Multi-Wavelength Constraints on the Cosmic Star Formation History from Spectroscopy: the Rest-Frame UV, H-alpha, and Infrared Luminosity Functions at Redshifts 1.9<z<3.4
Naveen A. Reddy, Charles C. Steidel, Max Pettini, Kurt L. Adelberger, Alice E. Shapley, Dawn K. Erb, Mark Dickinson
aa r X i v : . [ a s t r o - ph ] J un Received 2007 May 21; Accepted 2007 June 12
Preprint typeset using L A TEX style emulateapj v. 03/07/07
MULTI-WAVELENGTH CONSTRAINTS ON THE COSMIC STAR FORMATION HISTORY FROMSPECTROSCOPY: THE REST-FRAME UV, H α , AND INFRARED LUMINOSITY FUNCTIONS ATREDSHIFTS 1 . . Z . . Naveen A. Reddy , Charles C. Steidel , Max Pettini , Kurt L. Adelberger , Alice E. Shapley , Dawn K.Erb , and Mark Dickinson Received 2007 May 21; Accepted 2007 June 12
ABSTRACTWe use a sample of rest-frame UV selected and spectroscopically observed galaxies at redshifts1 . ≤ z < .
4, combined with ground-based spectroscopic H α and Spitzer
MIPS 24 µ m data, to derivethe most robust measurements of the rest-frame UV, H α , and infrared (IR) luminosity functions (LFs)at these redshifts. Our sample is by far the largest of its kind, with over 2000 spectroscopic redshiftsin the range 1 . ≤ z < . ∼ α line perturbations to the observed opticalcolors of galaxies, and contaminants. Taking into account the latter, we find no evidence for an excessof UV-bright galaxies over what was inferred in early z ∼ z ∼ z ∼
2. Corrected for extinction, the UV luminosity density(LD) at z ∼ z ∼ ∼ z ∼
6, primarily reflecting an increase in the number density of bright galaxies between z ∼ z ∼
2. Our analysis yields the first constraints anchored by extensive spectroscopy on the infraredand bolometric LFs for faint and moderately luminous ( L bol . L ⊙ ) galaxies. Adding the IRto the emergent UV luminosity, incorporating independent measurements of the LD from ULIRGs,and assuming realistic dust attenuation values for UV-faint galaxies, indicates that galaxies with L bol < L ⊙ account for ≈
80% of the bolometric LD and SFRD at z ∼ −
3. This suggeststhat previous estimates of the faint-end of the L bol LF may have underestimated the steepness of thefaint-end slope at L bol < L ⊙ . Our multi-wavelength constraints on the global SFRD indicatethat approximately one-third of the present-day stellar mass density was formed in sub-ultraluminousgalaxies between redshifts z = 1 . − . Subject headings: galaxies: evolution — galaxies: formation — galaxies: high redshift — galaxies:luminosity function — galaxies: starburst — infrared: galaxies INTRODUCTION
Constraining the star formation history and stellarmass evolution of galaxies is a central componentof understanding galaxy formation. Observationsof the stellar mass and star formation rate density,the QSO density, and galaxy morphology at bothlow ( z .
1) and high ( z &
3) redshifts indicatethat most of the activity responsible for shapingthe bulk properties of galaxies to their present form Based, in part, on data obtained at the W.M. Keck Observa-tory, which is operated as a scientific partnership among the Cal-ifornia Institute of Technology, the University of California, andNASA, and was made possible by the generous financial supportof the W.M. Keck Foundation. Also based in part on observationsmade with the
Spitzer Space Telescope , which is operated by theJet Propulsion Laboratory, California Institute of Technology, un-der a contract with NASA. California Institute of Technology, MS 105–24, Pasadena, CA91125 National Optical Astronomy Observatory, 950 N Cherry Ave,Tucson, AZ 85719 Institute of Astronomy, Madingley Road, Cambridge CB3OHA, UK McKinsey & Company, 1420 Fifth Avenue, Suite 3100, Seattle,WA 98101 Department of Astrophysical Sciences, Peyton Hall-Ivy Lane,Princeton, NJ 08544 Harvard-Smithsonian Center for Astrophysics, 60 GardenStreet, Cambridge, MA 02138 occurred in the epochs between 1 . z . HST , Spitzer , and
Chandra , allowed for the study of largenumbers of galaxies at z ∼
2. These developments haveprompted a spate of multi-wavelength surveys of highredshift galaxies from the far-IR/submm to IR, near-IR,optical, and UV, enabling us to examine the SEDs ofstar forming galaxies over much of the 7 decades offrequency over which stars emit their light either directlyor indirectly through dust processing (e.g., Steidel et al.2003, 2004; Daddi et al. 2004b,a; Franx et al. 2003;van Dokkum et al. 2003, 2004; Abraham et al. 2004;Chapman et al. 2005; Smail 2003).The first surveys that efficiently amassed large samplesof high redshift galaxies used the observed U n G R colorsof galaxies to identify those with a deficit of Lyman con- Reddy et al.tinuum flux (e.g., Steidel et al. 1995) in the U n band (i.e.,U “drop-outs”) for galaxies at z ∼
3. Those initial resultshave been adapted to select galaxies at higher redshifts( z >
4; e.g., Bouwens et al. 2005, 2004; Dickinson et al.2004; Bunker et al. 2004; Yan et al. 2003) and moder-ate redshifts (1 . . z .
3; Adelberger et al. 2004;Steidel et al. 2004). Combining these high redshift re-sults with those from
GALEX (e.g., Wyder et al. 2005),we now have an unprecedented view of the rest-frameUV properties of galaxies from the epoch of reionizationto the present, perhaps the only wavelength for whichstar forming galaxies have been studied across more than ∼
93% of the age of the Universe. The accessibility ofrest-frame UV wavelengths over almost the entire age ofthe Universe makes rest-frame UV luminosity functions(LFs) useful tools in assessing the cosmic star formationhistory in a consistent manner.The foray of observations into the epoch around z ∼ z ∼
2, and/or (3) they are estimated overa relatively small number of fields such that cosmic vari-ance may be an issue. While purely magnitude limitedsurveys allow one to easily quantify the selection func-tion, as we show below, Monte Carlo simulations com-bined with accurate spectroscopy can be used to quantifyeven the relatively complicated redshift selection func-tions and biases of color-selected samples of high redshiftgalaxies. This “simulation” approach allows one to assessa number of systematics (e.g., photometric imprecision,perturbation of colors due to line strengths, etc.) andtheir potential effect on the derived LF; these systematiceffects have been left untreated in previous calculations ofthe LFs at z ∼ − . . z . .
6, in multiple in-dependent fields. The selection criteria aim to identifyactively star-forming galaxies at z ∼ z ∼ z &
4) where no corresponding multi-wavelength data exist to assess the fraction of galaxiesthat are not recovered by color selection. In our case,applying the Monte Carlo method to joint photometricand spectroscopic samples of high redshift galaxies al-lows one to assess the systematic effects of photometricscattering and the intrinsic variation in colors due to lineemission and absorption with unprecedented accuracy.In this paper, we take advantage of a large sample ofspectroscopically-confirmed star-forming galaxies to ar-rive at the first completeness-corrected spectroscopic es-timate of the UV LF and star formation rate density(SFRD) at z ∼
2, computed across the many indepen-dent fields of our survey. We extend our results by usingspectroscopy of Lyman Break galaxies in many new in-dependent fields to recompute the UV LF and SFRD atulti-Wavelength LFs at 1 . . z . . z ∼ α , and infrared luminosity functions at red-shifts z ∼ −
3. The primary goal of this paper is tothen use these luminosity functions to evaluate the cos-mic star formation history in a consistent manner across4 decades of wavelength.The outline of this paper is as follows. In §
2, wedescribe the fields of our survey and the color criteriaused to selected candidate galaxies at z ∼
2. We thenproceed with a description of the spectroscopic followupand quantify the fraction of contaminants, including lowredshift ( z <
1) star forming galaxies and low and highredshift AGN and QSOs, within the sample. We con-clude § z ∼
2. In §
3, we detail the MonteCarlo method used to assess both photometric bias anderror, the effect of Ly α line perturbations on the observedrest-UV colors of galaxies, and the procedure used to cor-rect our sample for completeness. Our results pertainingto the intrinsic Ly α equivalent width and reddening dis-tributions of 1 . ≤ z < . § α LFs are pre-sented respectively in §
5, 6, and 7. Lastly, we discuss theimplications of our results for the luminosity and globalstar formation rate densities in §
8. A flat ΛCDM cosmol-ogy is assumed with H = 70 km s − Mpc − , Ω Λ = 0 . m = 0 . DATA: SAMPLE SELECTION AND SPECTROSCOPY
Fields
Our z ∼ V ≤ . . . z . .
8, ideally placed to study thecorrelation between z ∼ z ∼ z ∼ z ∼ z ∼
3. Fields where we have carried out LBG selectionare also listed in Table 1.One of the unique advantages of our analysis is thatwe use a large number of uncorrelated fields (14 and 29for the z ∼ z ∼ . ≤ z < .
4, in order to compute the LF, negating theneed for uncertain normalization corrections to accountfor clustering and cosmic variance. For example, we findevidence for significant large scale structure within sev-eral fields of the z ∼ z ∼ ∼ z ∼ ∼ z ∼ Photometry
Photometry was performed using a modified version ofFOCAS (Valdes 1982). Object detection was done at R band, and G − R and U n − G colors were computed byapplying the R -band isophotal apertures to the G and U n images (see Steidel et al. 2003, 2004 for further de-tails). The optical images have typical depth of R ∼ . ∼ ′′ diameter aperture (3 σ ).Field-to-field variations in photometry are dominated bysystematics due to the different instruments, filter sets,and slightly varying observing conditions when the fieldswere imaged. These field-to-field systematics are negli-gible compared to measurement errors. We have incor-porated some of these effects (e.g., seeing, airmass of theobservation, CCD response, and filter shape) in comput-ing the expected colors of galaxies with known intrinsicproperties. Modeling all the field-to-field variations inphotometry rapidly becomes a very complex problem,infeasible to resolve within a reasonable time frame. Theremaining biases (e.g., errors in the zeropoints used) arediscussed in § Color Selection
Even with a priori knowledge of the intrinsic proper-ties of all z ∼ TABLE 1Survey Fields α a δ b Field SizeField Name (J2000.0) (J2000.0) (arcmin ) N BXc N BX (1 . ≤ z < . d N LBGe N LBG (2 . ≥ z < . f Q0000 00 03 25 -26 03 37 18.9 ... ... 28 12CDFa 00 53 23 12 33 46 78.4 ... ... 100 30CDFb 00 53 42 12 25 11 82.4 ... ... 121 21Q0100 01 03 11 13 16 18 42.9 345 65 100 18Q0142 01 45 17 -09 45 09 40.1 287 72 100 20Q0201 02 03 47 11 34 22 75.7 ... ... 87 13Q0256 02 59 05 00 11 07 72.2 ... ... 120 42Q0302 03 04 23 -00 14 32 244.9 ... ... 191 29Q0449 04 52 14 -16 40 12 32.1 188 40 88 13B20902 09 05 31 34 08 02 41.8 ... ... 78 34Q0933 09 33 36 28 45 35 82.9 ... ... 211 47Q1009 10 11 54 29 41 34 38.3 306 33 137 25Q1217 12 19 31 49 40 50 35.3 240 26 65 11GOODS-N 12 36 51 62 13 14 155.3 909 138 210 62Q1307 13 07 45 29 12 51 258.7 1763 40 564 8Westphal 14 17 43 52 28 49 226.9 612 39 334 177Q1422 14 24 37 22 53 50 113.0 ... ... 453 g h TOTAL ... ... 3186.1 10013 1017 5452 1006 a Right ascension in hours, minutes, and seconds. b Declination in degrees, arcminutes, and arcseconds. c Number of BX candidates to R = 25 . d Number of spectroscopically confirmed BX candidates with redshifts 1 . ≤ z < .
7, excluding those whose spectraindicate an AGN/QSO. Note that the total numbers of galaxies, excluding AGN and QSOs, with spectroscopic redshifts1 . ≤ z < . e Number of LBG candidates to R = 25 . f Number of spectroscopically confirmed LBG candidates with redshifts 2 . ≤ z < .
4, excluding those whose spectraindicate an AGN/QSO. Note that the total numbers of galaxies, excluding AGN and QSOs, with spectroscopic redshifts2 . ≤ z < . g Number includes 180 galaxies with 25 . < R ≤ . h Number includes 10 galaxies with 25 . < R ≤ . have been designed to select high redshift galaxies is rest-frame UV color selection, initially used to target galax-ies at z ∼ . . z . . −
10m class telescopes. Color-selected high red-shift galaxy surveys will, as a consequence, have rathercomplex selection functions. The approach described in § . . z . . z ∼ z ∼ U n G R JK s photome-try and spectroscopic redshifts (Adelberger et al. 2004;Steidel et al. 2004). Initial spectroscopy of z ∼ . . z . . G − R ≥ − . U n − G ≥ G − R + 0 . G − R ≤ . U n − G ) + 0 . U n − G ≤ G − R + 1 . , (1)termed as “BX” selection (Adelberger et al. 2004;Steidel et al. 2004), with fluxes in units of AB magni-tudes (Oke & Gunn 1983). Candidates were selected to R = 25 . R -band that is 0 . z = 2 . z >
1) than at z ∼
3. Additionally, we exclude allsources with R <
19 that are saturated in our images,all of which are stars. The above criteria yielded 10013ulti-Wavelength LFs at 1 . . z . . ∼ − , uncorrected for contamination fromobjects with redshifts z < . § z ∼ z ∼ . . z . . G − R ≤ . U n − G ≥ G − R + 1 . . (2)These criteria form the superset of the individual sets ofcriteria for “C”, “D”, “M”, and “MD” candidate typesgiven in Table 4 of Steidel et al. (2003). Hereafter, wewill refer to all these different candidate types as LBGs.Candidates were selected to R = 25 .
5, except in the fieldQ1422 where the photometric depth allowed selection ofcandidates to R = 26 .
0. The number of z ∼ R = 25 . ∼
25% of the total R and K s -band counts to R = 25 . K s (AB) = 24 .
4, respectively.
Spectroscopic Followup
The spectroscopic followup of candidates is discussedextensively in Steidel et al. (2003) and Steidel et al.(2004). Of the 10013 BX candidates, we have targeted24% (2382 out of 10013) with spectroscopy, yielding 1711redshift identifications, or a 72% success rate averagedover all fields. As discussed in § z >
1, are h z i = 2 . ± .
32 and h z i = 2 . ± .
26, respectively. Preliminary versions ofthese histograms, along with sample spectra of BX galax-ies and LBGs, are presented in Steidel et al. (2003, 2004).Table 2 lists the spectroscopic fractions relevant for theBX and LBG samples.
Interloper Contribution and AGN
The region of color space defined by BX selection (e.g.,Figure 4) is also expected to include galaxies outside ofthe targeted redshift range, including star forming galax-ies at z . . z < . R < .
5, as indicated inTable 3. One can impose a rough magnitude cutoff toonly consider those candidates with
R ≥ .
5, but thiswould preclude the analysis of the bright-end of the BXand LBG luminosity distributions, as well as more de-tailed studies of the UV spectra of optically-bright ob-jects. Other options to reduce the contamination fraction
Fig. 1.—
Arbitrarily normalized spectroscopic redshift distribu-tions of galaxies with z > . include using the R− K color where the associated bandsno longer bracket strong spectral breaks for low redshiftsources. For example, the BzK criteria of Daddi et al.(2004b) can be used to reduce the foreground contami-nation fraction in color-selected samples.The interloper fractions are apt to decrease as the sur-vey progresses and we become more adept at excludingthem from masks based on other multi-wavelength data,such as their
R − K colors. However, until now, wehave not used any of the techniques discussed above toactively discriminate against placing possible interloperson slitmasks; doing so would complicate our ability toapply the observed contamination fractions to determinethe interloper rate among all BX/LBG sources. There-fore, the fractions in columns (4) and (7) of Table 3 areassumed to represent the overall fraction of interlopers asa function of R for the photometric samples. For the BXsample, most of the contamination at bright magnitudesarises from foreground galaxies. For the LBG sample,most of the contamination arises from stars. Applying abright magnitude limit of R = 23 . σ < − )AGN whose rest-UV colors are similar to those of highredshift star forming galaxies, but which show prominent(and in some cases broad) emission lines such as Ly α ,CIV, and NV. The detection rate of such sources is ∼ .
8% (similar to the rate found among UV-selected z ∼ R < . z > § TABLE 2Spectroscopic and AGN/QSO Fractions of the BX and LBG Samples
BX LBG R N phota N specb f specc f AGNd f AGN ( z ≥ . e N photf N specg f spech f AGNi f AGN ( z ≥ . j . − . . − . . − . . − . . − . . − . . − . < . < .
