The Astrophysical Journal, 672:198–206, 2008 January 1 A

# 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A.

THE FAINT-END SLOPES OF GALAXY LUMINOSITY FUNCTIONS IN THE COSMOS FIELD1

Charles T. Liu,2 Peter Capak,3 Bahram Mobasher,4 Timothy A. D. Paglione,5 R. Michael Rich,6
Nicholas Z. Scoville,3 Shana M. Tribiano,7 and Neil D. Tyson8
Received 2006 August 11; accepted 2007 August 9

ABSTRACT
We examine the faint-end slope of the rest-frame V-band luminosity function ( LF), with respect to galaxy spectral
type, of field galaxies with redshift z < 0:5, using a sample of 80,820 galaxies with photometric redshifts in the 2 deg2
Cosmic Evolution Survey (COSMOS) field. For all galaxy spectral types combined, the LF slope ranges from
1.24
to
1.12, from the lowest redshift bin to the highest. In the lowest redshift bin (0:02 < z < 0:1), where the magnitude
limit is MV P
13, the slope ranges from

1:1 for galaxies with early-type spectral energy distributions (SEDs)
to

1:9 for galaxies with low-extinction starburst SEDs. In each galaxy SED category (early-type, Sbc, Scd+Irr,
and starburst), the faint-end slopes grow shallower with increasing redshift; in the highest redshift bin (0:4 < z < 0:5),


0:5 and
1.3 for early types and starbursts, respectively. The steepness of
at lower redshifts could be qualita-
tively explained by LF evolution, or by large numbers of faint dwarf galaxies, perhaps of low surface brightness, that
are not detected at higher redshifts.
Subject headingg s: cosmology: observations — galaxies: dwarf — galaxies: evolution —
galaxies: fundamental parameters — galaxies: luminosity function, mass function — surveys
Online material: color figure

1. INTRODUCTION largest contiguous area of the sky yet observed with the Hubble
Space Telescope (HST ), but also deep multiwavelength imag-
The luminosity function ( LF) of galaxies varies substantially
ing across the entire 2 deg 2 COSMOS field (Capak et al. 2007).
with respect to many key physical parameters such as galaxy
COSMOS thus affords deep, homogeneous photometric and pho-
morphology, environment, color, star formation rate, surface bright-
ness, and redshift. These many differences serve as powerful diag- tometric redshift coverage for a sample of some 106 galaxies, com-
plementing well at higher redshifts the largest galaxy surveys of
nostics of the broad tapestry of galaxy evolution.
the relatively nearby universe with which galaxy LFs have been
The most important requirement for the accurate derivation of
galaxy LFs is large, complete samples of galaxies with reliable derived, such as the Two Degree Field Galaxy Redshift Survey
(2dFGRS; Colless et al. 2001; Croton et al. 2005) and the Sloan
photometry and redshift determinations. The Cosmic Evolution
Digital Sky Survey (SDSS; Blanton et al. 2003; Bell et al. 2003;
Survey (COSMOS; Scoville et al. 2007a) contains not only the
Abazajian et al. 2004).
1
Comprehensive analyses of the LF characteristics of the entire
Based on observations with the NASA / ESA Hubble Space Telescope, ob- COSMOS galaxy sample will ultimately be forthcoming upon the
tained at the Space Telescope Science Institute, which is operated by the Asso-
ciation of Universities for Research in Astronomy (AURA), Inc., under NASA completion of the spectroscopic portion of the survey (Lilly et al.
contract NAS 5-26555; also based on data collected at Kitt Peak National Obser- 2007) and the continued addition and refinement of the photo-
vatory, Cerro Tololo Inter-American Observatory, and the National Optical As- metric and photometric redshift measurements in multiple band-
tronomy Observatory, which are operated by AURA, Inc., under cooperative passes (Capak et al. 2007; Mobasher et al. 2007). Already,
agreement with the National Science Foundation; at the Subaru Telescope, which
is operated by the National Astronomical Observatory of Japan; with XMM- however, it is feasible to address important scientific questions
Newton, an ESA science mission with instruments and contributions directly about galaxy LFs with the current optical and near-infrared multi-
funded by ESA Member States and NASA; at the European Southern Obser- band data (Capak et al. 2007; Taniguchi et al. 2007; Scarlata et al.
vatory under Large Program 175.A-0839, Chile; at the Canada-France-Hawaii 2007) in conjunction with the HST I814 broadband Advanced
Telescope with MegaPrime/ MegaCam, operated as a joint project by the CFHT
Camera for Surveys (ACS) imaging (Scoville et al. 2007b).
Corporation, CEA / DAPNIA, the National Research Council of Canada, the
Canadian Astronomy Data Centre, the Centre National de la Recherche Scien- One particular component of the field galaxy LF, the faint end
tifique de France, TERAPIX, and the University of Hawaii; and the National (MAB >
18) of the LF at low to intermediate (0 P z P 0:5)
Radio Astronomy Observatory, which is a facility of the National Science Foun- redshifts, is uniquely well suited for analysis with the COSMOS
dation operated under cooperative agreement by AURA, Inc. data. This population of galaxies lies in an apparent magnitude
2
Astrophysical Observatory, Department of Engineering Science and Physics,
City University of New York, College of Staten Island, 2800 Victory Boulevard,
range somewhat beyond the 2dFGRS and SDSS survey limits;
Staten Island, NY 10314. they are too faint for spectroscopic observations and have red-
3
California Institute of Technology, MC 105-24, 1200 East California shifts too low to be included in surveys optimized for high-redshift
Boulevard, Pasadena, CA 91125. galaxies. Yet exploring these galaxies’ contribution to the overall
4
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore,
MD 21218.
LF is critical for understanding galaxy evolution during the latter
5
City University of New York, York College, 94-20 Guy R. Brewer half of cosmic history.
Boulevard, Jamaica, NY 11451. Recent comprehensive studies of field galaxy LF evolution
6
Department of Physics and Astronomy, University of California, Los have focused on the luminosity evolution of the bright end
Angeles, CA 90095. (MAB P
20). There, some consensus appears to be gradually
7 City University of New York, Borough of Manhattan Community College,

