovetsky 1971) andperform V /V max as a completeness test whilst including the Virgo Cluster and the LocalGroup. They showed that the effects of including such clusters had a minimal impact ontheir results. Furthermore, they calculated the space density Φ ( M ) via Schmidt’s /V max estimator, where they summed over all galaxies within absolute magnitude intervals.Felten (1976) made an extensive comparison of /V max with the classical test. This paperderives a generalised version of /V max between absolute magnitude ranges M < M Avni & Bahcall (1980): The top panel shows the construction of the generalised V/V max for two in-coherent overlapping samples (B and D) into two region independent samples, (B-C) and D (see Equation 55).The middle panel shows the construction of V e /V a for a coherent sample constructed from two overlappingsamples (B and D). The shaded region in the bottom panel represents V e for the case z ≤ z Bmax ( F ) inEquation 57.hedding Light on the Galaxy Luminosity Function 25 and shows that it is superior to that of the classical estimator by being an unbiased estima-tor.Avni & Bahcall (1980) generalised V /V max for multiple samples for two distinct cases:1. Firstly, for combining independent multiple samples that are still physically separated.2. Secondly, for combining independent samples in which the individual samples are over-lapping.In the first scenario they consider complete ‘incoherent’ samples which do not overlap onthe plane of the sky that could either be initially non-overlapping, or could be constructedfrom overlapping samples as illustrated on the top panel in Fig. 5. The term ‘incoherent’refers to combining samples in which one remembers for each object its parent sample.In this particular case the V /V max statistic can be constructed from overlapping samplesdividing the space into two distinct volumes (B-C) and D. For this method they show that acombined sample average of V ′ /V ′ max is given by (cid:28) V ′ V ′ max (cid:29) B − C , D = N B − C N B − C + N D (cid:28) V ′ V ′ max (cid:29) B − C + N D N B − C + N D (cid:28) V ′ V ′ max (cid:29) D (55)where V ′ represents the density-weighted volume, N B − C is the number of objects in sample(B-C) and N D is the number of objects in sample D.The second scenario considers the simultaneous analysis of independent complete ‘co-herent’ samples in which the individual samples are physically joined and a new statistic, V e /V a , is constructed (see illustration in the middle panel of Fig. 5). By this description,‘coherent’ refers to the method of combining independent samples. Here, a new variable V ′ a is defined as the density-weighted volume available to an object for being included in thecoherent sample. This new volume is defined as V ′ a [ F i ] = Ω B − C π V ′ [ z Bmax ( F i )] + Ω D π V ′ [ z Dmax ( F i )] (56)where Ω B − C and Ω D are the solid angles subtended on the sky and F i is flux of the object.The second new variable V ′ e is defined as the density-weighted volume enclosed by an objectin the coherent sample and is given by V ′ e [ z i , F i ] = Ω B − C π V ′ ( z i ) + Ω D π V ′ ( z i ) , z i ≤ z Bmax ( F i ) Ω B − C π V ′ [ z Bmax ( F i )] + Ω D π V ′ ( z i ) , z i > z Bmax ( F i ) (57)This first case in Equation 57 is illustrated in the bottom panel of Fig. 5. This leads to thesample average of V ′ e /V ′ a being defined as (cid:28) V ′ e V ′ a (cid:29) = 1 N T X i (cid:26) V ′ e [ z i , F i ] V ′ a [ F i ] (cid:27) (58)where N T is the total number of objects in the two combined samples.Hudson & Lynden-Bell (1991) recast V /V max for analysis of the diameter function of galax-ies. Therefore, for diameter-limited catalogues which have both a maximum and minimumdiameter cut-off they show that the completeness test can be written as VV max = θ − − θ − θ − − θ − , (59) where θ is the major diameter of a given galaxy, θ lim is the lower diameter limit of the sur-vey and θ max is the maximum diameter cut-off of the survey.Eales (1993) provided a more rigorous and generalised treatment for estimating errors forsurveys sampling smaller volumes than the quasar samples examined in Schmidt’s originalwork. Consequently, the error estimation laid out by Eales takes into account effects fromstrong clustering of galaxies and shot noise. The key to this approach draws from Peebles(1980) by incorporating the variance in the number of galaxies N within successive i slicesof redshift and absolute magnitude of a parent sample Var( N i ) = D ( N i − h N i i ) E = Z ϕ ( r ) dV + Z Z ϕ ( r ) ϕ ( r ) ξ ( | r − r | ) dV dV , (60)where r is a position vector and the integrals are over the comoving volume subtended bythe solid angle of the sample bounded by the redshift limits. The 2-point correlation functionis given by ξ ( r ) , which accounts for the error due to clustering and the selection function,and ϕ ( r ) , taking into account the contribution of shot noise. Eales goes on to show that thetotal error in the LF is then estimated by σ = φ var( N ) / N ! (61)In the same paper the generalised /V max estimator introduced by Felten (1976) is extendedto examine the evolution of the LF as a function of redshift. Similarly, van Waerbeke et al.(1996) looked specifically at the effects of pure luminosity evolutionary models on QSOsvia the V max estimator to constrain cosmological parameters.Qin & Xie (1997) generalised the now familiar Schmidt notation in terms of a new statisticcalled n/n max that is applicable to any kind of distribution of objects. This, therefore, wouldbe an improved measure of the traditional V /V max test where the estimator is weighted dif-ferently and the distribution in question is assumed to homogeneous. This fitting techniquedemonstrated that if the adopted LF is correct then the distribution of n/n max is uniform onthe interval [0,1] , n ( M, z ) n max [ M, z max ( M )] = z R Φ ( M, z ) dV ( z ) z max ( M ) R Φ ( M, z ) dV ( z ) , (62)and the authors showed that its expectation value h n/n max i is / .Following from this, another statistic, o/o max , based on the cumulative LF and indepen-dent from n/n max was introduced by Qin & Xie (1999): o ( M, z ) o max ( z ) = M R M min Φ ( M, z ) dM M max ( z ) R M min Φ ( M, z ) dM . (63)This statistic combined with that of Qin & Xie (1997) are designed to provide a sufficienttest for any adopted LF form. In the latter paper they apply both estimators to AAT sample hedding Light on the Galaxy Luminosity Function 27 data from the UVX survey (Boyle et al. 1990).Page & Carrera (2000) improved the method to take into account systematic errors intro-duced for objects close to the flux limit of a survey. As they point out, for evolutionary stud-ies of galaxies the traditional approach, as extended by Avni & Bahcall (1980) and Eales(1993), is very common (see e.g. Maccacaro et al. 1991; Ellis et al. 1996) but can distort theapparent evolution of extragalactic populations. Through the use of Monte Carlo simula-tions, with a sample of 10,000 objects and simulating an unevolving two-power law modelX-ray LF, they compare the /V max estimation of the differential LF given by φ /V max ( L, z ) = 1 ∆L N X i =1 V max ,i , (64)to their improved binned approximation of the φ est , which assumes that φ does not changesignificantly over the luminosity and redshift intervals ∆L and ∆z , respectively, and is de-fined as φ est = N L max R L min z max ( L ) R z min ( dV ) / ( dz ) dzdL, (65)where N is the number of objects within some volume-luminosity region.Miyaji, Hasinger, & Schmidt (2001) extend the Page & Carrera (2000) method for the studyof active galactic nuclei (AGN). To help reduce biases from binning effects they employ aparametric model to correct for the variation of the LF within each bin. With this weightingscheme, the binned LF is calculated by Φ ( L i , z i ) = Φ ( L i , z i ) model N obs i N model i , (66)where L i and z i represent the luminosity and redshift at the centre of the i th bin. Φ ( L i , z i ) m is the best-fit model evaluated at L i and z i . N obs i is the number of observed objects in eachbin and N m i is the number of objects estimated from the best-fit model. The method stillassumes homogeneity within each bin and obviously requires that the model accurately de-scribes the data. However, it is remarked that Poisson statistics can be used to compute theerrors. This approach has been more recently applied by e.g. Croom et al. (2009b) for theirstudies of quasi-stellar objects (QSOs).Chapman et al. (2003) uses /V max in order to construct the bivariate luminosity function(BLF) Φ ( L, C ) , in luminosity L and colour C . This method is discussed in greater detail in § dN/dz photo distribution into V max by adopting a deconvolution technique. This method willbe detailed in § V /V max estimator to probeevolution of the Sloan Digital Sky Survey (SDSS) DR7 Luminous Red Galaxies (LRGs)(see Eisenstein et al. 2001, for main sample selection). As pointed out by the authors, recentstudies of evolution in LRG samples employs a procedure that constructs pairs of samples Fig. 6 Image courtesy of Tojeiro & Percival (2010). In the above schematic three populations (i.e galaxieswith similar colours and luminosities) are considered. For V A in the population 1 scenario galaxies are givena weight equal to unity. However, a galaxy in V B in this population is down-weighted. If a galaxy is inthe population 2 sample and therefore confined to one slice only it would be given a weight of zero. Thepopulation 3 galaxies are those that can been see throughout the entire survey sample. This represents avolume-limited sample where the corresponding weight is always unity. - one being at high redshift and the other at low redshift. However, matching the individualgalaxy properties between the two samples traditionally requires the removal of galaxiesthat could not have been observed due to a varying selection function (see e.g. Wake et al.2006). To overcome this, a weighting scheme is constructed to the two redshift slices: V A (for high redshift) and V B (for low redshift). The weighting scheme down-weights galaxiesto keep the total weight of each galaxy population the same in the different redshift slices.Therefore, each galaxy in V A or V B is, respectively, weighted by w i = V A V A max ,i min ( V A max ,i V A , V B max ,i V B ) or, V B V B max ,i min ( V A max ,i V A , V B max ,i V B ) . (67)This is illustrated in Fig. 6. However, as the authors carefully note, this approach providesa weighting scheme only and is not a completeness estimator unlike the traditional Schmidttest. As such, incompleteness may still be inherent in the parent sample that is under test.Moreover, for a survey constructed by a magnitude-limited sample, the Schmidt estimatorwould be applied instead.Lastly, there has been a more recent paper by Cole (2011) in which he generalises /V max with a density corrected V dc , max estimator that takes into account effects from density fluc- hedding Light on the Galaxy Luminosity Function 29 tuations within a given volume. The development of such a method was used to providean algorithm which generates magnitude-limited random (unclustered) galaxy catalogues,which take into account both the correlation function and galaxy evolution.5.3 The C − methodAs already discussed, the drawback in the use of the /V max is the assumption that the dis-tribution of objects is spatially uniform. The increase in the number and variety of redshiftsurveys over the years has confirmed that galaxies have strong clustering properties. Natu-rally, this can introduce a bias in constructing the differential LF. However, it was not longbefore alternative approaches were developed that could circumvent this problem.The C − method was introduced by Lynden-Bell (1971), where he applied it to the quasardata of Schmidt (1968). It is a maximum likelihood procedure adapted from the survivalfunction and does not require any binning of data. Instead, the C − method estimates thecumulative luminosity function (CLF). It has the advantage over the classical and /V max methods as it does not require any assumptions about the distribution of objects within thedata-set. Furthermore, as remarked by Petrosian (1992), all non-parametric methods areessentially variations on the C − method in the limiting case of having one galaxy per bin.Its mathematical properties have also been examined rigorously in the statistical literature(see Woodroofe 1985; Chen et al. 1995).In practice the method recovers the CLF without normalisation with the use of a weightedsum of Dirac δ -functions (thus assuming no errors in magnitude). As we shall see, to accountfor errors in magnitude one can add a smoothing kernel into the procedure. To understandhow the method works let us firstly consider the following. In an ideal scenario where oneis not restricted by observational constraints such as faint and bright apparent magnitudelimits, constructing the cumulative luminosity function (CLF), Φ ( M ) would be a relativelytrivial task. However, these limitations, in reality, lead us instead to observe a sub-populationof galaxies sampling a CLF which we can refer to as X ( M ) (Subbarao et al. 1996; Willmer1997). Thus we find that dΦΦ > dXX (68)Let us now consider this observed distribution of galaxies in terms of the absolute mag-nitude, M , and distance modulus, Z , plane M − Z , and assume (as with all the methodsdescribed so far) separability between M and Z (see Fig. 7 ignoring the red coloured mark-ings for the moment). From this figure we can see the sharp faint apparent magnitude limit m flim of the survey indicated by the diagonal line. We can now write the probability densityfunction of M and Z for the observable population as dP = ρ ( Z ) dZ φ ( M ) dM Θ ( m flim − m ) (69)where, Θ ( m flim − m ) is a Heaviside function describing the magnitude cut, m flim defined as Θ ( x ) = (cid:26) x ≥ , x < , (70)However, to determine Φ ( M ) given the observed X ( M ) we can construct a subset re-gion of X we call C − ( M ) where we are now able to obtain dΦΦ = dXC − (71) Fig. 7 The construction of the C − method introduced by Lynden-Bell (1971). C k defines a region C − foreach galaxy ( M k , Z k ) in the sample. All the galaxies in this region are counted excepted for ( M k , Z k ). M min is imposed by the brightest galaxy in the survey data. Similarly, Z min is the minimum distance modulusdefined by the nearest galaxy at redshift z min . In this scenario, the integrated CLF can be written in form, Φ ( M ) = A exp M Z −∞ dX ( M ) C − ( M ) , (72)where, A is a normalisation factor and X ( M ) represents the parent set of points withinwhich one constructs the C − ( M ) subset. However, we require the differential luminosityand density distribution functions which can be represented by a series of Dirac δ functions,respectively, given as φ ( M ) = N X i =1 ψ i δ ( M − M i ) , (73) ρ ( Z ) = N X i =1 ρ i δ ( Z − Z i ) , (74) hedding Light on the Galaxy Luminosity Function 31 where ψ i and ρ i are the respective step sizes. The distance modulus, Z , is calculated by Z = m − M = 5 log ( d L ) + 25 (75)where d L is the luminosity distance to the object and m is the apparent magnitude. To thenconstruct the CLF the ( M k , Z k ) data are firstly sorted from the brightest to the faintestgalaxy such that M k +1 ≥ M k for k = 1 , N , and a region on the plane for each galaxylocated at M k defines the C − ( M k ) function such that C k ≡ C − ( M k ) , k = 1 , ..., N M min ≤ M < M ′ ,Z min ≤ Z ≤ Z ′ (76)as illustrated in Fig. 7. According to Jackson (1974), the superscript ‘minus’ is to emphasisethat the point at ( M k , Z k ) is not included when evaluating C − ( M k ) . The coefficients of theLF are determined from the recursion relation, ψ k +1 = ψ k C − k ( M ) + 1 C − k +1 ( M ) , (77)Therefore, the CLF is given by, Φ ( M k ) = M Z M min φ ( M ) dM = ψ M k 74 which can otherwise limit the use of C − towards the faint magnitudelimit of a survey. Subbarao et al. (1996) extended the method for photometric redshifts by considering, foreach galaxy, the probability distribution in absolute magnitude M resultant from the photo-metric redshift error. By adopting a Gaussian error distribution for the function z ∗ ( m i , M ) ,the redshift for the i th galaxy with an apparent magnitude m i , they showed that for a com-plete magnitude-limited sample the defined region C − ( M ) is now given as C − ( M ) = 0 . X i (cid:20) erfc (cid:18) z ∗ ( m i , M ) − z i σ i (cid:19) − erfc (cid:18) z ∗ ( m lim , M ) − z i σ i (cid:19)(cid:21) , (80)where erfc( x ) is the complementary error function.Willmer (1997) included Lynden-Bell’s approach applied to simulated data and theCfA - 1 survey (Huchra et al. 1983) when comparing various LF estimators, and is discussedin more detail in § C + estimator, which they then incorporated into their “Algorithm for Luminosity Function”(ALF) tool (see e.g Ilbert et al. 2004, 2005; Zucca et al. 2009). Essentially Equation 77 isredefined such that for each k th galaxy, the contribution to the LF is given by ψ ( M k ) = 1 − P k − j =1 ψ ( M j ) C + ( M k ) (81)where, galaxies for the ψ ( M j ) summation are sorted from the faintest to the brightest and C + ( M k ) is the number of galaxies in the region, C + ( M k ) , k = 1 , ..., N M < M k ,Z min ≤ Z ≤ Z max ,k ( M k ) (82)Thus to obtain a binned version of the LF, the quantity ψ ( M k ) is estimated for each galaxyand the returned values binned in absolute magnitude.5.4 The φ/Φ methodOriginally introduced by Turner (1979) and Kirshner et al. (1979) the φ/Φ method (as coinedby BST88) is a natural progression from the classical method ( § dM , N ( dM ) to the total number of galaxies brighter than M , N ( ≤ M ) within the maximum volume assuming a complete sample: N ( dM ) N ( ≤ M ) = dN ( ≤ M ) N ( ≤ M ) = φ ( M ) ρ ( z ) dM dV M R −∞ φ ( M ′ ) ρ ( z ) dM ′ dV = φ ( M ) dMΦ ( M ) ≈ d ln Φ ( M ) , (83)where ρ ( z ) is the density function and Φ ( M ) is the integrated LF. It is clear from this equa-tion that the density functions cancel, thus rendering the estimator independent of the distri-bution of galaxies. This estimator has been further developed slightly - Davis et al. (1980),Davis & Huchra (1982) to bin the data in equal distance intervals. However, as shown in hedding Light on the Galaxy Luminosity Function 33 de Lapparent et al. (1989) the approximation in Equation 83 introduces a bias in the determi-nation of the LF for large dM . To avoid this bias it has been common place to instead assumean analytical form for the LF as in Turner (1979), Kirshner et al. (1979), Davis & Huchra(1982) and de Lapparent et al. (1989). So although by its original construction this estima-tor is non-parametric, subsequent applications have found its place to be more useful as aparametric one.5.5 The step-wise maximum likelihood methodOne of the first embodiments of the step-wise MLE approach, was presented by Nicoll & Segal(1983) and can be thought of as a binned version of the Lynden-Bell’s C − method. This wasapplied in their analysis of chronometrical cosmology and also considers variations of pro-gressive truncation in apparent magnitude as well as multivariate complete samples.A more advanced version of Nicoll and Segal’s approach by Chołoniewski (1986) ap-plied the same Poisson probability distribution in the MLE of Marshall et al. (1983) (seesection 4). In this paper, the data are projected on the absolute magnitude M and distancemodulus Z plane and divided into equal sized cells such that M ∈ [ M i − , M i ] , i = 1 , ..., A, (84) Z ∈ [ Z j − , Z i ] , j = 1 , ..., B, The likelihood function can be represented as L = A Y i =1 B Y j =1 e − λ ij λ N ij ij N ij ! , (85)where λ ij = 1¯ n Φ i ( M ) ρ j ( Z ) dM dZ, (86)where N ij is the number of galaxies in the ( i, j ) cell, ¯ n is the mean density of the sample, Φ ( M ) is LF, and ρ ( Z ) is the density function where the separability between Φ ( M ) and ρ ( Z ) is assumed.The method which is would become known as the step-wise maximum likelihood method(SWML) was introduced by Efstathiou, Ellis, & Peterson (1988) (hereafter EEP88) and rep-resents the non-parametric version of the STY79 method. As its name implies the techniquedoes not depend on an analytical form for φ ( M ) . Instead the LF is in effect parameterisedas a series of N p step functions allowing us to define the following initial setup: Φ ( M ) = φ k , M k − ∆M < M < M k + ∆M , (87)where, k = 1 , ..., N p The differential LF can then be expressed as φ ( M ) = N X i =1 φ i W ( M i − M ) , (88) where W ( x ) represents two window functions, W ( x ) ≡ ( , − ∆M ≤ x ≤ ∆M , , otherwise . (89)Therefore, it can be shown that the expression for the step-wise likelihood is given by L ( { φ i } i =1 ,...,I | { M k } l =1 ,...,K ) = N obs Y k =1 K P l =1 W ( M l − M k ) φ lK P l =1 φ l H ( M lim ( z k ) − M l ) ∆M (90)where H ( x ) = , x ≤ − ∆M/ , ( x/∆M + 1 / , − ∆M/ ≤ x ≤ ∆M/ , , x ≥ ∆M/ . (91)In EEP88 the errors on the LF parameters are assumed to be asymptotically normally dis-tributed giving a covariance matrix, cov( φ k ) = [ I ( φ k )] − , (92)where I ( φ k ) is the information matrix (see Eadie et al. 1971). Strauss & Willick (1995);Koranyi & Strauss (1997) noted two drawbacks of the method. The first concerns discreti-sation of the LF using step functions. A bias is introduced into the selection function due tohaving discontinuous first derivatives. The authors instead suggest interpolating through thesteps and then calculating the selection function to reduce this bias. The second drawbackis the sensitivity of the LF to the choice of bin size. In Koranyi & Strauss (1997) they showby example, that if the total number of bins is too small, this can dramatically underestimatethe faint-end slope of the LF.Heyl et al. (1997) extended the use of the SWML method by generalising it in a similarway as Avni & Bahcall (1980) did for V /V max by combining various surveys with differentmagnitude limits, coherently. Moreover, this extension also provided an absolute normalisa-tion and was used to probe density evolution in the LF by spectral type. In the same paper,they compare this method against the /V max estimator (see § The move beyond the standard LF analysis has seen constructing bivariate LFs (BLF) whichhave proven useful to further constrain galaxy formation and evolution theory. Much of thework described in this section represent natural progressions of the STY79 and the EEP88MLE estimators that have combined various observables such as galaxy radius, colour,galaxy diameters and surface brightness with space density. A more general approach re-cently developed by Takeuchi (2010) shall be discussed in greater detail in § hedding Light on the Galaxy Luminosity Function 35 Φ ( M, log r ) . In this scenariothe number of galaxies in the interval dMd (log r ) dm can be written as n ( M, log r, m ) = φ ( M, log r ) f ( m ) ρ ( Z )10 . Z ) , (93)where Z = m − M is the distance modulus, f ( m ) is a completeness function which de-scribes selection effects and ρ ( Z ) is the density function. The likelihood function is thendefined as L = X i ln φ ( M i ) f ( m i ) ρ ( m i − M i )10 . m i − M i )+ ∞ R −∞ dM + ∞ R −∞ dm φ ( M ) f ( m ) ρ ( m − M )10 . m − M ) + X i ln S ( M i , log( r i )) , (94)The maximisation of the first component of the likelihood gives the shape of the LF φ ( M ) .However, the second component S ( M, log r ) requires performing the MLE to the distribu-tion of galaxies in the M − log r plane which has the assumed form, S ( M, log r ) = 1 √ πσ exp (cid:20) − (log r + aM + b ) σ (cid:21) . (95)6.2 ColourChapman et al. (2003) described a parametric approach to fit a bivariate distribution, Φ ( L, C ) ,in luminosity L and colour C , that utilises the /V max estimator. This paper is concernedspecifically with the local infrared-luminous galaxies from the IRAS µ m (see Fisher et al. 1995, for full survey description). They use the IR colour distri-bution defined as the R (60 , ≡ S µm /S µm flux ratio (where S refers to the waveband) to explore colour-added evolutionary models.Recalling that the V max estimator can be expressed as Φ ( L ) ∆L = X i V max ,i , (96)where ∆L is the luminosity range within which we have a number density of galaxies, Φ ( L ) .To reiterate, V max ,i represents the maximum volume that the i th galaxy can be located inand still be detected.They find that the population of sources in the IRAS sample is represented best by alognormal distribution such that G ( C ) = exp " − × (cid:18) C − C σ C (cid:19) . (97)The colour distribution is modelled analytically as R (cid:18) (cid:19) = C ∗ × (cid:18) L ∗ L TIR (cid:19) − δ (cid:18) L TIR L ∗ (cid:19) γ , (98)where L TIR is defined as the total infrared luminosity. δ and γ represent the slopes of thefaint-end and bright-end of the LF, respectively. The final BLF convolves the LF, Φ ( L ) , modelled as a two-power law with the colourfunction, Φ ( C ) , as defined above to give Φ ( L, C ) dLdC = Φ ( L ) × Φ ( C ) dLdC = ρ ∗ × (cid:18) LL ∗ (cid:19) (1 − α ) × (cid:18) LL ∗ (cid:19) − β × exp " − (cid:18) C − C σ C (cid:19) dLdC, (1 − α ) , (99)where the C is calculated using the observed IRAS broad-band fluxes. See also, Chapin et al.(2009), Marsden et al. (2010) for further extensions to this method.6.3 Galaxy diametersThe SWML has now become a very popular method for determining the LF and Sodre & Lahav(1993) extended it (and the STY79 estimator) to a bivariate distribution of magnitudes andgalaxy diameters (see also Santiago et al. 1996). In this variation the conditional probabilityof finding a galaxy with a diameter D and absolute magnitude M can be written as P ( D, M ) = P ( M ) P ( D | M ) = P ( D ) P ( M | D ) . (100)The bivariate distribution of diameters and magnitudes is then expressed as Ψ ( D, M ) d D d M = φ ( D ) ϕ ( M | D )d D d M, (101)where φ ( D ) is the diameter distribution function and ϕ ( M | D ) gives the luminosity distri-bution for galaxies with diameter D . In terms of the step-wise likelihood, the distributionfunction of Equation 101 is now parameterised as N D bins in diameter and N M bins inabsolute magnitude such that, Ψ ( D, M ) = Ψ jk , j = 1 , ..., N D , k = 1 , ..., N M . (102)It can be shown that the log likelihood is given by ln( L ) = N X i =1 N D X j =1 N M X k =1 W ijk ln[ Ψ jk C V ( D i /v ∗ i , M i + Z i )] (103) − N X i =1 ln N D X i =1 N M X m =1 H ilm Ψ lm ∆D∆M ! + constant , (104)where W ( x, y ) = , if − ( ∆D ) / ≤ x ≤ ( ∆D ) / and − ( ∆M ) / ≤ y ≤ ( ∆M ) / , otherwise . (105)In a general way the quantity C V defines the velocity completeness function C V ( θ m , m ) asthe fraction of galaxies with measured velocities v and with apparent diameters between θ and θ + d θ , and apparent magnitudes between m and m + d m . Z i is the distance modulus v ∗ i ≡ D i /θ m . Ball et al. (2006) adopted this approach to construct the BLF with surfacebrightness. hedding Light on the Galaxy Luminosity Function 37 µ e = m + 2 . π ( r e ) ] − 10 log(1 + z ) − k ( z ) − e ( z ) , (106)where r e is the true observed half-light radius and k ( z ) and e ( z ) are the respective K − andevolution corrections. To construct the bivariate SWML for this case they firstly define anobservable window function given by O i ( M, µ e ) = , if M bright ,i < M < M faint ,i and µ e high ,i < µ e < µ e low ,i , otherwise . (107)Taking the Sodre & Lahav (1993) approach they define W ijk to weight each galaxy to ac-count for incompleteness such that W ijk = N i N i ( Q z ≥ , if M j − ∆M ≤ M i < M j + ∆M and µ ek − ∆µ e ≤ µ ek < µ ek + ∆µ e , otherwise , (108)where N i is the total number of galaxies lying in the same m − µ e bin as galaxy i . Q z definesa redshift quality, where a Q z ≥ equates to a reliable redshift measurement. The quantity N i ( Q z ≥ is thus the number of galaxies which known redshifts in the same bin. Finally,to account for the fraction of the M j − µ ek bin which lies inside the observable window ofgalaxy i they define the visibility function to be, H ijk = 1( ∆M∆µ e ) M j + ∆M/ Z M j − ∆M/ dM µ ek + ∆µ e / Z µ ek − ∆µ e / dµ e O i ( m, µ e ) , (109)They then fit a functional form characterising the joint luminosity-surface brightness (BBD)distribution given by Φ ( M, µ e ) = 0 . √ πσ µ e φ ∗ . M ∗ − M )( α +1) e − . M ∗− M ) (110) × exp (cid:26) (cid:20) µ e − µ e ∗ − β ( M − M ∗ ) σ µ e (cid:21)(cid:27) , where the upper part of Equation 110 has the usual Schechter function parameters and thelower part of the equation has additional parameters, µ e ∗ , σ µ e and β , to characterise thesurface brightness data.Section 9 examines in more detail a new technique to the BLF by Takeuchi (2010) whichoffers a more generalised approach by using the copula to connect two distribution functionsand the Pearson correlation coefficient to explore the correlation between the two. The use of photometric redshifts (photo-z) is playing a more central role in probing thecosmological model (see e.g. Blake & Bridle 2005; Banerji et al. 2008). Measurements canbe performed quickly and to high redshift and are therefore specifically suited to large galaxysurveys. However, as previously discussed, the resulting precision is substantially worse thanmeasuring them spectroscopically. Consequently, the ability to accurately constrain the LF isseverely impeded. In what follows, I discuss some of the modifications made to the standardLF estimators to incorporate photo-z measurements that are not precisely known.7.1 Smoothing kernelAlready discussed in § C − method given by X ( M ) = 0 . X i erfc (cid:18) z ∗ ( m i , M ) − z i σ i (cid:19) , (111)which led to estimating the C − function as shown in Equation 80. This essentially trans-forms the traditional Lynden-Bell ‘discrete’ approach into a more ‘continuous’ one, whereerrors in the redshift distribution coupled with K -corrections can now be represented by asmoothing function and integrated over the redshift probability distribution for each galaxy.To assess the errors and explore biases they use bootstrapping Monte Carlo techniques.7.2 Deconvolution - a V max generalisationA paper by Sheth (2007) revisited the Schmidt V max estimator and extended its use forsurveys with measured photo-z by casting it as a deconvolution problem. To begin with theprobability of estimating a redshift z e given its true value z t is given by dN e ( z e ) dz e = Z dz dN ( z t ) dz p ( z e | z t ) . (112)By now considering the generalised case in which a catalogue has both a minimum limitingvolume, V min for which an object would be too bright to be included in the catalogue, and theusual maximum volume, V max , the number of galaxies, N , with true absolute magnitude, M t , in a magnitude-limited catalogue is given by N ( M t ) = φ ( M t )[ V max ( M t ) − V min ( M t )] . (113)This equation is consistent within the usual Schmidt framework, where the LF would nowbe constructed by a / [ V max ( M t ) − V min ( M t )] weighting scheme. However, we are nowrequired to consider the number of galaxies N e with an estimated absolute magnitude M e which is shown to be given by N e ( M e ) = Z dM t φ ( M t ) d L ( M max t ) Z d L ( M min t ) dd L dV com ( d L ) dd L × p ( M t − M e | d L , M t ) , (114) The subscript t refers throughout this section as the true measured value, which, in this case, would bederived from a spectroscopic redshifthedding Light on the Galaxy Luminosity Function 39 where d L is the luminosity distance and V com is the comoving volume. In order to simplifythe process it is assumed that p ( M t − M e | d L , M t ) does not depend on d L and so the numberof estimated absolute magnitudes can be reduced to N e ( M e ) = Z dMφ ( M t ) V ( V max , V min , M t ) , (115)where, V ( V max , V min , M t ) = d L ( M max t ) Z d L ( M min t ) dd L dV com ( d L ) dd L p ( M t − M e | d L , M t ) . (116)An iterative deconvolution algorithm originally developed by Lucy (1974) is finally ap-plied to estimate the redshift distribution and thus the luminosity function. In Rossi & Sheth(2008) they apply this method via mock catalogues to probe their effectiveness in areassuch as the size-luminosity relation which is often distance-dependent (see also Rossi et al.2010).7.3 Modifying the maximum likelihood estimatorUsing Monte Carlos simulations Chen et al. (2003) demonstrated that applying the standardSTY79 MLE to photo-z data produces an intrinsic bias into the shape of the LF. This mani-fests as a steeper slope at the bright