01 1617 442 0.27 0.01 0.0125 . − . < . < . . − . k k k k k Total 10007 1711 0.17 0.02 0.02 5448 1492 0.27 0.03 0.03 a Number of BX candidates. b Number of BX candidates with spectroscopic redshifts. c Fraction of BX candidates with spectroscopic redshifts. d Fraction of AGN/QSOs in BX sample with spectroscopic redshifts. e Fraction of AGN/QSO in BX sample with z spec ≥ . f Number of LBG candidates. g Number of LBG candidates with spectroscopic redshifts. h Fraction of LBG candidates with spectroscopic redshifts. i Fraction of AGN/QSOs in LBG sample with spectroscopic redshifts. j Fraction of AGN/QSO in LBG sample with z spec ≥ . k Numbers are for Q1422 field.
TABLE 3Interloper ( z < . ) Statistics of the BX and LBG Samples BX LBG R N z ≥ N ≤ z < . f ≤ z < . N z ≥ N ≤ z < . f ≤ z < . . − . . − . . − . . − . . − . . − . . − . < . . − . < . . − . < . TOTAL 1711 332 0.19 1492 96 0.06 a Number of sources with spectroscopic redshifts. b Number of sources with z < . c Fraction with z < . Spectroscopic Completeness
Assessing photometric and spectroscopic completenessis a key ingredient in determining the total completenessof our survey. The photometric completeness (i.e., thefraction of galaxies at redshifts 1 . . z . . § R = 25 . . . R . . § . . z . . Fig. 2.—
Apparent magnitude versus redshift for spectroscop-ically confirmed BX objects (in blue) and LBGs (in red) in theredshift range 1 . < z < .
0. AGN/QSOs in the BX and LBG sam-ples are denoted by the large open circles. The dashed horizontalline indicates the self-imposed R = 25 . It is instructive to note that, with respect to the red-shift distribution, there are two forms of completeness wemust concern ourselves with. The first is how well theredshift selection function for the spectroscopic sample re-flects the underlying selection function for the photomet-ric sample. We have just argued that the spectroscopicand photometric samples must have similar redshift dis-tributions. The second is how well the photometric selec-tion function reflects the underlying redshift distributionof all star-forming galaxies. As discussed in § . ≤ z < . . ≤ z < .
4. The modulation of this intrinsic (roughlyconstant) redshift distribution into the Gaussian distri-bution of Figure 1 can be modeled with great precisionby way of Monte Carlo simulations, as we demonstratebelow. Readers who wish to skip directly to the resultsmay proceed to § METHOD: INCOMPLETENESS CORRECTIONS
A primary aim of this analysis is to connect the ob-served properties of BX galaxies and LBGs to the un-derlying population of all star-forming galaxies at red-shifts z ∼ −
3. To this end, we have constructed aplausible population of galaxies with a range of redshifts(1 . . z . . . . z . . Monte Carlo Simulations
We employed a Monte Carlo approach both to (1) de-termine the transformation between the intrinsic proper-ties of a galaxy (e.g., its luminosity, reddening, and red-shift) and its observed rest-UV colors and (2) quantifythe effects of photometric errors in their measured rest-UV colors, similar to the method used in Shapley et al.(2001), Adelberger & Steidel (2000), and Steidel et al.(1999). Template galaxies with intrinsic sizes of 0 . ′′
05 to0 . ′′ ′′ ) of the optical images.Variations in the light profile used (e.g., exponential,de Vaucouleurs) have a negligible effect on the simula-tion results; the intrinsic size of the light emitting region( ∼ . ′′ − . ′′ HST
ACS observations; Law et al.2007) is almost always smaller than the seeing disk.The expected rest-UV colors of a galaxy with a par-ticular redshift and reddening are computed by assum-ing a Bruzual & Charlot (1996) template galaxy withconstant star formation for 1 Gyr and a Calzetti et al.(2000) extinction law. The BX selection criteria weredesigned to select z ∼ z ∼ − >
100 Myr) starbursts and the constantstar formation model described above should reproducethis behavior to the extent required by the simulations(e.g., Shapley et al. 2005). In particular, the rest-UV col-ors of galaxies are essentially constant after 10 years ofstar formation, once the mix of O and B stars stabilizes .The Calzetti et al. (2000) reddening law reproduces the average expected star formation rates of z ∼ − . L bol . . L ⊙ where the bulk of our sample lies (Reddy et al. 2006b).The use of a constant star forming model and the Calzettireddening law should therefore adequately parameterizethe SEDs of most optically-bright star-forming galaxiesat z ∼ −
3. An advantage of spectroscopic followup ofphotometrically selected BX galaxies and LBGs is thatwe can also constrain the effects of IGM opacity andLy α absorption/emission ( § E ( B − V ) distribution for spectroscopi-cally confirmed galaxies. Small variations in the assumed Note that because our selection, and hence simula-tions, are concerned with the rest-UV colors, adopting aMaraston et al. (2006) model (where most of the difference withthe Bruzual & Charlot (1996) model is in the rest-optical) shouldminimally affect our results. There is considerable leeway in the best-fit star formationhistories for the optical/IR SEDs of UV-selected z ∼ >
100 Myr(Shapley et al. 2005; Erb et al. 2006c).
Reddy et al.Schechter parameters of the LF do little to change theresults, since our main goal is to sufficiently populateredshift space and rest-UV color space with a realisticdistribution of objects. The results are also insensitiveto small variations in the assumed E ( B − V ) distributionas long as the range of E ( B − V ) chosen reflects thatexpected for the galaxies. A by-product of the luminos-ity function analysis is that we also compute the best-fitunderlying E ( B − V ) distribution. The validity of theassumed LF and E ( B − V ) distributions can be testedby comparing with the inferred LF and E ( B − V ) dis-tributions. Significant differences between the assumedand inferred distributions imply that the initial assump-tions for the LF and E ( B − V ) distribution were differ-ent from their true values. The colors were corrected foropacity due to the intergalactic medium (IGM) assuminga Madau (1995) model, and corrected for filter and CCDresponses and airmasses appropriate for the each field ofthe survey.The intrinsic rest-UV colors are randomly assigned tosimulated galaxies that are then added to the images inincrements of 200 galaxies at a time. This ensures thatthe image including all added (simulated) galaxies hasconfusion statistics similar to the observed image, sincethis will affect the photometric uncertainties and sys-tematics due to blending. We then attempt to recoverthese simulated galaxies using the same software used torecover the real data, and record whether a simulatedgalaxy is detected and what its observed magnitude andcolors are. We repeated this procedure until approxi-mately 2 × simulated galaxies were added to each ofthe U n , G , and R images of each field. This large numberof simulated galaxies is necessary in order to sufficientlypopulate each bin of luminosity, reddening, and redshift.The end product of the simulations are sets of trans-formations for each field that give the probabilities thatgalaxies with intrinsic luminosities ( L ′ ), reddenings ( E ′ ),and redshifts ( z ′ ) will be observed to have luminosities L ,reddenings E , and redshifts z (or alternatively, the prob-abilities that galaxies with true properties L ′ E ′ z ′ will bemeasured with a particular set of rest-UV colors). Photometric Uncertainties
We have used the results of the Monte Carlo simu-lations ( § R and 0.2 mag in U n − G and G − R color todetermine the uncertainties in the recovered magnitudesand colors of objects in each field. Systematic bias inthe G − R color was estimated by computing the quan-tity ∆[ G − R ] = ( G − R ) meas − ( G − R ) true which wastypically . .
04 mag with uncertainty estimated to be σ (∆[ G −R ]) ∼ .
09 mag. The typical random uncertain-ties in U n − G and R are ∼ .
15 mag and ∼ .
13 mag,respectively. These quantities were determined usingthe same method as presented in Shapley et al. (2005),Steidel et al. (2003), and Shapley et al. (2001). The un- certainties were generally larger for objects faint in R (Steidel et al. 2003). The field-to-field results were con-sistent with each other (i.e., the typical biases and un-certainties from field-to-field were within 0 . . § α perturbations to the colors. Ly α Equivalent Width ( W Ly α ) Distribution The presence of Lyman alpha absorption and/or emis-sion can perturb the observed rest-UV colors of z ∼ − α equivalent width W Ly α . Toinvestigate these effects, we measured the W Ly α for 414spectroscopically confirmed BX galaxies with redshifts1 . ≤ z < . . ≤ z < . W Ly α distribution is shown in Figure 4.The two shaded “zones” in Figure 4 reflect the redshiftranges where Ly α falls within the U n and G -bands. Thisfigure demonstrates how galaxies that are targeted bythe BX criteria can be shifted out of the BX selectionwindow due to Ly α emission or absorption. Fig. 3.— (a) Rest frame Ly α equivalent width ( W Ly α ) distri-bution for 482 spectroscopically observed z ∼ W Ly α distribution for subsets in redshift. Weuse the convention that W Ly α > ξ values in-dicate the probability that the distributions are drawn from thesame parent population as the W Ly α distribution for galaxies at2 . ≤ z < .
48, the redshift range where Ly α does not affect therest-UV colors. Since our ultimate goal is to determine how the W Ly α distribution perturbs the intrinsic colors of galaxies (i.e.,the colors we would measure in the absence of absorp-tion and/or emission), we must first determine whetherthe measured W Ly α distribution reflects the intrinsicdistribution for the parent population of galaxies. Inother words, we must check if our color selection crite-ulti-Wavelength LFs at 1 . . z . . Fig. 4.—
Perturbation of U n G R colors from Ly α absorption andemission. The trapezoid is the BX selection window defined byEquation 1. The U n G R colors of a template galaxy with constantstar formation for >
100 Myr (after which the UV colors are es-sentially constant) and E ( B − V ) = 0 .
13 (the mean for the z ∼ z = 1 to 3. The lowerand upper shaded regions correspond to redshift ranges where theLy α line falls in the U n and G -bands, respectively. In the absenceof photometric errors and assuming all galaxies can be describedby the SED assumed here, galaxies with redshifts 1 . . z . . . . z . .
93 will fall in the dark gray regions with a prob-ability of 64% based on the W Ly α distribution in Figure 3. Themedium and light gray regions correspond to scattering probabil-ities of 30% and 6%, respectively. Arrows labeled “abs” indicatethe direction in which the colors will be perturbed with increasingLy α absorption. ria introduces significant biases into the measured W Ly α distribution. We can begin by examining some char-acteristics of the measured W Ly α for BX galaxies andLBGs, summarized in Table 4. The BX distribution has h W Ly α i ∼ − W Ly α for LBGs. While the measurements of W Ly α for individ-ual galaxies may be highly uncertain, the difference inthe average W Ly α suggests that the high redshift (LBG)population has a higher incidence of Ly α in emission thanthe low redshift (BX) population. This disparity betweenthe lower and higher redshift populations can be betterappreciated by examining column (3) of Table 4 thatshows that the fraction of galaxies with W Ly α ≥
20 ˚A( f
20) is almost twice as high among LBGs as it is forBX galaxies.The change in f
20 is even more apparent when weconsider BX galaxies in different redshift ranges: f
20 forBX galaxies with redshifts between z ≈ . z = 2 . z =1 . z ≈ . α emission at higher redshifts, but is this trend intrinsic tohigh redshift galaxies, or is it introduced as a result ofcolor selection bias, as Figure 4 suggests?We can test for systematics induced by the color cri-teria by examining f
20 for BX galaxies at redshifts2 . ≤ z < .
48, where Ly α lies outside the U and G -bands. These galaxies have a similar f
20 to that of z < .
17 galaxies (Table 4), implying that the frac-tion of absorption versus emission line systems culledby the BX criteria is similar between the z < .
17 and
TABLE 4Measured W Ly α Distributions
Sample h W Ly α i a f ( W Ly α ) ≥
20 ˚A b BX (ALL: 1 . ≤ z < . − . ≤ z < . − . ≤ z < . − . ≤ z < .
70) 2 ˚A 0.20LBG (2 . ≤ z < .
4) 9 ˚A 0.23 a Mean rest-frame W Ly α . b Fraction with W Ly α ≥ the 2 . ≤ z < .
48 samples (this assumes that thereis little evolution in the W Ly α distribution between the z < .
17 and 2 . ≤ z < .
48 subsamples). Focusingon the high redshift subsample with 2 . ≤ z < . α in absorption,yet their f
20 is similar to that of LBGs. In other words,the 2 . ≤ z < .
70 subsample has an f
20 value thatdoes not indicate a preferential selection of absorptionover emission line galaxies relative to that of the lowerredshift subsamples. Rather, the f
20 value is larger thanthose for the lower redshift subsamples and is similar tothat of the LBGs. These conclusions are supported by aKolmogorov-Smirnov test, the results of which are sum-marized in Figure 3. Namely, ξ in the figure indicatesthe probability that the W Ly α distributions for the to-tal sample, the sample with 1 . ≤ z < .
17, and thesample with 2 . ≤ z < .
70, are drawn from the sameparent population as the sample with 2 . ≤ z < . α does not effect the U n G R colors. Galaxieswith 2 . ≤ z < .
70 have a W Ly α distribution that de-viates significantly from the one at 2 . ≤ z < . α line profiles would then suggest thatthe BX color criteria do not significantly modulate theintrinsic W Ly α distribution of z ∼ For thepurposes of our simulations, we make the approximationthat the observed W Ly α distribution for BX galaxies canbe applied to our simulated galaxies to obtain the aver-age perturbation of their rest-UV colors.Since the Ly α line falls in the G -band for galaxies inthe entire redshift range 2 . ≤ z < .