199 Chambers Street, New York, NY 10007. emerging about the extent of that evolution. Dahlen et al. (2005),
8
American Museum of Natural History, Central Park West at 79th Street, Willmer et al. (2006), and Scarlata et al. (2007) all find a brightening
New York, NY 10024. of MB; by 1 mag in the range 0 P z P1. In shorter wavelength
198
 FAINT-END SLOPES OF GALAXY LFs 199

bandpasses, the luminosity evolution is more pronounced, while

at longer wavelengths it appears to be weaker and possibly even
present in the negative sense, dimming in the near-infrared J band
from z  0:4 to 0.9 (Dahlen et al. 2005).
The evolution of the faint-end slope, however, remains highly
uncertain. At low redshifts, results from the SDSS (Blanton et al.
2005; Baldry et al. 2005), the 2dFGRS (Croton et al. 2005), and
other large data sets (see, e.g., Brown et al. 2001; Budavari et al.
2005; Driver et al. 2007) roughly agree, for example, on a mod-
erate slope of

1:1. Beyond redshifts of a few tenths, the
faint-end slope becomes very difficult to address, mainly because
the number of low-luminosity galaxies detected in galaxy surveys
decreases dramatically with increasing redshift. Despite a number
of efforts to measure evolution in the faint-end slope at redshifts
less than z  1 (Wolf et al. 2003; Ilbert et al. 2005; Zucca et al.
2006), very little is known for galaxies fainter than M 
18.
This is in part because the relationship between the bright-end
Fig. 1.—Spectral energy distributions from Mobasher et al. (2007) used to
and faint-end characteristics of galaxy LFs is not straightforward, compute COSMOS galaxy spectral types and photometric redshifts. This figure
often resulting in a trade-off between the precision LF evolution is a reproduction of Fig. 1 of Mobasher et al. [See the electronic edition of the
measurements at the two ends. Baldry et al. (2005) and Willmer Journal for a color version of this figure.]
et al. (2006), for example, each select fixed LF faint-end slopes
based on galaxies brighter than M 
18 and use that constraint
tive comparisons with the results of other large field galaxy sur-
throughout their bright-end evolution measurements.
veys. Studies of the bright-end evolution of the galaxy LF from
Whether or not the faint-end slope evolves with redshift,
the COSMOS survey are given elsewhere (e.g., Scarlata et al.
however, it is clear that its steepness varies widely for galaxies of
2007), and a more detailed breakdown of the 0 P z P 0:5 galaxy
different morphological and spectral types, indicating substantial
population by redshift, galaxy spectral type, and galaxy morphol-
differences in the evolutionary histories of galaxies. Broadly
ogy will be presented in a future paper (C. Liu et al. 2008, in
speaking, irregular galaxies, ‘‘blue’’ galaxies, and strongly star-
preparation). Throughout this paper, we adopt a flat cosmology
forming galaxies, three significantly overlapping galaxy sub-
with
 ¼ 0:7,
m ¼ 0:3, and H0 ¼ 70 km s
1 Mpc
1.
populations, evolve more strongly than galaxies of other types.
Such galaxies are characterized by very steep faint-end LF slopes
2. GALAXY SAMPLE AND PHOTOMETRIC REDSHIFTS
and substantial evolution in luminosity and/or number density
at even moderate redshifts of z P 0:5 (Marzke et al. 1994; Lilly In our analysis, we use a compilation of the COSMOS optical/
et al. 1995; Ellis et al. 1996; Liu et al. 1998; Bromley et al. 1998; near-infrared data (Capak et al. 2007; Mobasher et al. 2007),
Lin et al. 1999). More recent work has further confirmed and quan- which includes observations with the HST ACS (I814 ), the Sub-
tified this trend at higher redshifts (see, e.g., Chen et al. 2003; aru Telescope (B, V, r 0, i 0 , z 0 , and NB816), the Canada-France-
Gabasch et al. 2004; Pérez-González et al. 2005; Dahlen et al. Hawaii Telescope (CFHT; u  and i  ), and the 4 m KPNO Mayall
2005). At low redshifts, evidence is mounting that composite and CTIO Blanco Telescopes (Ks ), as well as supplementary data
parameterizations may more accurately reflect the shape of the from the Sloan Digital Sky Survey. The data from the differ-
LF than the usual single-function ones (de Lapparent et al. 2003, ent telescopes were all matched to a common pixel scale and
2004; Blanton et al. 2005), and that, faintward of M 
18, the smoothed to the same point-source function. SExtractor (Bertin
power-law slope of the LF may differ quantitatively from the slope & Arnouts 1996) was then used in dual mode to generate a pho-
brightward of that threshold (Madgwick et al. 2002; Norberg et al. tometric catalog, selected using the Subaru i 0 and CFHT i  im-
2002; Blanton et al. 2005). This could arise, for example, due to ages. The limiting 3  AB magnitude is i 0 ¼ 26:03. A detailed
large numbers of low-luminosity ‘‘blue’’ galaxies ( Wolf et al. description of the imaging data, photometry, and photometric
2003) or low surface brightness galaxies ( Impey et al. 1996; calibration is given in Capak et al. (2007).
Blanton et al. 2005) that may have previously evaded detection. Photometric redshifts for individual galaxies were computed
In this paper, we present measurements of the faint-end slopes using the methods described in Mobasher et al. (2007). Six basic
of the rest-frame V-band luminosity functions of galaxies in the galaxy spectral types adapted from the four template types (E,
COSMOS survey at 0 P z P 0:5, focusing in particular on the Sbc, Scd, and Im) from Coleman et al. (1980) and the starburst
change in that slope as a function of redshift and galaxy spectral templates SB2 and SB3 of Kinney et al. (1996) were used. These
type. The depth and breadth of the COSMOS multiband pho- templates are presented graphically in Figure 1. These galaxy
tometry allows for reliable identifications of galaxy redshift and spectral types are derived from empirical data and represent the
spectral type, with robust redshift error estimates for each galaxy, range of non-AGN galaxy spectral energy distributions (SEDs)
to a limit of mAB  25 in the optical passbands. Even so, the rel- from redder to bluer colors; these starburst SEDs, for example,
atively large and varying redshift uncertainties of photometric represent very blue galaxies not significantly reddened or ob-
redshifts can present substantial quantitative challenges and sys- scured by dust. Interpolation was used between the six spectral
tematic biases (SubbaRao et al. 1996; Liu et al. 1998; Chen et al. types to produce a grid of 31 possible galaxy SED fits.
2003; Dahlen et al. 2005). We use Monte Carlo simulations to Using the multiband photometry, the COSMOS photomet-
characterize these biases and to recover the faint-end slopes of ric redshift code ( Mobasher et al. 2007) was used to derive a
galaxy-type–specific LFs. photometric redshift zp for each galaxy, with 68% and 95% con-
This work represents an initial study of the general properties fidence intervals computed above and below that value. To de-
of these type-specific faint-end LF slopes, to provide quantita- termine the accuracy of the code, the zp -values were compared
200 LIU ET AL. Vol. 672