end of the LF caused by intrinsic fainter galaxies beingscattered more into the brighter end than brighter galaxies being scattered into the faint endof the LF. To attempt to reduce this bias they convolve for each photometric redshift ofgalaxy i , the Gaussian kernel with the redshift likelihood function to give p i ( z − z i ; z i ) = ∞ Z σ z √ π L iz ( z ′ − z i ; z i ) exp (cid:20) − ( z − z ′ ) σ z (cid:21) dz ′ . (117)Thus, they show that the total likelihood function is then represented by L = Y i P i ( m i , z i ; M ∗ )= Y i z f Z ψ i p i ( z i − z ′ ; z i ) dz ′ Θ m max Z m min z f Z ψ i p i ( z i − z ′ ; z i ) dzdm − , (118)where Θ is the fraction of the angular area sensitive enough to detect the galaxy m i at aredshift z i and, ψ i = 10 . [ M ∗ − M i ( m i ,z ′ ) ] (1+ α ) exp (cid:16) − . [ M ∗ − M i ( m i ,z ′ ) ] (cid:17) . (119)Applying this modification to the same simulated data showed an improvement in the re-covered input LF with a reduction in the systematic uncertainties of ∼ . mag.Following from this, and in the same paper as his deconvolution technique for V max , Sheth (2007) improves upon the work of Chen et al. by deriving a new likelihood. To begin withthe general likelihood is expressed as L ( a ) = Y i φ ( L i | z i , a ) S ( z i , a ) , (120)where S is the selection function, a represents the parameters that describe the shape of theLF and φ ( L i | z i , a ) is the LF at a redshift z .If we now denote the estimated photometric redshift as z e and thus the estimated ap-parent luminosity as l e . Also, if l t and z t are the respective true apparent luminosity andredshift then the number N of estimated redshifts within a flux-limited catalogue is shownto be, N ( z e , a ) = Z dz dV com dz l max t Z l min t dl t πd L ( z t ) × φ ( L t | a ) p ( z e | z t , l t ) , (121)where, V com is the comoving volume. The absolute luminosity, L t = l t πd L ( z ) ,where d L ( z ) is the luminosity distance. The joint distribution of z e and l e is then given by l e N ( l e , z e , a ) = Z dz dV com dz πd L ( z t ) × φ ( L t | a ) p ( z e | z t , l t ) . (122)Thus the probability distribution by which the likelihood is to be maximised is such that L ( a ) = N Y i =1 N ( l i,e , z i,e , a ) N ( z i,e , a ) , (123)The modification by Christlein et al. (2009) attempts to define a likelihood function simi-lar in approach to that of M83 (as described in Equation 43) which, as suggested by theauthors, would replace the photometric redshift of Sheth by a more robust observable. Justas M83 gridded the luminosity-redshift information, this method defines a parameter spacethat encompasses absolute magnitude M , SED and redshift z of a galaxy. This so-called photometric space is treated as an n -dimensional flux space where each dimension corre-sponds to one filter band in which a flux, f , is measured. This space is gridded into cellssuch that each cell either contains 0 or 1 galaxies. Equation 43 can then be written as L = N Y i λ ( f i ) df e − λ ( f i ) df Y j e − λ ( f j ) df , (124)where λ ( f i ) df is the expectation of the number of galaxies in the i th cell in photometricspace. To map between the LF parameters and photometric space it is assumed that the pho-tometric data are normally distributed centred on the expectation value for a given absolutemagnitude M , SED and redshift z . Thus λ ( f i ) is defined as λ ( f i ) = X SED Z dM Z dz (cid:18) dV c dz (cid:19) p σ ( f i | M ; SED; z ) Q n ∆ n Φ ( M ; SED; z | P ) , (125) hedding Light on the Galaxy Luminosity Function 41 where P represents the LF parameters, dV c /dz is the differential comoving volume and p σ represents the fraction of galaxies that contribute to the galaxy density at a point f i . Thestandard Schechter LF of the form in Equation 9 was adopted to which an evolutionary termwas added. To constrain a galaxy to a given z , SED and M the above equation is modifiedas λ i = X SED Z dM Z dz (cid:18) dV c dz (cid:19) δ ( z − z t ) × δ (SED − SED t ) × δ ( M − M t ) × ( M ; SED; z | P ) , (126)where z t , SED t and M t are the true values to which a given galaxy is constrained. Thestandard Schechter LF of the form in Equation 9 was adopted which an evolutionary termadded. R u ss e ll J ohn s t on Table 1: Summary of the traditional LF estimators. Estimator Type Developed by Section Pros ConsMaximum Likelihood (MLE) Parametric Sandage et al. (1979) § 4, Page 17 No binning required ( No built-in goodness of fitNo built-in normalisation Extensions to MLE: Provide normalisation Davis & Huchra (1982) § L modelled by Poisson probabilities Marshall et al. (1983) [M83] § § § § Bivariate LF: Galaxy radius Chołoniewski (1985) § /V max ) Chapman et al. (2003) § Photo z: Gaussian kernel Chen et al. (2003) § § § Classical Method ( Φ = N/V ) Non-Parametric Christensen (1975)Schechter (1976)Felten (1977) § ( Biased by clusteringReturns bias LF estimate / V max & V / V max Non-Parametric Schmidt (1968) § No assumption of form of LFEasy to applyMany improvements madeProvides test of evolutionProvides test of completeness Biased by clustering Extensions to V max : Combining multiple data-sets Avni & Bahcall (1980) § § § h e dd i ng L i gh t on t h e G a l a xy L u m i no s it y F un c ti on43 Table 1 – continued from previous pageEstimator Type Developed by Section Pros Cons n/n max & o/o max variants Qin & Xie (1997, 1999) § § § § Bivariate LF: Colour (utilises MLE) Chapman et al. (2003) § Photo z: Deconvolution technique Sheth (2007) § The C − method Non-Parametric Lynden-Bell (1971) § No parametrisation of the LF requiredrecovers the CLFIndependent of clustering effects No built-in normalisation Extensions to the C − method: Combining multiple data-sets Jackson (1974) § § § § C + variant Zucca et al. (1997) § Photo z: Gaussian error distribution Subbarao et al. (1996) § The φ/ Φ method Non-Parametric ( Turner (1979) , Kirshner et al. (1979) § ( Biased estimatorCorrelated errors Extensions to φ/ Φ : Adopting parametric form of the LF e.g. de Lapparent et al. (1989) § R u ss e ll J ohn s t on Table 1 – continued from previous pageEstimator Type Developed by Section Pros ConsThe Step-Wise Max Likelihood Non-Parametric Nicoll & Segal (1983) , Chołoniewski (1986)Efstathiou et al. (1988) § No assumption of form of LFrobust estimatorIndependent of clustering effects ( No built-in normalisationselection f n bias Extensions to the SWML: Combining multiple data-sets Heyl et al. (1997) § § n biasInterpolating through steps Koranyi & Strauss (1997) § n bias Bivariate LF: Galaxy diameters Sodre & Lahav (1993) § § hedding Light on the Galaxy Luminosity Function 45 Thus far all the traditional LF estimators have been summarised along with their most rele-vant modifications and extensions. This aim of this section is to highlight some of the key re-sults from papers by Heyl et al. (1997) (H97), Willmer (1997) (W97), Takeuchi, Yoshikawa, & Ishii(2000) (TYI00) and Ilbert et al. (2004) (I04), which have provided detailed comparisonsand, therefore, useful insights of selected LF estimators. Such comparisons are vital and en-able us to gain a practical sense of the limitations and any intrinsic bias of these techniquesin a more controlled environment.8.1 Heyl et al. (1997)As already discussed in the previous section, H97 extended the EEP88 SWML (which wasreferred to as the SSWML to make the distinction) which could account for multiple sam-ples with varying magnitude limits. The paper was also concerned with exploring the roleof evolution in galaxy surveys. This discussion is focussed on the first half of this paperwhere they compare the SSWML with the /V max estimator using simulated data. TheMonte Carlo (MC) samples drew magnitudes from a Schechter function adopting a standardcosmology. For simplicity, K -corrections were not considered. With this basic set up, thefollowing scenarios were explored. In the first scenario they constructed a sample of 1800 galaxies, binned equally over 6 ap-parent magnitude ranges with an overall magnitude range between . < m < . mag.To give a simplistic model of density evolution they doubled the density of galaxies beyond z = 0 . .Their findings in this case showed both estimators were very consistent with each other,with the exception of the SSWML returning smoother results. It was concluded that neitherindicated any intrinsic bias with respect to the input LFs. Potential intrinsic biasing due to clustering (particularly in the /V max method) was thenexplored using three MC samples with over densities built in at z = 0 . and with eachsample having clustered fractions of 45%, 65% and 85%. By exploring various bin widthsizes they found that for very narrow bin widths both estimators were biased at the faint-endslope of the LF, over predicting the density. However, as the bin widths increased /V max remained unchanged but the SSWML method showed more robustness to the clusteringshowing a systematic decrease in bias. For the three catalogues both estimators did show anincrease in steepness of the LF with the SSWML being less affected (see Figure 8). It wastherefore concluded that the SSWML was a superior estimator to that of V max for surveyswith strong clustering.8.2 Willmer (1997)In this paper, the author compares LF parameter values derived from the Chołoniewski(1987) version of Lynden-Bell’s C − method, the MLE of STY79, the Turner (1979) φ/Φ Fig. 8 Plots courtesy of Heyl et al. (1997). Selected results from H97 (Case II in our text) where increasinglevels of clustering in the MC samples are shown. The top-left panel has no clustering, the remaining panelsare the result of 45% (top right), 65% (bottom left) and 85% (bottom right) clustering. method, the SWML estimators of Chołoniewski (1986) and EEP88, and finally Schmidt’s /V max estimator. All were applied in two different cases: the first using MC simulations(based on the CfA survey) and the second to the CfA-1 (Huchra et al. 1983) survey data. In this case a total of 1000 MCs were drawn from a Schechter function (Schechter 1976)using a homogeneous redshift distribution and inputting three different values of α (i.e. α in = − . , − . and -1.5 ) in order to probe sensitivity of the LF estimators to the in-clination of the faint-end slope. Finally, the MCs were designed to have a similar surveygeometry as the CfA-1 redshift survey.It was demonstrated that all of the estimators under comparison recovered the α in valuesextremely well with the exception of /V max (see Table 2). Overall the STY79 and C − hedding Light on the Galaxy Luminosity Function 47 Table 2. Extract from Willmer (1997) reproduced by permission of the AAS. Medianvalues for recovered parameters M ∗ and α , derived from 1000 CfA-1 like Monte Carlosimulations. M ∗ α M ∗ α M ∗ Input Values -1.50 -19.10 -0.70 -19.10 SWML -1.45 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± /V max ∆z -1.67 ± ± ± ± /V max ∆M -1.50 ± ± ± ± − -1.51 ± ± ± ± Table 3. Extract from Willmer (1997) reproduced by permission of the AAS. Schechterfunction parameters for CfA1 survey, using same limits for all methods Method α M ∗ NotesSWML -1.20 ± ± ∆M = 0.25STY -1.11 ± ± ± ± ∆M = 0.25Turner -1.11 ± ± ∆z = 500 km s − max -1.59 ± ± ∆M = 0.251/V max -1.70 ± ± ∆z = 500 km s − C − -1.20 ± ± methods were shown to be the most robust providing the best results. They show that despitehaving a homogeneous sample, /V max indicated bias, giving higher values for the faint-endslope compared to the other estimators under test. However, as we shall see in the followingsection, TYI00 applied the modified Eales (1993) V max variant to a homogeneous sampleand demonstrated its consistency with other estimators. In Willmer et al. (2006) they applythis variant to the Deep Extragalactic Evolutionary Probe 2 (DEEP2) survey data to avoidpotential bias. The estimators were then tested against the actual survey data from CfA-1 (see Table 3).The results indicated that all the estimators gave consistent results with α = − . and M ∗ = − . mag with the exception once again of the V max . In this case they noted thatthe density measured was lower than STY79 by a factor of 2.Overall, it was found that he STY79 and the C − methods gave the best most robustresults. However, they did note that STY79 fit underestimated the faint-end slope whereasthe bias observed in /V max showed overestimating the faint-end slope. Therefore, for sce-narios where samples have a steeper slope, the C − method would be a preferred choice ofestimator. /V max estimatorwhich traces evolution with redshift. The second method was a variation on the C − es-timator which they refer to as the Lynden-Bell–Choloniewski–Catditz–Petrosian (LCCP) method, which incorporates the Chołoniewski (1987) extension and the smoothing kernelintroduced by Caditz & Petrosian (1993) (see Equation 79). The third and forth methodswere the Chołoniewski (1987) and the EEP88 respective variations of the SWML estima-tors. They consider the following three distinct scenarios. The four LF estimators were applied to three different types of mock catalogue using thefollowing LFs: – Uniform distribution. – Power-law that increases towards faint magnitudes, – Power-law that decreases towards fainter magnitudes, – Gaussian distribution, – Schechter function with α = − . , implying a flat faint-end slope, – Schechter function with α = − . , implying a steep faint-end slope.The MC simulations were created with the following spatial density features: – A homogeneous distribution. This was largely to explore any bias that may have beeninherent in the /V max as originally claimed by W97. The sample was created neglect-ing the K -correction, with a redshift limit of z = 0 . and an apparent magnitude limit m lim = 13 . mag. – Two inhomogeneous samples. In the first, half the galaxies in the sample lie within adense cluster at a distance of 0.8 Mpc with a radius of 0.8 Mpc. The second mocks anoverall under-density with a spherical void with a radius of 1.6 Mpc at a distance of 0.8Mpc. Both scenarios have the same overall density as the homogeneous sample.An MC catalogue was then generated with sample sizes of 100 and 1000 to observe theeffects of Poisson noise when one is sampling from small data-sets.The results from this part of the study revealed that in the homogeneous case all estima-tors gave results consistent with each other, thus contradicting earlier claims by W97, wherethey found a bias in their results when applying the original Schmidt /V max .In contrast, the clustered MC sample showed that /V max was heavily biased produc-ing an overestimation in the recovered LF - as would be expected. Moreover, the resultsshowed the other three estimators were not adversely affected by this inhomogeneity anddemonstrated good agreement with all the input LFs. However, larger error bars were ob-served when applying the Choloniewski method which implied shot noise was beginning todominate for the smaller of the sample sizes. An excerpt of these results in Figure 9. In this case they apply three of the four LF estimators to a MC 2dFGRS sample prepared byCole et al. (1998) - it is unclear why the LCCP method was excluded from this part of the hedding Light on the Galaxy Luminosity Function 49 Fig. 9 Extract from Takeuchi, Yoshikawa, & Ishii (2000) reproduced by permission of the AAS. An Examplefrom TYI00 (Case I) of the effects from applying the estimators on a homogenous MC sample (left) and aclustered MC sample (right). On the left hand plots we observe all four estimators are consistent with eachother with no indication of intrinsic bias. However, the clustered sample on the right-hand panel shows howthe expected bias of /V max affects determination of the faint end. From this panel, there is no evidence ofclustering intrinsically biasing the remaining estimators. The input luminosity function in both scenarios forthis example was a uniform distribution. analysis. In general they found that all estimators gave consistent results compared with theinput LF (see the top panels in Figure 10). They did, however, report slight deviations in the /V max estimator which were considered to be due to the clustering in the sample. All four estimators were applied to the photometric redshift HDF (Fern´andez-Soto et al.1999; Williams