4, we cannot ex-amine trends in the W Ly α distribution for the LBGs inthe same way we did for the BXs. However, in § W Ly α distribution for LBGs shouldapproximately reflect the intrinsic W Ly α distribution for z ∼ § W Ly α distributions are correct. Quantifying Incompleteness
Effective Volume ( V eff ) Method The fraction of galaxies with a given set of binned prop-erties that satisfy the color criteria can be computed di-rectly from the results of the Monte Carlo simulations. Shapley et al. (2003) have demonstrated that W Ly α is in factdependent upon the rest-frame UV colors and magnitudes of galax-ies. However, the small biases that these trends may have on theobserved W Ly α do not have a significant impact on the derived LFsat z ∼ z ∼ L ), redshift ( z ), and reddening ( E ( B − V )) of a galaxy.Under the assumption that these properties are indepen-dent of each other, and if we let the indices i , j , and k run over the range of values of L , z , and E ( B − V ),then the true number of galaxies in the ijk th bin can beapproximated as n trueijk ≃ n obsijk / ¯ p ijk (3)where ¯ p ijk are the mean probabilities that a galaxy inthe ijk th bin is (a) detected and (b) satisfies the colorcriteria (e.g., Adelberger 2002). These probabilities ¯ p ijk are simply ¯ p ijk = 1 n ijk n X p ijkn (4)where p ijkn is the probability that the n th simulatedgalaxy in the ijk th bin will be detected as a candidate,and n ijk is the total number of simulated galaxies in the ijk th bin. The values p ijkn take into account the prob-ability that the colors of the n th simulated galaxy willbe perturbed by the W Ly α distribution of Figure 3 andstill be selected as a BX object. They also fold in theprobability that a non-candidate simulated galaxy willfall in the BX selection window. Dividing by n ijk nor-malizes the mean probabilities ¯ p ijk and accounts for boththe fraction of galaxies whose photometric errors scatterthem out of the BX selection window and galaxies thatare not detected in the simulations. If the true comov-ing volume corresponding to the j th bin in redshift is V j ,then the effective volume associated with the j th bin in z is V eff j ≡ V j ik X ¯ p ijk = V j × ξ j , (5)where ξ j are commonly referred to as “completeness co-efficients”: ξ j ≡ ik X ¯ p ijk . (6)The photometric properties of each field are unique dueto differences in the observing conditions, and this willaffect the computed ξ j . We can then determine the com-pleteness coefficients for each field and then perform aweighted-average of them (i.e., weighted according to thefield size) to obtain mean completeness coefficients, ¯ ξ j . Maximum Likelihood Method ( V lik ) While the procedure just described can be used tomake an initial guess as to the shape of the reddeningand luminosity distributions, it can lead to spurious re-sults, particularly for objects whose true colors are suchthat they lie outside of or close to the edges of the BXselection window. Equation 3 is approximately true onlyif the average measured properties of a galaxy are thesame as the true (simulated) properties, and this willcertainly not be the case for galaxies that are preferen-tially scattered into the BX window due to photometricerrors or the presence of Ly α absorption/emission (e.g.,Adelberger 2002). The approach described above willalso not take into account photometric bias and the pref-erential scattering of objects from one bin to another if Fig. 5.—
Cartoon illustration of how the probability that agalaxy with intrinsic (true) properties L ′ E ′ z ′ may not have a one-to-one correspondence with bins of observed (measured) properties LEz (or measured colors ugr ). the bin sizes are comparable to (or smaller than) thephotometric errors (Adelberger 2002).Figure 5 further illustrates these issues. In the sim-plest case, galaxies that fall within a particular bin oftrue properties (say, L ′ E ′ z ′ ) will, on average, have mea-sured properties corresponding to bin L E z . In thiscase, we can use the approach of § V eff method) to simply divide the observed number of galax-ies in bin L E z by the probability that galaxies in bin L ′ E ′ z ′ will be observed in bin L E z (call that prob-ability p ′ ′ ′ → ), as shown by the leftmost arrow inFigure 5. However, we can point to several examplesthat suggest that there may not be a one-to-one corre-spondence between bins of intrinsic and observed prop-erties, as illustrated by the remaining arrows in Figure 5.First, in order to accurately compute the luminosity andreddening distributions at z ∼
2, we cannot make ourbin sizes much larger than the typical photometric errorssince the observed range of U n − G and G − R colors forgalaxies at a single redshift ( . . . U n − G and G −R . There-fore, galaxies that ought to fall within a particular bin ofmeasured properties will be scattered into adjacent bins.This would not be a problem if each bin of measuredproperties gained and lost an equal number of galax-ies, but since the luminosity and reddening distributionsare peaked, photometric errors will scatter galaxies awayfrom the peak and into the wings of the distributions.Second, the distribution of errors in colors is not sym-metric with respect to the true values such that there isa systematic tendency to scatter galaxies into redder binsmore often than bluer ones (Steidel et al. 2003). Third,the presence of Ly α absorption in a galaxy’s spectrumwill, depending on the redshift, cause us to overestimatethe reddening. Finally, there will be some galaxies whosetrue properties are such that on average they lie outsidethe selection windows, and only get scattered into thesample because of photometric errors, such as might bethe case for galaxies lying close to the color selectionboundaries.Because of these systematic effects, the number ofgalaxies within a particular bin of measured propertieswill be some weighted combination of the numbers ofgalaxies within intrinsic bins that contribute to that mea-ulti-Wavelength LFs at 1 . . z . . L , reddening E , and redshift z within a field of size ∆Ωis ¯ n ( L, E, z ) dVd Ω dz = µ Z dL ′ dE ′ dz ′ f ( L ′ ) g ( E ′ ) h ( z ′ ) p L ′ E ′ z ′ → LEz dVd Ω dz , (7)where µ is related to the total comoving number densityof galaxies; p L ′ E ′ z ′ → LEz is the (transitional) probabilitythat galaxies with intrinsic L ′ E ′ z ′ will be measured tohave LEz ; and f(L’), g(E’), and h(z’) are the intrinsicdistributions of luminosity, reddening, and redshift, re-spectively, normalized such that Z L max L min dLf ( L ) = Z E max E min dEg ( E ) = Z z max z min dzh ( z ) = 1 (8)(Adelberger 2002). Our goal is to determine the intrinsicdistributions f ( L ), g ( E ), and h ( z ), but inverting Eq. 7to solve for these distributions is intractable. One al-ternative is to compute the likelihood ( L ) of observingour data, which is expressed as a list of galaxies withobserved L i E i z i , for a given set of f gh distributions: L ( L i E i z i ) ∝ exp (cid:20) − µ ∆Ω Z dLdEdz ¯ n ( L, E, z ) dVd Ω dz (cid:21)Y i ¯ n ( L i E i z i ) . (9)The discrete form of Eq. 9, extended to incorporate eachof l different fields, can be expressed as L ( n ijkl ) ∝ exp − X ijkl ¯ n ijkl Y ijkl ¯ n n ijkl ijkl , (10)where ¯ n ijkl is the mean number of galaxies in the i th binof luminosity, j th bin of reddening, and k th bin of redshiftin the l th field that the assumed values of f i , g j , and h k imply; and n ijkl is the observed number of galaxies in thesame bin (e.g., Adelberger 2002). The discrete version ofEq. 7 is¯ n ijkl = µ ∆Ω l X i ′ j ′ k ′ f i ′ g j ′ h k ′ V k ′ p l,i ′ ,j ′ ,k ′ → ijk , (11)where ∆Ω l is the size of the l th field, V k ′ is the comov-ing volume in Mpc arcmin − corresponding to bin k ′ inredshift, and p l,i ′ ,j ′ ,k ′ → ijk is the probability that a galaxyin the l th field in the i ′ j ′ k ′ bin of luminosity, reddening,and redshift, will have measured properties correspond-ing to bin ijk . Assuming that the data quality does notvary significantly from field-to-field, we can simplify theprobabilities such that¯ p i ′ j ′ k ′ → ijk ≡ X l ∆Ω l p li ′ j ′ k ′ → ijk / X l ∆Ω l . (12) Maximizing the likelihood as expressed in Eq. 10 is equiv-alent to minimizing − ln L ∝ X ijk ¯ n ijk − X ijk n ijk ln ¯ n ijk , (13)and is more amenable to computation than Eq. 10. Implementation of the Maximum-Likelihood Method
We first used the Monte Carlo simulations to determinethe transitional probabilities that relate the true lumi-nosities, reddenings, and redshifts of galaxies to their ob-served rest-UV colors. Following the discussion of § W Ly α according to the distributions shown inFigure 3 for the BX sample and the distribution shownin Figure 8 of Shapley et al. (2003) for LBGs (see alsoFigure 8). We took advantage of both the U n − G and G −R colors in our analysis of the z ∼ E ( B − V ) distribution,something that was not possible at z ∼ U n either due to severe blan-keting by the Ly α forest or the suppression of continuumflux shortward of the Lyman limit.Figure 6 is useful in visualizing the transitional prob-abilities, in this case for the BX sample, where we showthe relative probability distribution for galaxies between1 . < z < . E ( B − V ) distributions assumed in computing thetransitional probabilities. This distribution reflects bothphotometric error and Ly α perturbation of the expectedrest-UV colors. One noticeable feature of Figure 6 isthe divergent behavior of the selection function for low( z . .
0) and high ( z & .
7) redshift galaxies, wherehigher redshift galaxies have redder U n − G colors fora given SED. This can be understood, in part, by ex-amining Figure 4. If z ∼ z > . α perturbation. First, we find no evidence that photomet-ric errors increase for galaxies at higher redshifts. Sec-ond, the (1+ z ) dependence of the observed W Ly α will re-sult in a larger color change (for a fixed rest-frame W Ly α )for higher redshift galaxies than for lower redshift galax-ies, such that the scattering probability distribution cov-ers a larger area in color space, making it less likely for aparticular source to fall within the BX selection window.Finally, the U n − G color changes more rapidly for higherredshift galaxies where Ly α forest absorption begins toincreasingly affect the U n -band. All of these effects couldexplain the relatively small number of z > . z > . z ∼ z > . z > . . . z . . Fig. 6.—
Relative probability distribution for galaxies with in-trinsic colors ( U n − G ) true and ( G − R ) true to be detected andselected as BX objects (solid line same as in Figure 4). The dis-tribution is weighted according to the incidence of galaxies witha particular set of intrinsic colors as determined from the LF and E ( B − V ) distributions used to compute the transformation be-tween intrinsic and observed colors. The distribution is non-zeroexterior to the BX window (trapezoid) as a result of photometricerror and Ly α line perturbations of the colors. Galaxies with ex-pected (or intrinsic) U n − G colors bluer than required to satisfy BXcriteria are particularly prone to selection as discussed in § . < z ≤ .
48 are expected to lie.These galaxies’ colors are unaffected by Ly α line perturbations. space as is evident from Figure 4. Small variations in col-ors as a result of photometric errors or Ly α absorptioncan shift a large numbers of such galaxies into the BXselection window. This effect can be viewed in Figure 6,where there is a high relative probability for galaxies withblue U n − G colors (the “BM” galaxies; e.g., Figure 10of Adelberger et al. 2004) to satisfy BX selection, partlydue to the effect of Ly α absorption in these systems (cf.,Figure 3b). The highest density region in this figure (be-tween the two white curves of Figure 6) occurs in thesame color space expected to be occupied by galaxies atredshifts where the Ly α line does not affect the U n G R colors (2 . < z ≤ . α absorption/emission,and photometric error ( § f gh ) can lead to spurious results given the large parameter space and possibility of numerous lo-cal minima in likelihood space. A reasonable approach isto then make some simplifying assumptions, such as fix-ing the redshift distribution to be constant and assumingan LF computed using the method of § g ) as parameterized by the E ( B − V )color excess, using values of p i ′ j ′ k ′ → ijk relevant for thespectroscopic sample. In other words, p i ′ j ′ k ′ → ijk willgive the probability that a galaxy with true properties inthe i ′ j ′ k ′ th bin will be measured with properties in the ijk th bin and be spectroscopically observed. The proba-bility that a candidate lying within a particular bin of R magnitude will be spectroscopically observed is approxi-mated using the spectroscopic fractions listed in Table 2.These spectroscopic fractions are then multiplied by theprobability that an object is a star-forming galaxy (i.e.,not an AGN/QSO) using the AGN/QSO fractions in therelevant magnitude range (Table 2). At this stage, wemust rely on the spectroscopic sample since we can onlyestimate E ( B − V ) for galaxies with redshifts. Then,keeping g fixed to the best-fit E ( B − V ) distribution, wetake advantage of the full photometric sample to mini-mize Eq. 13 with respect to the luminosity distribution( f ). The revised estimate of f can then be held fixed torefine our estimate of g . The process goes through severaliterations where at the last stage we vary f , g , and h si-multaneously. The results of this procedure indicate thatour initial assumption of a constant redshift distribution(i.e., number of galaxies in each of the redshift bins isroughly constant) is a reasonable one to make. The red-shift distributions predicted for BX galaxies and LBGsgiven the maximum-likelihood f gh distributions are ex-cellent matches to the observed redshift distributions ofBX galaxies and LBGs, as shown in Figure 7. Fig. 7.—
Expected redshift distributions (lines) given our best-fit reddening and luminosity distributions, compared with the ob-served redshift distributions of BX galaxies (left panel) and LBGs(right panel), indicated by the shaded histograms.
Uncertainties in the luminosity and E ( B − V ) distri-bution were estimated by generating many fake realiza-tions of our observed data from the catalogs of simulatedgalaxies, and recomputing the best-fit f gh . The disper-sion in measurements of f gh are taken to be the 1 σ errors. It is important to note that the errors in our es-timates are due to a combination of Poisson noise andfield-to-field dispersion. Unlike all other previous esti-mates of the z ∼ − systematic effects mentioned in § . . z . . Obs z~2 Exp z~3 Obs z~3 Exp z~3 Obs z~3 Obs z2+3 Exp z~3 Obs z~3 Obs z~3 Exp z~2 Obs z~2 Exp z~2 Obs z~2 Obs z2+3 Exp z~2 Obs z~2
Fig. 8.—
Comparisons between expected and observed W Ly α distributions for different assumptions of the intrinsic W Ly α distributionsbetween redshifts 2 . ≤ z < . . ≤ z < . W Ly α distribution derived assuming many realizations of the LF and E ( B − V ) distribution(see text). The parameter ξ denotes the likelihood that the observed and expected distributions are drawn from the same distribution. Weuse the convention that W Ly α > α emission. RESULTS: INTRINSIC W LY α AND E ( B − V )DISTRIBUTIONS Validity of Assumed W Ly α Distributions
An important question is whether the distribution ofLy α emission and absorption profiles of galaxies changesas a function of redshift. Such trends with redshift mayindicate fundamental differences in the ISM of galax-ies and/or changing large-scale environments as a func-tion of redshift. Can we do better job of determin-ing whether the intrinsic W Ly α distribution of galaxieschanges as a function of redshift? Ideally, we would haveliked to include the W Ly α distribution as another freeparameter in the maximum-likelihood method discussedin § f gh ), we would be maximizing four. However, thiswould needlessly complicate our ability to determine themaximum-likelihood f gh distributions, especially since the luminosity distribution ( f ) is insensitive to smallchanges in the W Ly α distribution. As a compromise, we can investigate how different assumptions of the intrin-sic W Ly α distributions of galaxies affect the distributionsthat we expect to measure.Figure 4 illustrates how the color selection criteria canmodulate the observed W Ly α distribution of galaxies,such that the observed distribution may be different thanthe intrinsic distribution. The Monte Carlo simulationsdiscussed in § W Ly α distributions for the BX and LBGsamples with those expected based on the transitionalprobabilities. The results of this comparison are sum-marized in Figure 8, which shows W Ly α for various as-sumptions of the input W Ly α distribution. We considerthree cases. In the first case, we assume that the intrinsic W Ly α distribution of galaxies at z ∼ z ∼
2. In the second case, we assume that4 Reddy et al.the intrinsic W Ly α distribution at z ∼ z ∼ z ∼
3. In the third case, we assume that theintrinsic W Ly α distribution at z ∼ z ∼
3. Three analogous casesare considered for the z ∼ W Ly α distribution for 2 . ≤ z < . intrinsic distribu-tion at 2 . ≤ z < . W Ly α distribution for lower redshift (1 . ≤ z < .
7) galaxies(green rectangles, labeled “Obs z ∼ intrinsic distribution can be testedby comparing the expected distribution (red rectangles)with the actual measured distribution (blue rectangles).The vertical sizes of the rectangles for the observed dis-tributions reflect Poisson errors. Uncertainties in the ex-pected distributions (red rectangles) are computed byconstructing many samples of galaxies drawn randomlyfrom the maximum-likelihood luminosity and E ( B − V )distributions ( § § W Ly α distributions measured foreach of these simulated samples.The top left panel of Figure 8 shows that assumingan intrinsic distribution of W Ly α for galaxies at redshifts2 . ≤ z < . W Ly α for BX-selected (1 . ≤ z < .
7) galaxiesresults in an expected distribution at 2 . ≤ z < . measured . In this case, the expected distributionexhibits a larger fraction of galaxies with absorption anda deficit of emission-line galaxies when compared withthe measured W Ly α distribution. Therefore, the intrin-sic W Ly α for 2 . ≤ z < . . ≤ z < .
7) galaxies. The bottom left panel of Fig-ure 8 tells a similar story. Assuming 1 . ≤ z < . W Ly α identical to that measured forLBGs (2 . ≤ z < .
4) results in an expected distributionfor 1 . ≤ z < . . ≤ z < .
7. Therefore, the intrinsic W Ly α distri-bution for 1 . ≤ z < . α in absorption than what isobserved for higher redshift (2 . ≤ z < .
4) galaxies.The middle panels of Figure 8 show what happensif we assume that the intrinsic W Ly α distribution at1 . ≤ z < . . ≤ z < . In the bottom middle panel,the expected and observed distributions are not signifi-cantly different, so it is plausible that the intrinsic distri-bution of W Ly α for redshift 1 . ≤ z < . . ≤ z < . α in absorption. Decreasing the fraction of galaxies Since the comoving number densities of galaxies at redshifts1 . ≤ z < . . ≤ z < . W Ly α distribution as an equally weightedsum of the measured distributions in these two redshift ranges. with absorption in the intrinsic distribution (e.g., by as-suming some non-equally weighted combination of W Ly α distributions at low and high redshifts) may result in abetter match for the observed z ∼ z ∼ W Ly α distribu-tions at 1 . ≤ z < . . ≤ z < . assumed intrinsic distributions, even if the assump-tions are erroneous (compare red and green rectangles inleft and middle panels of Figure 8; a Kolmogorv-Smirnov(KS) test indicates a &
50% probability that the intrinsicand expected distributions are drawn from the same pop-ulations). These observations suggest that the BX andLBG color selection criteria do not alter significantly theparent W Ly α distribution of galaxies. We already dis-cussed in § U n G R colors are unaffected by Ly α ; we havejust shown it to be true for LBGs also.In all cases shown in Figure 8, we have quantified thedisparity in the observed and expected W Ly α distribu-tions by computing the statistic ξ as follows. We gener-ated 10000 realizations of the expected W Ly α distribution(in the same way as we did to compute the uncertaintiesin the expected distribution; see above). We then per-formed a KS test to determine the probability ( p KS ) thateach of these realizations are drawn from the same distri-bution as the observed W Ly α distribution. The quantity ξ is then defined as the ratio of the number of realiza-tions where p KS < . ξ indicate that the expected andobserved distributions are less likely to have been drawnfrom the same parent distribution. The values of ξ aregiven in each panel of Figure 8, and support our con-clusion that the color criteria do not significantly alterthe intrinsic W Ly α distributions. In summary, we findevidence that the fraction of emission-line galaxies ( f W Ly α distributions for galaxies at lower and higher red-shift (Figure 8). E ( B − V ) Distributions
A useful by-product of the maximum-likelihoodmethod ( § E ( B − V ),corrected for incompleteness. Figure 9 shows the best-fit E ( B − V ) distributions, compared with the observeddistributions, for galaxies at redshifts 1 . ≤ z < . . ≤ z < . V eff method ( § E ( B − V ) distributions thatare within 10% of the observed distributions and there-fore deviate significantly from our best-fit distributionsat z ∼ z ∼
3. The analysis indicates that the true E ( B − V ) distributions are slightly bluer, on average,than observed. Table 5 lists the mean and dispersionof E ( B − V ) for the observed and maximum-likelihooddistributions for galaxies at redshifts 1 . ≤ z < . . . z . . Fig. 9.—
Comparison of best-fit and observed E ( B − V ) distributions for galaxies at redshifts 1 . ≤ z < . . ≤ z < . G − R colors, and assuming a constant star formationmodel attenuated by the Calzetti et al. (2000) law. The width of each bar reflects the Poisson error in the corresponding bin. The redlines and yellow shaded regions indicate the mean and 1 σ errors on the maximum-likelihood (best-fit) distributions. The blue dashedlines indicate the distribution without correcting for Ly α perturbation to the observed colors. Dashed and solid vertical lines, respectively,denote the average E ( B − V ) for the observed and best-fit distributions. The E ( B − V ) distribution data are summarized in Table 5. . ≤ z < . E ( B − V ) distribution for the BX sampleis expected to be slightly biased toward redder spectralshapes than the intrinsic values because our photomet-ric method makes the colors appear slightly redder thanthey really are — and thus E ( B − V ) is redder — par-ticularly for fainter galaxies ( § α absorption in a galaxy’s spectrum will,depending on the redshift, cause the G -band magnitudeto appear fainter than the true broadband magnitude(corrected for line effects), such that E ( B − V ) will beoverestimated. This latter effect can be visualized for1 . ≤ z < . α (dashedblue line) is systematically redder than the corrected dis-tribution (solid red line), owing to the presence of Ly α absorption among, and the low f
20 value of, the major-ity of galaxies at 1 . ≤ z < . α are less apparent inthe E ( B − V ) distribution for 2 . ≤ z < . α perturbations.Before proceeding with a discussion of E ( B − V ) as anindicator of dust reddening and the variation of E ( B − V )with redshift and apparent magnitude, we remind thereader that we can only correct for the incompletenessof objects whose colors are such that they are scatteredinto the selection windows. In other words, there areundoubtedly galaxies at these redshifts that will neverscatter into the BX/LBG selection windows, for exam-ple, those galaxies that are optically-faint either becausethey have little star formation or are very dusty star-bursts (e.g., DRGs and SMGs). Therefore, the E ( B − V )for such dusty galaxies will not be reflected in the dis-tributions shown in Figure 9. Typically, such very dustygalaxies would have E ( B − V ) > .