Fig. 2.—HST ACS I814 and Subaru B images of a representative selection of very low luminosity galaxies in the COSMOS field. Each image is 1500 across, and each
image pair is labeled by the COSMOS catalog ID number, photometric redshift, and apparent and absolute V magnitudes of the object at the image center.

with spectroscopic redshifts (zs ) in 868 galaxies with z < 1:2 type, spiral, and starburst galaxies have values of rms( z) ¼ 0:034,
and iAB P 24 that had secure redshift measurements from the 0.030, and 0.042, respectively.
zCOSMOS survey (Lilly et al. 2007). The galaxy spectral types We chose to compute our luminosity functions using the
(20% early-type, 63% spiral, and 17% starburst) are evenly dis- V-band data for this study. The saturation limit for bright objects
tributed with redshift in the spectroscopic sample, and about half in that image was V  18:5, and the 3  faint-detection limit was
of the sample (45%) is at zs  0:5. V  26:4, so we conservatively chose as our apparent magnitude
The detailed statistics of the zp and zs comparisons are de- range 19:0 < mV < 25:0. Details of the photometry are described
scribed in Mobasher et al. (2007), primarily in terms of the param- in Capak et al. (2007). An extinction-corrected, rest-frame absolute
eter z ¼ (zp
zs )/(1 þ zs ). As given in Table 4 and Figure 5 of MV magnitude based on the derived zp and using the K-correction
Mobasher et al. (2007), the nominal dispersion between these for the best-fit spectral type was computed for each galaxy. Ground-
photometric redshifts and spectroscopically determined redshifts and space-based images of a representative subsample of the faint-
was rms(z) ¼ 0:033 for non-AGN galaxies. With respect to the est galaxies (MV 
16 and fainter) in this sample are presented
different spectral types, Mobasher et al. (2007) showed that early- in Figure 2.
No. 1, 2008 FAINT-END SLOPES OF GALAXY LFs 201