et al. 1996) to probe evolution of the LF shape. The overall sample sizeunder test consisted of 946 galaxies. Whilst they provide extensive details of the intrinsicevolutionary properties found in this study, overall they reported that all four 4 estimatorsgave consistent results with a distinct evolutionary trend in the LF with redshift. An exampleof these results is shown in the bottom panels of Figure 10. It has been noted that rudimen-tary determination of K -corrections adopted by TYI00 in this analysis may have adverselyaffected the overall accuracy of LF determination for all of the estimators applied in theHDF results (Takeuchi, private communication). However, as we shall see in the followingsection, Ilbert et al. (2004) provided a rigorous study of the effect of K-correction bias onLF estimators.Finally, an interesting aside in this analysis compares the algorithms for efficiency. Theyfound that the Chołoniewski method was the fastest, whereas the EEP88 method requiredmore iterations and V max required the maximum volume to be calculated for each galaxywhich both contributed to a slower computational time. Fig. 10 Extracts from Takeuchi, Yoshikawa, & Ishii (2000) reproduced by permission of the AAS. Toppanels: results from Case II where 2dFGRS MC samples were tested. They showed that considering thesamples in redshift space (left) and real space (right) had little adverse effect on determining the LF. Bothpanels show consistency between the estimators under test compared to the input LF. Bottom panels: exampleof results from Case III where the estimators were applied the HDF data. In the context of this review it canbe seen that in the redshift ranges considered in both panels, once again, all estimators seem to show overallconsistency with each other. hedding Light on the Galaxy Luminosity Function 51 limits within which a given population could be observed at a given redshift for a fixedapparent magnitude limit. Figure 11 illustrates such a scenario.A relationship can be established that relates the redshift z , the effective wavelengthof the selection filter, λ s , and the effective wavelength of the reference filter, λ Ref whichallows us to probe the absolute magnitude limit, M faintlim , for a given reference filter, as afunction of redshift for different galaxy types. In this example the galaxy types are irregu-lars, spirals and ellipticals which can be generally referred to ranging, respectively, from the‘blue’ populations to ‘red’ populations. The three panels in Figure 11 represent three differ-ent cases exploring the relationships shown in Equation 127 within a given redshift range, z low ≤ z < z high , corresponding to . ≤ z < . . z low < λ s /λ Ref ∼ λ s /λ Ref > λ s /λ Ref (127)Each case corresponds to observing the three galaxy populations within different referencefilters which can be summarised as follows.1. z low < λ I AB /λ UV HST :In the top panel of Figure 11 M faintlim is defined for the UV Ref filter and is conse-quently brighter for blue galaxies than for red. Thus, in the top panel of the figure thefaint irregular SEDs becoming increasingly unobservable within the absolute magnituderange where the other spiral and elliptical galaxy types can still be detected. In this fil-ter within the redshift range considered, both irregular- and spiral-type galaxies are notobservable beyond absolute magnitudes fainter than M faintlim = − . . However, the el-liptical galaxies remain observable out to M faintlim = − . . Therefore, one would expectthe global estimation of the LF would become biased between − . . M faintlim . . .2. z low ∼ λ I AB /λ B HST :In the middle panel of Figure 11 M faintlim is now defined for the B Ref filter and is approx-imately the same for all galaxies for the given B-band reference filter within the redshiftrange considered. In this case one would expect minimal bias effect as most galaxy pop-ulations would remain observable out to the faintest absolute magnitude limit.3. z low > λ I AB /λ I HST :Finally, observing galaxies in the I Ref filter as show in the bottom panel of Figure 11implies that M faintlim is now brighter for red galaxies than for blue and therefore faint redgalaxies are missing from the sample. Opposite to Case 1, elliptical galaxies are nowunobservable within the absolute magnitude range where irregulars would still be de-tected. However, as can be seen, this bias should be less prevalent than in Case 1 sincethe maximum absolute magnitude range this effects is only in order of ∼ mag. Thus,where all SEDs have the same absolute magnitude limit, no bias should be present inthe recovered LF. Fig. 11 Extracts from Ilbert et al. (2004) reproduced by permission. To illustrate how the bias creeps intothe global LF consider the three cases in the above panels. In the top panel a galaxy distribution is shownconsisting of irregulars (crosses), spirals (solid triangles) and ellipticals (open circles). For each galaxy typeSED, the corresponding faint observable absolute magnitude limit M faintlim is shown as a function of redshift.The vertical dashed lines represent the redshift region . ≤ z < . within which the bias is studied. Thesection filter is I with I AB ≤ mag and the reference filters are respectively shown from top to bottom asUV FOCA (2000 A), B HST (4500 A), I HST (8140 A). To explore the impact of this bias, I04 apply the /V max , C + , STY79 and the EEP88 es-timators to both real and simulated data across these reference filters. For this review onlyresults pertaining to the UV and B filters are summarised which are, respectively, shownas the top and bottom panel sets in Figure 12. The simulated mocks were drawn from themulti-colour mock catalogues that are based on an empirical approach using observed LFsto derive redshift and apparent magnitude distributions [see I04 and Arnouts et al. (1999)for more details]. In essence, they generate two sets of mock catalogues which have beenclassified into three main spectral classes: irregulars, spirals and ellipticals. In the first setof mocks (corresponding to the left-hand panels in each set of Figure 12) they use only oneSED per spectral class. This is to compare against the second set of more realistic mockswhere objects have been drawn from an interpolated set of 72 SEDs (the middle panels ineach set of Figure 12). In this case one spectral class can correspond to multiple SEDs andtherefore the bias may be evident within a single spectral class.For the real data sample, they used the photometric catalogue and photometric redshiftsfrom the Hubble Deep Field (HDF) North and South surveys (Arnouts et al. 1999, 2002) hedding Light on the Galaxy Luminosity Function 53 Fig. 12 Extracts from Ilbert et al. (2004) reproduced by permission. Examples of how this form of obser-vational bias affects the recovery of the global LF for different estimators in two different reference filters-UV (top panel set) and B-HST [4500A] (bottom panel set). For each panel set the results for the simulatedmock data are shown in the left and middle plots. The LFs from HDF-North and -South are shown in theright-hand plots of each set. The resulting LF estimates for each estimator are indicated on each panel set.The LFs corresponding to the three input SEDs used are shown as dotted lines on the left-hand plots of eachpanel set: from the steepest to the shallowest slope, irregulars, spirals and ellipticals LF, respectively. Theglobal simulated LF (i.e. the sum of the three input LFs) is shown as a solid line.4 Russell Johnston (the right-hand panels in each set of Figure 12). For consistency, they have used the exactlythe same set of 72 SEDs as in the mock sample.Before discussing their findings, the remainder of the setup of Figure 12 is discussedfirst. The upper and lower plots in each panel set are the recovered LFs in the respectiveredshift ranges . ≤ z < . and . ≤ z < . . The first of these ranges correspondsto the range shown in Figure 11. The left-hand plots show the input LFs for each spectraltype in the mock samples, indicated by the dotted lines. From the steepest to the shallowestslope correspond to irregulars, spirals and ellipticals. In each plot the global input LF isshown as a solid line. Finally each LF estimator is indicated by a: dashed line (STY79),triangles (SWML EEP88), squares ( C + -method) and circles ( /V max ). In each case theglobal LF is determined. Due to the nature of how each LF estimator is constructed, the resulting bias affects themin different ways. Examining firstly the top panel set corresponding to the reference filter,UV, we can see clearly that all LF estimators underestimate the global input LF. In the mockdata for the first redshift range (top panels), the C + -method and /V max methods follow thebrighter end of the LF well but then show a sharp drop beyond M ∼ − , at which pointthey seem to recover the input LF of the elliptical samples. A similar trend is observed for themore distant redshift range. This bias seems entirely consistent with Case 1 described aboveand shown in the top panel of Figure 11. Whilst the SWML also shows a drop off at thismagnitude it is less biased, recovering more the shape of the input elliptical LF as opposedto the LF itself. Contrastingly, the parametric STY79 method shows only a marginal short-fall in the recovered LF at faint magnitudes, and, moreover, in the left-hand bottom panel itseems to recover the input global LF exactly.Whilst the HDF data in the right panels do show a drop off for all estimators toward thefaint end of the LF, the impact is significantly less than in the mock data. It was thought thatthis was due to the very small sample size of the data and thus it was concluded that in thisreference frame the global LF is not wholly recoverable. As already discussed and shown in the bottom panel of Figure 11, in this reference framethe absolute magnitude limits for the spectral types are very similar across the redshift rangeindicated. Thus in the top plots of the middle panel set in Figure 12 it is clear that all LFestimators recover the global LF extremely well. In the bottom plots of this set we can ob-serve a slight overestimation in the SWML and STY79 estimators toward the faint end. Thisis consistent with the middle panel of Figure 11 where at this redshift range, the absolutemagnitude limits for each spectral type begin to show stronger divergence.The HDF LF is fairly recovered in the filter with only slight underestimation in STY79at . ≤ z < . and slight overestimation at . ≤ z < . .This type of analysis into the robustness of the traditional LF estimators provides a usefultool for probing the deep survey sample where samples may be more prone to biasing fromvariable magnitude limits resulting from large K -corrections across spectral types. The au-thors conclude that since the STY79 and SWML methods differ in their biasing (beyond hedding Light on the Galaxy Luminosity Function 55 Poisson errors) compared to the /V max and C + estimators can be a good indicator thateither one has bias in recovering the global LF.Several possible ways to reduce this bias entering into LF estimation are offered. Theseinclude selecting galaxies subsamples in the closest rest-frame filter to the reference filteras performed by e.g. Poli et al. (2003). However, this requires multi-colour information tobe able to derive the same rest-frame band LF at different redshifts. Another possible routeapplies more to large survey data. This would require estimating the global LF within theabsolute magnitude range in which all galaxy types are detected. Of course, this approachwould require the cutting of a percentage of the data. Despite the continuing popularity of such methods as /V max , MLE and the SWML, therehas been renewed exploration into more innovative statistical approaches. These new meth-ods are largely motivated by the potential hazards and pitfalls inherent with the currenttraditional approaches. I now examine in more detail three such approaches to estimatingthe LF that have emerged over the last few years. The first is a semi-parametric approach bySchafer (2007), the second is a Bayesian approach by Kelly et al. (2008), and the third, byTakeuchi (2010), applies the copula to construct the bivariate LF.9.1 Schafer (2007) - A semi-parametric approachThis approach is statistically rigorous and considers data-sets that are truncated i.e. fluxlimited. There can be potential advantages for this method summarised as follows.1. No strict parametric form is assumed for the bivariate density.2. No assumption of independence between redshift and absolute magnitude is made.3. No binning of data is required.4. A varying selection function can be incorporated.By not assuming separability Schafer decomposes the bivariate density φ ( z, M ) into, log φ ( z, M, θ ) = f ( z ) + g ( M ) + h ( z, M, θ ) , (128)where h ( z, M, θ ) has a parametric form that incorporates the dependency between red-shift, z , absolute magnitude, M , and the real valued parameters, θ , by folding in, for ex-ample, evolutionary models. The functions f ( z ) and g ( M ) are, however, determined non-parametrically. He then incorporates an extended form of the maximum likelihood approachcalled the ‘local’ likelihood estimator for the density estimation and applies this to 15,057quasars from Richards et al. (2006). This semi-parametric approach has the advantage ofallowing the user to estimate evolution of the LF with redshift without assuming a strictparametric form for the bivariate density. The only parametric form required is that whichmodels the dependence between redshift and absolute magnitude. Moreover, it should benoted that this method assumes a complete data-set. This approach is a non-parametric extension of the MLE where one assumes the data X =( X , X , ..., X n ) are observations of independent, identically distributed random variablesfrom a distribution with density f . The MLE ( ˆ f MLE ) for f is defined as the f ∈ F , where F denotes the class of candidates for f , and is maximised as n X j =1 log f ( X j ) − (cid:26) n (cid:20)Z f ( x ) dx − (cid:21)(cid:27) (129)From this, one can localise the likelihood criterion and thus construct the final local likeli-hood ˆ f LL estimator by smoothing the local estimates giving ˆ f LL ( x ) ≡ "X u ∈G K ∗ ( x, u, λ ) ˆ f u ( x ) u ∈G K ∗ ( x, u, λ ) (130)where G forms a grid u ∈ G of equally spaced values (between -3 and 3 in the authorsexample) of a Gaussian density with mean zero and variance of unity. The term, K ∗ ( x, u, λ ) ,is therefore a kernel function such that X u ∈ G K ∗ ( x, u, λ ) = 1 ∀ x. (131)By making G sufficiently large, the amount of smoothing is completely dominated by thekernel function parameter, λ . The local density likelihood is incorporated into the case for flux-limited survey data whereone can include the dependence between the redshift, z , and absolute magnitude, M . A firstorder approximation of h is made from Equation 128 such that h ( z, M, θ ) = θzM. (132)After an extensive derivation, a global criterion for the likelihood is found to be given by L ∗ ( f, g, z, M, θ ) ≡ n X j =1 w j X u ∈ G K ∗ ( z j , u, λ ) a u ( z j )+ X u ∈G K ∗ ( M j , v, λ ) b v ( M j ) + h ( z j , M j , θ ) − Z A ( exp( h ( z, M, θ )) " X u ∈ G K ∗ ( M, v, λ ) exp( b v ( M )) × " X u ∈ G K ∗ ( M, v, λ ) exp( a v ( M )) dM dz )! , (133)where a u ( z ) and b v ( M ) are degree p polynomials which form part of the smoothing termof K ∗ for local estimates. A defines the region outside of which the data are truncated on hedding Light on the Galaxy Luminosity Function 57 Fig. 13 Extract from Schafer (2007) reproduced by permission of the AAS. Estimates of the luminosity func-tion at different redshifts (black solid curves and error bars), compared with estimates from Richards et al.