45, although a sig-nificant fraction also show bluer E ( B − V ) comparable to those of BX/LBGs (Chapman et al. 2005). Becausethese dusty star-forming and quiescent galaxies are inlarge part optically-faint, not accounting for them in ouranalysis should minimally affect our E ( B − V ) distribu-tion for optically-bright galaxies (Figure 9). Further, aswe show in § R < . z ∼ − E ( B − V ) as a Proxy for Dust Reddening Up until now, we have been using E ( B − V ) (the rest-frame UV slope) to parameterize the range of spectralshapes observed among high redshift galaxies. A numberof studies have shown that E ( B − V ) also has a physicalinterpretation: it correlates very well with the redden-ing, or dust obscuration, of most high redshift galax-ies (e.g., Calzetti et al. 2000; Adelberger & Steidel 2000;Reddy et al. 2006b). Here we define reddening as theattenuation of luminosity by dust which can be parame-terized, for example, by the quantity L bol /L UV . Combin-ing Spitzer
MIPS data for a sample of spectroscopically-confirmed redshift 1 . . z . . K -corrections could be computed accurately, Reddy et al.(2006b) showed that E ( B − V ) not only correlates with L bol for galaxies with L bol . . L ⊙ , but that the cor-relation is identical to that established for local galaxies(Calzetti et al. 2000; Meurer et al. 1999).The E ( B − V ) for relatively dust-free (or very young)galaxies is dominated by intrinsic variations in the SEDsof high redshift galaxies, and so E ( B − V ) is not a directindicator of reddening for these galaxies (which is why wemeasure a non-negligible number density of galaxies with E ( B − V ) < L bol & L ⊙ that are optically-bright ( R < .
5) andsatisfy the rest-UV color criteria, but have E ( B − V ) thatseverely underpredict their attenuations and bolometric6 Reddy et al. TABLE 5Normalized E ( B − V ) Distributions E ( B − V ) BX (Measured) 1 . ≤ z < . . ≤ z < . − . − . a < .
01 0 . ± .
02 0 . ± .
01 0 . ± . − . a . ± .
01 0 . ± .
02 0 . ± .
01 0 . ± . . ± .
01 0 . ± .
03 0 . ± .
02 0 . ± . . ± .
02 0 . ± .
02 0 . ± .
02 0 . ± . . ± .
01 0 . ± .
03 0 . ± .
01 0 . ± . . ± .
01 0 . ± .
02 0 . ± .
01 0 . ± . < .
01 0 . ± .
01 0 . ± .
01 0 . ± . h E ( B − V ) i . ± .
07 0 . ± .
12 0 . ± .
07 0 . ± . a We measure a non-negligible number of galaxies with E ( B − V ) < E ( B − V ) of dust-free and/orvery young galaxies is dominated by intrinsic variations in the SED (see text). luminosities (Reddy et al. 2006b). Finally, we remindthe reader that we cannot account for the E ( B − V )of objects that have a zero probability of being scat-tered into our sample. Nonetheless, our completeness-corrected estimates of the E ( B − V ) distributions suggestan average attenuation between dust-corrected and un-corrected UV luminosity, L cor UV /L UV , of ∼ −
5. Thisis similar to the value measured from (1) stacked X-ray data for BX galaxies and LBGs (Nandra et al. 2002;Reddy & Steidel 2004), and (2) MIPS luminosities anddust-corrected UV and H α luminosities (Reddy et al.2006b; Erb et al. 2006b). It is also the same value advo-cated by Steidel et al. (1999) in correcting observed UVluminosities for dust extinction among z ∼ Comparison of Reddening Distributions with Redshift
Remarkably, we find very little evolution in the red-dening distribution between redshifts 1 . . z . .
4, de-spite the roughly 730 Myr timespan between the meanredshifts for the low ( h z i = 2 .
30) and high ( h z i = 3 . E ( B − V ) since the BX criteria were designed to se-lected galaxies with a similar range of spectral propertiesas LBGs. Even so, the incompleteness corrections mod-ulate two very different observed E ( B − V ) distributionsfor the lower and higher redshift samples to the pointwhere they are virtually identical. The difference in thefraction of large W Ly α emission systems between the twosamples ( § W Ly α emission systems could be young and relatively dust-freegalaxies ( § E ( B − V ) and dust attenuation be-tween z ∼ z ∼ §
8, the lack ofevolution in E ( B − V ) implies that the extinction prop-erties of bright ( R ≤ .
5) star-forming galaxies are notchanging significantly between redshift z ∼ z ∼ z .
2) and higher ( z & Reddening Distribution as a Function of Rest-FrameUV Magnitude
Before turning to a discussion of the LF, we mustfirst determine whether the best-fit E ( B − V ) distribu-tion shows any systematic changes as a function of rest- Fig. 10.—
Comparison of maximum-likelihood E ( B − V ) distri-butions for galaxies at redshifts 1 . ≤ z < . . ≤ z < . frame UV magnitude, since such changes can, in prin-ciple, affect the shape and normalization of the LF. Asa first test, we restricted the maximum-likelihood analy-sis ( § . ≤ R ≤ .
5, and we did not find any significanttrend in the E ( B − V ) distribution as a function of mag-nitude to R = 25 . E ( B − V ) can be used as a proxy for thereddening of galaxies ( § Spitzer
MIPS data for a sample of BX-selected galaxiesin the GOODS-N field; these data give us an independentprobe of the dust emission in z ∼ L bol /L UV , where L bol ≡ L IR + L UV (infrared plus UV lu-minosity), as a function of observed optical magnitude,from the MIPS analysis of the GOODS-North field byReddy et al. (2006b). The open red circles indicate rest-UV-selected objects at 1 . ≤ z ≤ .
6, most of which areBX galaxies, detected at 24 µ m, and the large pentagonand crosses denote the average stack and distributionin R magnitude, respectively, for galaxies undetected Here we define L IR as the total luminosity between 8 and1000 µ m. ulti-Wavelength LFs at 1 . . z . . Fig. 11.—
Distribution of attenuation factors, parameterized as L bol /L UV , as inferred from Spitzer
MIPS data, as a function ofapparent optical magnitude R for rest-UV-selected galaxies withredshifts 1 . . z . .
6. Also indicated is the stacked averagefor 48 galaxies undetected at 24 µ m (large blue pentagon) andunconfused with brighter sources; the distribution in R magnitudefor a larger sample of 73 galaxies undetected at 24 µ m is shown bythe arbitrarily normalized crosses. Total stacked 24 µ m and X-rayaverages are indicated by the solid red circles and green points,respectively, that include all galaxies. at 24 µ m. The total 24 µ m stacked averages includ-ing both detected and undetected galaxies at 24 µ m areshown by the solid red circles. Similarly, the L bol /L UV inferred from X-ray stacked averages (computed in man-ner similar to that presented in Reddy & Steidel 2004where L bol is determined from the X-ray flux) includingall galaxies, irrespective of direct detection in the Chan-dra dis-persion in attenuation factor increases towards faintermagnitudes, as evidenced by the larger spread of 24 µ mdetection galaxies and as would be expected if optically-faint galaxies have contribution from both heavily dust-obscured objects as well as those with intrinsically lowstar formation rates, the results of Figure 11 suggest thatthe average extinction correction (based on the stackedpoints) is approximately constant over the range in R magnitude considered here. These results confirm thetrends noted by Adelberger & Steidel (2000), who usedlocal templates to deduce that the observed UV lumi-nosities of galaxies at redshifts z = 0, z ∼
1, and z ∼ L bol /L UV (e.g., Fig-ure 17 of Adelberger & Steidel 2000). We have confirmedthis trend explicitly at redshifts 1 . . z . .
6. The ob-served (unobscured) UV luminosity (i.e., the emergentluminosity after attenuation by dust) to R = 25 . z ∼ − unobscured UV luminosity, at least to R = 25 .
5, there is a very strong dependence of the dust- We note that Reddy et al. (2006b) excluded objects from theiranalysis that were directly detected in the
Chandra corrected
UV luminosity (or IR or bolometric luminos-ity) on the attenuation factors of galaxies (Reddy et al.2006b; Adelberger & Steidel 2000; see also § R = 25 .
5. We return to the issue of how a varyingreddening distribution affects our calculation of the totalluminosity density in § RESULTS: UV LUMINOSITY FUNCTIONS
Preferred LFs
To provide the closest match between rest-frame wave-lengths, and thus avoid cosmological K -corrections, weused R -band as a tracer of rest-frame 1700 ˚A emissionat the mean redshift of the LBG sample, h z i ∼ . G and R -band ( m G R ) as a tracer of rest-frame 1700 ˚A atthe mean redshift of the BX sample, h z i ∼ .
30, where m G R is simply the magnitude corresponding to the av-erage of the G and R fluxes. Absolute magnitudes werecomputed using the standard relation: M AB (1700˚ A ) = m − d L /
10 pc) + 2 . z ) , (14)where M AB (1700˚ A ) is the absolute magnitude at rest-frame 1700 ˚A, d L is the luminosity distance, and m is theapparent magnitude at R -band at z ∼ G R -band at z ∼
2. We have made the reasonableassumption that the SED K -correction is approximatelyzero for the average rest-UV SED of BX-selected galax-ies after a star formation age of 100 Myr for the typicalreddening ( E ( B − V ) ∼ .
15) of galaxies in our sample.The maximum-likelihood rest-frame 1700 ˚A luminos-ity functions for z ∼ z ∼ E ( B − V ) distribution (as determined from thespectroscopic sample; Figure 9) fixed. The extension ofthe spectroscopically-determined E ( B − V ) distributionto the photometric sample is a reasonable approximationgiven that (a) the spectroscopic and photometric samplesare likely to have the same redshift distribution ( § E ( B − V ) distribution remains unchangedas a function of R magnitude to R = 25 . § α perturbations. Errors in theluminosity functions reflect both Poisson counting statis-tics and field-to-field variations; the latter are accountedfor by examining the dispersion in the LF as a function ofmagnitude for each of the fields of the survey (see § z ∼ α , characteristic absolute magnitude M ∗ (or characteristic luminosity L ∗ ), and characteristic num-ber density φ ∗ are estimated by simulating many realiza-tions of the LF as allowed by the errors (assuming theerrors follow a normal distribution), fitting a Schechterfunction to each of these realizations, and then determin-ing the dispersion in measured values for α , M ∗ , and φ ∗ for these realizations.Based on integrating our maximum-likelihood LFs, thefraction of star-forming galaxies with redshifts 1 . ≤ z < . M AB (1700˚ A ) < − .
33 (i.e., R = 25 .
58 Reddy et al. -2.1 -1.8 -1.5-22-21
Fig. 12.—
Rest-frame UV luminosity functions at z ∼ z ∼ HST , respectively) computed in our analysis, compared with z ∼ z ∼ z ∼ α and M ∗ for the z ∼ α and M ∗ for z ∼ TABLE 6Rest-Frame UV Luminosity Functions of . . z . . Galaxies φ Redshift Range M AB (1700˚ A ) ( × − h . Mpc − mag − )1 . ≤ z < . − .
83 — − .
33 0 . ± . − .
33 — − .
83 0 . ± . − .
83 — − .
33 0 . ± . − .
33 — − .
83 0 . ± . − .
83 — − .
33 1 . ± . − .
33 — − .
83 2 . ± . − .
83 — − .
33 3 . ± . . ≤ z < . − .
02 — − .
52 0 . ± . − .
52 — − .
02 0 . ± . − .
02 — − .
52 0 . ± . − .
52 — − .
02 0 . ± . − .
02 — − .
52 0 . ± . − .
52 — − .
02 1 . ± . at z = 2 .
3) that have colors that satisfy BX criteria is ≈ . ≤ z < . M AB (1700˚ A ) < − . R = 25 . z = 3 .
05) that have colors that satisfy theLBG criteria is ≈ R = 25 . total fraction of galaxies with 1 . ≤ z < . R < . either the BX or LBG criteriais 0 . V eff versusmaximum-likelihood method, the significance (or lackthereof) of the Schechter parameters, and field-to-fieldvariations. We conclude this section by examining howphotometric redshifts can introduce non-trivial biases inthe computation of the LF.ulti-Wavelength LFs at 1 . . z . . Comparison of the V eff and Maximum-LikelihoodMethods The maximum-likelihood technique was used to de-rive the LFs presented here. However, because manypublished LFs are derived using the less accurate V eff method, particularly for dropout samples at high red-shift, it is useful to determine how close (or how far)such determinations are from reality by comparing withour maximum-likelihood value. Fig. 13.—
Comparison of the maximum-likelihood LF at z ∼ V eff determinations from our work (open circles) andSawicki & Thompson (2006) (open triangles). Figure 13 compares the LFs at z ∼ V eff ( § § V eff determinationat z ∼ V eff LF with respect to themaximum-likelihood value. These biases are particularlyapparent for criteria that target a narrow range in colorspace, such as the BX criteria, where photometric scatteror perturbations due to Ly α can be as large or larger thanthe width of the color selection windows (e.g., Figure 4,6). For example, we would have inferred a significantlyshallower faint-end slope of α = − . ± .
15 had werelied on the LF derived from the V eff method. Forthe LBG criteria, we find little difference in the V eff andmaximum-likelihood determinations of the LF, mostlydue to the fact that the LBG selection window covers alarger area of color space and outlier objects (with col-ors placing them near the selection boundaries) for whichthe bias is the largest will make up a significantly smallerfraction of the sample. Schechter Parameters
The spectroscopic sample allows us to accurately con-strain the LF, taking into account sample completeness, Note that our V eff determination is slightly different from thatof Sawicki & Thompson (2006), despite the use of the exact samefilter set and color criteria between the two studies, since our V eff determination includes the effects of Ly α line perturbations to therest-UV colors and incorporates the maximum-likelihood E ( B − V )distribution in the LF calculation. interloper fraction, and line perturbations, for galaxieswith R < .
5. It is brighter than this limit that weconsider our LF to be most robust. The results for z ∼ R = 25 . z ∼ ∼ z ∼ . ≤ z < . α = − . ± .
11, with a characteristic magnitude of M ∗ AB (1700˚ A ) = − . ± . z ∼ U -dropout galaxies in HDF-N where the red-shift distribution was modeled using the color criteriaof Dickinson (1998) and assuming the range of intrinsicspectral shapes of LBGs found by Adelberger & Steidel(2000) (open squares in Figure 12). Based on the com-bined Keck spectroscopic and HDF-N U -dropout sam-ples, Steidel et al. (1999) found a steep faint-end slope α = − . ± .
13. Further refinement of the incomplete-ness corrections by Adelberger & Steidel (2000) resultedin a combined fit to the ground-based spectroscopic andHDF U -dropout samples of α = − . ± .
11. Fittingonly the ground-based (spectroscopically determined)points from our analysis at z ∼ α = − . ± . M ∗ AB (1700˚ A ) = − . ± .
16. Combining ourdata with the HDF-N U -drop points, and excluding thefaintest HDF point that may suffer from incompleteness(Steidel et al. 1999), results in a fit with α = − . ± . M ∗ AB (1700˚ A ) = − . ± .