3. LUMINOSITY FUNCTIONS for z < zp and

As with all surveys that rely primarily on photometric rather PG (z þ z; z; u ) dz
than spectroscopic redshifts, the application of the COSMOS Nzþz ¼
AG (z þ z; z; u )
galaxy sample to the derivation of galaxy LFs requires great
care in order to account for both random and systematic errors for z > zp.
in the redshift determinations. Just as methods of computing This ‘‘fuzzing’’ of a galaxy’s luminosity distribution in red-
photometric redshifts have evolved and improved (Koo 1986; shift space is straightforwardly achieved numerically, with a
Connolly et al. 1995; Liu & Green 1998; Benı́tez 2000; Mobasher choice of dzT to minimize random magnitude errors. For this
et al. 2004, 2007), so too have the techniques with which to quan- COSMOS data set, we used dzl ¼ 0:02l and dzu ¼ 0:02u .
tify and compensate for the effects of relatively large redshift error This divides each galaxy into a Gaussian-weighted luminosity
bars in LF calculations (SubbaRao et al. 1996; Liu et al. 1998; distribution with 300 bins from z
3l to z þ 3u . The entire
Chen et al. 2003; Dahlen et al. 2005; Pérez-González et al. 2005). distribution for each galaxy is normalized to unity.
In this work, we adapt the method used in Liu et al. (1998), In the standard 1/Vmax method, each galaxy contributes a weight
updating it with additional components similar to those used in to the luminosity function that is equal to the inverse of the ac-
more recent studies (e.g., Chen et al. 2003; Dahlen et al. 2005; cessible volume within which it can be observed. The accessible
Pérez-González et al. 2005) to reproduce the faint-end slope of volume, referred to here as Vmax , is the total comoving volume
the LFs of COSMOS galaxies. Our strategy is based on the within the redshift boundaries of the sample in which the given
1/Vmax method (Schmidt & Green 1986), which is well described galaxy could be and fall within the selection criteria of the sam-
by Chen et al. (2003) as a maximum likelihood method with ple. In our case, the relevant criteria are the bright and faint appar-
which to estimate a luminosity function without assuming any ent magnitude limits and the effective solid angle of the COSMOS
parametric form. We account for photometric redshift errors by survey.
weighting the galaxies as probability-smoothed luminosity dis- In the case of a probability-weighted luminosity distribution
tributions at the redshifts where they are measured. for individual objects, it is straightforward to compute Vmax for
each fractional galaxy; correspondingly, its contribution to the
3.1. The Modified 1/Vmax Method luminosity function is (1/Vmax )Nzþz . Assembling the luminosity
Consider a galaxy with an apparent magnitude of mf in a pass- function is then a matter of summing those contributions within
band f and with a redshift of z  . If  ¼ 0, then the absolute absolute magnitude bins.
magnitude is
3.2. Redshift Limits and Sample Size
Mf ¼ mf
5 log ½dL (z)
25:0
kf (z); Since the primary goal of this work was to examine the LF
faint-end slope, the upper redshift boundary was determined
where dL (z) is the luminosity distance in units of Mpc and kf (z) mainly by our desire to sample with high completeness to at least
is the K-correction at that redshift, in that passband, for the spec- as faint as MV 
16:5 in the entire redshift range. For a typical
tral energy distribution of the galaxy. The contribution of that galaxy in the sample, depending on the galaxy’s K-correction,
galaxy to the luminosity distribution is then a delta function of this corresponds roughly to z P 0:4. But because each galaxy’s
amplitude unity at redshift z. luminosity is calculated as a probability-weighted distribution,
In the case in which  > 0 and the error distribution is Gaussian, there is a statistically significant contribution to the LF at more
the galaxy can be thought of as adding a series of fractional con- than a full magnitude beyond the formal absolute magnitude limit.
tributions to the luminosity distribution in the redshift space sur- Thus, with the caveat that we are beyond that limit, we were also
rounding z. Such a fraction at, for example, redshift z þ z and able to derive faint-end slopes of the LFs of galaxies in the redshift
with a differential redshift width of dz would have an absolute range 0:4 < z  0:5.
magnitude of Similarly, in the low-redshift range, we are also able to mea-
sure fainter in absolute magnitude than the formally faintest de-
Mf0 ¼ mf
5 log ½dL (z þ z)
25:0
kf (z þ z) tectable galaxy. However, to avoid large systematic magnitude
errors and biases from structure in the local universe, we set a
lower redshift bound of z > 0:02 for deriving the LFs. This means
and an amplitude of
that we have statistically meaningful luminosity contributions to
the LFs down to an absolute magnitude of MV 
12:8 for
PG (z þ z; z; ) dz
Nzþz ¼ ; 0:02 < z  0:1.
AG (z þ z; z; ) Within our apparent magnitude limits of 19:0 < V < 25:0,
the COSMOS survey photometric and zp catalog (Capak et al.
where PG and AG are the Gaussian probability function and its 2007) contains 49,161 galaxies in the redshift range 0:02 <
integral, respectively (see, e.g., Bevington & Robinson 1992). z  0:5. Below and above this redshift range, however, there are
For the photometric redshifts of the COSMOS survey, the red- galaxies within the apparent magnitude limits whose probability-
shift error distribution is not Gaussian, but rather can be modeled smoothed luminosity distributions contribute to the light within
as two half-Gaussians (Capak et al. 2007; Mobasher et al. 2007), that range. Using 68 to denote the width of the 68% confidence
where the 68% confidence interval on the lower and upper limits interval for zp, we thus also include the contributions of all other
are l and u , respectively. For a galaxy with a photometric red- galaxies whose luminosity distribution tails fall within 368 . (For
shift zp , the amplitude of each fractional contribution to the lumi- example, a galaxy with zp ¼ 0:51 and 68 ¼ 0:05 would con-
nosity distribution would be tribute the portion of its luminosity distribution from the range
0:36  z  0:50, while a galaxy with zp ¼ 0:61 and 68 ¼ 0:05
PG (z þ z; z; l ) dz would contribute from the range 0:46  z  0:50.) There are
Nzþz ¼
AG (z þ z; z; l ) 31,659 galaxies in the catalog that make such a partial contribution;
202 LIU ET AL. Vol. 672
TABLE 1
Galaxy Samples and zp Confidence Levels

Number of Galaxies Redshift Constraints 68 Constraintsa Median 68/(1 + z) Mean 68/(1 + z) rms 68/(1 + z)

41,237..................................... ... 0.1 0.042 0.041 0.053

49,161..................................... 0.02  z  0.50 ... 0.043 0.050 0.073
80,820..................................... ... ... 0.052 0.081 0.239

Note.—Galaxy spectral types as defined by Mobasher et al. (2007).

a
The quantity 68 is the width of the 68% confidence interval for zp.

even though many of those galaxies add only a tiny fraction of a of the magnitude and redshift limits of our survey, and for each
galaxy into the z  0:5 redshift range, we include them in our anal- galaxy detected, we randomly added an error to the redshift of
ysis for statistical completeness. Thus, a total of 80,820 galaxies that galaxy that was consistent with the measured dispersion of
are included in the sample used to derive the LFs. the COSMOS photometric redshift catalog, rms(z) ¼ 0:033.
To check how the inclusion of these galaxies might affect the Finally, we assigned to that galaxy a value of 68 equal to the me-
distribution of the zp accuracy in the sample as a whole, we dian value of 68 of the 80,820 galaxy sample; i.e., 0:052(1 þ zs ).
created a subset of the 80,820 galaxy sample in which the galax- The LF for this simulated galaxy sample was then computed using
ies have a value of 68 of at most 0.1. This subset of 41,237 gal- the modified 1/Vmax method described above.
axies (51% of the entire sample) effectively contains objects whose We generated 150 such simulated LFs for each value of
tested.
values of 68 are no more than 3 times the rms(z) of the spec- To illustrate the results of these simulations, we present three sets
troscopically tested accuracy of the COSMOS zp code. We then of them in Figure 3. The dashed lines show the input faint-end
computed the mean, median, and rms values of 68 /(1 þ z) for slopes (
¼
0:7,
1.1, and
1.5) for the simulations. Of the
the 41,237, 49,161, and 80,820 galaxy samples. We give the 150 simulations, we exclude the four outliers furthest above and
results in Table 1. As might be expected, the median, mean, and four furthest below the input value of
; the remaining 142 simu-
rms values all increase as the constraints in redshift and 68 are lations (i.e., 95%) yield results that fall within the envelope
lifted and the sample sizes grow. The median 68 /(1 þ z) value bounded by the solid lines above and below each dashed line.
increases only modestly on an absolute numerical basis (from The values of
represented by each of those solid and dashed
0.042 to 0.043 to 0.052); the mean and rms values increase more lines are given in the figure.
substantially. As Figure 3 shows, the ‘‘fuzzing’’ of the galaxies due to zp
uncertainties does cause systematic biases of the calculated faint-
3.3. Simulations end LF slope. At
’
1:1, the bias is negligible, and the fuzz-
Computing LFs using ‘‘fuzzy’’ galaxies with photometric red- ing basically just increases the uncertainty of the measured slope.
shifts is clearly vulnerable to a set of systematic errors that would For steeper input values of
, however, the output value of
is
not be present for a galaxy sample with secure spectroscopic quantitatively biased toward steeper values, and for shallower
redshifts. In this work, we use the standard galaxy LF parameter- input values of
, the opposite is true. The bias increases as the
ization of Schechter (1976), where M  is the characteristic magni- input values of
grow more extreme; an input of
¼
1:9, for
tude,  is the characteristic number density, and
is the faint-end example, produces output values of
in a 95% envelope in the
slope (Lin et al. 1996): range
2:3 P
P
1:8, whereas an input of
¼
0:4 pro-
duces a corresponding output of
0:5 P
P þ0:1.