(2006) (light grey solid curves and error bars). Error bars represent one standard error and account for statis-tical errors only. the ( z, M ) plane. The quantity w j is a weighting to take incompleteness into account and isdefined as the inverse of the selection function.Estimating the LF in this way has the advantage of allowing the user to estimate theevolution of the LF without assuming a strict parametric form of the bivariate density. Themethod was applied to quasar data from Richards et al. (2006) and compared to their resultswhich applied the Page & Carrera (2000) version of /V max (see page 27). These resultsshown in Figure 13 are in good agreement with Richards. § The form of the likelihood function that they adopt for the LF estimation is derived from abinomial distribution. Whilst they highlight that the traditional approach of using a Poissondistribution is incorrect, they show that as long as the survey’s detection probability is small,both approaches yield the same results. We recall the relation of the LF to the probabilitydensity of ( L , z ) can be written in the following separable form: p ( L, z ) = 1 N φ ( L, z ) ρ ( z ) , (134)where L is the luminosity, z is redshift and N is the normalisation set as the total number ofobjects in the observable Universe. From this starting point, the authors assume a parametricform for φ ( L , z ), with parameters θ and show that the observed data likelihood function isgiven by, p ( L obs , z obs , I | θ, N ) ∝ C Nn [ p ( I = 0 | θ )] N − n Y i ∈A obs p ( L i , z i | θ ) , (135)where p ( L, z | θ ) = N Y i =1 p ( L i , z i | θ ) (136)is the likelihood function for all N sources in the universe. L obs and z obs denote the sourcesobserved in a given survey. By adding in sample selection, the probability that the surveymisses a source, given by the parameters θ , is p ( I = 0 | θ ) = Z Z p ( I = 0 | L, z ) p ( L, z | θ ) dL dz (137)In Equation 135, I is a vector of size N taking on values I i if i th source is included in survey otherwiseFinally the term C Nn = N ! /n !( N − n )! is the binomial coefficient and A obs is the set of n included sources. Equation 135 thus represents observed data likelihood function given anassumed LF that is parameterised by θ . hedding Light on the Galaxy Luminosity Function 59 Fig. 14 From Kelly et al. (2008): Illustration of the prior distribution for the mixture Gaussian function(GF) approach to describing the LF with K = 5 GFs. For the two cases shown, each has the resultingmarginal distributions of log z and log L on the respective top and right side of each plot. The left-hand panelillustrates unimodal prior assumed in the paper where the GFs are close together. When the GFs are furtherapart (right-hand panel) the resulting LF is multimodal. It is then shown that, for a given set of observed data, the joint posterior probability distri-bution of θ and N for the LF is given by p ( θ, N | L obs , z obs ) ∝ p ( N | θ, n ) p ( θ | L obs , z obs ) , (138) To model the LF they adopt a similar approach as applied by Blanton et al. (2003b), wherea mixture of Gaussian functions were used to accurately estimate the LF. In this case, tominimise the number of Gaussian functions (GFs) required to describe the LF, they considerlog of the joint distribution of L and z . Equation 134 is now re-written as p ( L, z ) = p (log L, log z ) Lz (ln 10) , (139)Moreover, unlike Blanton et al. (2003b), they allow the widths of GFs to vary and do not fixthe centroids to like on a grid of values. This also adds to minimising the number of GFsrequired. The mixture of K Gaussian functions can be written as p (log L i , log z i | π, µ, Σ ) = K X k =1 π k π | Σ k | / × exp (cid:20) − 12 ( x i − µ k ) T X − k ( x i − µ k ) (cid:21) , (140)where θ = ( π, µ, Σ ) , P Kk π k = 1 , x i = (log L i , log z i ) , µ k is the 2-element mean positionvector for the k th Gaussian, P k is the 2 × k th Gaussian, and x T denotes the transpose of x . The model for the LF is given by φ ( L, z | θ, N ) = NLz (ln 10) (cid:18) dVdz (cid:19) − (141) × K X k =1 π k π | P k | / exp (cid:20) − 12 ( x − µ k ) T X − k ( x − µ k ) (cid:21) Another crucial aspect to using the mixture model is the form of its prior distribution. Essen-tially, it is assumed, in general, the LF will be unimodal i.e. it does not show distinct peaks(see Figure 14). Consequently, only the prior distribution has a parametric form allowingthe GF widths to be closer together and thus aiding the convergence of the Markov ChainMonte Carlo (MCMC). The form of this prior is given by p ( π, µ, Σ, µ , A ) ∝ K Y k =1 p ( µ k | µ , Σ ) p ( Σ k | A ) ∝ K Y k =1 Cauchy ( µ k | µ , T )Inv − Wishart ( Σ k | A ) , (142)where Cauchy ( µ k | µ , T ) refers to a two-dimensional Cauchy distribution as a functionof µ k , with mean vector µ and scale matrix T . The Inv − Wishart ( Σ k | A ) is the inverseWishart density as a function of Σ k , with 1 degree of freedom and scale matrix A .Finally, the total joint posterior distribution is shown to be given by p ( θ, N, µ , A | log L obs , log z obs ) ∝ p ( N | θn ) p ( θ, µ , A | log L obs , log z obs ) (143)MCMC is then applied using the Metropolis-Hastings algorithm (MHA) (Metropolis & Ulam1949; Metropolis et al. 1953; Hastings 1970) for obtaining random draws of the LF from theposterior distribution. Given a suitably large enough number of Gaussian functions it is flex-ible enough to give an accurate estimate of any smooth and continuous LF. They found that K ∼ true Schechter function form, but was not as accurate as if one had fitted the correct para-metric model. Moreover, they showed that one could just as easily substitute the mixturemodel for any parametric form of the LF into the likelihood function and posterior distribu-tion. The Bayesian mixture of Gaussian functions model is able to accurately constrain theLF, even below the survey detection limit.9.3 Takeuchi (2010) – Copula and correlation for the Bivariate LFTakeuchi adopts a technique for BLF construction that uses the copula function to con-nect two distribution functions and defines nonparametric measures of their dependence. Todemonstrate this technique the FUV-FIR BLF is constructed by adopting the Saunders et al.(1990) LF shown in Equation 13 for the IR, and a Schechter function shown in Equation 5for the UV. However, to summarise this technique some of the framework underpinning thecopula definition is firstly introduced. FUV-FIR refers to the far ultraviolet and the far infraredhedding Light on the Galaxy Luminosity Function 61 Fig. 15 Extract from Kelly et al. (2008) reproduced by permission of the AAS. The input LF derived from asimulated sample is shown as a solid red line. The dashed line represents the posterior median estimate of theLF based on the mixture of Gaussian functions model. The shaded region is 90 % of the posterior probability.The solid vertical line shows the flux limit of the sample at each redshift. Sklar (1959) created a new class of functions which he called copulae . In the most generalcase, Sklar’s theorem states that if x , x , ..., x n variables of H , an n -dimensional distribu-tion function with marginal distribution functions (or ‘marginals’) F , F , ..., F n , then thereexists an n -copula C such that H ( x , x , ..., x n ) = C [ F ( x ) , F ( x ) , ..., F n ( x n )] (144)The above relation has the property that if the marginal distributions F are continuous thenthe resulting copula C is unique. However, if this is not the case then the copula is uniqueon the range of values of the marginal distributions on Range F × Range F × .... × Range F n , where Range F i is the range of the function F i .The implementation of the copula in this work reduces the above to a 2-dimensionalbivariate case where H is defined as H ( x , x ) = C [ F ( x ) , F ( x )] . (145)This theorem gives the basis that any bivariate distribution function with given marginals canbe expressed in this form. Therefore, in essence, the copulae connect the joint distributionfunctions to their one-dimensional margins. Copula is Latin for “a link” or “tie”2 Russell Johnston The copula itself is defined on the unit n -cube[0,1] n having uniformly distributed marginaldistributions. For the 2-dimensional case, the copula is a function C : [0 , → [0 , whichhas the following properties:1. C is grounded: that is, ∀ u, v ∈ [0 , , C ( u, 0) = 0 and C (0 , v ) = 0 C is 2-increasing: thus, ∀ u , u , v , v ∈ [0 , such that u ≤ u and v ≤ v , yielding, C ( u , v ) − C ( u , v ) − C ( u , v ) + C ( u , v ) ≥ ∀ u, v ∈ [0 , , C ( u, 1) = u and C (1 , v ) = v (146) Lastly, the Fr´echet–Hoeffding lower and upper bounds that are defined for every copula C and every (u,v) ∈ [0 , are such that max( u + v − , ≤ C ( u, v ) ≤ min( u, v ) (147) There are a number of forms the copula can take and Takeuchi explores two of them: theFarlie-Gumbel-Morgenstern (FGM) copula and a Gaussian copula. For illustration of themethod and simplicity only the Gaussian form is considered here. To measure the depen-dence between the two distribution functions F ( x ) and F ( x ) the linear Pearson product-moment correlation function is applied. However, it is noted that to explore the non-lineardependencies, the Kendal τ (Kendall 1938) or Spearman’s ρ s (Spearman 1904) estimatorswould be preferred.The Gaussian copula C G is defined as C G ( u , u ; ρ ) = Φ [ Φ − ( u ) , Φ − ( u ); ρ ] (148)where Φ is the standard normal CDF (with mean zero and variance of unity) and Φ is thebivariate CDF with correlation ρ . By differentiating C G w.r.t u and u one obtains the C G density function shown to be given by c G ( u , u ; ρ ) = 1 √ det Σ exp (cid:26) − h Ψ − T ( Σ − − I ) Ψ − i(cid:27) (149)where Φ − ≡ [ Φ − ( u ) , Φ − ( u )] T and I is the identity matrix. Σ is the covariance matrixgiven by Σ = (cid:18) ρρ (cid:19) (150)Thus if we now denote the univariate LFs as φ (1)1 ( L ) and φ (1)2 ( L ) then the bivariate PDF φ (2) ( L , L ) is described by a differential copula as φ (2) ( L , L ) ≡ c h φ (1)1 ( L ) , φ (1)2 ( L ) i φ (1)1 ( L ) φ (1)2 ( L ) (151) It should be noted that Equation 151 is the correct form for the expression of φ (2) ( L , L ) . The equiv-alent expression given by Equation 32 in Takeuchi (2010) contained a typographical error.hedding Light on the Galaxy Luminosity Function 63 Fig. 16 Extract from Takeuchi (2010): The analytical BLF constructed with the Gaussian copula with modelLFs of UV- and IR-selected galaxies. The BLFs are normalised so that integrating over the whole ranges ofL1 and L2 gives 1. The linear correlation coefficient ρ varies from 0.0 - 0.9 from top left to bottom right. Thecontours are logarithmic with an interval ∆ log φ (2) = 0 . drawn from the peak probability. Thus, under the Gaussian copula the BLF is finally given by φ (2) ( L , L ; ρ ) = 1 √ det Σ exp (cid:26) − h Ψ − T ( Σ − − I ) Ψ − i(cid:27) × φ (1)1 ( L ) φ (1)2 ( L ) (152)where Φ − ≡ h Φ − (cid:16) φ (1)1 ( L ) (cid:17) , Φ − (cid:16) φ (1)2 ( L ) (cid:17)i T (153)Figure 16 shows a summary of results from the above application of the method using theGaussian copula to construct the FUV-FIR BLF. The marginal distributions in this appli-cation are given as a standard Schechter LF (Equation 5 on page 3) for the FIR LF and the log-Gaussian LF (Equation 13 on page 4) as defined by Saunders et al. (1990) for theFIR LF. It is demonstrated that if the linear correlation between two variables is strong theGaussian copula is a useful way in which to construct this type of distribution functionsince it connects two marginal distributions and is directly related to the linear correlationcoefficient between the two variables. 10 From Tests of Independence to Completeness Estimators This review has had a thorough look at many of the non-parametric and parametric methodsused to determine LFs. However, as discussed § Φ ( M, Z ) = φ ( M ) ρ ( Z ) (154)The quantity Z now refers to the distance modulus which is used instead of redshift through-out this section. The distance modulus is related to redshift and magnitude in the followingway, Z = m − M = 5 log [ d L ( z, Ω m , Ω Λ , H )] + 25 (155)This is a fundamental and crucial assumption that is generally accepted over small red-shift bins (or shallow redshift surveys). However, what is sometimes overlooked is whetherthe assumption of separability is valid. That is, after all corrections have been made fore.g. evolution, K -correction etc., are the observables absolute magnitude M and distancemodulus Z , of Equation 154 statistically independent?10.1 Efron and Petrosian independence testIn order to test this assumption of independence (or separability) between two variables,Efron & Petrosian (1992) developed a simple ranked-based non-parametric test statistic fordata-sets with a single truncation. Their method is, in principle, an extension of the C − method. For this technique Efron and Petrosian construct the Kendall (Kendall 1938) teststatistic, τ , based on the rejection of independence between the random variables M and Z under test. Although they give a very detailed and formal explanation, only the mainpoints are summarised here. To illustrate this test refer back to Figure 7 on page 30, whichschematically shows the construction of the C − method.If, for the moment, we imagine the most simplest scenario where we are not hamperedby an apparent magnitude limit m flim , then M and Z would be independent assuming wehave a complete sample. Consequently, the rank, R i , of M i , would be uniformly distributedbetween 1 and the number of galaxies in the set, N gal , with respected expectation value andvariance, E = 12 ( N + 1) , V = 112 ( N − . (156)The quantity T for each galaxy is then defined as T i = ( R i − E ) V , (157) hedding Light on the Galaxy Luminosity Function 65 Fig. 17 Schematic illustrating the construction of the Efron and Petrosian (1992) test of independence.In order for this test to be applied correctly it is crucial to account for the presence of the faint apparentmagnitude limit, m flim which introduces a correlation between M and Z . Thus, drawing from the basic ideasof the C − method, it is possible to construct a separable region, S , for each galaxy located at ( M i , Z i ) withinwhich one can now can construct the τ statistic of independence. such that R i is now normalised to have mean of zero and a variance of unity. The hypoth-esis of independence is then rejected or accepted depending on the value of T i . One thenconstructs confidence levels of rejection by combining T i into the single statistic τ such that τ = P i ( R i − E ) rP i V (158)where, by definition, a τ of 1 indicates a 1- σ correlation and conversely, a τ of 0 indicatesthat the variables are completely uncorrelated.Now, if the magnitude limit m flim is re-introduced as in Figure 17, the above method hasto be modified to correctly account for the magnitude limit breaking the separability between M and Z . This requires the construction of subsets for each galaxy located at ( M i , Z i ) tocorrectly calculate the ranks R i of the entire set of observables. Therefore, within each set,all objects which could have been observed up to m flim limit are counted such that for eachgalaxy an area S is defined: S ≡ ∞ < M i < M lim and < Z i < Z max ( M i ) , as illustrated in Figure 17. Within this subset R i is uniformly distributed between 1 and N i ( S ) , where N i ( S ) is the number of objects in each subset. With this amendment to themethod, the construction of the τ statistic remains the same with τ once again being definedas, τ = P i ( R i − E ) rP i V (159)where the expectation value E and variance V remain unchanged from Equation 156. Thestatistic τ thus gives us an unbiased measure of the correlation of the data-set whilst properlyaccounting for the apparent magnitude limit.In Efron & Petrosian (1999) they extend this method for data which have a double trun-cation i.e. a survey that has distinct faint and bright flux limits. The natural progression wasto explore cases where the value of | τ | ≥ 1, which implies random variables M and Z cannotbe considered separable. In this case they assume that there is some form of luminosity evo-lution in the underlying data. And so by adopting a parametric form for luminosity evolutionthat varies with an evolutionary parameter, usually referred to as β , they were able to showthat for a particular value of β , τ ( β ) would equal 0.Maloney & Petrosian (1999) apply these methods on various quasar samples to deter-mine the density functions and the luminosity evolution. These techniques have also beenused by Hao et al. (2005) for AGN’s to test the correlation between the host galaxy and