12. Not surprisingly,these values are in excellent agreement with those foundby Adelberger & Steidel (2000), primarily because thesame faint (HDF) data are used to determine the faint-end slope. The best-fit Schechter function at z ∼ α and M ∗ are useful in parameterizing the gen-eral shape of the LF, we caution against over-interpretingtheir validity when accounting for faint galaxies that arebeyond current spectroscopic capabilities. The absenceof spectroscopic constraints on the asymptotic faint-endslope and a less than exponential fall-off of bright sourcesboth conspire to make α steeper (i.e., more negative).However, these parameters are useful in describing a lo-cal approximation to data points that are not far from L ∗ . For convenience, the best-fit parameter values fromour analysis are listed in Table 7. In the subsequent anal-ysis, we will assume α = − . z ∼ −
3. Note that if the steeper faint-end slopes inferred from our shallower ground-based data( α ∼ − .
85) accurately reflect reality, then this will serveto increase the total UV luminosity density of galaxieswith SFRs between 0 . ⊙ yr − by ∼
20% and ∼
50% at z ∼ z ∼
3, respectively, relative to thevalues obtained with α = − . Field-to-Field Variations
Access to multiple uncorrelated fields allows us tojudge the effects of large scale structure on the derivedLF. The dispersion in normalization between the lumi-nosity function in bins of R derived in individual fieldsis a strong function of R , as illustrated in Figure 14 forthe z ∼ z ∼ TABLE 7Best-fit Schechter Parameters for UV LFs of . . z . . Galaxies
Redshift Range α M ∗ AB (1700˚ A ) φ ∗ χ . ≤ z < . − . ± . − . ± .
23 (1 . ± . × − . ≤ z < . − .
60 (fixed) − . ± .
08 (3 . ± . × − . ≤ z < . − . ± . − . ± .
16 (1 . ± . × − . ≤ z < . − . ± . − . ± .
12 (1 . ± . × − field-to-field, within a given magnitude bin, are deter-mined by weighting the LF values by the field size suchthat LF determinations from larger fields are given moreweight than LF determinations from smaller fields. Thefractional dispersion in normalization is then defined asthe ratio of the dispersion of these weighted values andthe weighted mean value of the LF in each bin. Thisfractional dispersion in normalization is much larger atthe bright-end for R < . z ∼ z = 2 . . ≤ z < . z = 2 . z ∼ z ∼ . . z . . systematic effects of photometric bias and Ly α perturbations arealready reflected in the derived LFs. Effect of Photometric Redshifts
In light of recent literature regarding photometric es-timates of the UV LFs at redshifts z ∼ − z phot ). We derived the photometric redshifterror, defined as ∆ z ≡ z phot − z spec , (15)for a sample of 925 star-forming galaxies with spectro-scopic redshifts 1 . < z spec < . Fig. 14.—
Fractional dispersion in normalization of the z ∼ sufficient multi-wavelength data to warrant SED analy-sis. This is by far the largest spectroscopic sample atthese redshifts, and it enables us to investigate how ∆ z varies as a function of both redshift and magnitude since,in principle, the error will depend both on the relativeplacement of spectral breaks across the photometric fil-ters (i.e., the redshift of the galaxy) and on the qualityof the photometry and significance of the detection (i.e.,the apparent magnitude of the galaxy).Photometric redshifts were estimated using the Hy-perZ code of Bolzonella et al. (2000). We only consid-ered galaxies with detections in at least the followingbands: G , R , and K s . At least half of the resulting 925objects also have detections in either the J -band and/or Spitzer
IRAC bands. All but 52 of the 925 objects aredetected at U n ; the remaining 52 are all “C” candidates( § z > .
7. We considered a variety ofstar formation histories, reddening, and redshifts whenfitting the data using HyperZ. Figure 15 compares thephotometric and spectroscopic redshifts for the sampleof 925 galaxies. The biases and dispersions in photo-metric redshifts for galaxies fainter and brighter thanM ∗ = − .
97 in different redshift ranges are listed inTable 8, both including and excluding catastrophic out-liers with z phot < .
6. Even excluding outliers with z phot < . σ (∆ z ) ∼ . − .
42. Further, in all cases over the red-shift ranges where we compute the LFs, 1 . ≤ z < . . ≤ z < .
4, we find significant photometric biasesranging from ∆ z ∼ . − .
5, in the sense that z phot issystematically under-estimated (i.e., luminosity is over-ulti-Wavelength LFs at 1 . . z . . Fig. 15.— ( Left: ) Comparison between the photometric redshifts derived using HyperZ (Bolzonella et al. 2000) and spectroscopic redshiftsfrom our ground-based survey for a sample of 925 star-forming galaxies. The solid line denotes the unity relationship, and dashed linesdemarcate the region over which the UV LF is computed. (
Right: ) Comparison of the UV LF derived using the photometric redshiftdistribution in the left panel (open circles) with our spectroscopic determination (solid circles).
TABLE 8Biases and Dispersions in Photometric Redshift Errors( ∆ z = z phot − z spec ) for Star-Forming Galaxies at . ≤ z ≤ . a All Galaxies Excluding z phot ≤ . M > − . M ≤ − . M > − . M ≤ − . . ≤ z < . . ± .
40 0 . ± .
48 0 . ± .
35 0 . ± . . ≤ z < . − . ± . − . ± . − . ± . − . ± . . ≤ z < . − . ± . − . ± . − . ± . − . ± . a ∆ z ≡ z phot − z spec . estimated).A simulation was constructed to examine the effect ofthese biases and dispersions on the LF, similar to themethod presented in Marchesini et al. (2007), but modi-fied to (a) allow for many realizations of the intrinsic LFand (b) account for photometric redshift errors using theempirical data of Figure 15 and Table 8. To accomplishthis, we first generated many realizations of the UV LF asallowed by the (normally-distributed) errors of the spec-troscopically determined LF. We then drew magnitudesrandomly from a Schechter distribution determined byfitting a Schechter function to each of these realizationsof the intrinsic LF. Since the probability of a galaxy lyingat redshift z , p ( z ) ∝ dVdz ∝ d ( z )(1 + z ) p Ω Λ + Ω M (1 + z ) , (16)is roughly constant over the redshift interval 1 . ≤ z ≤ .
5, we drew redshifts from a uniform distribution. Theresult is a list of simulated redshift and magnitude pairs,( z spec , M ), for galaxies. A photometric redshift was as-signed to each galaxy by randomly drawing a redshiftfrom the distribution of z phot (in left panel of Figure 15)within a box of width δz spec = 0 . z spec ,using the z phot distribution for galaxies either brighteror fainter than M ∗ = − .
97 depending on the magni- tude M of the simulated galaxy. The absolute magni-tude of the galaxy was then recomputed assuming thephotometric redshift. We then reconstructed the LF at1 . ≤ z < . . ≤ z < .
7, the net result is that we would haveover-estimated the intrinsic LF had we relied on photo-metric redshifts. While the difference in the photometricand spectroscopic LFs on the faint-end is small, it be-comes quite significant for galaxies brighter than M ∗ .This systematic difference arises from the fact that thechange in absolute magnitude (∆ M ) for a fixed ∆ z andapparent magnitude will be larger for galaxies scatteredfrom low to high redshift than for galaxies scattered fromhigh to low redshift. For example, a galaxy at redshift z spec = 1 . z phot = 2 .
0, implying | ∆ z | = 0 .
4, results in ∆ M ≈ .
44. However, a galaxy atredshift z spec = 3 . z phot = 2 . | ∆ z | as above) results in ∆ M ≈ .
26. The neteffect is that the bright-end of the LF is systematicallyinflated with respect to the faint-end.There are three further issues to note. First, it hasbecome common in the literature to estimate photomet-2 Reddy et al.ric redshift errors independent of fitting the stellar pop-ulations of galaxies by simply shifting prescribed fixedtemplates until a best-fit redshift is reached. Redshifterrors derived in this manner will underestimate the trueerror in redshift obtained by marginalizing over the un-certainties of fitting those templates to the broadbandphotometry. Second, the simulation performed here ben-efited from the a priori knowledge that all the galaxiestruly lie at the correct redshifts 1 . ≤ z < .
7. Photo-metric redshift scatter (e.g., Figure 15) will generally belarger than that presented here since there will undoubt-edly be some very low redshift galaxies ( z < .
4) thatare scattered into the range 1 . ≤ z < .
7. Third, weremind the reader that the photometric redshift errorsderived here are for optically-bright ( R < .
5) objectswith spectroscopic redshifts. It is likely that the photo-metric redshift errors will be larger than assumed herefor very faint galaxies where the photometric uncertain-ties may be larger. This, in turn, may bias the faint-endslope more severely than reflected in our simulations. Westress that the results of the photometric redshift simula-tion presented here (Figure 15) are unique to our sample.As a result, the biases in the LF may be different for sur-veys that incorporate a different number of photometricfilters with differing photometric data quality, althoughthe photometric redshift accuracy found here is similar tothat presented in Shapley et al. (2005) and Reddy et al.(2006b) using more (different) bands. In any case, thissection illustrates how photometric redshifts can inducenon-trivial biases in the LF.
UV LF Summary
To summarize, this section has focused on our measure-ments of the UV LF at z ∼ z ∼
3. Our methodfor computing the LFs takes into account a number ofsystematic effects including contamination from low red-shift interlopers and AGN, Ly α line perturbations to theobserved colors of galaxies, and photometric scatter. Alarge number of independent fields allows us to controlfor sample variance. Further, spectroscopic redshifts en-able us to precisely correct for the effect of IGM opacityon the rest-UV colors. Our method for computing theLFs uses a maximum-likelihood technique to account forthe systematic scattering of galaxies in parameter (e.g.,luminosity, reddening, and redshift) space. Given thisdetailed treatment, we consider the UV LF at z ∼ z ∼ R < . § z ∼ − z ∼ −
3, but also present H α LFs atsimilar redshifts, the latter of which may be useful forcurrent and future emission line studies. RESULTS: REST-FRAME 8 µ M, INFRARED, ANDBOLOMETRIC LUMINOSITY FUNCTIONS
As suggested in the previous analysis, correcting therest-UV LF for the effects of dust extinction is a key component in recovering the star formation rate density.Aside from our knowledge of the E ( B − V ) distributionat high redshift ( § − R < . L IR /L UV ) around 4 −
5, supporting the average correc-tion advocated by Steidel et al. (1999). Further progresswas made by taking advantage of the unique sensitivityof the
Spitzer
MIPS instrument, allowing us to directlydetect for the first time the dust emission from L ∗ galax-ies at z & . average trends establishedby previous X-ray stacking studies, and further demon-strated that moderate luminosity galaxies (10 . L bol . . ) at z ∼ E ( B − V ) distributionof most z ∼ § L IR /L UV .In this section, we present our constraints on the 8 µ m,infrared, and bolometric luminosity functions of z ∼ − E ( B − V ) distribution and the distribution of 24 µ m fluxes) forevaluating the dust attenuation of galaxies. In § E ( B − V ) distribution toinfer the IR LF. In subsequent sections, we combine ourUV LF with the observed 24 µ m properties of galaxiesto infer the IR LF. The two methods are compared indetail in § Extinction-Corrected Measures of the LuminosityFunction
As a first step, we can use the Meurer et al. (1999)relation to recover the dust-corrected LF. The methodproceeded with the following steps:1. We first generated many realizations of themaximum-likelihood UV LF and E ( B − V ) distributionat z ∼
2, assuming normal LF and E ( B − V ) distribu-tion errors. We randomly chose an LF and E ( B − V )distribution from these many realizations, to create anLF/ E ( B − V ) pair, {L , E} .2. Because the LF and E ( B − V ) distributions do notchange significantly over the redshift range 1 . ≤ z < . E ( B − V ) distribution is insensitiveto absolute magnitude down to our spectroscopic limit( § intrinsic redshift z ,magnitude M , and reddening E ( B − V ) of a galaxy areindependent variables. Redshifts were drawn randomlyfrom a uniform distribution. Magnitudes were drawnfrom the range − . M (1700˚ A ) . − . . . z . . L from the {L , E} pair. The faint limit of M (1700˚ A ) = − . ∼ . ⊙ yr − using the Kennicutt (1998) calibration.We drew galaxies down to this low limit of unobscuredluminosity because such galaxies can be scattered to binsof higher luminosity after correcting for extinction. Sim-ilarly, E ( B − V ) values were drawn randomly from the E ( B − V ) distribution E from the {L , E} pair, excludingnegative E ( B − V ) values that reflect unphysical redden-ing values. The result is a list of galaxies associated witha triplet ( z , M , E ( B − V )).3. The rest-frame 1700 ˚A specific luminosity of eachgalaxy is calculated as L = 4 πd (1 + z ) 10 − . . m ) , (17)where d L is the luminosity distance at redshift z and m is the apparent magnitude of the galaxy with ab-solute magnitude M at redshift z (Eq. 14). We thencalculate νL ν at 1700 ˚A to yield the UV luminosity.The E ( B − V ) for the galaxy is used in conjunctionwith the Calzetti et al. (2000) relation to derive the dust-corrected UV luminosity.4. To determine the IR luminosity corresponding tothis dust-corrected UV luminosity, we assumed that theUV and IR emission are tied directly to the SFR of thegalaxy. The IR luminosity is assumed to be the lumi-nosity which, when added to the unobscured UV lumi-nosity, yields the same SFR that would have been ob-tained from the dust-corrected UV luminosity, assumingthe Kennicutt (1998) relations. These IR luminositiesare then perturbed by a normal distribution with sigmaof 0 . E ( B − V ) and IR luminosity; e.g.,Meurer et al. 1999).5. These IR luminosities are then binned to producean IR LF. This is the IR LF corresponding to the {L , E} pair selected in step (1).Steps (1)-(5) are repeated many times, each time ran-domly drawing different {L , E} pairs. Aside from uncer-tainties in the rest-frame UV faint-end slope, there aretwo other systematics that can bias our determinationof the IR LF: (1) a change in the faint-end slope of therest-frame UV LF and (2) a change in the attenuation ofUV-faint galaxies. We now discuss these two systematiceffects in detail.First, we must determine how changing the numberdensity of such faint objects, determined by the faint-endslope α , affects the IR LF. In principle, we could simplyfix α = − . z ∼ α to vary freely in the Schechter fits to therealizations. However, this method will cause us to un-derestimate the errors on the faint-end of the IR LF. Toobtain a truer estimate at the faint-end, we allowed α tovary freely around a normal distribution with a mean of h α i = − . σ ( α ) = 0 .
11, sim-ilar to the dispersion in α that we measure when fittingthe UV LF (Figure 12 and Table 7). For simplicity, weassume that α varies according to h α i = − . ± .
11 at z ∼ R > . E ( B − V ) we considered two cases. In thefirst case, we assume that the E ( B − V ) distributionis constant to arbitrarily faint rest-UV magnitudes. Inthe second case, we assume that the E ( B − V ) distri-bution is constant to R = 25 . § h E ( B − V ) i = 0 .
04 (with same dispersion) for galax-ies fainter than R = 25 . E ( B − V ) distribution). We have as-sumed this value of E ( B − V ) = 0 .
04 because it is similarto the E ( B − V ) observed for very faint ( . . ∗ ) UV-selected galaxies inferred from dropout samples at higherredshifts (Bouwens et al. 2007, submitted). Because R = 25 . E ( B − V )distribution will suddenly change fainter than this limit.Rather, the distribution is likely to gradually fall towardslower E ( B − V ), or bluer rest-frame continuum spectralslopes ( β ), proceeding to fainter galaxies, assuming thatsuch fainter galaxies have lower star formation rates andare less dusty than R < . E ( B − V ) distribution will very likely fall betweenthe two extremes assumed above. We will return to thispoint shortly. Fig. 16.—
Infrared luminosity functions at z ∼ z ∼
3, cal-culated assuming the Meurer et al. (1999) and Kennicutt (1998)relations to convert unobscured UV luminosity and E ( B − V ) toinfrared luminosities. The width of the shaded regions reflect theuncertainty in the rest-frame UV-slope and the attenuation distri-bution for R > . For now, our IR LFs estimated from the E ( B − V )distributions are shown in Figure 16 and tabulated inTable 9. The uncertainty in the LFs include uncertaintyin the rest-frame UV faint-end slope and the uncertaintyin the E ( B − V ) distribution for R > . E ( B − V ) is held constant. The lower limit of each LF We make the reasonable assumption that very dustyULIRGs with faint UV luminosities make up a small fractionof the total number of UV-faint galaxies. Assuming other-wise would imply a significantly larger number of IR luminousgalaxies than are presently observed in shallow IR surveys (e.g.,P´erez-Gonz´alez et al. 2005; Caputi et al. 2007).