(M ) ¼  ð0:4 ln 10Þexp
10
0:4(M
M ) 10
0:4ð M
M Þ(1þ
) dM : From all the simulation results, standard bootstrap methods
were used to estimate the expected systematic offsets in
for an
ð1Þ

In any given redshift bin, objects near the peak of the LF—
that is, near M  —will have part of their light distributed toward
brighter magnitudes, and objects at the bright and faint ends of
the galaxy sample will have their light scattered still further. This
will cause an overestimate of those parts of the LF that contri-
bute the least light. In addition, as the luminosity contributions
of the galaxies are distributed across a large redshift range, light
from galaxies inside a given redshift bin will sometimes be scat-
tered out of that bin, and since lower z redshift bins have smaller
volumes than higher z ones, there is the risk that more light
would be added into lower redshift bins than would be removed
from them. This could bias the distribution of derived absolute
magnitudes in those bins.
We quantify and correct for these errors using Monte Carlo
simulations in the manner described by Liu et al. (1998). For an
arbitrary fixed M  and  , we created populations of galaxies Fig. 3.—Examples of the results of simulations testing the biasing effect of
that followed Schechter functions with faint-end slopes in the probability-weighted luminosity functions on the measurement of
. The dashed
lines represent the
-values of the input LFs, and the solid lines show the
range
2:2 <
<
0:2. For each value of
, we populated a boundaries wherein 95% of the output LFs are contained. The three numbers
simulated COSMOS survey volume with the corresponding gal- above each set of lines are the
-values of the upper bound, input, and lower
axy population. We then ‘‘detected’’ these galaxies on the basis bound LFs, respectively.
No. 1, 2008 FAINT-END SLOPES OF GALAXY LFs 203

sure those parameters in this work. Accurate measurement of

those values for the COSMOS survey are presented, for a slightly
different galaxy spectral type classification scheme, by Scarlata
et al. (2007).
3.4. Probability-weighted Luminosity Functions
The measured luminosity functions for the entire galaxy sam-
ple, corrected for this bias in
, are presented in Figure 4. Only
the faint ends of the LFs, operationally defined here as galaxies
fainter than MV ’
18, are presented. LFs were calculated in
five redshift bins: 0:02 < z  0:1, 0:1 < z  0:2, 0:2 < z  0:3,
0:3 < z  0:4, and 0:4 < z  0:5. In addition, we divided the
galaxies into four subsamples according to galaxy spectral type:
type 1 (early-type galaxies), type 2 (Sbc), types 3 and 4 combined
(Scd + Irr), and types 5 and 6 combined (low-extinction star-
Fig. 4.—Faint-end portions of the V-band galaxy luminosity functions, (MV ), bursts). For each subsample, LFs were also calculated in the same
in the redshift range 0:02  z  0:5. Each LF is offset by a constant for clarity. five redshift bins. The results are presented in Figure 5.
The best-fit weighted least-squares faint-end power-law slope for each LF has There are three primary sources of errors in the LFs: (1) the
been overplotted (dotted lines) and is labeled with its redshift bin and slope. systematic error in the LF slope and absolute magnitude deter-
minations, described in the text above and characterized using
actually observed galaxy population. These offsets were then simulations; (2) a Poisson-like error that derives naturally from
used to correct the computed luminosity functions in order to re- the modified 1/Vmax method, which is the reciprocal of the square
cover the original faint-end slopes of the galaxy samples in each root of the total number of fractional galaxies in each bin; and
redshift bin. This correction affects the overall slope only and is (3) a non-Gaussian error as a function of absolute magnitude,
not intended to remove any inherent, non–power-law structure due to the asymmetric uncertainty in the photometric redshift de-
in the observed LF. Also, this correction strategy was optimized termination of each galaxy. The second source of error, because
to recover the faint-end slope, rather than the characteristic mag- of our large galaxy sample size, is much smaller than the third
nitude M  or the number density normalization  (as was done, source of error. We computed the error from that third source
for example, in Chen et al. 2003). We thus do not attempt to mea- with a standard bootstrap technique, using 150 random samplings

Fig. 5.—Faint-end portions of V-band luminosity functions, (MV ), for galaxies divided by spectral type. Symbols are the same as those in Fig. 4.
204 LIU ET AL. Vol. 672
TABLE 2 has
¼
1:05  0:01 for a slightly redshifted r-band galaxy LF
Luminosity Function Slope Fits for galaxies brighter than Mr 
17 ( Blanton et al. 2003) and