nu-clear luminosity. Kocevski & Liang (2006) applied this to constrain luminosity evolutionarymodels of Gamma Ray Bursters (GRBs).10.2 The T c and T v completeness estimatorsIn Rauzy & Hendry (2000) the authors drew upon the ideas from Efron & Petrosian (1992)to develop a non-parametric method to fit peculiar velocity field models. This methodologywas later extended in Rauzy (2001) as a completeness test. The motivation for this statisticalmethod was to introduce a non-parametric test that could determine the true completenesslimit in apparent magnitude of a magnitude-redshift survey whilst retaining as few modelassumptions as possible and being independent of spatial clustering.The fundamental approach of the Rauzy completeness is the same as Efron and Petrosiani.e assumption of separability between the absolute magnitude, M , data and the distancemodulus Z data. As demonstrated in the previous section, the presence of a faint apparentmagnitude limit introduces a correlation between the variables M and Z for observablegalaxies. Thus to retain the assumption of separability and construct a completeness statistic,the author defines a separable region defined by the random variable ζ . The left-hand panelof Figure 18 illustrates how the variable ζ is constructed such that it satisfies hedding Light on the Galaxy Luminosity Function 67 Fig. 18 Schematic construction the random variable, ζ , for the Rauzy (2001) completeness test (left-handpanel) and the Johnston, Teodoro, & Hendry (2007) extension (right-hand panel). In the left-hand plot theconstructed regions S and S are defined for ‘trial’ faint apparent magnitude limit m f ∗ for a typical galaxy at ( M i , Z i ) . Also shown is the true faint apparent magnitude limit m flim , within which the rectangular regions S and S contain a joint distribution of M and Z that is separable. The right-hand plot now shows theextension where one accounts for the presence of a secondary bright apparent magnitude limit. The S and S regions are uniquely defined for a slice of fixed width, δZ , in corrected distance modulus, and for ‘trial’faint apparent magnitude limit m f ∗ . Also shown are the true bright and faint apparent magnitude limits m blim and m flim , within which the rectangular regions S and S once again contain a joint distribution of M and Z that is separable. ζ = F ( M ) F [ M lim ( Z )] ≡ S S ∪ S , (160)where F ( M ) is the Cumulative Luminosity Function (CLF) defined as F ( M ) = Z M −∞ f ( x ) dx. (161)It follows immediately from its definition that ζ has the property of being uniformly dis-tributed between 0 and 1. The random variable ζ can be estimated directly from the data,without prior knowledge of the functional form of the CLF and is independent of the spa-tial distribution of galaxies. Therefore, the estimator ˆ ζ i can be combined for each observedgalaxy into a single statistic, T c , which can then be used to test the magnitude completenessof a given sample for adopted trial magnitude limits m f ∗ . T c is defined as T c = N gal X i =1 (cid:18) ˆ ζ i − (cid:19), N gal X i =1 V i , (162)where V i is the variance. If the sample is complete in apparent magnitude, for a giventrial magnitude limit, then T c should be normally distributed with mean zero and vari-ance unity. If, on the other hand, the trial faint magnitude limit is fainter than the truelimit, T c will become systematically negative, due to the systematic departure of the ˆ ζ i distribution from uniformity on the interval [0 , . The random variable ζ , was applied by Fig. 19 Schematic construction of the rectangular regions S and S , defined for a typical galaxy at ( M i , Z i ) , which feature in the estimation of the variant completeness test statistic, T v . The left-hand panelillustrates how S and S are constructed for a survey with only a faint magnitude limit m flim , and are shownfor a trial faint limit m f ⋆ . The right-hand panel shows the case where the survey also has a true bright limit m blim (which we assume for simplicity is known), and the rectangles are constructed for trial bright and faintlimits m blim and m flim , respectively. Note that the construction of the rectangles is unique for a ‘slice’ offixed width, δM , in absolute magnitude. Rauzy, Hendry, & D’Mellow (2001) in conjunction with the Kolmogrov-Smirnov test andthe correlation coefficient ρ , to devise a method to fit luminosity function models to observ-ables. This method was applied to the Southern Sky Redshift Survey (SSRS2) sample ofda Costa et al. (1988).The Rauzy completeness test has since been applied to a wide variety of data exploringe.g. the completeness limits of the HI mass function in the HIPASS survey (Zwaan et al.2004), the HI flux completeness of ALFALFA survey data (Toribio et al. 2011), and fornearby dwarf galaxies in the local volume (Lee et al. 2009). A recent study by Devereux et al.(2009) adopted the Rauzy method to determine the completeness of multi-wavelength se-lected data.In Johnston, Teodoro, & Hendry (2007) they extended its usefulness to survey data thatare characterised either by two distinct faint and bright flux limits and/or the case wherethe presence of a bright limit is harder to detect. This is illustrated in the right-hand panelof Figure 18. In addition to this they constructed a new statistic called T v which was con-structed instead from the cumulative distribution function of the corrected distance modulusfor observable galaxies and denoted by H ( Z ) , such that H ( Z ) = Z Z −∞ h ( Z ′ ) dZ ′ . (163) hedding Light on the Galaxy Luminosity Function 69 Fig. 20 The Johnston, Teodoro, & Hendry (2007) T c and T v completeness test statistics applied to a SDSSEarly Type galaxy sample consisting of 30,000 objects. Right panel: shows the distribution of sources in M and Z which is well defined by the respective faint and bright apparent magnitude limits m blim =14.5and m flim =17.45 mag. Left panel: beginning the test at the bright magnitude limit, T c and T v are estimatedfor each trial apparent magnitude limit, m ∗ , moving toward and beyond the faint limit of the survey inincrements of 0.1. The y -axis indicates the completeness limits, where, for a complete sample, it is expectedto fluctuate about zero. The true limit of the survey will be indicated by a systematic drop in T c or T v belowpre-determined limits, in this case − σ . The vertical dashed lines indicate the respective limiting magnitudesof the survey sample, m blim and m flim . Thus, a new random variable, τ , was defined that mirrored the ζ variable and was constructedfor the two cases, Case I : τ = H ( Z ) H [ Z max ( M )] ≡ S S ∪ S . (164) Case II : τ = H ( Z ) − H [ Z min ( M − δM )] H [ Z max ( M )] − H [ Z min ( M − δM )] ≡ S S ∪ S , (165)as illustrated, respectively, on the left- and right-hand panels in Figure 19. As with the T c statistic, one can combine the estimator ˆ τ i for each observed galaxy into a single statistic, T v , which we can use to test the magnitude completeness of our sample for an adopted trialmagnitude limit m f ∗ . T c is defined as T v = N gal X i =1 (cid:18) ˆ τ i − (cid:19), N gal X i =1 V i . (166) T v has the exact properties as T c such that if the sample is complete in apparent mag-nitude, for a given trial magnitude limit m ∗ , T v should be normally distributed with meanzero and variance unity.Figure 20 illustrates how the test works in practice when applied to real survey data. Thesample under test comprises of early-type galaxies selected from the SDSS as described inBernardi et al. (2003b, 2005). To summarise, the galaxies have been selected within themagnitude range . < m r < . within a redshift range . < z < . . The resulting distribution in absolute magnitude M and distance modulus Z is shown on the right-handpanel of Figure 20. It should be noted that the magnitudes have been K-corrected. The left-hand panel shows the resulting completeness test for T c (black solid line) and T v (red solidline). Since this survey sample is well defined by a faint and bright apparent magnitudelimit, the Johnston, Teodoro, & Hendry (2007) estimators are applied, adopting a respectivefixed δZ and δM width of 0.2 for T c and T v . For both estimators the test begins with atrial magnitude limit equal to the bright limit of the sample m ∗ = 14 . . T c and T v arethen estimated for increasing values of m ∗ in increments of 0.1. Note that the test does notregister a result for either estimator until there are enough objects to be counted for the ζ and τ random variables. In this example this corresponds to an m ∗ = 14 . . As the figure shows,both T c and T v fluctuate about zero within approximately the | | σ limits. Both test statisticsthen systematically drop sharply below − σ at the actual magnitude limit of the sample. Forthis sample under test the completeness test indicates that there are no residual systematicsbetween . < m r < . and that the data are therefore complete in apparent magnitudeup to the published survey limit.In Teodoro, Johnston, & Hendry (2010) extensions to the method were made to gen-eralise the approach. This extension based the completeness calculation on a minimum (orconstant) signal-to-noise level computed directly from the ζ (and/or τ ) defined regions. Withthe introduction of a secondary bright limit (Figure 18-right) one is now restricted with thevolume subsample size for each galaxy. In studying the distribution of galaxies in the ( M, Z )plane one is trying to understand the underlying luminosity function of a given population ofgalaxies as well as the way that such a function is sampled. To do so one uses a finite numberof galaxies to estimate whatever measurements. This makes them prone to shot noise andalso to the existence of some spurious features of the global properties of the data-set. 11 Applications This review has focussed primarily on the statistical methodology surrounding the luminos-ity function in the context galaxy populations. This section now focuses on selected areaswithin astronomy where the LF has had a strong impact.It should firstly be noted that the LF has not been confined just to the study of galaxypopulations. The stellar LF has had an equally long history (see e.g. Salpeter 1955; Sandage1957; Schwarzschild & H¨arm 1958; Limber 1960; Simoda & Kimura 1968; Miller & Scalo1979, for early studies). In particular, the stellar LF has proven useful when constrainingstellar evolution rates (e.g. Sandquist et al. 1996; Zoccali & Piotto 2000; Hargis et al. 2004),age and distance determination to globular clusters as well as constraining stellar evolu-tion processes (see e.g. Bahcall & Yahil 1972; Renzini & Fusi Pecci 1988). More recentlythe Century Survey Galactic Halo project utilised the spectroscopic redshift surveys, SDSS(York et al. 2000) and 2MASS (Cutri et al. 2003), to survey blue horizontal branch (BHB)stars within the Galactic halo (Brown et al. 2003). In Brown et al. (2005) they applied theEEP88 step-wise LF estimator to construct the BHB LF and compare with globular clusterdata. Other studies by e.g. Covey et al. (2008) explores the LF and mass functions of lowmass stars extracting data from the same surveys. The Hubble Space Telescope has also beenpivotal for the study of young star clusters within the Milky Way, and their correspondingLF studies have proved useful in probing the underlying mass function (e.g. Zhang & Fall1999; Gieles et al. 2006; Devine et al. 2008; Portegies Zwart et al. 2010). The white dwarfluminosity function (WDLF) is another galactic source which provides useful constraints hedding Light on the Galaxy Luminosity Function 71 on star-formation rate and history of the Galactic disk (e.g. Liebert et al. 1988; Wood 1992;Harris et al. 2006).In terms of extragalactic sources, there are extensive LF studies covering a broad rangewavelengths which include: UV selected Lyman break galaxies (LBGs) to probe the veryhigh redshift Universe out to z ∼ (e.g. Steidel et al. 1996, 1999; Madau et al. 1996;Sawicki & Thompson 2006; Bouwens et al. 2008b; Stanway et al. 2008); HI selected galax-ies (e.g. Zwaan et al. 2001); and the radio LF (e.g. Willott et al. 2001; Auriemma et al.1977; Kaiser & Best 2007); mid- and far-infrared (Hacking et al. 1987; Saunders et al. 1990;Pozzi et al. 2004; Caputi et al. 2007; Rodighiero et al. 2010); sub-mm (e.g Dunne et al. 2000;Vaccari et al. 2010); X-ray (e.g. Ueda et al. 2003; Hasinger et al. 2005; Brusa et al. 2009;Aird et al. 2010); and gamma ray bursters (GRBs) (e.g. Schmidt 2001; Wanderman & Piran2010).The remainder of this section discusses in more detail some of the recent highlights re-sulting from optical LF studies at low and intermediate redshifts as well the LF contributionto probing the evolutionary processes of QSOs.11.1 The optical luminosity functionThe review by BST88 gave a very comprehensive overview of the impact of galaxy LFstudies up to the end of the 1980s. This section now examines the work carried out byvarious groups from the 1990s to present day, and summaries the developments in probingthe optical LF from low redshift ranges ( z . . ) up to the more intermediate redshifts z . . . Prior to the emergence of large redshift surveys such as the Two Degree Field Galaxy Red-shift Survey (2dFGRS) and the Sloan Digital Sky Survey (SDSS), obtaining large completesamples of the local Universe was largely limited to the technological constraints of obtain-ing spectroscopic redshifts. Nevertheless, throughout the 1990s surveys such as Stromlo-APM Redshift Survey (S-APM) (Loveday et al. 1992), Las Campanas (LCRS) (Lin et al.1996) and the ESO Slice Project (ESP) (Zucca et al. 1997) made significant progress toconstrain the LF to . < z < . of field galaxies.In 1992 the S-APM survey measured b J magnitudes of 1769 galaxies out to b J =17 . mags at a redshift z ∼ . . They applied the MLEs of STY79 and EEP88 to esti-mate the LF and constrained Schechter parameters M ∗ = − . ± . , a faint end slopeof α = − . ± . and a normalisation φ ∗ = (1 . ± . × − Mpc − . LF resultsfrom the CfA survey were then published by Marzke et al. (1994b) using a sample of 9063galaxies out to m z ≤ . and within a range − ≤ M z ≤ − . By applying the STY79and the EEP88 methods they determined an overall M ∗ = − . ± . , α = − . ± . anda normalisation of φ ∗ = (4 . ± . × − Mpc − (see also Marzke et al. 1994a, for theirextended analysis to morphological types). By 1996, the LCRS had imaged 23,690 objectsfrom which 18,678 galaxies were selected out to a limiting r -band magnitude of r . . and an average redshift of z = 0 . . It was one of the first to use multi-fibre spectroscopy,allowing between 50 and 112 redshifts to be measured simultaneously. They too adopted thesame LF estimators to the data and measured LF parameters roughly consistent with S-APMsurvey. Table 4 shows a summary of the results to compare to other redshift surveys. Fig. 21 Reproduced from Norberg et al. (2002) by permission. The top panel shows the 2dFGRS LF resultsusing the non-parametric SWML and the parametric MLE estimators and compared to the SDSS LF resultsfrom Blanton et al. (2001). The bottom panel shows comparisons with Zucca et al. (1997) and Loveday et al.