TABLE 9 µ m, IR, and Bolometric Luminosity Functions of . . z . . Galaxies a . ≤ z < . . ≤ z < . . Mpc − decade − ) (h . Mpc − decade − )log[ νL ν (8 µm )] 8.25 — 8.50 (3 . ± . × − (2 . ± . × − . ± . × − (2 . ± . × − . ± . × − (1 . ± . × − . ± . × − (1 . ± . × − . ± . × − (1 . ± . × − . ± . × − (6 . ± . × − . ± . × − (4 . ± . × − . ± . × − (2 . ± . × − . ± . × − (1 . ± . × − . ± . × − (8 . ± . × − . ± . × − (4 . ± . × − . ± . × − (2 . ± . × − log L IRb . ± . × − (1 . ± . × − . ± . × − (1 . ± . × − . ± . × − (1 . ± . × − . ± . × − (7 . ± . × − . ± . × − (4 . ± . × − . ± . × − (3 . ± . × − . ± . × − (2 . ± . × − . ± . × − (1 . ± . × − . ± . × − (6 . ± . × − . ± . × − (3 . ± . × − . ± . × − (1 . ± . × − . ± . × − (6 . ± . × − log L bolb . ± . × − (2 . ± . × − . ± . × − (2 . ± . × − . ± . × − (1 . ± . × − . ± . × − (9 . ± . × − . ± . × − (6 . ± . × − . ± . × − (3 . ± . × − . ± . × − (2 . ± . × − . ± . × − (1 . ± . × − . ± . × − (7 . ± . × − . ± . × − (4 . ± . × − . ± . × − (1 . ± . × − . ± . × − (7 . ± . × − Errors include systematic uncertainty in attenuation distribution for R > . b The values listed in this table are derived assuming the Caputi et al. (2007) calibrationbetween νL ν (8 µm ) and L IR . corresponds to the second case where E ( B − V ) suddenlydecreases to have a mean of h E ( B − V ) i = 0 .
04 for galax-ies fainter than R = 25 .
5. In general, the systematic un-certainties related to a changing attenuation distributionfor UV-faint galaxies will dominate the uncertainties inthe faint-end slope ( § Distribution of Dust Attenuation Factors
As alluded to above, the distribution of rest-frame5 − . µ m luminosities ( L − . µ m ) of z ∼ Spitzer /MIPS can be used to assess the in-frared luminosity function independent of any assump-tion regarding the relationship between rest-frame UVslope and extinction, as per the previous discussion. Tothis end, we must quantify the distribution of dust at-tenuation among z ∼ A MIR ), far-IR ( A IR ),and bolometric attenuation ( A bol ) factors as the ratio be-tween νL ν (8 µm ), L IR , and L bol , respectively, and L .Following the calibration of Reddy et al. (2006b), we can relate these A factors to each other: A IR ≈ . A MIR A bol ≡ L IR + L L ≡ A IR + 1 . (18)Note that these attenuation factors, A , are distinguishedfrom the rest-frame UV attenuation factor which is theratio between the dust-corrected and unobscured UV lu-minosities.The normal distribution of E ( B − V ) for galaxies at z ∼ − A will abide by a log-normal distribu-tion. We modeled the shape of the log A distributionby considering the measured log A of rest-UV-selectedgalaxies with bolometric luminosities L bol < . L ⊙ .From Reddy et al. (2006b), h log A IR i ≈ .
67, implying L IR /L ≈ .
7, for the combined sample of 24 µ mdetected and undetected rest-UV-selected galaxies to R = 25 .
5. This mean attenuation implies the Gaussianfit shown in Figure 17, compared with the distributionof log A for 24 µ m detected galaxies.ulti-Wavelength LFs at 1 . . z . . Fig. 17.—
Distribution of measured log A IR for 24 µ m de-tected galaxies with R < . L bol < . L ⊙ ,indicated by the red hashed histogram. The solid curve de-notes the Gaussian fit to the inferred distribution of all U n G R -selected galaxies with R < .
5, irrespective of 24 µ m detectionlimit, and the vertical line indicates the mean of the distribution, h log A IR i ≈ .
67. Log A IR for bright SMGs from the analysis ofReddy et al. (2006b) is also shown. To construct a fair representation of the attenuationfactors of high redshift galaxies, there is another is-sue that is pertinent. Namely, the distribution abovedoes not take into account the non-negligible fraction of z ∼ − on average , than those of typical galaxies at theseredshifts (e.g., Chapman et al. 2003; Reddy et al. 2005;van Dokkum et al. 2003). Virtually all of these galaxieshave luminosities L bol & . L ⊙ (Reddy et al. 2006b)and ≈
50% of those that are also bright at submil-limeter wavelengths, f µ m & L bol . . L ⊙ andbecause most galaxies with the largest attenuation fac-tors have L bol & . L ⊙ , we will only consider the L bol < L ⊙ regime when computing the IR LF. Wecombine our spectroscopically constrained estimate ofthe IR LF with higher luminosity data from the liter-ature in order to compute the total luminosity and starformation rate densities in § Attenuation of Rest-UV Faint Galaxies
As discussed in § R -band limit for spectroscopycan affect significantly our inferences of the faint-endof the IR LFs. To remind the reader, in construct-ing the IR LFs that we would have inferred based onthe UV continuum slope, we considered two cases. Inthe first, the distribution of E ( B − V ) is held fixed tothe maximum-likelihood value (e.g., Figure 10) irrespec-tive of optical magnitude. In the second, we assumedthe maximum-likelihood value for those galaxies with R < .
5; for those fainter than this limit, we assumeda mean h E ( B − V ) i = 0 .
04 (corresponding to β ∼ − . µ m, IR, and bolometric LFs based on MIPS 24 µ mdata, we assumed two cases similar to the ones consid- Fig. 18.—
Illustration of the two cases we consider for the at-tenuation distribution of galaxies as a function of rest-frame UVmagnitude. In both cases, the height of the bars reflect a 1 σ dispersion of 0 .
53 dex. ered above. In the first, we assume that all galaxiescan be ascribed to the attenuation distribution shownin Figure 17. In the second, we assume that the atten-uation distribution shifts to a mean of h log A IR i = 0,with the same dispersion as before, for galaxies with25 . ≤ R < .
5, then shifts again to a lower meanof h log A IR i = − R = 27 . R = 25 . total IR LD andSFRD estimates that are comparable, if not lower, thanestimates based on samples of the brightest ( R < . K s ( V ega ) <
22) objects at these redshifts, mostof which have luminosities comparable to LIRGs andULIRGs (Reddy et al. 2005). Since the total IR LD andSFRD must be at least as large as that contributed byLIRGs and ULIRGs, the attenuation distribution of sub-6 Reddy et al.
Fig. 19.— µ m and IR luminosity functions at redshift 1 . ≤ z < .
7, compared with predictions in the higher redshift range2 . ≤ z < .
4. The right panel also demonstrates the similarity in the IR LF derived from the rest-frame UV slope (Figure 16) withthat derived using the attenuation factors calculated from
Spitzer /MIPS observations of 1 . . z . . ULIRG galaxies cannot be much lower than what we haveconsidered here. We will return to this point in § Rest-Frame µ m and IR Luminosity Functions Combining our UV LFs with the attenuation distri-bution derived from MIPS 24 µ m observations (usingthe same method described in § µ m observations to probe star-forming popu-lations at z ∼ µ m luminosity function; we did this using the8 µ m attenuation factors observed for z ∼ We then use the relationship be-tween A IR and A MIR (Eq. 18) to infer the IR luminosityfunction.Our inference of the 8 µ m and IR LFs at 1 . ≤ z < . L IR . L ⊙ ) areshown in Figure 19 and listed in Table 9. For later com-parison, we have assumed the relation between 8 µ m andIR luminosity given by Caputi et al. (2007). The up-per and lower limits of the shaded regions in the figurecorrespond to the two different cases of attenuation dis-tributions discussed in the previous section. We cannotdirectly measure the rest-frame mid-IR luminosities ofgalaxies at 2 . ≤ z < .
4, but we show the predicted8 µ m and IR LFs at these redshifts assuming (a) thesame attenuation distribution and (b) the same relation-ship between mid-IR and total IR luminosities found for z ∼ . . L IR . L ⊙ fromthe rest-frame UV slope, or E ( B − V ), is consistent withthe one inferred from the MIPS-determined attenuationfactors of these galaxies. This similarity reflects the sig-nificant correlation between E ( B − V ) and attenuation For ease of comparison with previous literature, we expressthe mid-IR attenuation factor A IR in terms of νL ν (8 µm ) ratherthan L − . µ m as was used in Reddy et al. (2006b). The relation-ship between the two for the typical mid-IR SED of star-forminggalaxies is νL ν (8 µm ) ≈ . L − . µ m . TABLE 10Contributions of the IR LD at z ∼ L IR log IR LD10 — 10 L ⊙ . ± .
17 (9%)10 — 10 L ⊙ . ± .
09 (23%)10 — 10 L ⊙ . ± .
06 (39%) > L ⊙ . ± .
32 (25%) a Total ( × < L IR < ∞ ) b : . ± . a From Caputi et al. (2007). b The lower limit of L IR = 6 × L ⊙ roughly corre-sponds to an SFR of 0 . ⊙ yr − . for galaxies with moderate luminosities, and such galax-ies are typical of the redshift z ∼ − ∼ L ∗ ).Because our data are most sensitive to galaxies with L IR . L ⊙ , we must incorporate direct measure-ments of the IR LF for high luminosity objects, such asthose from Spitzer mid-IR surveys. There are severalpublished values of the IR LF for ULIRGs; here we as-sume the most recent determination from Caputi et al.(2007). Table 10 lists the contribution of galaxies indifferent luminosity ranges to the IR LD, where wetake the total LD to be that of galaxies with L IR > × L ⊙ . This limit corresponds to galaxies with SFRsof ∼ . ⊙ yr − . The total LD changes negligibly byintegrating to zero luminosity. Table 10 shows that —despite the large systematic uncertainties at the faint-endinduced by variations in the attenuation distribution —a significant fraction of the IR LD at z ∼ § Bolometric Luminosity Functions
Finally, to gain an accurate picture of the distri-bution of total energetics of star-forming galaxies, wemust consider the combined contribution from unob-scured (UV) luminosity and obscured (IR) luminosity.While the bolometric luminosity should closely followulti-Wavelength LFs at 1 . . z . . L bol ≈ L IR & × L ⊙ ), Reddy et al. (2006b) show thatsuch an assumption is no longer valid for galaxies with L bol . × L ⊙ (at z ∼ L ⊙ galaxy at z ∼ . L ⊙ galaxy at z ∼ Figure 20 shows the bolometricluminosity functions of star-forming galaxies at redshifts1 . ≤ z < .
4, and values are listed in Table 9. Thebolometric LF is larger in all luminosity bins consideredthan the IR LFs given that objects will shift from lower tohigher luminosity bins after accounting for the emergentUV luminosity of high redshift galaxies. We note thatin computing our prediction for the bolometric LF at2 . ≤ z < .
4, we have assumed the same distribution ofattenuation factors that was found for z ∼ § . L bol . L ⊙ ) galaxiesto the global luminosity density. Fig. 20.—
The bolometric luminosity function at redshift1 . ≤ z < .
7, calculated using the sum of the UV (unobscured)and IR (obscured) luminosities of galaxies. Our prediction of thebolometric LF at 2 . ≤ z < . While the tight correlation between L bol and attenuation hasbeen observed both locally and at high redshift, the normaliza-tion of the relationship increases at higher redshift. This means,for example, that galaxies at z ∼ L bol ) than z = 0 galaxies (Reddy et al. 2006b).Hence, the fraction of total luminosity emerging in the UV is largerat higher redshift than it is locally for galaxies of a given bolomet-ric luminosity. Note that this observation is still consistent withthe finding that dustier systems dominate the luminosity densityat z ∼ − § RESULTS: H α LUMINOSITY FUNCTION
We briefly discuss our derivation of the H α LFs here,as they may be useful for current and future high red-shift emission line studies. The key ingredient that al-lows us to convert our UV LF into an estimate of theH α LF is the correlation between dust-corrected UV andH α estimates of star formation rates. Erb et al. (2006b)found a significant (6 . σ ) correlation with 0.3 dex scat-ter between the extinction corrected UV estimates andH α estimates of the SFRs, assuming the Calzetti et al.(2000) relation, for a sample of 114 rest-frame UV se-lected galaxies at z ∼
2. We can invert the relationshipbetween extinction-corrected SFR and H α line emissionin order to infer the H α LF. The spectroscopic H α obser-vations used to establish the correlation between H α andUV-determined SFRs are described in detail in Erb et al.(2006a,b,c). Method
The method used to estimate the H α LF of z ∼ § E ( B − V ) distributionsand randomly selected magnitudes and E ( B − V ). Todetermine the H α luminosity corresponding to this dust-corrected UV luminosity, we assumed that the UV andH α emission are tied directly to the SFR of the galaxy,where the SFR is calibrated using the Kennicutt (1998)relations. It is then easy to show that L H α [ergs s − ] ≈ . × L [ergs s − Hz − ] . (19)The resulting H α luminosities are then perturbed by0 . α estimates(Erb et al. 2006b). For consistency with previous deter-minations of the H α LF at lower redshifts, it is usefulto derive an H α luminosity function uncorrected for ex-tinction at z ∼
2. To accomplish this, we assume thatthe E ( B − V ) value, which is derived from the rest-frameUV colors, reflects the nebular reddening of the galaxy(see also Erb et al. (2006b)). Applying the Calzetti et al.(2000) relation to the intrinsic H α luminosity, and as-suming the same value of E ( B − V ), yields an estimateof the observed H α luminosity. Note that because theattenuation A λ /E ( B − V ) is a factor of ≈ α , 6563 ˚A, than at 1700 ˚A, it is notcorrect to simply convert the observed UV luminosityto a SFR and then back to an observed H α luminosity:one must take into account the differential extinction, fora given E ( B − V ), between these two wavelengths. Thedust-corrected (intrinsic) and uncorrected (observed) H α luminosities are then binned to produce a dust-correctedand observed H α LF, respectively. We considered thesame two cases for the E ( B − V ) distribution of UV-faint galaxies as in § α LFs at z ∼ Predicted H α LFs at z ∼ Erb et al. (2006b) show that assuming the same E ( B − V ) indust-correcting the H α estimates, as opposed to a smaller nebularreddening as advocated by Calzetti et al. (2000), results in betteragreement with dust-corrected UV and stacked X-ray estimates. Fig. 21.—
Dust-corrected and observed (uncorrected for extinction) H α LFs inferred for star-forming galaxies with redshifts 1 . ≤ z < . . ≤ z < . E ( B − V ) distribution for R > . TABLE 11H α Luminosity Functions of . . z . . Galaxies a . ≤ z < . . ≤ z < . φ Dust-Corrected φ Observed φ Dust-Corrected φ log L ( Hα/ergs s − ) (Mpc − decade − ) (Mpc − decade − )41.00 — 41.25 (4 . ± . × − (4 . ± . × − (1 . ± . × − (2 . ± . × − . ± . × − (3 . ± . × − (1 . ± . × − (1 . ± . × − . ± . × − (2 . ± . × − (9 . ± . × − (1 . ± . × − . ± . × − (1 . ± . × − (6 . ± . × − (6 . ± . × − . ± . × − (1 . ± . × − (3 . ± . × − (4 . ± . × − . ± . × − (6 . ± . × − (2 . ± . × − (2 . ± . × − . ± . × − (3 . ± . × − (1 . ± . × − (1 . ± . × − . ± . × − (2 . ± . × − (7 . ± . × − (8 . ± . × − . ± . × − (1 . ± . × − (3 . ± . × − (3 . ± . × − . ± . × − (6 . ± . × − (1 . ± . × − (1 . ± . × − . ± . × − (2 . ± . × − (3 . ± . × − (5 . ± . × − . ± . × − (9 . ± . × − (2 . ± . × − (1 . ± . × − Errors include the systematic uncertainty in the E ( B − V ) distribution for R > . The correlation between UV and H α SFRs has onlybeen tested directly at redshifts 2 . . z . .
6, wherethe H α line falls within the K -band, making it accessi-ble to near-IR spectrographs, such as Keck II/NIRSPEC(McLean et al. 1998). It is useful, nonetheless, to predictthe form of the H α LF at z ∼ α SFRs holds at these higherredshifts. This is a reasonable assumption to make giventhat, to R = 25 .
5, the z ∼ z ∼ E ( B − V )(Figure 10), and galaxies in these respective samples havesimilar average dust attenuation factors, represented asthe ratio of the dust-corrected UV and unobscured UVluminosities, of ∼ − α LFs at z ∼
3, computed using the steps above andusing the combined ground-based and HDF samples togenerate the UV LF realizations, are shown in Figure 21and listed in Table 11. We briefly present a comparison of our H α LFs and luminosity densities with others fromthe literature in § DISCUSSION
We have presented the most robust estimates of therest-frame UV LFs and moderate luminosity regime ofthe IR LFs of star forming galaxies at redshifts 1 . ≤ z < . §
5, 7, 6). We have demonstrated how pho-tometric redshifts over this redshift range can introducenon-trivial biases in the LF, underscoring the need forspectroscopy where it is feasible ( § α line perturbations to theintrinsic rest-frame UV colors of galaxies ( § α emission and redshift and its dependence onthe physical properties of galaxies. In § z ∼ . . z . . α , and IR LFs with those of previousstudies, and the evolution in the LF and luminosity den-sity, are discussed in § § W Ly α Distribution as a Function of Redshift
The analysis of § W Ly α distribution of galaxies at 1 . ≤ z < . W Ly α ≥
20 ˚A ( f
20) was larger at higher redshifts. Thistrend was recognized by comparing f
20 between (a) BXgalaxies and LBGs and (b) BX galaxies at z ≤ .
48 and z > .