’
1:3 at fainter magnitudes (Blanton et al. 2005). Blanton
Galaxy Spectral Typea Redshift Range

et al. (2005) have further shown that, with the appropriate con-
T1–T6 (All) ................................. 0.02  z < 0.1
1.24  0.07 version of the 2dFGRS bJ data, they and the SDSS g-band LFs
0.1  z < 0.2
1.18  0.07 have consistent low-luminosity slopes. At higher redshift, our
0.2  z < 0.3
1.09  0.08 results are consistent with those of Scarlata et al. (2007), who in-
0.3  z < 0.4
1.12  0.08 dependently derived
¼
1:26  0:15 in the range 0:2 < z <
0.4  z < 0.5
1.12  0.10 0:4 for a portion of the COSMOS survey area. Our results are
T1 (early-type) ............................. 0.02  z < 0.1
1.10  0.08 also consistent with results from other surveys given in the litera-
0.1  z < 0.2
0.81  0.09 ture; for example, with the V-band LF derived from the VIRMOS-
0.2  z < 0.3
0.60  0.12
VLT Deep Survey (VVDS; Ilbert et al. 2005), where
¼ 1:21 
0.3  z < 0.4
0.53  0.16
0.4  z < 0.5
0.52  0.20
0:04 in the range 0:2 < z < 0:4.
T2 (Sbc) ....................................... 0.02  z < 0.1
1.16  0.07 The formal errors in our
measurements are higher than those
0.1  z < 0.2
1.13  0.07 of most of these other studies; this is probably mainly because of
0.2  z < 0.3
1.01  0.10 the systematic slope uncertainties that we attempt to account for
0.3  z < 0.4
0.75  0.15 with our simulations. Our overall agreement, however, appears
0.4  z < 0.5
0.92  0.16 to confirm that we have properly accounted for the errors that
T3 + T4 (Scd + Irr) ..................... 0.02  z < 0.1
1.46  0.07 result from representing galaxies as probability-smoothed lumi-
0.1  z < 0.2
1.37  0.08 nosity distributions.
0.2  z < 0.3
1.28  0.09
0.3  z < 0.4
1.19  0.10
0.4  z < 0.5
1.11  0.15
4.1. LFs as a Function of Galaxy Spectral Type
T5 + T6 (starbursts)..................... 0.02  z < 0.1
1.88  0.18 The luminosity functions in the four galaxy spectral type bins
0.1  z < 0.2
1.65  0.14 we used—SED templates of early-type, Sbc, Scd+Irr, and low-
0.2  z < 0.3
1.53  0.10 extinction starbursts—follow the well-known pattern of steeper
0.3  z < 0.4
1.35  0.11
values of
for bluer galaxies. In our low-redshift bin,
increases
0.4  z < 0.5
1.27  0.15
from
1:10  0:08 in early types to
1:88  0:18 in starbursts;
a
Galaxy spectral types as defined by Mobasher et al. (2007). the trend continues with increasing redshift, showing a similar
steepening of
from
0:52  0:20 to
1:27  0:15.
As with the full galaxy sample, these type-specific results are
(with duplication allowed ) of the observed data set to determine consistent with the findings of previous work in the literature.
the 68% confidence intervals for  in each magnitude bin. Of course, due to differing galaxy selection criteria and redshift
Each LF segment was fitted to a power-law slope using weighted binning, exact comparisons are not always possible. Generally
least squares. To avoid possible biasing of the faint-end slope speaking, however, for local galaxies, our red/early-type and inter-
by galaxies near M  , which as we mentioned above is not well mediate spiral galaxy LFs are consistent with SDSS and 2dFGRS
determined by our technique, we only use data fainter than MV ¼ results, and previous samples of z P 0:1 galaxy populations have

18:3 in our fits. In one case, we make a more restrictive mag- also shown very steep values of
for the bluest and most irreg-
nitude cutoff: the lowest redshift bin in the starburst spectral type. ular galaxies:
¼
1:87 for the CfA Redshift Survey (Marzke
There the measured LF drops discontinuously at MV ¼
17. We et al. 1994),
1.84 for the Las Campanas Redshift Survey
suspect that this has occurred because M  may be quite faint for (Bromley et al. 1998),
1.81 for the SSRS2 (Marzke et al. 1998),
this galaxy type and redshift bin, thus distorting our measure- and
1.9 from the Deep Multicolor Survey (Liu et al. 1998) and
ment of the LF brightward of that point. So to ensure that the
the SDSS ( Nakamura et al. 2003). At higher redshifts, compar-
fit to that LF is not correspondingly distorted, we use only data isons with COMBO-17 (Wolf et al. 2003), the VVDS (Zucca
fainter than MV ¼
16:8 for that particular fit. In both Figures 4 et al. 2006), and the COSMOS survey itself (Scarlata et al. 2007)
and 5, the measured LF data points are plotted as symbols, and show broad consistency across the various galaxy type and red-
the best-fit
-values are plotted as dotted lines. The faint-end shift intervals.
slope fits are summarized in Table 2.
4.2.
versus z: Evolution or Selection?
4. DISCUSSION
In the context of the broad consistency of our results with those
Because of the combined large area and depth of the COSMOS in the literature, perhaps the most striking result in this work is the
survey, the luminosity functions presented in this work provide a clear trend, with every galaxy spectral type, of a flattening of the
glimpse of
across a substantial range of redshift in a single, faint-end slope with increasing redshift. From our lowest redshift
consistent data set. With this view, our results show that for all bin to our highest—i.e., from z  0 to z  0:5—the change in
spectral types combined,
¼ 1:24  0:07 for the local (0:02 < slope 
¼ 0:58, 0.24, 0.35, and 0.61, respectively, for early
z  0:1) universe. As the redshift increases,
flattens out some- types, Sbc, Scd+Irr, and low-extinction starbursts.
what, and it is
1:12  0:10 in our highest redshift bin (0:4 < On the surface, this trend may not appear to be consistent with
z  0:5). previous work. Much of the work to derive the evolution of the
Our local LF is consistent with results from the two largest lo- galaxy LF parameters M  and   as a function of redshift (e.g.,
cal galaxy surveys to date, which have comparable (105 gal- Lin et al. 1999; Wolf et al. 2003; Baldry et al. 2005; Willmer et al.
axies) sample sizes to our study here. The 2dFGRS survey found 2006) in fact depends on the assumption that
does not evolve
that, for the bJ -band galaxy luminosity function,
¼
1:21  with redshift, or at most weakly evolves to z P 1. We have fo-
0:03 (Norberg et al. 2002), and it was
1:18  0:02 for the cused here on the measurement of
rather than those other
redshift range 0:02 < z < 0:25 (Croton et al. 2005). The SDSS parameters; that, and the fact that all of our measurements have
No. 1, 2008 FAINT-END SLOPES OF GALAXY LFs 205