(1992). One year later, the ESP survey published LF results that pushed approximately 2 mag-nitudes fainter, out to b J = 19 . mag but with a smaller sample of 3342 galaxies out to amedian redshift at z ∼ . . To estimate the LF they use the STY79 MLE and the modi-fied non-parametric Lynden-Bell C − estimator called, ‘ C + ’, as described by Equation 81on page 32. Whilst the M ∗ and φ ∗ values were consistent with the previous two surveys,the faint-end slope was considerably different, yielding a slope of α = − . . The discrep-ancy in the faint-end slopes was thought to be more an effect of incompleteness and samplevariance toward the faint end in LCRS and S-APM.However, as multi-fibre technology improved surveys like the 2dFGRS (Colless 1998)marked a new era of mapping the local Universe. With a final catalogue of ∼ , measured spectroscopic redshifts out to a z max ∼ . and a limiting magnitude of b J . . mag, the 2dFGRS was, at this point, the largest redshift survey compiled. Initial LFanalysis was performed by Folkes et al. (1999) using a preliminary sample of 5869 galaxies hedding Light on the Galaxy Luminosity Function 73 in the range . ≤ z ≤ . to b J = 19 . mag. They applied the standard Schmidt V max and STY79 estimators and constrained overall values of M ∗ = − . and α = − . , whichshowed a much closer consistency with the ESP LF parameters. As the survey reachedits conclusion, later studies by Madgwick et al. (2002) (examining the LF as a functionof spectral type) and Norberg et al. (2002), using respective sample sizes of 75,000 and110,500 galaxies, would confirm these original results with global LF parameters of M ∗ = − . , α = − . and , φ ∗ = (1 . ± . × − Mpc − and a luminosity density of j b J = ( . ± . ) × hL ⊙ Mpc − from Norberg et al. (2002).Finally, the advent of SDSS (York et al. 2000) brought, for the first time, a single surveyimaging the sky in five bands ( u ′ , g ′ ,r ′ , i ′ , z ′ centred at 3540, 4770, 6230, 7630 and 9130 ˚ A respectively). It overtook the 2dFGRS in terms of the number of objects and remains, forthe time being, the largest photometric and spectroscopic survey compiled to date. Withthe final catalogue containing ∼ , , spectroscopically confirmed galaxy redshifts,SDSS represents the most accurate map of the nearby Universe out to z . . . Particularto this survey was the use of non-standard filters (see Fukugita et al. 1996, for more details)which required converting to make direct comparisons to existing surveys. The first LFanalysis reported by SDSS was by Blanton et al. (2001) using the a sample 11,275 galaxiesof the commissioning data. As well as computing LFs all the SDSS bands, they were ableto use the five-band colour information to convert their magnitudes into previous surveybands to make an accurate comparison of previous LF results. They converted to LCRS r -band and b J (to compare to 2dF). These results are also shown on Table 4. Generally,their measured LF parameters were closer to that of 2dF than LCRS. However, they notedtheir luminosity densities were 1.4 times higher to that of 2dF, as measured by Folkes et al.(1999) reporting j r = ( . ± . ) × hL ⊙ Mpc − , concluding that previous surveys hadmissed a considerable fraction of luminosity density in the Universe. Figure 21 shows LFcomparisons of the S-APM, ESP and the SDSS Blanton et al. (2001) survey samples takenfrom the Norberg et al. (2002) paper.Around two years later, Blanton et al. (2003b) compiled a sampled from the second datarelease SDSS-DR2 consisting of 147,986 galaxies to measure the LF at z=0.1 in all SDSSbands. The main difference in their analysis with the previous paper was the incorporationof evolutionary model into the LF. This resulted in an r ∗ -band Schechter function fit with M ∗ = − . ± . and α = − . ± . , and a newly computed luminosity density of j r = ( . ± . ) × hL ⊙ Mpc − which was now consistent with the 2dF.The Millennium Galaxy Catalogue (MGC) (Liske et al. 2003) used B -band imagingon the Wide Field Camera on the 2.5m Issac Newton Telescope to produce higher qualityphotometry compared to 2dF and SDSS. The survey was designed to be fully containedwithin the 2dF and SDSS-DR1 fields and imaged 10,095 galaxies out to a limiting magnitude B MGC < . mag within a redshift range . < z < . . In Driver et al. (2005)they applied their bivariate Step-Wise LF estimator (as detailed in § M ∗ =19.60 ± α = − . ± φ ∗ = ± M ∗ value was in line with 2dF, the recovered faint-end slopewas flatter. Their resulting luminosity density of j b J = ( . ± . ) × hL ⊙ Mpc − wassimilar to that of 2dF.A more recent analysis by Montero-Dorta & Prada (2009) has included 516,891 galaxiesin the r ∗ -band from SDSS-DR6, achieving a much greater redshift completeness than theprevious analyses. Their results made accurate constraints on both the bright and faint endsof the LF with r ∗ -band Schechter fits of M ∗ = − . ± . and α = − . ± . .They reported notable differences in the bright-end of the best-fit LF of an excess at the M u h − . . The excess then weakens moving in the g -band and fades away toward the z -band. It was concluded that this excess may be the result of QSOs. Our recent ability to probe toward higher redshifts (z . . ) with significant enough numberstatistics, has opened a new window to the evolutionary processes of different galaxy popu-lations through the study of their LFs. However, it should be noted that making comparisonsbetween various surveys has been a tricky process largely due to the ways in which galaxytypes have been defined e.g. by spectral type or by morphology (see e.g. Ilbert et al. 2006,for a discussion). Nevertheless, over the last 15-20 years, an overall consistent picture ofgalaxy evolution at this redshift range seems to be reaching convergence (for reviews seeKoo & Kron 1992; Ellis 1997; Faber et al. 2007).Through the mid to late 1990s work in this area was largely led by groups such asthe Canada-France Redshift Survey (CFRS) (Lilly et al. 1995), Autofib I & II (Ellis et al.1996; Heyl et al. 1997), the Hawaii Deep Fields (Cowie et al. 1996), the Canadian Networkfor Observational Cosmology survey (CNOC1 & 2) (Lin et al. 1997, 1999) and the NorrisSurvey of the Corona Borealis Supercluster (Small et al. 1997; Lin et al. 1999). Whilst thesesurvey samples typically ranged from hundreds to just under 2000 galaxies, a clear pictureof evolution between ‘red’ early-type galaxies and ‘blue’ star-forming late-type galaxieshad emerged in which star-forming galaxies were observed to have more rapid and strongerevolutionary properties compared to early-type galaxies. Table 5 provides a summary theseand other surveys related to this study.The CFRS, using a sample of 730 I -band selected galaxies, published results that seemedto lend support to the monolithic collapse scenario (e.g. Eggen et al. 1962; Searle & Zinn1978) in which galaxies originate from large regions of primordial baryonic gas which thencollapses to form stars within the central region of a dark matter halo, thus allowing the mostmassive galaxies to form first. They found that the LF of red galaxies showed no evidence foreither number density or luminosity evolution over the range < z < . . However, theirresults for the blue population of galaxies indicated evolution in the form of a steepening inthe faint-end slope of the LF.The Autofib Redshift Survey explored both the global evolution properties of the B-bandLF (Autofib I Ellis et al. 1996) and evolution by spectral type (Autofib II Heyl et al. 1997)out to z ∼ . and covering a range in magnitude . < b J < . . It was also this paperthat saw extensions made to both the SWML method of EEP88 and the MLE of STY79 forthe combining of multiple samples (see e.g. § z ∼ . . The early-type spirals showed some evidence for evolution with slight steepening of the faint endslope of the LF as a function of redshift. Overall there seemed no significant change in bothluminosity and number density for the early-type spiral galaxies. However, the late-typespirals showed significant evolution over the same redshift range in both luminosity andincreased number density. Moreover, using [O II ] emission as an indicator of star-formation,they also found a steep increase in such emission implying a rapid increase in star formationrate as a function of redshift.At around the same time, the CNOC redshift surveys were being carried out, whichby the end of the 1990s utilised a sample of ∼ galaxies to probe evolution of theLF between . < z < . within the magnitude range . < R C < . (Lin et al.1997, 1999, respectively known as CNOC1 and CNOC2). At this time it was the largest hedding Light on the Galaxy Luminosity Function 75 intermediate-redshift galaxy survey. They classified galaxies into early-types, intermediate-types and late-types, estimating rest-frame B , R C , and U band LFs. Applying both theSTY79 and EEP88 estimators they found that when looking back in redshift they observedtheir early-type sample showing an increasing luminosity evolution but with a reduced num-ber density. Their intermediate-types showed no evidence for number evolution but indicateda slight increase in luminosity. The late-type sample showed strong evidence for numberevolution with stronger star formation at higher redshifts. Overall, they reported distinct sep-aration of luminosity (early-types) and density evolution (late-types). It was noted that dueto differences in both the way galaxies were classified and evolutionary models adopted,direct comparison of CNOC2 and Autofib was difficult. However, in general the CNOC2group found their results qualitatively consistent.The COMBO-17 (Wolf et al. 2003) survey marked a significant increase in the numberof galaxies surveyed from . < z < . , totalling 25,431 measured photometric redshiftsout to a limiting magnitude R < . . They provided a thorough study of evolution as afunction of LF and comoving luminosity density at different rest-wavelengths with compar-isons made to the CFRS, CNOC2 and CADIS (Fried et al. 2001) surveys. The galaxies weresplit into four types based on templates from Kinney et al. (1996) where Types 1,2,3 & 4were, respectively, categorised as E-Sa, Sa-Sbc, Sbc-SB6 & SB6-SB1. After accounting forthe differences in galaxy-type classification, they found their results to be consistent withCNOC2. They reported evolution of the LF was strongly SED type dependent in both den-sity normalisation and characteristic luminosity. For their early-type sample (Type 1) theyfound a faint-end slope of α = 0 . ± . which then steepened to α = 1 . ± . forbluer galaxies (Type 4). For the latest-type galaxies they found no real change in the den-sity normalisation, φ ∗ over their redshift range, but instead, found a progressively faint M ∗ .Conversely, their sample of early-type galaxies they found an order magnitude increase in φ ∗ from z = 1 . to present day. M ∗ , however, did become progressively fainter, albeit to asmaller degree compared to the latest-type galaxies.Bell et al. (2004) described a model independent method for segregating galaxies intoearly- and late-types by identifying the bimodal rest-frame color distributions of galaxiesout to z ∼ . This would prove an invaluable approach for future studies.The VIMOS-VLT Deep Survey (VVDS) (e.g. Ilbert et al. 2005) combined spectroscopicredshifts in the Chandra Deep Field South and photometric redshifts from COMBO-17 andmulti-colour images from the HST/ACS Great Observatories Origin Deep Survey (GOODS)to probe evolution out to z ∼ . . Analysis of this survey was performed in two differentways: in Zucca et al. (2006) they classified galaxies according to spectral type, in Ilbert et al.(2006) they instead look at galaxy morphology for classification.As with COMBO-17 Zucca et al. (2006) split their sample population of 7713 galax-ies into four spectral types, but based their classifications on different templates using CCWColeman et al. (1980). In this way the four types were classified as: E/S0 (Type 1), early spi-rals (Type 2), late spirals (Type 3) and irregulars (Type 4). For their LF studies they appliedtheir ALF algorithm and compared their results with previous surveys such as CNOC2 andCOMBO-17. The comparisons indicated general agreement with CNOC2 in terms of neg-ative density evolution for early-types and a strong positive density evolution for late-typegalaxies. However, unlike CNOC2, they did not observe luminosity evolution for early-types. Their comparison to COMBO-17 did yield notable differences in the shapes and nor-malisation of the LFs. For example, the main discrepancy that was reported concerned theevolution of the Type 1 galaxies. In COMBO-17 they saw very strong density evolutionwhich decreased with increasing redshift. Such evolution was not observed in the VVDSsample. It was thought this may be due to differences in galaxy classification. As a test they consolidated their Type 1 and 2 populations and re-estimated their LFs. Whilst thisseemed to reduce the discrepancy the differences in the slopes and normalisation of the LFsremained.In Ilbert et al. (2006) their classification scheme of the VVDS data was morphologi-cal, based on structural features of galaxies within which three main populations were es-tablished - disk-dominated populations, red bulge-dominated populations and blue bulge-dominated populations. As in Zucca et al. (2006) they applied their ALF algorithm to com-pute the LFs and presented results from the rest-frame B -band LF out to z = 1 . . In generalthey observed a strong dependency on the shape of the LF according to morphological type.At redshift range . < z < . the red bulge-dominated population LFs that indicateda shallow faint-end slope of α = 0 . ± . . This was was in stark contrast to the diskpopulation which showed a much steeper slope of α = − . ± . . Due to the irregulargalaxies being included in the disk population sample, it was expected that strong evolutionwould be observed. Instead, the opposite was found. It was thought the effects such as cos-mic variance and the domination of spiral galaxies in this sample were possible reasons forthis effect. With the red bulge-dominated population they found that in general there was adistinct number density evolution with the age of the Universe consistent with the hierarchi-cal scenario, and indicating that E/S0 galaxies are already established at z ∼ . Finally theblue bulge-dominated population was found to have strong evolution with a brightening of ∼ . mag between . . z . . .At around the same time as VDDS the Deep Extragalactic Evolutionary Probe 2 (DEEP2)(e.g. Willmer et al. 2006) survey team reported their B -band LF results of ∼ , galax-ies out to z ∼ . As well as examining the global LF they also provide analysis for sub-divided blue and red populations (using colour bimodality). As with previous studies theirfindings showed red and blue galaxies evolving differently. In general, the blue galaxiesshow strong luminosity evolution but little number evolution, whereas the red galaxies showstrong number evolution with little change in luminosity over the redshift range. Their re-sults showed similar trends to previous surveys such as COMBO-17.Faber et al. (2007) combined data from DEEP2 and COMBO-17 to produce a catalogueof 39,000 galaxies which helped reduce effects from cosmic variance and Poisson noiseby 7% and 15% per redshift bin. Using the colour bimodality approach as first employedby Bell et al. (2004), galaxies were classified into blue and red sub samples. They foundthat for a fixed number density moving toward higher redshifts, the blue population showeda brightening in magnitude. The red galaxies, however, showed almost no evolution for afixed absolute magnitude. Their Schechter LF fits showed good agreement with the originalanalysis in both DEEP2 and COMBO-17 at all redshifts. When combining their distantSchechter LF parameters with local estimates they concluded that the number density ofblue galaxies has remained the same since z = 1 , whilst the red galaxies have shown anincrease in number density by a factor of ∼ since z ∼ . This result is restricted togalaxies near L ∗ B and does not extend to more luminous galaxies.Finally, the more recent and ongoing COSMOS survey (Scoville et al. 2007) and the red-shift follow-up zCOSMOS (Lilly 2007; Zucca et al. 2009) have published LF studies haveusing the 10k bright sample which, so far, consists of 10, 644 objects (Zucca et al. 2009).This paper estimates both the global LF (gLF) in the range . < z < . and the LF asfunction of both spectrophotometric and morphological type. In terms of spectrophotometrictyping, galaxies were divided into similar categories as in Zucca et al. (2006) with Type 1 =E/S0, Type 2 = early spirals, Type 3 = late spirals and Type 4 = irregular and starburst galax-ies. As in previous studies, the ALF algorithm was applied to estimate all LFs. To allow forbetter constraints on luminosity evolution, the parameter α was fixed to the value obtained hedding Light on the Galaxy Luminosity Function 77 in the redshift range . < z < . . Overall, they found that evolutionary properties inboth luminosity and number density was consistent with previous VVDS studies. To betterconstrain evolutionary trends their Type 3 and 4 populations were combined. With this