48, with results summarized in Table 4. In thelatter case, the fact that we see a trend in f
20 withredshift even for galaxies selected using a single set ofcolor criteria (BX) further strengthens our conclusionsthat f
20 increases with redshift, irrespective of selectionbiases (see Figure 3).To interpret this trend in a physical context, we as-sembled the stellar population parameters for galaxieswith measured W Ly α where SED modeling was avail-able from Shapley et al. (2005) and Reddy et al. (2006a).The resulting sample includes 139 galaxies, 14 with W Ly α ≥
20 ˚A. We used KS-tests to determine whetherthe SED parameters (star formation histories, ages, stel-lar masses, and star formation rates) for galaxies with W Ly α <
20 ˚A are drawn from the same parent pop-ulation as those with W Ly α ≥
20 ˚A. Doing this, wefound no significant differences in the star formationhistories, ages, stellar masses, and star formation ratesof galaxies between these two samples. This result isnot surprising given (1) the small sample size analyzedhere, (2) the significant systematic degeneracies betweenSED parameters (e.g., Shapley et al. 2005; Erb et al.2006c; Papovich et al. 2001), and (3) the large uncer-tainty in the measured W Ly α for individual galaxies.Galaxies with W Ly α ≥
50 ˚A have an average age of ∼ ±
300 Myr whereas those below this limit havean average age of ∼ ±
600 Myr. The difference inaverage age is not significant given the large dispersionin ages measured for the two samples.Nonetheless, several previous studies at z & α emitting galaxies are young ( .
50 Myr), low-metallicity systems with small stel-lar masses (e.g., Stanway et al. 2007; Pentericci et al.2007; Dow-Hygelund et al. 2007; Pirzkal et al. 2006;Finkelstein et al. 2007; Lehnert & Bremer 2003), par-ticularly in relation to galaxies without Ly α in emis-sion at the same redshifts ( z ∼
6; Stanway et al. 2007;Dow-Hygelund et al. 2007). Results at z ∼ α emission than highstellar mass and high metallicity galaxies (Erb et al.2006a). Among UV-continuum selected samples, bothDow-Hygelund et al. (2007) and Stanway et al. (2007)find a fraction of z ∼ W Ly α .
25 ˚A sim-ilar to the fraction found at z ∼ W Ly α >
100 ˚A) compared to the z ∼ α profiles andthe physical properties of galaxies is well-known to bequite complicated, with a sensitivity to ionizing flux, dust obscuration, and velocity of outflowing material (e.g.,Tapken et al. 2007; Reddy et al. 2006b; Hansen & Oh2006; Shapley et al. 2003; Adelberger et al. 2003). De-spite these complications, the advantage of our largespectroscopic analysis is that we can very accuratelyquantify the trends between Ly α emission and redshift( § z ∼ . ≤ z < . α in emission, then the frequencyof such Ly α emitting galaxies should increase with in-creasing redshift as the average galaxy age (and averagedust-to-gas ratios; see Reddy et al. 2006b), decreases. VVDS-Inferred Excess of Bright M (1700 ˚ A ) . − . Galaxies at Redshifts . . z . . I -band magnitude limited VIMOS VLT Deep Sur-vey (VVDS) that indicated a significant excess of bright( M (1700˚ A ) . − .
0) galaxies with redshifts 2 . . z . . We suggest two reasons for the discrepancy betweenour LF and that of the VVDS. First, the frequency of ob-jects with redshifts outside the redshift range 2 . ≤ z < . f c ) for M (1700˚ A ) . − . f c . The VVDS redshifts used to com-pute the LF fall within 4 categories. Flag-1 and Flag-2 objects are considered to have the least secure red-shifts, whereas flag-3 and flag-4 objects are more secure(Paltani et al. 2007). Paltani et al. (2007) then deter-mine a contamination fraction of f c ∼ .
54 for the 254flag-1/2 sources and assume a value of f c = 0 for the12 flag-3/4 sources that are used in their LF compu-tation. Weighting the fractions according to the num-ber of sources then yields a net contamination rate of[254 × .
54 + 12 × / [254 + 12] ≈ . The actual contamination rate among VVDS objectsmust be larger than f c = 0 .
52 for several reasons. First,the BX and LBG criteria account for the U n G R col-ors of ∼ −
80% of VVDS objects claimed to lieat 2 . ≤ z < . same region of color space encom-passed by either the BX or LBG criteria). Yet, 77%of spectroscopically-confirmed candidates in the BX and Although the VVDS UV LF of Paltani et al. (2007) is com-puted in a slightly different redshift range, 3 . z .
4, from thatconsidered for our z ∼ . ≤ z < . M (1700˚ A ) . − . z ∼ z ∼ This gives the same result as the “photometric rejection”method, where 137 of 266 objects are likely to lie outside the red-shift range 2 . ≤ z < .
4, as discussed in Paltani et al. (2007). M (1700˚ A ) . − . Second, VVDS objects that do not satisfy the BX andLBG color criteria because of their redder G − R col-ors lie in the same region of color space as low red-shift star-forming galaxies (e.g., Adelberger et al. 2004;Reddy et al. 2005). Without additional K s -band datato exclude these low-z interlopers (e.g., BzK selectionof Daddi et al. 2004b), the contamination rate amongthese red G − R objects is likely to be at least as largeas the rate among objects that are targeted by criteriaspecifically designed to selected galaxies at 2 . ≤ z < . R ≤
24 (roughly corresponding tothe VVDS magnitude limit) with matching U n G R pho-tometry in our catalog. Of these matches, there are 755sources that satisfy the following conditions: (a) spectro-scopically observed in the TKRS, (b) do not satisfy eitherthe BX or LBG criteria, and (c) have U n − G > . G −R < .
8. These limits define the region in color spaceof the ≈
20% of VVDS objects that do not satisfy the BXor LBG criteria. Of these 755 bright sources that wereobserved in TKRS, 581, or ≈ z . .
4. This is a strictlower limit to the contamination rate since there will be(a) some 1 . < z < . z < . α line. In summary, our value of f c = 0 . f c = 0 .
52 claimedby Paltani et al. (2007), yet is based on our sample of285 secure spectroscopic redshifts versus only 12 secureredshifts and 254 unsecure redshifts of the VVDS sur-vey. For the reasons discussed above, the real contam-ination rate among VVDS objects is likely to be largerthan f c = 0 .
77. Even so, assuming this fraction wouldconservatively lower the brightest VVDS LF points bya factor of at least two. We already alluded to in § . ≤ z < .
4, in otherwords QSOs and other bright AGN. We have explicitlyexcluded AGN from our LF determination, as describedin § For the BX sample, most of the contamination at bright mag-nitudes arises from foreground galaxies. For the LBG sample, mostof the contamination arises from stars. Note that although nominally the BX criteria are designed toselect galaxies with z < .
7, we include them in the discussion heresince a significant fraction of VVDS objects claimed to lie at z & than ∼
60% in the brightest luminosity bin of the VVDSanalysis (Table 2). In fact, of the 14 spectroscopically-confirmed LBGs in our sample with M (1700˚ A ) . − . . Fig. 22.—
Comparison of our z ∼ . ≤ z < . . ≤ z < .
4, as determined from ourspectroscopic sample, results in much better agreement betweenthe two LFs, as indicated by the filled squares.
To conclude our comparison, the two primary causesfor the excess of bright galaxies inferred by the VVDS is(a) their underestimate of the fraction of objects that lieoutside the redshift range 2 . ≤ z < . . ≤ z < . . ≤ z < .
4, allow us to very accurately quantify the magnitudeof both sources of contamination, all within a combinedsurvey area that is roughly twice as large as the VVDSfield. Figure 22 demonstrates that applying the contami-nation fractions determined from our spectroscopy to theVVDS points (after factoring out the VVDS contamina-tion correction) results in a better agreement betweenthe VVDS and our LF. Taking all these results into con-sideration, we find no convincing evidence for an excessof bright galaxies at 2 . ≤ z < . Evolution of the UV and H α Luminosity Functionsand Densities
UV LFs
Figure 12 summarizes our determinations of the rest-frame UV LFs at redshifts 1 . ≤ z < .
4, compared withthe Steidel et al. (1999) LF at z ∼
4. As our methodulti-Wavelength LFs at 1 . . z . . Fig. 23.— ( Left: ) Comparison of a few
UV LFs from the literature. The local UV LF (derived from GALEX data) is shown (Wyder et al.2005) along with the z ∼ B , V , and I dropoutsamples of Bouwens et al. (2007; submitted). ( Right: ) Unobscured (uncorrected for extinction) UV luminosity density, integrated to afixed luminosity of L lim = 0 . L ∗ z =3 , from the following sources: Wyder et al. (2005) at z ∼
0, Schiminovich et al. (2005) at z ∼ . − . z ∼ . z ∼ −
6. Our determinations at z ∼ z ∼ TABLE 12Summary of Total UV, H α , and IR Luminosity Densities at . ≤ z < . a H α LD IR LD b Redshift Range (ergs s − Hz − Mpc − ) (ergs s − Mpc − ) (L ⊙ Mpc − )1 . ≤ z < . . ± . × (4 . ± . × (1 . ± . × . ≤ z < . . ± . × (2 . ± . × (6 . ± . × Uncorrected for extinction and integrated to 0 . L ∗ z =3 . b Values assume the Caputi et al. (2007) calibration between νL ν (8 µm ) and L IR . The IRLD at z ∼ of constraining the reddening and luminosity distribu-tions takes into account a number of systematic effects(e.g., contamination fraction particularly at the brightend of the LF, photometric bias and errors, Ly α lineperturbations to the observed colors ) that have notbeen considered in previous analyses (e.g., Gabasch et al.2004; Le F`evre et al. 2005) or were only partially con-sidered (Steidel et al. 1999; Adelberger & Steidel 2000;Sawicki & Thompson 2006), we regard our LFs as themost robust determinations at z ∼ z ∼ R = 25 . M ∗ (at z ∼
2) between redshifts 1 . ≤ z < .
5: the number den-sity of galaxies brighter than M ∗ = − .
97 appears tobe constant over the ∼ . z ∼ z ∼ .
3. This lack of evolution in the bright-end ofthe UV LFs does not specifically address how a galaxyof a particular luminosity will evolve. For example, thelack of evolution at the bright end ( M AB (1700˚ A ) . − z ∼ z ∼
2, but will not necessarily be absent fromthe z ∼ Our determinations of the LFs are insensitive to small changesin the assumed W Ly α distributions, such as those caused by trendsin W Ly α with apparent magnitude and color. z ∼ z ∼
2. In any case, the lack of evolutionat the bright-end of the UV LF implies that whateverUV-bright galaxies at z ∼ − z ∼ z ∼
2. The net effect is that the numberdensity of galaxies with ( M AB (1700˚ A ) . −
21) remainsessentially constant.For galaxies fainter than M ∗ , we do find evidence fora small evolution between z ∼ z ∼
2: the num-ber density of galaxies fainter than M ∗ = − .
97 is sys-tematically larger at z ∼ z ∼ R = 25 . M AB (1700˚ A ) ∼ − z ∼ z ∼
3. What is clear is that the number den-sity of − . M AB (1700˚ A ) . −
19 galaxies at z ∼ atleast as large as the the corresponding number density at z ∼
3. To put these results in context, Figure 23 sum-marizes our results at z ∼ z ∼ Figure 23 demonstrates the evolution of the UV LF.The number density of bright galaxies increases from z ∼ z ∼ M ∗ by The figure is not meant to be comprehensive with respect toall determinations of the UV LF, particularly at z & Fig. 24.— ( Left: ) Comparison of our inferred dust-corrected H α luminosity function at z ∼ z ∼
3, withthe direct H α LF determinations at lower redshift from Tresse et al. (2002); Tresse & Maddox (1998). (
Right: ) Extinction-corrected H α luminosity density: open squares are from Hopkins (2004) and include determinations from Gallego et al. (1995); P´erez-Gonz´alez et al.(2003); Sullivan et al. (2000); Tresse & Maddox (1998); Tresse et al. (2002); Glazebrook et al. (1999); Hopkins et al. (2000); Yan et al.(1999); Moorwood et al. (2000). The point at z = 2 .
75 is the H β determination from Pettini et al. (1998). The large red pentagons denotevalues from this work. ∼ . − . z ∼ z ∼
4. To quantify this further, we have calculated therest-frame 1700 ˚A unobscured UV luminosity density(LD). The luminosity density and error are estimatedby simulating many realizations of the UV LF consistentwithin the normally-distributed LF errors and evaluatingthe mean luminosity-weighted integral of the LFs fromeach of these realizations. The mean value and dispersionof these integrated values give the mean luminosity den-sity and error, and values are listed in Table 12. We haveassumed a faint-end slope of α = − . ± .
11 (i.e., as in § z ∼
2. For consistency, all the LFs are integratedto a luminosity limit of L lim = 0 . L ∗ z =3 . This corre-sponds to a luminosity of L lim = 3 . × ergs s − Hz − at 1700 ˚A, is equivalent to ≈ . L ∗ z =2 , and correspondsto an unobscured SFR of ∼ . ⊙ yr − assuming theKennicutt (1998) relation. The right panel of Figure 23summarizes the integrated (unobscured) UV LD as afunction of redshift including several published values,showing the significant evolution between z ∼ z = 0.While the observed evolution in the UV LF is hardlysurprising, we have placed this evolution during the mostactive epoch of star formation ( z ∼ −
4) on a securefooting with our extensive spectroscopic analysis and de-tailed completeness corrections. Our analysis covers alarger number of uncorrelated fields than what has typ-ically been considered in previous studies, thus enablingus to mitigate the effects of sample variance. We notethat part of the evolution of the unobscured UV LF maybe a result of extinction, as we will discuss shortly. H α LFs
Figure 24 compares our inference of the H α LF at z ∼ z ∼
3, with the direct determina-tions at lower redshifts by Tresse & Maddox (1998) andTresse et al. (2002). The evolution of the dust-correctedH α LF qualitatively mimics the evolution observed in theUV LF (Figure 23) for redshifts z . −
3. We see a fac-tor of two decline in the number density of moderatelyluminous galaxies with 10 < L ( Hα ) < . ergs s − from z ∼ z ∼ .
7. The decline of moderately lumi- nous galaxies is at least a factor of 4 −
10 between z ∼ z ∼ . z ∼ z ∼ L ( Hα ) . ergs s − is primarilya result of the fact that galaxies on the faint-end of theUV LF (where we observed the same small systematicexcess; see Figure 12) are scattered to correspondinglymore luminous bins of H α luminosity after correcting forextinction. The significance of the systematic excess at z ∼ z ∼ α LFs at z &
2. What is certain is that thenumber (and luminosity) density of moderately H α lumi-nous galaxies at z ∼ at least as large as the number at z ∼
3. In contrast, the significance of the increased fre-quency of moderately luminous galaxies at z ∼ z ∼ . α luminosity density shows a decline from z ∼ z = 0. To illustrate this, we have compiled estimates ofthe H α LD from Hopkins (2004) in the right panel of Fig-ure 24, including our new inferences at z ∼ z ∼ z ∼ z ∼ α LFs. Quantitatively, the H α LD per comoving Mpcdecreases by a factor ∼
25 between z ∼ β observations have been performed fora handful of objects at z ∼ α LF at z ∼
3. Since our H α results on the SFRD are degeneratewith those estimated from the UV and IR LFs, we willnot discuss the H α LFs any further.