come from a single data set, plus the fact that each individual weighted probability-smoothed luminosity distributions and us-
determination of
is consistent with previous work, supports the ing a modified 1/Vmax method. A total of 49,161 galaxies have
likelihood that this observed flattening trend is real. photometric redshifts that fall in this redshift range; within and
The question is, do these changing slopes represent true evo- outside this range, a total of 80,820 galaxies contribute to the
lution in
, or do they reflect our ability to detect different galaxy derived LFs. Extensive Monte Carlo simulations were used to
populations as a function of redshift? The latter possibility can be characterize and account for the systematic and random errors of
discussed in the context of, among others, de Lapparent et al. this technique.
(2003) and Blanton et al. (2005), who suggest that the faint end For all galaxy spectral types, the LF slope ranges from
1.24
of the field galaxy LF is comprised of a composite population of to
1.12 from the lowest redshift bin to the highest. In the lowest
dwarf and nondwarf galaxies, each with its own functional form. redshift bin (0:02 < z < 0:1), where the magnitude limit is
This would mean that a single power law is not quite sufficient to MV P
13, the slope ranges from
¼
1:10  0:08 for galax-
describe the LF faint end accurately. Blanton et al. (2005) further ies with early-type spectral energy distributions (SEDs) to
¼
suggest, through detailed examination of the faint galaxy pop-
1:88  0:18 for galaxies with low-extinction starburst SEDs.
ulation in the SDSS, that a large fraction of these dwarf galaxies In each galaxy SED category (early-type, Sbc, Scd+Irr, and star-
may have very low surface brightnesses and are thus not in- burst), the faint-end slopes grow shallower with increasing red-
cluded in most faint-end LF measurements. shift; in the highest redshift bin (0:4 < z < 0:5),
¼
0:52 
Our dependence on photometric redshifts places an important 0:20 and
1:27  0:15 for early types and starbursts, respectively.
caveat on the interpretation of our data: by using ‘‘fuzzy’’ galax- All of our derived type-specific LFs, across our redshift ranges,
ies, any second-order deviations from a power law at the faint are broadly consistent with the findings of previous authors. Our
end of the LF may well have been smoothed out and are thus not results thus show a flattening trend for
with increasing redshift
recoverable from our LF measurements. So if a deviation from a for each spectral type. It is unclear, however, if this is evidence
single faint-end power law does occur at very low luminosities, of evolution of
in the galaxy LF or of preferential selection of
we cannot address that issue with this work. dwarf galaxies in the local universe. The steepness of
at lower
Due to the substantial depth of the COSMOS survey imaging, redshift could be qualitatively explained, for example, by large
it is likely that we have successfully measured a larger fraction numbers of faint dwarfs, perhaps of low surface brightness, that
of low surface brightness dwarf galaxies than have wider area, shal- are not detected at higher redshifts. We will address this question
lower surveys such as SDSS or 2dFGRS. The steepness of our in a future paper, when the full set of COSMOS data, and in par-
low-redshift LFs may reflect this. Even the COSMOS survey ticular, spectroscopic redshifts, has been obtained.
depth, however, does not allow us to measure values of
fainter
than MV 
17 at z ¼ 0:5, so we cannot say if the flattening of

in our higher redshift bins is due to the nondetection of these
dwarfs. There may be some circumstantial evidence, however, to The HST COSMOS Treasury program was supported through
support that picture. For example, Dahlen et al. (2005) measured NASA grant HST-GO-09822. We wish to thank Tony Roman,
for the GOODS survey a value of
¼
1:37 for the rest-frame B Denise Taylor, and David Soderblom for their assistance in the
band in the range 0:1 < z < 0:5, which is somewhat steeper than planning and scheduling of the extensive COSMOS observa-
most LF measurements in this range. However, the GOODS sur- tions. We gratefully acknowledge the contributions of the entire
vey is very deep, so in this broad redshift bin, a large detected COSMOS collaboration, which contains more than 70 scientists.
fraction of faint, low surface brightness dwarfs near z k 0:1 could More information on the COSMOS survey is available at http://
have driven
to a steeper value for the full range. www.astro.caltech.edu/~ cosmos. It is a pleasure to acknowledge
When the spectroscopic portion of the COSMOS survey the excellent services provided by the NASA IPAC/ IRSA staff
(Lilly et al. 2007) is completed, we will be able to address this (Anastasia Laity, Anastasia Alexov, Bruce Berriman, and John
question in more detail, as we deconvolve the faint end of the gal- Good ) in providing online archive and server capabilities for the
axy LF as a multivariate function of color, morphology, luminos- COSMOS data sets. The COSMOS Science meeting in 2005
ity, and redshift. May was supported in part by the NSF through grant OISE-
0456439. We thank Paris Bogdanos and James Cohen for image
5. CONCLUSIONS processing and data formatting assistance. C. Liu, T. Paglione,
and S. Tribiano gratefully acknowledge support from a City Uni-
Using the COSMOS multiband photometry and photometric
versity of New York CCIR grant, as well as the hospitality and
redshift catalog, we have constructed faint-end rest-frame V-band
support of the Hayden Planetarium and Department of Astro-
luminosity functions for the galaxy population at 0:02 < z  0:5
physics at the American Museum of Natural History.
in the COSMOS survey volume. Since we are using photometric
redshifts, we have computed these LFs by treating galaxies as Facilities: HST (ACS), Subaru, KPNO, CTIO, CFHT.