theType 1 galaxies showed both luminosity and number evolution with a brightening of M ∗ by ∼ . mag and a decrease in φ ∗ from the lowest redshift bin [0.1 to 0.35] to the highest[0.75 to 1.00]. Whilst the Type 2 population showed an overall brightening of ∼ . mag,there was no significant number evolution observed. Finally, the Type 3 + 4 galaxies showedevolution in both number and luminosity with a brightening of ∼ . mag and an increaseof φ ∗ with redshift.Their morphological studies saw galaxies classified into three groups: early-types, spi-rals and irregulars, and were analysed over the same redshift range as the spectrophotometrictypes. The summary of these results showed that early-type galaxies dominate the bright-end of the LF at low redshifts ( z < . ), whilst spirals tended to dominate at the faint-end.Their sample showed that at the redshift range . < z < . a positive luminosity evo-lution was observed in both the spiral and early-type populations. At the highest redshiftrange, z > . , the irregulars were observed to show strong increase in number evolutiontoward high redshifts.11.2 The QSO LFQuasi-stellar objects (QSO) were first identified in 1963 by Maarten Schmidt and earlypioneering work by e.g. Schmidt (1963, 1968, 1972); Schmidt & Green (1983) confirmedthat they had strong evolutionary properties with a steep increase in space densities withincreasing redshift. Central to understanding their evolutionary process is estimating thequasar luminosity function (QLF) as a function of redshift.Research in this field has garnered renewed interest in recent times due, in part, to sur-veys such as the 2dF and the SDSS which increased the number of QSO detections byorders of magnitude, vastly improving on number statistics, providing more comprehensivecatalogues to fainter limiting magnitudes. Furthermore, recent observational evidence byKormendy & Richstone (1995); Magorrian et al. (1998); Ferrarese & Merritt (2000) has es-tablished a relationship between the evolution of galaxies with their central supermassiveblack holes. Thus, the QSO luminosity function (QLF) can potentially provide strong con-straints on SMBH formation and evolution (Haiman & Loeb 2001; Yu & Tremaine 2002),constraints on structure formation models coupled with the environmental processes suchas AGN feedback (e.g. Small & Blandford 1992; Haehnelt & Rees 1993; Efstathiou & Rees1988; Kauffmann & Haehnelt 2000).Work by Boyle et al. (1988a,b) originally showed that at low redshifts the QLF can bewell described by a broken double power-law of the form Φ ( M, z ) = Φ ∗ . M − M ∗ ]( α +1) + 10 . M − M ∗ ]( β +1) (167)with a break at M ∗ and a bright-end slope α steeper than the faint-end slope β .The work performed during the 1980s and early 1990s using parametric fits to data hadbuilt a picture for the QSO LF at redshifts z . . which was described by a steep brightend slope ( α ∼ − . ) and a very flat faint-end slope ( β ∼ − . ) and showed strong evolu-tionary properties that could be described by fitting pure luminosity evolution models (PLE; see § z ∼ z < . .The AGN LF has also been studied extensively in the X-ray. Whilst early measurementsof the X-ray AGN LF by Boyle et al. (1993) showed evidence that evolution was followingthe standard PLE model, more recent results have also reported the AGN downsizing. Inthis scenario lower luminosity AGN peak in their comoving space density at lower redshifts( z . ), whilst higher luminosity AGN peak at higher redshifts z ∼ (e.g. Ueda et al. 2003;Barger et al. 2005; Hasinger et al. 2005; Babi´c et al. 2007; Brusa et al. 2009)The advent of the The 2-degree Field QSO redshift survey (2QZ) (Boyle et al. 2000;Croom et al. 2001, 2004) provided a substantial increase in the number of QSOs catalogu-ing a total of 23,338 within the magnitude range . < b J < . out to . < z < . .However, it was not until the 2dF-SDSS LRG and QSO (2SLAQ) survey (Richards et al.2005; Croom et al. 2009a) that evidence of downsizing was also observed. The 2SLAQ sur-vey combined photometry from SDSS-DR1 (e.g. Gunn et al. 1998; Stoughton et al. 2002;Abazajian et al. 2003) to produce a catalogue of 10,637 QSOs. 2SLAQ provided a sam-ple 1 magnitude deeper than 2QZ allowing to probe the faint end of the QLF out to b J < . mag and to a redshift of z < . . By applying a similar approach to construct the QLFas in Croom et al. (2004), they provided constraints on both the faint and bright end of theQLF, finding evidence for number evolution in the form of downsizing which had not beenobserved in earlier studies by e.g. Richards et al. (2005) (using early results from 2SLAQ)and Wolf et al. (2003) (using the COMBO-17 survey). Using this catalogue, Croom et al.(2009b) showed how the activity in low-luminosity active galactic nuclei (AGNs) peaks at alower redshift than that of more luminous AGNs.Accurately constraining the QLF at z ≥ is much more of a challenge where lownumbers of low-luminosity objects at high redshift have meant faint end of the QLF ismuch more poorly understood. As such, whether or not downsizing is observed at earlierepochs remains a pertinent question. However, there have been made a lot of efforts tobuild larger catalogues (e.g. Fan et al. 2000, 2001, 2003, 2004; Richards et al. 2006; Goto2006; Siana et al. 2008; Glikman et al. 2010; Willott et al. 2010). There have been a numberof studies of the faint-end slope at z ∼ (e.g. Hunt et al. 2004; Bongiorno et al. 2007;Fontanot et al. 2007; Siana et al. 2008). However, recent work by Ikeda et al. (2011) andinitial studies by Glikman et al. (2010) have both provided constraints on the QLF at z ∼ that, at first, yielded conflicting results.Glikman et al. (2010) using both Deep Lens Survey Wittman et al. (2002) and the NOAODeep Wide-Field Survey (Jannuzi & Dey 1999) found 23 QSOs candidates between . Survey Data m lim N gal Redshift LF Method ( z ) LF studies out to z . . CFRS (Lilly et al. 1995) . < I AB < . < z < . STY79, EEP88Autofib I (Ellis et al. 1996) . < b J < . < z . . 75 1 /V max , EEP88Hawaii Deep Fields (Cowie et al. 1996) K = 20 . , I = 23 , B = 24 . . < z . . /V max CNOC1 (Lin et al. 1997) variable 389 0 . < z < . STY79, EEP88Autofib II (Heyl et al. 1997) . < b J < . < z . . STY79, EEP88NORRIS (Small et al. 1997) . r . . < z ≤ . EEP88HST imaging of CFRS & Autofib 341 z . . /V max (Brinchmann et al. 1998)CNOC2 (Lin et al. 1999) . < R C < . ∼ . < z < . STY79, EEP88CADIS (Fried et al. 2001) I = 30 . . < z . . /V max , STY79CFGRS (Cohen 2002) R < . . < z < . STY79DEEP Groth Strip Survey (Im et al. 2002) . < I < . z . . /V max , STY79COMBO-17 (Wolf et al. 2003) R ≤ . ∼ , 000 0 . < z < . /V max , STY79ESO-S (de Lapparent et al. 2003) R C ≤ . . < z < . STY79, EEP88VIMOS VLT Deep (Ilbert et al. 2006) I AB = 24 . ∼ . < z < . /V max , C + , STY79, EEP88VIMOS VLT Deep (Zucca et al. 2006) I AB = 24 . . . z . . /V max , C + , STY79, EEP88DEEP2 (Willmer et al. 2006) R AB ∼ . ∼ , z . . /V max (Eales 1993), STY79zCOSMOS (Zucca et al. 2009) ≤ I ≤ . z . . /V max , C + , STY79, EEP882 Russell Johnston 12 Concluding Remarks This review is an attempt to consolidate all the most innovative statistics developed forestimating the LF from their early beginnings 75 years ago to present day. Figures 22 and23 show a time-line diagram which charts the genealogical progress of all these estimators(a larger high-resolution version is available on request).Within the non-parametric regime it was discussed that whilst the traditional numbercount classical approach is straightforward in its construction, it is limited by its assumptionof spatial homogeneity - a limitation also shared with the /V max estimator. This intrinsicbias of V max has been demonstrated by W97 and TYI00 when direct comparisons to otherLF estimators showed overestimation of the LF particularly at the faint-end slope.In terms of the predictive power of the V /V max counterpart I would add a cautionarynote. Probing evolution in data can pose degenerate qualities making it difficult to deter-mine whether a significant departure from the expectation value of V /V max ( ≈ / ) is dueto inhomogeneity effects (introduced by clustering) or evolution, or simply an indication ofunderlying incompleteness of the catalogue from other sources of contamination. Neverthe-less, this seems not to have deterred its steady growth in popularity within the astronomicalcommunity as is evident from its numerous extensions to multiple surveys with varying fluxlimits, diameter-limited surveys, fitting generic LFs and adaptation to photometric redshiftsetc... In fact, /V max and V /V max remain one of the most widely applied non-parametricestimators for analysing the statistical properties of extragalactic sources. This is most likelydue to the ease with which it can be applied.The construction of the φ/Φ estimator and, in particular, the C − method offered a wayto effectively circumvent the assumption of homogeneity by the cancellation of the densityterms within their construction. The C − method could be considered as a breakthrough sincethe method required sampling the CLF and therefore no binning of the data was required(see Andreon et al. 2005, for an interesting discussion on the pitfalls of binning). And yet,despite the fact that it was pioneered just three years after Schmidt’s estimator and eightyears before φ/Φ , it did not grow in popularity as other tests did. Petrosian (1992) alsodemonstrated that all non-parametric methods are essentially variations of the C − methodunder the limit of having one galaxy per bin. It is therefore a slight mystery as to why thisapproach has not been applied more often. Perhaps this is due to other maximum likelihoodestimators (MLE) that developed as a result.Probably the most notable of the MLEs is the so-called STY79 estimator named afterits developers, Sandage, Tammann and Yahil in 1979. Its main appeal is that one must adopta analytical form for the LF and estimate its parameters via a maximum likelihood process.The form of the LF one chooses is rather arbitrary and in general the Schechter function(Schechter 1976) is favoured due to its robustness to various survey data. Approaching theproblem this way avoids many of the problems associated with binning techniques and alsoavoids issues of density inhomogeneities. However, one needs to assess the goodness-of-fitand determine the normalisation via independent means.EEP88 extended the ideas of Lynden-Bell’s C − method and the STY79 MLE by devel-oping the non-parametric case of the MLE by replacing the analytical form of the LF with aseries of step functions. This step-wise maximum likelihood (SWML) method has becomean equally favoured estimator in recent times. It, like V max has seen various extensions fore.g. bivariate distributions and combining multiple surveys to improve its usage.Also examined in detail were the more recent LF estimators that have emerged withinthe last few years that depart from the traditional approaches. The first one, developed bySchafer (2007), states key potential advantages for its construction that include making no hedding Light on the Galaxy Luminosity Function 83 strict assumption of separability between the probability densities ρ ( z ) and Φ ( M ) and in-corporating a varying selection function into the method. The second by Kelly et al. (2008)takes a Bayesian approach adopting a mixture of Gaussian functions for the prior distribu-tion. Finally, the more recent Takeuchi (2010) bivariate construction of the LF applied thecopula which connects two distribution functions. This parametric technique was applied tothe FUV-FIR BLF.The final part of the review shifted focus to one of the basic assumptions of most LFestimators - the assumption of separability between the luminosity function φ ( M ) and thedensity function ρ ( z ) . By building on Lynden Bell’s C − method, Efron & Petrosian (1992)constructed test statistic that can serve as a test of correlation between the assumed inde-pendent variables M and z and furthermore be used a to constrain pure luminosity models,where any such evolution would introduce correlations in the ( M, z ) distribution. It was thendemonstrated how Rauzy (2001) built upon the ideas in EP92 and turned them into a sim-ple but powerful robust non-parametric test of magnitude completeness. This completenesstest does not require any modelling of the LF and unlike V /V max is independent of thespatial distribution of sources. Moreover, it returns a differential measure of completenesswhere one can assess the level of incompleteness over a range of magnitudes by applyingsuccessive trial apparent magnitude limits.What has probably become obvious through this review is that choosing the correct LFestimator for your data is not a straight forward task. However, recent statistical advancesmean that there are a myriad of options available. This decision usually comes down to tasteand experience, and perhaps more crucially, how far one is willing to extend their analysisbeyond simple applications to obtain rigorous, if more computationally challenging, results.For now, the practice of applying several complementary LF estimators to a given data-setprovides a reasonable, if time consuming, approach.In closing, the notion that we have entered into an era of precision cosmology has beencited for more than a decade (Turner 1998) and can perhaps be justified when applied toCMBr measurements (e.g. Spergel et al. 2003; Larson et al. 2011). However, in the study ofgalaxy redshift surveys, the era of precision cosmology would appear to be approaching butwill require improvement in both the quality and size of our data-sets and, crucially, in ourstatistical toolbox before we can claim that is has truly arrived. Whether the more recentmethods will become as popular as the traditional ones remains to be seen, since shiftstowards seemingly more complex techniques can take time to catch on. Nevertheless, it isencouraging that this branch of astronomy continues to develop and embrace new statisticalmethods. With with the next generation of galaxy redshift surveys on the horizon, it will beinteresting to see how the current approaches to constraining the luminosity function willadapt and which new methods will come to prominence. Acknowledgements There are a number of people who deserve a lot of thanks. Firstly, I would like tothank David Valls-Gabaud for encouraging me to write this review and also taking the time to read theseveral drafts along the way. The following people also deserve my gratitude for reading various parts of thereview and/or providing useful comments and criticisms (in alphabetical order): Steve Crawford, Nick Cross,Andreas Faltenbacher, Martin Hendry, Chris Koen, Roy Maartens (for his title suggestion!), Prina Patel, MatSmith, Tsutomu Takeuchi, Luis Teodoro and Christopher Willmer. I would like to extend special thanks toJeremy Heyl, Olivier Ilbert, Brandon Kelly, Peder Norberg, Chad Schafer, Tsutomu Takeuchi, Rita Tojeiroand Christopher Willmer for being kind enough to allow me to include some of their figures - it saved me alot of time and helped bring more clarity to this review. I would also like to acknowledge the current supportof the National Research Foundation (South Africa) and the funding body Engineering and Physical SciencesResearch Council (EPSRC) whilst at Glasgow University (UK). Finally, thank you to Joel Bregman at A&ARfor his guidance and the anonymous referee for his/her helpful comments.4 Russell Johnston Fig. 22 Part 1 of a schematic charting the development of all the major statistical methods that estimate thegalaxy LF. A larger high resolution chart is available on request. Fig. 23 shows Part 2.hedding Light on the Galaxy Luminosity Function 85 Fig. 23 Part 2 of a schematic charting the development of all the major statistical methods that estimate thegalaxy LF. A larger high resolution chart is available on request.6 Russell Johnston References Aarseth, S. J. 1963, MNRAS, 126, 223Aarseth, S. J., Turner, E. L., & Gott, III, J. R. 1979, Astrophys. J., 228, 664Abazajian, K., Adelman-McCarthy, J. K., Ag¨ueros, M. A., et al. 2003, A J, 126, 2081Abazajian, K. N., Adelman-McCarthy, J. K., Ag¨ueros, M. 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