Average Extinction and IR and Bolometric LFs
Evolution of Dust Obscuration with Redshift
As mentioned in § z ∼ z = 0 may be partly a result of systematic differ-ences in extinction with redshift. It is already knownthat galaxies of a fixed bolometric luminosity have anaverage attenuation factor at z ∼ ∼ −
10 times smaller than the attenuation factors of lo-ulti-Wavelength LFs at 1 . . z . . Fig. 25.—
Trend between bolometric luminosity and dust attenuation based on the analyses of Reddy et al. (2006b) at z ∼ z ∼
1, and Buat et al. (2006) at z ∼
0. The thickness of the lines show schematically the contribution ofgalaxies in different ranges of bolometric luminosity to the bolometric luminosity density, with the relative fractions listed. Fractionalcontributions at z ∼ z ∼ z ∼ cal galaxies (Reddy et al. 2006b; Burgarella et al. 2007;Buat et al. 2007). Figure 25, adapted from Reddy et al.(2006b) and the GALEX results of Buat et al. (2006) andBurgarella et al. (2007), illustrates the offset between the z ∼ z ∼
1, and z = 0 trends between L bol and attenua-tion. Reddy et al. (2006b) interpreted this trend as a re-sult of the increasing extinction per unit SFR (or increas-ing dust-to-gas ratio) as galaxies age. The dependenceof attenuation on bolometric luminosity at low redshiftshas been discussed by many authors (e.g., Buat et al.2005; P´erez-Gonz´alez et al. 2003; Afonso et al. 2003;Sullivan et al. 2001; Hopkins et al. 2001a,b). The analy-ses of Reddy et al. (2006b); Adelberger & Steidel (2000)and Figure 25 demonstrate that this dependence contin-ues unabated from z = 0 to z ∼ − z ∼ z = 0,then the offset shown in Figure 25 implies that, whenintegrating the UV (or H α ) LF to a fixed luminosity,the extinction correction will be larger at lower redshifts.However, as Figures 23 and 24 demonstrate, there is a very strong evolution in the LF between z ∼ z ∼
0. We find a similar evolution in the IR LFs and IRLDs, shown in Figures 26 and 27, respectively, where wesummarize the comparison between the z ∼ z ∼ L bol versus attenuation trends between z ∼ z = 0, as well as the relative numbers of galaxies indifferent luminosity ranges that contribute to the LD.For example, LIRGs at z ∼ −
10 timesless dusty than LIRGs at z = 0, but there are manymore LIRGs at z ∼ z ∼
0. The contribution ofgalaxies in different luminosity ranges to the bolometricLD at z ∼ z ∼
1, and z = 0 are shown schematicallyin Figure 25.If the IR LF was unevolving between z ∼ z = 0,then the fact that galaxies of a fixed L bol are 8 − z ∼ z ∼ Fig. 26.—
Comparison of our inference of the 8 µ m (left) and IR LFs (right) at z ∼ z ∼
3, with directmeasurements from the literature: z ∼ z ∼ µ m LF at z ∼ . z ∼ & × L ⊙ galaxies by a factor of > the average extinction correction needed to recover thebolometric LD from the UV LD would be roughly 8 − larger at z ∼ z ∼
2. However, becauseof the evolution in number density, the average correc-tion at z ∼ −
10 times larger than the valueat z ∼
2. Schiminovich et al. (2005) find an averageattenuation factor for UV-selected samples at z . h L bol /L UV i ∼
7, which is only 1 . . z ∼ −
3. Itis interesting to note that even taking into account theevolution in number density, the average correction at z ∼ z ∼
2, despitea greater fraction of the LD in dustier galaxies at highredshift. This result may be partly due to the redshiftevolution in L bol versus attenuation trend (Figure 25),but the larger correction at z ∼ z ∼ Evolution of the Dust-Obscured and BolometricLuminosity Densities
There are several important conclusions to draw fromour analysis of the IR luminosity density. First, the un-obscured UV LD drops by a factor ∼
23 between z ∼ z ∼
0, whereas the IR LD drops by a factor of ∼
14 between z ∼ independent determinations of the av-erage extinction corrections at z ∼ h L bol /L UV i ∼ . z ∼ h L bol /L UV i ∼ z ∼ z = 0(see § z ∼ z ∼
3, a result consistent withthat of the UV LD analysis. This should not come asa surprise since the same incompleteness-corrected rest- frame UV LF was used to infer the IR LF. Nonetheless,our spectroscopic analysis puts the constraints on the H α and IR LD on a more secure footing. In particular, ouranalysis provides the first spectroscopic constraints onthe sub-ULIRG regime of the IR LFs at z & . ≤ z < . z ∼ L IR < L ⊙ (i.e., compare our IR LF points with the extrapolationof P´erez-Gonz´alez et al. (2005) and Caputi et al. (2007)for galaxies with L IR < L ⊙ in Figure 26). Whenintegrating the IR LF to account for all galaxies, wefind a total IR LD of ∼ . × L ⊙ Mpc − , abouta factor of 2 larger than the previous determination at z ∼
2. Note that the P´erez-Gonz´alez et al. (2005) de-termination of the IR LD at z ∼ . − § µ m and IR lu-minosity that has been shown to overproduce by fac-tors of > µ m stacking analysis(Papovich et al. 2007). This result is also supported byDaddi et al. (2007b), who also find a systematic excessof rest-frame 8 µ m emission relative to UV and radioemission for a sample of z ∼ z ∼ . . z . . Fig. 27.—
IR luminosity density as a function of redshift, including data from Yun et al. (2001) at z = 0, Flores et al. (1999) at0 . . z . .
0, Barger et al. (2000) at z ∼
2, and Chapman et al. (2005) at z &
2; all shown with open squares. Results from Caputi et al.(2007) and P´erez-Gonz´alez et al. (2005) are shown by the open circles and green lines, respectively. Our points at 1 . ≤ z < . . ≤ z < . νL ν (8 µm ) and L IR . The Reddy et al.(2006b) calibration would raise the points by ≈ z = 1. The inferred contribution of LIRGsfrom our analysis at z ∼ − ibration between 8 µ m and IR luminosity, results in anIR LD from ULIRGs that is roughly a factor of 2 − L ⊙ . Our sampleand analysis yield the first constraints on the IR andbolometric LFs for moderately luminous galaxies at red-shifts 1 . ≤ z < .
4, thus allowing us to evaluate theunobscured SFRD over a larger range of intrinsic lumi-nosity and redshift than previously possible. Our resultssuggest that the luminosity density contributed by sub-ULIRGs with L IR < L ⊙ at z ∼ z ∼ α = − . R > . ∗ galaxies on a more secure footing. However, as we dis-cuss in the next section, comparison of our total IR LDswith those corresponding to SFRD values estimated inprevious studies (e.g., Reddy et al. 2005) suggest the IRLD cannot be much lower than the value derived here,effectively placing a constraint on the attenuation of sub-ULIRGs.Lastly, we note that UV emission comprises a non-negligible fraction of the bolometric luminosity of LIRGs.This is an effect that becomes more pronounced at higherredshift as the average dust attenuation of galaxies ofa given bolometric luminosity decreases, as already dis-cussed. The bolometric LFs derived in § L ∼ L = 10 L ⊙ , and adding the6 Reddy et al.contribution of ULIRGs from Caputi et al. (2007). As-suming that the fraction of bolometric luminosity emer-gent in the UV in ULIRGs is negligible, then the to-tal bolometric LD at z ∼ ≈ . × L ⊙ Mpc − .Roughly 80% of this bolometric LD arises from galaxieswith L bol . L ⊙ . This bolometric LD is more than25% larger than what we would have inferred from theIR LF alone because the former includes the contributionof the LD emergent at UV wavelengths.To summarize, there are essentially four points worthkeeping in mind from our analysis of the IR and bolomet-ric LD. First, the evolution of the UV LD shows a markeddifference from the evolution of the dust-corrected H α and IR LDs between z ∼ z ∼
0. The differencecan be explained by an evolution in the average extinc-tion correction between z ∼ z ∼
2. Second, wefind that the IR LD at z ∼ z ∼
3, implying that the decline in SFRD tothe local value must have occurred after z ∼
2. Third,while our analysis becomes increasingly incomplete forthe most luminous galaxies at z ∼ −
3, it does providethe first spectroscopic constraints on the moderate lu-minosity (e.g., LIRG) regime of the IR LF. Even takinginto account the significant uncertainties associated withthe dust obscuration of UV-faint galaxies, these resultssuggest that previous studies have significantly underes-timated the contribution of galaxies with L IR . L ⊙ to the IR luminosity density. Finally, taking into ac-count the emergent UV luminosity density of galaxies,we find that sub-ULIRG galaxies comprise roughly 80%of the total bolometric LD at z ∼
2. In the next section,we will discuss these results in the context of the globalSFRD.
Constraints on the Global Star Formation RateDensity
We have converted the results of Figure 23 and 27 tostar formation rates (SFRs) using the Kennicutt (1998)relations and assuming a Salpeter (1955) IMF from 0 . ⊙ , with the results summarized in Figure 28. The UV points at low redshift ( z .
1) are correctedassuming a factor of 7 attenuation (Schiminovich et al.2005). Our UV points are corrected by an average factorof 4 . z ∼ − z ∼ at least as large as the value at z ∼
3. Applying a factor of 4 .
5, as suggested by stackinganalyses (e.g.,Reddy & Steidel 2004; Reddy et al. 2006b;Nandra et al. 2002), to correct our UV estimates for ex-tinction yields values that are in general accord with the Assuming the more realistic Chabrier (2003) IMF will reducethe SFRD by a factor of ∼ . IR estimates. On the one hand, we might not have ex-pected such good agreement given that the factor of 4 . total UV LD may be lower dependingon the extinction of UV-faint galaxies. However, we notethat the IR-estimated SFRD includes the contributionfrom ULIRGs (whereas the UV-estimated SFRD doesnot explicitly take them into account), so the differencein the two estimates is less than we might have expected.The important point is that despite the significant uncer-tainties regarding the attenuation of UV-faint galaxies,applying a factor of 4 . z ∼ BzK (Daddi et al. 2004b), DistantRed Galaxy (DRG; Franx et al. 2003) criteria, and sub-millimeter selection (e.g., Chapman et al. 2005). Takinginto account the overlap between galaxies selected usingthese methods, Reddy et al. (2005) compute an SFRDat z ∼ . ± .
03 M ⊙ yr − Mpc − for galaxieswith R < . K s ( AB ) < .
8, including the contribu-tion from submillimeter-bright ( S µ & ∼ . ⊙ yr − Mpc − .This in turn implies that the attenuation of sub-ULIRGscannot be so low as to bring down our total SFRDestimate to the point where it is in violation of thecensus-computed SFRD. As an example, integrating theCaputi et al. (2007) IR LF at z ∼ ∼ . +1 . − . × L ⊙ Mpc − , corresponding to an SFRD of0 . +0 . − . M ⊙ yr − Mpc − . This “total” value is alreadycomparable to, if not smaller, than the census value of0 . ± .
03 M ⊙ yr − Mpc − (Reddy et al. 2005); the lat-ter can be treated as a lower limit to the SFRD. Theimplications are that the faint-end of the IR LF is un-likely to be as shallow as that predicted from previousIR surveys, that such UV-faint galaxies are likely to havenon-negligible dust attenuation, and that the contribu-tion of sub-ULIRGs to the total LD and SFRD must belarger than previously inferred. Finally, we note thatthere are no observational constraints on the presenceof dusty LIRGs at z ∼ − and fall below the detection threshold of the Spitzer surveys. However, large numbers of such UV-faint dustyLIRGs would serve to increase the average attenuation ofUV-faint galaxies and would strengthen our conclusionsregarding the increased contribution of sub-ULIRGs tothe total LD and SFRD at z ∼ − L IR ≈ L bol . L ⊙ account for ≈ . ≤ z < .
7. The fraction rises to ∼
80% if we take into account the emergent UV luminos-ulti-Wavelength LFs at 1 . . z . . Fig. 28.—
Summary of UV (blue open circles) and IR (red open squares) estimates of the global star formation rate density as a functionof redshift. The UV points have been corrected for extinction in a differential manner (i.e., as a function of redshift): determinations at z . z ∼ z ∼ .
5; the high redshift ( z & z ∼ − z ∼ z ∼ § z ∼
2, are indicated by the red open pentagons. The errors on our UV and IR determinations at z ∼ z ∼ ∼
20% (derived from the errors on the luminosity density) and are smaller than the size of the large pentagons. ity density of sub-ULIRGs. Our analysis suggests thatmuch of the star formation activity at z ∼ z ∼ z ∼ ∼ .
16 M ⊙ yr − Mpc − (i.e., the average of our IR estimates of the SFRD at z ∼ z ∼
3) between z = 3 . z = 1 . ∼ L bol . L ⊙ isroughly constant ( ≈ z = 3 .
05 and z = 2 . Finally, including the z & α or IR data ex-ist at corresponding redshifts), suggests that the SFRDfalls off at these early epochs (e.g., Bouwens et al. 2006 We assume a Salpeter IMF in these calculations. A ChabrierIMF will reduce the stellar mass density estimates by a factor of ∼ .
8, but the relative contribution of sub-ULIRGs between z =3 .
05 and z = 2 . z ∼ − and references therein). Assuming a constant IMF im-plies that the SFRD at z ∼ ∼ z ∼ . z ∼ z ∼ z ∼ −
2, at which point the latter be-comes the dominant effect, leading to a decrease in thenumber density of bright galaxies between z ∼ z = 0 (Figure 23). As discussed above, this reversalin the evolution of the UV LF is likely due to gas ex-haustion resulting from any number of processes (e.g.,SN-driven outflows, AGN feedback, see also Bell et al.8 Reddy et al.2005). Once gas exhaustion has occurred and star forma-tion proceeds quiescently, at least a couple of mechanismshave been proposed to explain why gas is unable to coolonto galaxies with the largest stellar masses, includingAGN feedback (Croton et al. 2006; Scannapieco et al.2005; Granato et al. 2004) and dilution of infalling gasdue to virial shock (Dekel & Birnboim 2006). It is inter-esting to note that it is around this epoch, z ∼
2, thatAGN activity appears to peak (e.g., Hopkins et al. 2007;Fan et al. 2001; Shaver et al. 1996). CONCLUSIONS
We have used the largest existing sample of spectro-scopic redshifts in the range 1 . ≤ z < . α , and infrared (IR) wavelengths.The sample of rest-frame UV selected galaxies includes ∼ ∼ . ∼ . ≤ z < .
4. The largespectroscopic database yields critical constraints on thecontamination fraction of our sample from objects atlower redshifts ( z < .
4) and AGN/QSOs; statistics thatare vital to accurately estimate the bright-end of the LFs.We use our extensive sample to correct for incomplete-ness and recover the intrinsic rest-frame UV LF at z ∼ z ∼
3. Combining this result with H α and Spitzer
MIPS data in several of our fields enables us to infer theH α and IR LFs of star-forming galaxies. The principleconclusions of this work are as follows:1. The fraction of star-forming galaxies with rest-frame Ly α equivalent widths W Ly α >
20 ˚A in emission( f
20) increases with redshift in the range 1 . ≤ z < . . ≤ z < .
17 to 23% for Lyman-break galaxies at2 . ≤ z < .
40. If the general expectation is that youngand less dusty galaxies show Ly α in emission, then thetrend of increasing f
20 with redshift reflects the decreasein average galaxy age and metallicity with increasing red-shift.2. Based on integrating our maximum-likelihood LFs,the fraction of star-forming galaxies with redshifts 1 . ≤ z < . M AB (1700˚ A ) < − .
33 that have colors thatsatisfy BX criteria is ≈ . ≤ z < . M AB (1700˚ A ) < − .
02 that have colors that satisfy theLBG criteria is ≈ . ≤ z < . R < . z ∼ z ∼
2. Correctingfor extinction implies the dust-corrected UV luminositydensity at z ∼ z ∼
3, and roughly 9 times larger than the value at z ∼ z .
2, where gasexhaustion is likely to dominate the evolution of UV-bright galaxies.3. The incompleteness-corrected estimates of the E ( B − V ) distribution indicate very little evolution inthe average dust extinction of galaxies between z ∼ z ∼
2, and that such a distribution is approximatelyconstant to our spectroscopic limit of R = 25 .
5. Theseresults are in agreement with stacked X-ray analyses of z ∼ − z .
2) and high ( z &
2) redshift than what one wouldhave predicted from the observed UV LD (e.g., see alsoBouwens et al. 2006).4. Factoring in the contamination rate of our samplefrom galaxies at lower redshifts and AGN/QSOs withredshifts 2 . ≤ z < .
4, we find no evidence for an excessof UV-bright galaxies over what was inferred in the ini-tial LBG studies of Steidel et al. (1999) and Dickinson(1998), as has been recently claimed.5. The incompleteness-corrected rest-frame UV se-lected sample and deep
Spitzer
MIPS data in multi-ple fields are combined to yield the first spectroscopicconstraints on the faint and moderate luminosity sub-ULIRG ( L IR . L ⊙ ) regime of the total infraredluminosity functions at z ∼ z ∼
3. We usethis information to show that the number density of L IR . L ⊙ galaxies has been significantly under-estimated by previous studies that have relied on shal-lower IR data. After accounting for the emergent UVluminosity, and assuming a realistic range of attenua-tion for UV faint galaxies, we find that ≈
80% of thebolometric (IR+UV) luminosity density at z ∼ L bol < L ⊙ . Assuming a con-stant SFRD of 0 .
16 M ⊙ yr − Mpc − between z = 3 . z = 1 . z ∼ − L bol < L ⊙ galaxies at theseredshifts were responsible for approximately one-third ofthe buildup of the present-day stellar mass density.6. Our estimate of the total SFRD at z ∼ R > .
5) galaxies.Assuming more extreme changes in the dust attenuationof UV-faint galaxies than considered here would be re-quired to reconcile the steep and shallow faint-end slopesof the UV and IR LFs, respectively, implying that thevast majority of UV-faint galaxies would be forming starsfrom chemically pristine gas. However, such an extremescenario would result in a total
SFRD that is compara-ble to, if not smaller than, the lower limit from censusstudies (Reddy et al. 2005). The implications are thatsome significant fraction of sub-ULIRGs must have non-negligible dust extinction, the faint-end of the IR LFmust be steeper than what previous studies have sug-gested, and the contribution of sub-ULIRGs to the totalSFRD must be significantly larger than previously in-ferred.We acknowledge useful conversations with RychardBouwens, Emeric Le Floc’h, Casey Papovich, MarcinSawicki, and Lin Yan. We thank Emeric Le Floc’h forproviding data from Le Floc’h et al. (2005) in electronicformat, Rychard Bouwens for a careful reading of themanuscript, and the referee for helpful suggestions toimprove the clarity of the paper. This work would notulti-Wavelength LFs at 1 . . z . . REFERENCESAbraham, R. G., et al. 2004, AJ, 127, 2455Adelberger, K. L. 2002, PhD thesis, California Institute ofTechnologyAdelberger, K. L., Erb, D. K., Steidel, C. C., Reddy, N. 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