REFERENCES
Abazajian, K., et al. 2004, AJ, 128, 502 Budavari, T., et al. 2005, ApJ, 619, L31
Baldry, I., et al. 2005, MNRAS, 358, 441 Capak, P., et al. 2007, ApJS, 172, 99
Bell, E. F., McIntosh, D. H., Katz, N., & Weinberg, M. D. 2003 ApJS, 149, 289 Chen, H.-W., et al. 2003, ApJ, 586, 745
Benı́tez, N. 2000, ApJ, 536, 571 Coleman, G. D., Wu, C.-C., & Weedman, D. W. 1980, ApJS, 43, 393
Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393 Colless, M., et al. 2001, MNRAS, 328, 1039
Bevington, P. R., & Robinson, D. K. 1992, Data Reduction and Error Analysis Connolly, A. J., Csabai, I., Szalay, A. S., Koo, D. C., Kron, R. C., & Munn, J. A.
for the Physical Sciences (2nd ed.; New York: McGraw-Hill ) 1995, AJ, 110, 2655
Blanton, M. R., Lupton, R. H., Schlegel, D. J., Strauss, M. A., Brinkmann, J., Croton, D. J., et al. 2005, MNRAS, 356, 1155
Fukugita, M., & Loveday, J. 2005, ApJ, 631, 208 Dahlen, T., Mobasher, B., Somerville, R. S., Moustakas, L. A., Dickinson, M.,
Blanton, M. R., et al. 2003, ApJ, 592, 819 Ferguson, H. C., & Giavalisco, M. 2005, ApJ, 631, 126
Bromley, B. C., Press, W. H., Lin, H., & Kirshner, R. P. 1998, ApJ, 505, 25 de Lapparent, V., Arnouts, S., Galaz, G., & Bardelli, S. 2004, A&A, 422, 841
Brown, W. R., Geller, M. J., Fabricant, D. G., & Kurtz, M. J. 2001, AJ, 122, Driver, S. P., Popescu, C. C., Tuffs, R. J., Liske, J., Graham, A. W., Allen, P. D.,
714 & De Propris, R. 2007, MRAS, 379, 1022
206 LIU ET AL.
de Lapparent, V., Galaz, G., Bardelli, S., & Arnouts, S. 2003, A&A, 404, 831 Marzke, R. O., Huchra, J. P., & Geller, M. J. 1994, ApJ, 428, 43
Ellis, R. S., Colless, M., Broadhurst, T., Heyl, J., & Glazebrook, K. 1996, Mobasher, B., et al. 2004, ApJ, 600, L167
MNRAS, 280, 235 ———. 2007, ApJS, 172, 117
Gabasch, A., et al. 2004, A&A, 421, 41 Nakamura, O., Fukugita, M., Yasuda, N., Loveday, J., Brinkmann, J.,
Ilbert, O., et al. 2005, A&A, 439, 863 Schneider, D. P., Shimasaku, K., & SubbaRao, M. 2003, AJ, 125, 1682
Impey, C. D., Sprayberry, D., Irwin, M. J., & Bothun, G. D. 1996, ApJS, 105, Norberg, P., et al. 2002, MNRAS, 332, 827
209 Pérez-González, P. G., et al. 2005, ApJ, 630, 82
Kinney, A. L., Calzetti, D., Bohlin, R. C., McQuade, K., Storchi-Bergmann, T., Scarlata, C., et al. 2007, ApJS, 172, 406
& Schmitt, H. R. 1996, ApJ, 467, 38 Schechter, P. 1976, ApJ, 203, 297
Koo, D. C. 1986, ApJ, 311, 651 Schmidt, M., & Green, R. F. 1986, ApJ, 305, 68
Lilly, S. J., Tresse, L., Hammer, F., Crampton, D., & Le Fèvre, O. 1995, ApJ, Scoville, N. Z., et al. 2007a, ApJS, 172, 1
455, 108 ———. 2007b, ApJS, 172, 38
Lilly, S. J., et al. 2007, ApJS, 172, 70 SubbaRao, M. U., Connolly, A. J., Szalay, A. S., & Koo, D. C. 1996, AJ, 112,
Lin, H., Kirshner, R. P., Shectman, S. A., Landy, S. D., Oemler, A., Tucker, D. L., 929
& Schechter, P. L. 1996, ApJ, 464, 60 Taniguchi, Y., et al. 2007, ApJS, 172, 9
Lin, H., Yee, H. K. C., Carlberg, R. G., Morris, S. L., Sawicki, M., Patton, D. R., Willmer, C. N. A., et al. 2006, ApJ, 647, 853
Wirth, G., & Shepherd, C. W. 1999, ApJ, 518, 533 Wolf, C., Meisenheimer, K., Rix, H.-W., Borch, A., Dye, S., & Kleinheinrich, M.
Liu, C. T., & Green, R. G. 1998, AJ, 116, 1074 2003, A&A, 401, 73
Liu, C. T., Green, R. G., Hall, P. B., & Osmer, P. S. 1998, AJ, 116, 1082 Zucca, E., et al. 2006, A&A, 455, 879
Madgwick, D. S., et al. 2002, MNRAS, 333, 133
Marzke, R. O., da Costa, L. N., Pellegrini, P. S., Willmer, C. A., & Geller, M. J.
1998, ApJ, 503, 617